Dynamic Behavior of Materials, Volume 1

Conference Proceedings of the Society for Experimental Mechanics Series

For other titles published in this series, go to www.springer.com/series/8922

Tom Proulx Editor

Dynamic Behavior of Materials, Volume 1 Proceedings of the 2011 Annual Conference on Experimental and Applied Mechanics

Editor Tom Proulx Society for Experimental Mechanics, Inc. 7 School Street Bethel, CT 06801-1405 USA [email protected]

ISSN 2191-5644 e-ISSN 2191-5652 ISBN 978-1-4614-0215-2 e-ISBN 978-1-4614-0216-9 DOI 10.1007/978-1-4614-0216-9 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011929862 © The Society for Experimental Mechanics, Inc. 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Dynamic Behavior of Materials represents one of eight volumes of technical papers presented at the Society for Experimental Mechanics Annual Conference & Exposition on Experimental and Applied Mechanics, held at Uncasville, Connecticut, June 13-16, 2011. The full set of proceedings also includes volumes on Mechanics of Biological Systems and Materials, Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, MEMS and Nanotechnology; Optical Measurements, Modeling and, Metrology; Experimental and Applied Mechanics, Thermomechanics and Infra-Red Imaging, and Engineering Applications of Residual Stress. Each collection presents early findings from experimental and computational investigations on an important area within Experimental Mechanics. The Dynamic Behavior of Materials conference track was organized by: Vijay Chalivendra, University of Massachusetts Dartmouth; Bo Song, Sandia National Laboratories; Daniel Casem, U.S. Army Research Laboratory This Volume represents an ever growing area of broad interest to the SEM community, as evidenced by the increased number of papers and attendance in recent years. This track was initiated in 2005 and reflects our efforts to bring together researchers interested in the dynamic behavior of materials and structures, and provide a forum to facilitate technical interaction and exchange. The Sessions within this track are organized to cover the wide range of experimental research being conducted in this area by scientists around the world. The following general technical research areas are included:     

Composite Materials Dynamic Failure and Fracture Dynamic Materials Response Novel Testing Techniques Low Impedance Materials

vi

     

Metallic Materials Response of Brittle Materials Shock and Blast Loading Optical Techniques for Imaging High Strain Rate Material Response Simulation & Modeling of Dynamic Response & Failure Dynamic Response of Transparent Materials

The contributed papers span numerous technical divisions within SEM. It is our hope that these topics will be of interest to the dynamic behavior of materials community as well as the traditional mechanics of materials community. The track organizers thank the authors, presenters, organizers and session chairs for their participation and contribution to this track. We are grateful to the SEM TD chairs who cosponsored and organized sessions in this track (e.g., Composite Materials, Optical Techniques for Imaging High Strain Rate Events). The SEM support staff is also acknowledged for their devoted efforts in accommodating the large number of submissions this year. The opinions expressed herein are those of the individual authors and not necessarily those of the Society for Experimental Mechanics, Inc. Bethel, Connecticut

Dr. Thomas Proulx Society for Experimental Mechanics, Inc

Contents

1

Punch Response of Gels at Different Loading Rates M. Foster, P. Moy, R. Mrozek, J. Lenhart, T. Weerasooriya, U.S. Army Research Laboratory

2

A Kolsky Torsion Bar Technique for Characterization of Dynamic Shear Response of Soft Materials X. Nie, W. Chen, R. Prabhu, J.M. Caruthers, Purdue University; T. Weerasooriya, U.S. Army Research Laboratory

3

Loading Rate Effect on the Tensile Failure of Concrete and Its Constituents Using Diametrical Compression and Direct Tension S. Weckert, Defence Science and Technology Organisation; T. Weerasooriya, C.A. Gunnarsson, U.S. Army Research Laboratory

4

Influence of Strain-rate and Confining Pressure on the Shear Strength of Concrete P. Forquin, Université Paul Verlaine-Metz

29

5

Dynamic Tensile Properties of Steel Fiber Reinforced Concrete R. Chen, National University of Defense Technology; Y. Liu, X. Guo, Beijing Institute of Technology; K. Xia, University of Toronto; F. Lu, National University of Defense Technology

37

6

Effect of Liquid Environment on Dynamic Constitutive Response of Reinforced Gels S. Padamati, V.B. Chalivendra, A. Agrawal, P.D. Calvert, University of Massachusetts Dartmouth

43

7

Ballistic Gelatin Characterization and Constitutive Modeling D.S. Cronin, University of Waterloo

51

8

Strain Rate Response of Cross-linked Polymer Epoxies Under Uni-axial Compression S. Whittie, P. Moy, A. Schoch, J. Lenhart, T. Weerasooriya, U.S. Army Research Laboratory

57

9

Strength and Failure Energy for Adhesive Interfaces as a Function of Loading Rate T. Weerasooriya, C.A. Gunnarsson, R. Jensen, U.S. Army Research Laboratory; W. Chen, Purdue University

67

10

Fracture in Layered Plates Having Property Mismatch Across the Crack Front U.H. Bankar, A. Rajesh, P. Venkitanarayanan, Indian Institute of Technology Kanpur

77

11

Stress Variations and Particle Movements During Penetration Into Granular Materials H. Park, W.W. Chen, Purdue University

85

12

Sand Particle Breakage Under High-pressure and High-rate Loading Md.E. Kabir, W. Chen, Purdue University

93

1

11

13

viii 13

Experimental and Numerical Study of Wave Propagation in Granular Media T. On, K.J. Smith, P.H. Geubelle, J. Lambros, University of Illinois at Urbana-Champaign; A. Spadoni, C. Daraio, California Institute of Technology

95

14

Communication of Stresses by Chains of Grains in High-Speed Particulate Media Impacts W.L. Cooper, Air Force Research Laboratory

99

15

Effects of Thermal Treated on the Dynamic Facture Properties Using a Semi-circular Bend Technique T.B. Yin, University of Toronto/Central South University; X.B. Li, Central South University; K.W. Xia, S. Huang, University of Toronto

16

Development and Characterization of a PU-PMMA Transparent Interpenetrating Polymer Networks (t-IPNs) K.C. Jajam, S.A. Bird, M.L. Auad, H.V. Tippur, Auburn University

17

Dynamic Ring-on-Ring Equibiaxial Flexural Strength of Borosilicate Glass X. Nie, W. Chen, Purdue University

123

18

Stress-strain Response of PMMA as a Function of Strain-rate and Temperature P. Moy, C.A. Gunnarsson, T. Weerasooriya, U.S. Army Research Laboratory; W. Chen, Purdue University

125

19

Dynamic Behavior of Three PBXs with Different Temperatures J.L. Li, National University of Defense and Technology/Chinese Academy of Engineering and Physics; F.Y. Lu, R. Chen, J.G. Qin, P.D. Zhao, L.G. Lan, S.M. Jing, National University of Defense and Technology

135

20

Dynamic Compressive Properties of A PBX Analog as a Function of Temperature and Strain Rate J. Qin, Y. Lin, F. Lu, National University of Defense Technology; Zh. Zhou, Beijing Institute of Technology; R. Chen, J. Li, National University of Defense Technology

21

Dynamic Response of Shock Loaded Architectural Glass Panels P. Kumar, A. Shukla, University of Rhode Island

147

22

A Dynamic Punch Method to Quantify the Dynamic Shear Strength of Brittle Solids S. Huang, K. Xia, F. Dai, University of Toronto

157

23

A Sensored Projectile Impact on a Composite Sandwich Panel M. Mordasky, W. Chen, Purdue University

165

24

Cut Resistance and Fracture Toughness of High Performance Fibers J.B. Mayo, Jr., Tuskegee University/U.S. Army Research Laboratory; E.D. Wetzel, U.S. Army Research Laboratory

167

25

Kolsky Tension Bar Techniques for Dynamic Characterization of Alloys B. Song, H. Jin, B.R. Antoun, Sandia National Laboratories

175

26

Prediction of Dynamic Forces in Fire Service Escape Scenarios M. Obstalecki, J. Chaussidon, P. Kurath, G.P. Horn, University of Illinois at Urbana-Champaign

179

27

Tensile Behavior of Kevlar 49 Woven Fabrics over a Wide Range of Strain Rates J.D. Seidt, T.A. Matrka, A. Gilat, G.B. McDonald, The Ohio State University

187

109

117

141

ix 28

The Effect of Loading Rate on the Tensile Behavior of Single Zylon Fiber C.A. Gunnarsson, T. Weerasooriya, P. Moy, Army Research Laboratory

195

29

Statistical Analysis of Fiber Gripping Effects on Kolsky bar Test J.H. Kim, N.A. Heckert, S.D. Leigh, H. Kobayashi, W.G. McDonough, R.L. Rhorer, K.D. Rice, G.A. Holmes. National Institute of Standards and Technology

205

30

Perpendicular Yarn Pull-out Behavior Under Dynamic Loading J. Hong, Purdue University; J. Lim, Hyundai Motor Company; W.W. Chen, Purdue University

211

31

Dynamic Response of Homogeneous and Functionally Graded Foams When Subjected to Transient Loading by a Square Punch C. Periasamy, H. Tippur, Auburn University

32

Dynamic Strain Measurement of Welded Tensile Specimens Using Digital Image Correlation K.A. Dannemann, R.P. Bigger, S. Chocron, Southwest Research Institute; K. Nahshon, Naval Surface Warfare Center Carderock Division

217

33

Ultra High Speed Full-field Strain Measurements on Spalling Tests on Concrete Materials F. Pierron, Arts et Métiers ParisTech; P. Forquin, Paul Verlaine University

221

34

Contact Mechanics of Impacting Slender Rods: Measurement and Analysis A. Sanders, I. Tibbitts, D. Kakarla, University of Utah; S. Siskey, J. Ochoa, K. Ong, Exponent, Inc.; R. Brannon, University of Utah

229

35

Solenoid Actuated, Rail Mounted, Aircraft Payload Release Mechanisms C.L. Reynolds, Dynetics, Inc.; J.A. Gilbert, University of Alabama in Huntsville

237

36

Finite Element Modeling of Ballistic Impact on Kevlar 49 Fabrics D. Zhu, McGill University; B. Mobasher, S.D. Rajan, Arizona State University

249

37

Optimal Pulse Shapes for SHPB Tests on Soft Materials M. Scheidler, J. Fitzpatrick, R. Kraft, U.S. Army Research Laboratory

259

38

Dynamic Tensile Characterization of Foam Materials B. Song, H. Jin, W.-Y. Lu, Sandia National Laboratories

269

39

On Measuring the High Frequency Response of Soft Viscoelastic Materials at Finite Strains S. Teller, R. Clifton, T. Jiao, Brown University

273

40

The Blast Response of Sandwich Composites With a Graded Core: Equivalent Core Layer Mass vs. Equivalent Core Layer Thickness N. Gardner, A. Shukla, University of Rhode Island

281

41

Effects of High and Low Temperature on the Dynamic Performance of the Core Material, Face-sheets and the Sandwich Composite S. Gupta, A. Shukla, University of Rhode Island

289

42

Influence of Texture and Temperature on the Dynamic-tensile-extrusion Response of High-purity Zirconium D.T. Martinez, C.P. Trujillo, E.K. Cerreta, J.D. Montalvo, J.P. Escobedo-Diaz, Los Alamos National Laboratory; V. Webster, Case Western Reserve University; G.T. Gray, III., Los Alamos National Laboratory

213

297

x 43

Modeling and DIC Measurements of Dynamic Compression Tests of a Soft Tissue Simulant S.P. Mates, R. Rhorer, A. Forster, National Institute of Standards and Technology; R.K. Everett, K.E. Simmonds, A. Bagchi, Naval Research Laboratory

307

44

Measurement of R-values at Intermediate Strain Rates Using a Digital Speckle Extensometry J. Huh, Y.J. Kim, H. Huh, Korea Advanced Institute of Science Technology

317

45

Study of Strain Energy in Deformed Insect Wings H. Wan, H. Dong, Y. Ren, Wright State University

323

46

Experimental Study of Cable Vibration Damping A. Maji, Y. Qiu, University of New Mexico

329

47

Dynamic Thermo-mechanical Response of Austenite Containing Steels V.-T. Kuokkala, Tampere University of Technology; S. Curtze, Tampere University of Technology/Oxford Instruments Nano Analysis; M. Isakov, M. Hokka, Tampere University of Technology

337

48

Investigation into the Spall Strength of Cast Iron G. Plume, C.-E. Rousseau, University of Rhode Island

343

49

Development of Brick and Mortar Material Parameters for Numerical Simulations C.S. Meyer, U.S. Army Research Laboratory

351

50

Electrical Behavior of Carbon Nanotube Reinforced Epoxy Under Compression N. Heeder, A. Shukla, University of Rhode Island; V. Chalivendra, University of Massachusetts Dartmouth; S. Yang, K. Park, University of Rhode Island

361

51

Effect of Curvature on Shock Loading Response of Aluminum Panels P. Kumar, University of Rhode Island; J. LeBlanc, Naval Undersea Warfare Center; A. Shukla, University of Rhode Island

369

52

Deformation Measurements and Simulations of Blast Loaded Plates K. Spranghers, Vrije Universiteit Brussel; D. Lecompte, Royal Military Academy; H. Sol, Vrije Universiteit Brussel; J. Vantomme, Royal Military Academy

375

53

The Blast Response of Sandwich Composites With Bi-axial In-plane Compressive Loading E. Wang, University of Illinois Urbana-Champaign; A. Shukla, University of Rhode Island

383

54

Dynamic Response of Porcine Articular Cartilage and Meniscus under Shock Loading Y.-C. Juang, L. Tsai, National Kaohsiung University of Applied Sciences; H.R. Lin, Southern Taiwan University

393

55

Dynamic Response of Beams Under Transverse Impact Loadings D. Goldar, Sharda University

399

56

Constitutive Model Parameter Study for Armor Steel and Tungsten Alloys S.J. Schraml, U.S. Army Research Laboratory

409

57

A Scaled Model Describing the Rate-dependent Compressive Failure of Brittle Materials J. Kimberley,G. Hu, K.T. Ramesh, Johns Hopkins University

419

58

Experimental Verification of Negative Phase Velocity in Layered Media A.V. Amirkhizi, S. Nemat-Nasser, University of California, San Diego

423

xi 59

Gas Gun Impact Analysis on Adhesives in Sandwich Composite Panels M. Mordasky, W. Chen, Purdue University

425

60

Damage Analysis of Projectile Impacted Laminar Composites B.S. Nashed, J.M. Rice, Y.K. Kim, V.B. Chalivendra, University of Massachusetts Dartmouth

427

61

Rate Sensitivity in Pure Ni Under Dynamic Compression K.N. Jonnalagadda, Indian Institute of Technology Bombay

439

62

Temperature Effect on Drop-weight Impact of Woven Composites Y. Budhoo, Vaughn College of Aeronautics and Technology; B. Liaw, F. Delale, The City College of New York

443

63

Dynamic Mode-II Characterization of a Woven Glass Composite W.-Y. Lu, B. Song, H. Jin, Sandia National Laboratories

455

64

Rate Dependent Material Properties of an OFHC Copper Film J.S. Kim, Korea Railroad Research Institute; H. Huh, Korea Advanced Institute of Science and Technology

459

65

Zirconium: Probing the Role of Texture Using Dynamic-tensile-extrusion C.P. Trujillo, J.P. Escobedo-Diaz, G.T. Gray, III., E.K. Cerreta, D.T. Martinez, Los Alamos National Laboratory

467

Punch Response of Gels at Different Loading Rates Mark Foster [email protected] Paul Moy [email protected] Randy Mrozek [email protected] Joe Lenhart [email protected] Tusit Weerasooriya [email protected] Army Research Laboratory Weapons and Materials Research Directorate Bldg 4600 Deer Creek Loop Aberdeen Proving Ground, MD 21005-5069 ABSTRACT Synthetic soft polymer gels have many advantages over protein-based gels that are derived from animal collagen and bones such as stability at room temperature and prolonged shelf life. In addition, the ability to tailor the formulations and processes of synthetic gel to control mechanical properties both isotropically or anisotropically is another essential feature in order for gels to mimic the spectrum of biological tissues. However, it is impractical to physically characterize all aspects of every gel available. To do so would require production of a significant amount of material to accommodate all the varying tests needed for a comprehensive study. A novel punch test was developed as a simple solution to obtain mechanical responses at different loading rates without the production of a large amount of sample material. The gels used in this effort are 10% and 20% ballistic gelatin, the commercially marketed PermaGel™, and triblock copolymer gels. The experimental setup is discussed, and the results are presented and compared to a previous study that discussed the tensile behavior of these soft materials. INTRODUCTION Traditionally, ballistic gelatin is used extensively to examine the penetration depth of firearms. While useful as a base material for qualitative comparison, the applications of this biologically derived gelatin become very limited when used as a tissue simulant for quantitative testing. Next generation armor and protection systems require an understanding of injury mechanisms that has not yet been realized [1-3]. However, ballistic gelatins inconsistent viscoelastic properties, high sensitivity to temperature, and aging effects all prove detrimental to producing data to validate proposed material models [4]. Synthetic gelatins are therefore a promising solution because networks of cross-links and polymer chains can be tuned to resemble the mechanics of biological tissues. Unfortunately, such a wide range of controllable properties necessitates laborious and repetitive testing for full characterization.

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_1, © The Society for Experimental Mechanics, Inc. 2011

1

2 Much prior work has been performed on soft tissue stimulants. Methods have been established in constrained and unconstrained compression at a variety of strain rates [2,5,6]. Often this work is to measure a shear modulus for use in various constitutive models. While existing models can be useful at determining material response at lower rates of strain, a fully nonlinear model is required at higher rates [7]. Wu et al. [6] also demonstrated that friction can cause a non-uniform deformation state in unconfined compression tests which contradicts assumptions in traditional models. In a study on the effects of temperature, aging time, and strain rate to the penetration depth of ballistic gel, Cronin and Falzon [4] found that tensile strain dominates over shear strain during failure. This also contradicts many current models of tissues that are based on shear failure criteria. Microscale indentation and rheology tests have been performed, proving that viscoelastic and hyperelastic models are applicable at small (physiological) deformations, but large strain behavior of nonlinear tissues warrants further analysis [8-11]. Other methods of modeling human tissue behavior involve porcine, bovine, rat or mouse tissue. Snedeker et al. [8] performed impact experiments on porcine and human kidneys, and found large differences in stress to failure. Fracture tear tests have also found basic energy dissipation and tear resistance values [12]. Moy et al. [13,14] have used digital image correlation techniques to measure strain field qualities in notched gelatin tensile specimens. This technique provided a measurement of the maximum tensile strain and energy required for a crack to propagate through gelatin. This interest led to the development of a simple punch penetration test to screen the multitude of possible gelatin formulations into a smaller group of gels that resemble the mechanical behaviors of ordnance gelatins. Various gelatin materials were subjected to a constant-rate displacement by a 6.35 mm hemispherical penetrator tip. These materials included synthetic polymer gels, ballistic gels, and Permagel™, a commercially available ballistic gel replacement. Then puncture data was compared to gelatin fracture tear data from Moy et al. [13] to ensure that the basic puncture test can be used as a screening process to find suitable tissue surrogates without an indepth investigation into each material. Materials Three different synthetic polymer gels were made from different concentrations of Poly(styrene-b-ethylene-cobutylene-b-styrene) (SEBS) G1652 as-received from Kraton Polymers (Houston, TX, USA) and mineral oil asreceived from Aldrich Chemical (Milwaukee, WI, USA). These three gels were mixed in sheets with concentrations of 70, 80, and 90 percent mineral oil to SEBS polymer. Ballistic gelatin was made from Bloom 250 Type A ordnance gelatin mix as-received from GELITA USA (Sioux City, IA). Each batch was mixed in water according to manufacturer directions, poured into a sheet, refrigerated at 3.89°C, and tested the following day. The two types of ballistic gel typically used for bullet penetration testing are either 10 or 20 percent by mass gelatin, and both are accounted for in this work [4]. Permagel™ is a transparent material designed to have similar properties to 10% ballistic gelatin, but without its inherent disadvantages. The material was used as-delivered from USALCO (Browns Mills, NJ) and was cut from a large base block, molded into sheets and allowed to cool before testing. The group of gelatins tested is included in Table 1.

3 Table 1: Collection of Materials Tested Material Concentration Source Mineral Oil/SEBS polymer

Permagel Bloom 250A Ordnance Gel

70/30 80/20 90/10 N/A (by weight to water) 10% 20%

Aldrich Chemical/ Kraton Polymers

USALCO GELITA, USA

Test Methods A 4mm thick specimen was bonded to one side of a standard 7/16 inch washer, excess material was trimmed and the specimen was tested to full failure with a 6.35 mm hemispherical indenter. All specimens were inspected for bubbles or debris prior to gluing. An elevated acrylic table fixture was used to provide clearance for full specimen failure and give ample space for observation. A custom 45° triangle mirror fixture was then designed to fit underneath the elevated fixture to observe the specimen during the indentation. Figure 1 shows the test fixture. After the specimens fully adhered to the washer, the entire sample was placed on the top of the fixture centered over the middle 12.7 mm diameter hole in the acrylic plate. While the washer did not provide clamping pressure by any means, it did give an open space on the top gelatin surface to fill with lubricant. Friction between the acrylic and the gelatin provided ample resistance to prevent any slippage during the experiment.

Figure 1: Punch Test Fixture with 45° Mirror An Instron 8871 servo-hydraulic load frame was used to control and directly measure the load and displacement required to indent and to puncture the gel specimens. Three different displacement rates were addressed: 12.7, 127 and 1270 mm/min. These gave a wide range of material characteristics while remaining within the capabilities of the load frame. The overall machine setup is in Figure 2. Lubrication was used in order to minimize friction between the gel and indenter. For ballistics gel an olive oil was used, but silicone oil of similar viscosity was used for the Permagel™ and synthetic polymer gels, to ensure the lubricant did not affect the gel material being tested. Prior to each experiment, the indenter was lowered until it just contacted the gelatin surface.

4

Figure 2: Machine Setup with Accompanied Gel Punch Fixture

RESULTS AND DISCUSSION Without lubrication, friction between the indenter and the gelatin was quite evident in the preliminary testing. It was observed that when oil was omitted specimens would exhibit a “cork” type failure characteristic of a shearing failure instead of the desired Mode I tearing failure. Load and displacement data obtained for all experiments at each extension rate can be seen in Figures 3a through c. Figures 4a to 4c provides a magnified view of the gels at the lower load range. This data includes all testing, and it is clear from the amount of overlap that the test is inherently repeatable. The gelatins displayed a close agreement between certain pairs of material, such as the close agreement between 80/20 and Permagel™ for all three rates. However, ballistic gels do not exhibit the relaxation portion of the load-displacement curve that the other 4 materials expressed around 8 mm displacement. This can clearly be seen in the 1270 mm/min data where the 20% ballistic gel deviates from 70/30 data curve at around 7.5 mm extension. Note that prior to this point there is a close agreement between the 20% ballistic gel and 70/30 curves. The ballistic gels also show lower overall extensions than the other materials. When the 10% ballistic and Permagel™ gelatins are compared across the three rates studied, the load response converges as the displacement rate increases. This compels the idea that Permagel™ could be used as a replacement for the 10% ballistic gelatin, but only at higher rates.

5

Figure 3: Load vs. Extension for all Gel Materials Tested at (a) 12.7 mm/min, (b) 127 mm/min, and (c) 1270 mm/min Displacement Rate

6 6

4 80/20 90/10 Permagel 10% Ballistic

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(4c) Figure 4: Expanded Load vs. Extension for Punch Experiments at the Lower Load Region at (a) 12.7 mm/min, (b) 127 mm/min, and (c) 1270 mm/min Displacement Rate The maximum displacement to failure was compared in Figure 5 across the three displacement rates of 12.7, 127, and 1270 mm/min. Two distinct trends are clear. Most interesting is the distinct difference in trend between the mineral oil and the Permagel™/ballistic gel. The synthetic mineral oil gels exhibit a much larger stretch ratio than the ballistic gelatins or Permagel™ at a higher rate of displacement, which suggests a different strengthening mechanism may occur at these higher displacement velocities. This could easily be attributed to the structure of polymer chains and amount of crosslinking in the mineral oil gelatins which do not occur in the others. As the manufacturer claims Permagel™ is similar to the ballistic gels but requires a larger overall extension to fully penetrate, which is shown from the data at all rates of displacement.

7 60

Maximum Extension (mm)

50 40 30 20 10 0

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100 Displacement Rate (mm/min) 70/30 80/20

90/10 Permagel

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Figure 5: Maximum Extension as a Function of Displacement Rate for all Gel Punch Experiments A similar analysis was performed with the maximum load to failure in Figure 6. The 80/20 formulation ratio of mineral oil to SEBS polymer closely agrees with the Permagel™ at all displacement rates studied. Unlike the 70/30 gelatin which gave a much larger penetration resistance. Despite a similarity in extension to failure between the 70/30 and Permagel™ at lower displacement rates, the punch test showed a vast difference in stiffness of the 70/30 gelatin. While miniscule, the loads of the 90/10 polymer gel resemble the 10% ballistic gel across all displacement rates. 30

Maximum Load (N)

25 20 15 10 5 0

10

100 Displacement Rate (mm/min) 70/30 80/20

90/10 Permagel

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Figure 6: Maximum Load as a Function of Displacement Rate for all Gel Punch Experiments

8 Each load-extension curve was integrated to determine the overall energy required to fully puncture the gelatin. These values were then correlated to a prior work by Moy et al. which tested the fracture characteristics of both ballistic gel concentrations, and Permagel™ [13]. A Mode I test was used with a dogbone shaped specimen that had a small pre-crack. The crack propagation was then inspected using high speed cameras with DIC to acquire surface strains near and around the crack tip. The load-extension curve was also integrated to obtain fracture energy values. This comparison is shown in Figure 7 a, b, and c. It is important to note the difference in scaling between the two data sets, for they are not in perfect alignment in any case. However, considering the differences between the two test methods, the results are quite similar in trend. It is very easy to see the agreement in the 20% ballistic gel for example, where the fracture energy is close to a factor of 5x higher than the punch energy. This comparison validates the punch method presented here as a simple screening process as it is a much less demanding test.

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Figure 7a: Comparison between Punch Energies and Fracture Energies for Permagel™ [13]

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Figure 7b: Comparison between Punch Energies and Fracture Energies for 10% Ballistic Gel [13]

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Figure 7c: Comparison between Punch Energies and Fracture Energies for 20% Ballistic Gel [13] CONCLUSIONS A significant number of synthetic gelatins offers multiple solutions for potential replacement of ballistic gelatin. More importantly, polymer gels can be tailored to have characteristics to actual biological tissues. To adequately examine the load response of every gel available would be very demanding. In an effort to find a suitable tissue surrogate to validate theoretical models for next-generation armor and protection systems, several gelatin materials were compared in a simplified puncture test. Ballistic gels, mineral oil gels, and the commercially available Permagel™ were punctured to full failure using a hemispherical indenter across extension rates of 12.7, 127 and 1270 mm/min. Similar initial load responses were seen in the 10% ballistic, 80/20 mineral oil/SEBS, and

10 Permagel™ at each rate despite the ballistic gelatins lower extension and maximum load to failure. Through a comparison with previous mode I gel fracture data, the punch technique provides an easy screening method to classify a gelatin as a possible tissue simulant. REFERENCES 1. Song, B., Ge, Y., Chen, W., and Weerasooriya, T. Radial Inertia Effects in Kolsky Bar Testing of ExtraSoft Specimens. Experimental Mechanics, 47, pp. 659-670. 2007. 2. Saraf, H., Ramesh, K.T., Lennon, A.M., Merkle, A.C., and Roberts, J.C. Mechanical Properties of Soft Human Tissues Under Dynamic Loading. Journal of Biomechanics, 40, pp. 1960-1967. 2007. 3. Van Sligtenhorst, C., Cronin, D. S., and Brodland, G. W. High Strain Rate Compressive Properties of Bovine Muscle Tissue determined using a split Hopkinson bar apparatus. Journal of Biomechanics, 39, pp 1852-1858. 2006. 4. Cronin, D.S., and Falzon, C. Characterization of 10% Ballistic Gelatin to Evaluate Temperature, Aging and Strain Rate Effects. Proceedings of the 2010 SEM Annual Conference, Indianapolis, IN. 2010. 5. Kwon, J., and Subhash, G., Compressive Strain Rate Sensitivity of Ballistic Gelatin. Journal of Biomechanics, 43. pp 420-425. 2010. 6. Wu, J.Z., Dong, R.G., and Schopper, A.W. Analysis of Effects of Friction on the Deformation Behavior of Soft Tissues in Unconfined Compression Tests. Journal of Biomechanics, 37, pp 147-155. 2004. 7. Cronin, D.S., and Falzon, C. Dynamic Characterization and Simulation of Ballistic Gelatin. Proceedings of the 2009 SEM Annual Conference. Albuquerque, N.M. 2009. 8. Snedeker, J.G., Barbezat, M., Niederer, P., Schmidlin, F.R., and Farshad, M. Strain Energy Density as a Rupture Criterion for the Kidney: Impact Tests on Porcine Organs, Finite Element Simulation, and a Baseline Comparison Between Human and Porcine Tissues. Journal of Biomechanics, 38, pp 993-1001. 2005. 9. Lin, D.C., Shreiber, D. I., Dimitriadis, E. K., and Horkay, F. Spherical Identation of Soft Matter Beyond the Hertzian Regime: Numerical and Experimental Validation of Hyperelastic Models. Biomechanics and Modeling in Mechanobiology, 8 (5), pp 345-358, 2009. 10. Wu, J.Z., Dong, R.G., Smutz, W.P., and Schopper, A.W. Nonlinear and Viscoelastic Characteristics of Skin under Compression; Experiment and Analysis. Biomedical Materials and Engineering. 13 (4), pp 373-385, 2003. 11. Clark, A.H., Richardson, R.K., Ross-Murphy, S.B., and Stubbs, J.M. Structural and Mechanical Properties of Agar/Gelatin Co-gels. Small Deformation Studies. Macromolecules. 16, pp 1367-1374, 1983. 12. Furukawa, H., Kuwabara, R., Tanaka, Y., Kurokawa, T., Na, Y., Osada, Y., and Gong, J.P. Tear Velocity Dependence of High-Strength Double Network Gels in Comparison with Fast and Slow Relaxation Modes Observed by Scanning Microscopic Light Scattering. Macromolecules. 41, pp 7173-7178, 2008. 13. Moy, P. Gunnarsson, C. A. and Weerasooriya, T. Tensile Deformation and Fracture of Ballistic Gelatin as a Function of Loading Rate. Proceedings of the 2009 SEM Annual Conference. Albuquerque, NM. 2009. 14. Moy, P. Foster, M. Gunnarsson, C. A. and Weerasooriya, T. Loading Rate Effect on Tensile Failure Behavior of Gelatins under Mode I. Proceedings of the 2010 SEM Annual Conference. Indianapolis, IN. 2010.

A Kolsky Torsion Bar Technique for Characterization of Dynamic Shear Response of Soft Materials Xu Nie1*, Weinong Chen1, Rasika Prabhu2, James M. Caruthers2, Tusit Weerasooriya3 1

AAE&MSE schools, Purdue University. 2ChE school, Purdue University. 3Army Research Laboratory * Corresponding author: Xu Nie, 701 W. Stadium Ave. West Lafayette, IN 47907-2045 Email: [email protected]

ABSTRACT A novel Kolsky torsion bar technique is developed and successfully utilized to characterize the high strain rate shear response of a rate-independent end-linked polydimethylsiloxane (PDMS) gel rubber with a shear modulus of ~10 KPa. The results show that the specimen deforms uniformly under constant strain rate and the measured dynamic shear modulus well follows the trend determined by dynamic mechanical analysis (DMA) at lower strain rates. Contrastive Kolsky compression bar experiments are also performed on the same gel material with annular specimens. The dynamic moduli obtained from compression experiments, however, are an order of magnitude higher than those predicted by the torsional technique, due to the pressure caused by the radial inertia and end constraints. INTRODUCTION Characterization of dynamic response of soft biological tissues has seen a tremendous rise in the past decade. Among all the published non-oscillatory high rate results, dynamic uniaxial compression/tension has generated the most popular group of data [1], and its experimental conditions have also been extensively investigated [2]. There are two major issues associated with the axial loading conditions when the strain rate is high: 1. Dynamic stress (or force) equilibrium across the specimen length, and 2. Radial inertia induced pressure by strain rate and strain acceleration. A preliminary solution to minimize the inertia effect is to punch a hole in the center of the specimen, for which the pressure was greatly reduced by creating a stress-free inner surface. However, for materials as soft as human brain tissues whose elastic moduli are typically in the range of 0.1-10 KPa, even the reduced pressure in an annular sample can be sufficiently high to overshadow the intrinsic material response. To separate the pressure from the intrinsic mechanics response of soft materials, a pure shear loading condition is desired. In this paper, we present a newly developed desktop Kolsky torsion bar technique for the characterization of high rate shear mechanical properties of soft materials. The effectiveness of this torsion bar technique was demonstrated by our calibration experiments on the end-linked polydimethylsiloxane (PDMS) gel rubber. EXPERIMENTS AND RESULTS A typical oscilloscope record of the modified Kolsky torsion bar experiment is shown in Fig. 1. The trace noted as “incident bar signal” is measured by the strain gages mounted on the incident bar, while the other trace is taken from the torque sensor which connects to the external ring adapter. Since the gel material under investigation has extremely low wave impedance compared to that of the incident bar, most of the incident wave is reflected back. Consequently, the reflected pulse would not see any noticeable difference, both in shape and amplitude, from the incident pulse. We used the incident wave to calculate the shear strains in the T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_2, © The Society for Experimental Mechanics, Inc. 2011

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specimen so that a better-quality signal can be used directly from the oscilloscope reading. The stress-strain curves of five different samples loaded at shear strain rate of ~1000s-1 are displayed in Fig. 2. Although some discrepancies were found on the five measured stress-strain curves, all of them exhibited linear elasticity when the strain is beyond 8%. In order to compare the shear modulus with those obtained from DMA tests, and thus evaluate the validity of our Kolsky torsion bar experiment, the tangential of these stress-strain curves in Fig. 2 were measured in the strain range from 8% to the maximum strain on each curve. As mentioned before, the purpose of conducting current dynamic torsional experiments on soft materials is to directly acquire their shear constitutive properties, which in the past were mostly inferred from the dynamic compression results. To compare the measured modulus value from uniaxial compression experiments with those obtained from DMA and Kolsky torsion bar experiments, dynamic compressive experiments on this same PDMS gel were also conducted at comparable strain rates. The results are plotted in Fig. 3. The dynamic shear elastic modulus of gel measured with torsion bar technique follows the trend of DMA test results, while the same material exhibited much higher modulus value (about an order of magnitude) when it was under dynamic compression. Such a large discrepancy between the two dynamic testing techniques and the analysis of the discrepancy reveal that the Kolsky torsion bar experiment is necessary to characterize the shear behavior of extra soft materials under high strain rate loading conditions.

Fig. 1 The original signals of torsional experiments

Fig. 2 Shear stress-strain curve of PDMS at 1000/s by Kolsky torsion bar technique

Fig. 3 Comparison of shear modulus obtained by different testing techniques at different shear rates REFERENCE: [1] Bo Song, Weinong Chen, Yun Ge and Tusit Weerasooriya, “Dynamic and quasi-static compressive response of porcine muscle”, Journal of Biomechanics, 40, 2999-3005, 2007 [2] Bo Song and Weinong Chen, “Dynamic stress equilibration in split Hopkinson pressure bar tests on soft materials”, Experimental Mechanics, 44, 300-312, 2004

Loading Rate Effect on the Tensile Failure of Concrete and Its Constituents using Diametrical Compression and Direct Tension

Samuel Weckert1 [email protected] Tusit Weerasooriya2 [email protected] C. Allan Gunnarson2 [email protected] 1

Defence Science and Technology Organisation Edinburgh, South Australia, 5111 2

Army Research Laboratory Weapons and Materials Research Directorate Bldg 4600 Deer Creek Loop Aberdeen Proving Ground, MD 21005-5069 ABSTRACT The loading rate effect on the tensile failure strength of concrete and its constituent materials has been investigated. Concrete is inherently weaker in tension than compression so tensile failure represents the dominant failure mode. Understanding the failure characteristics of concrete, particularly at high loading rate, is important for developing modeling capabilities, in particular for predicting spallation damage and fragmentation. Several concretes, and their constituents, have been investigated at different loading rates to understand the tensile failure behavior as a function of loading rate. In this paper, experimental procedures that were used are discussed, and results from two different tensile testing methods, direct tension and diametric compression (Brazilian/split-tension), are presented for several of these materials. INTRODUCTION Tensile failure is a vulnerable failure mode for concrete as it is much weaker in tension compared with other modes of failure such as compression. Typically the tensile strength is an order of magnitude less than the compressive strength. High strain rate tensile testing of concrete is important for weapons effects problems, such as penetration and explosive loading, where the loading rates are very high and tensile failure can occur as spallation damage in a target. Materials can behave differently at high strain rates, so material characterization in this regime is important for developing accurate material models for simulations. Direct tension experiments produce a nominally uniaxial tensile stress state, however, it can be difficult to implement because of issues associated with gripping the sample. This is particularly the case for brittle 1

This work was undertaken as a collaborative effort between Australia and the U.S. while Sam Weckert was on a 6 month attachment at ARL in the High Rate Mechanics and Failure branch under the Scientists and Engineers Exchange Program in 2010.

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14 materials, such as concrete, where it is not possible to use conventional grips or threaded joints. High strength adhesives can be used to grip normal concrete; however they are not strong enough for testing the new generation of high strength concretes. Notched specimens can be used to reduce the cross-sectional area [1], however, this creates a stress concentration which leads to an under prediction of the tensile strength. Dog-bone specimen geometries are also possible, however these are difficult implement in brittle materials. The diametric compression test, also known as the Brazilian or Split-tension test, offers an alternate test method to indirectly obtain the material tensile response. This test induces tensile stresses within the specimen by point diametric compression of a disc shaped sample. This permits the use of simpler compression testing apparatus to obtain tensile material response data. Other tensile test techniques also include three or four point bend tests and high rate specific spallation experiments [2]. This report presents the tensile strength for various concretes and their constituent materials (mortar and aggregate) using the diametric compression technique. Tests are conducted at high loading rate using a Split Hopkinson Pressure Bar (SHPB) apparatus and intermediate and low loading rates using an Instron hydraulic test machine for comparison. This allows an investigation of the loading rate effects for each of these materials. Direct tension tests at high loading rate have also been conducted for several of the concrete materials to allow a comparison of tensile strengths from the direct tension and diametric compression test methodologies. MATERIALS The tensile strengths of five different materials were investigated in this study: 1. SAM35 concrete: a 3500psi (~24MPa) minimum quasi-static unconfined compressive strength concrete produced by the US Engineering and Research Development Center (ERDC) [3]. It contains small limestone aggregate components up to approximately 8mm in size. 2. Mortar: prepared from a commercially available mix - Drypack basic mortar sand and cement, Adelaide Brighton Cement Limited, Australia. 3. Granite: charcoal black granite, Starrett, True Stone Tech Division, MN, US. 4. Ultra High Performance Concrete (UHPC): a reactive powder concrete reinforced with steel fibers of length 12.7mm and diameter 0.2mm, randomly distributed through the concrete at 6.2% by weight [4], samples obtained from Australia. 5. Alcatraz concrete: obtained from a tourist commercial vendor at the Alcatraz prison in San Francisco, California. The SAM-35 represents a common concrete mix and the mortar and granite are representative of typical concrete constituents (however, the SAM-35 contains limestone aggregate not granite). The UHPC is a new generation high strength concrete which was tested to evaluate its enhanced characteristics. The diametric compression tests used disc shaped specimens with a nominal thickness of ¼” (6.35mm). The mortar, UHPC and Alcatraz concrete specimens had a nominal diameter of 20mm, whereas the SAM-35 and granite specimens had a nominal diameter of 1” (25.4mm). Direct tension tests at high loading rate were also performed on the mortar and SAM35 concrete using cylindrical samples with ¾” diameter and ¾” length. The size of the specimen geometry was dictated by the available experimental equipment and was fairly small relative to the size of the aggregrate components in the concrete and the steel fiber distribution for the UHPC. In addition, the material heterogeneity is further amplified in the diametric compression test where only a portion of the sample is in tension. Consequently it is expected that there will be considerable scatter in the results and so a minimum of five tests were conducted for each material at each loading rate. EXPERIMENTAL METHODOLOGY Diametric compression test technique The diametric compression (Brazilian/split tension) test uses a circular disc sample, which is point loaded at diametrically opposite points in compression. This test methodology produces a biaxial stress state where a tensile stress is induced perpendicular to the compressive stress along the loading axis and the material fails in tension. The stress state produced with this loading condition is discussed in detail in [5]. The diametric

15 compression test methodology gives the tensile failure strength of the material, however, no pre or post-peak stress-strain response can be obtained. A problem with the point loading used in this test methodology, particularly for brittle materials, is that the sample is subject to high stress concentrations at the external load contact points. Thus failure may initiate at these contact points rather than in the induced tensile region in the bulk of the specimen, which invalidates the test. To reduce the stress concentrations at the contact points, wooden bearing strips are recommended for distributing the load in quasi-static diametric compression tests [6,7], however, these are not suitable for high rate tests because of the reflections of stress waves and material impedance effects. For high rate diametric compression tests, other researchers have suggested several techniques to overcome this problem, both with the objective of spreading the load over a small area at the sample sides to reduce the stress concentrations. The first method is to cut flat areas onto the sides of the sample at loading points [8]. The second method is to maintain a circular disc shaped sample, but use concave curved input/output bars for loading the sample [9], as shown in figure 1. It is this second method, which was adopted at all loading rates for the tests presented here.

Figure 1: Curved input/output bars for loading the disc specimen

Figure 2 shows the normal equation used to obtain the tensile strength, σ, for a diametric compression test.

Figure 2: Compressive loading and tensile stress diagram/equation for diametric compression setup

where P is the axial compressive load, D is the specimen diameter and t is the specimen thickness. A modification to this equation is used in [9] to account for the spreading of the load at the contact points. However, the modified equation introduces a contact width parameter between the sample and the curved loading platens, which is difficult to measure. Thus the modified equation is not used for the work presented here. For a contact width of 2.5mm for a 25mm diameter sample, the stress is only reduced by 4% by the modified equation, so the implication of ignoring this correction factor is relatively small. Low and intermediate loading rate experiments The low and intermediate rate diametric compression tests were performed using a 5000lb Instron hydraulic test machine. The tests were conducted with a constant compressive displacement rate of 0.001mm/s for the low rate and 1mm/s for the intermediate rate experiments. Instrumentation for these tests included the load cell and displacement transducer in the test machine and high-speed video to record the material loading and failure process. High loading rate experiments

16 The high rate diametric compression tests were performed on a compression Split-Hopkinson Pressure Bar (SHPB) with 1¼” diameter aluminum input and output bars. Background on the SHPB and test methodology is provided in [10]. The SHPB input and output bars were instrumented with semiconductor strain gauges. The semiconductor gauges have a much higher sensitivity compared with traditional metal foil strain gauges and are essential for measuring the small strains associated with testing concrete in tension. A comparison of the signals from the two gauge types is shown in figure 3, which illustrates the noise reduction using the higher sensitivity semiconductor gauges.

Figure 3: Comparison of metal foil and semiconductor strain gauges - voltage signal (left); and strain signal (right)

A 24” long striker bar, accelerated by compressed nitrogen, was used to impact the input bar to produce the compressive incident pulse. This was used to load the sample at a compressive displacement rate of approximately 1000mm/s. The striker bar had a flat impact end, however a small amount of silicon grease on the impact face was used for shaping the incident pulse. This has the effect of damping high frequency components (ringing) associated with the impact and increases the pulse rise time to load the specimen more gradually. Ramping the load in this way is critical for allowing time for the stress to equilibrate, through multiple wave reverberations, in the sample. This is particularly important for brittle materials, which only undergo minimal strain before failure and equilibrium needs to be achieved before this time for a valid fracture strength to be reported. Pulse-shaping of the incident pulse for SHPB experiments is discussed further in [11]. Figure 4 shows a detail of the high loading rate experimental setup.

Figure 4: High rate loading experimental setup

17 Figure 5 shows experimental results performed here with and without pulse shaping. The shape of the incident pulse is changed significantly and the comparison between the transmitted and incident minus reflected pulses, which is proportional to the force on either side of the sample (ie. the stress equilibrium condition), is greatly improved.

Figure 5: Test with no pulse-shaping (left); and with pulse-shaping using silicon grease (right)

The specimen loading and failure process for the high rate tests was imaged using a Shimadzu Hypervision HPV2 high-speed camera. This camera produces 102 frames at 312x260 resolution at a frame rate of up to 1 million frames per second. The camera was synchronized to the strain gauge on the output bar, so the images could be related to the stress history in the sample from the transmitted pulse. The high-speed images provided a visual assessment that the crack was initiating in the centre of the specimen rather than at the loading contact points. Tensile strain and strain rate Using the diametric compression technique it is easy to measure the compressive displacement rate associated with a test. This can be calculated from the reflected pulse for the SHPB high rate tests and is available from the machine displacement transducer for the intermediate and low rate tests. However, in measuring the tensile strength of the material, it is the tensile strain and strain rate, which is of interest. For diametric compression tests, other researchers have instrumented the sample with a strain gauge [8,12] to measure the tensile strain. However, by gluing a strain gauge to the sample where it is expected to fail, this has the potential to reinforce the material and influence the tensile failure strength and crack initiation defect point. Other non-invasive techniques for measuring the tensile strain include full field optical techniques such as moire interferometry [13, 14] and Digital Image Correlation (DIC) [9,13,15,16]. The DIC technique involves high speed imaging of a random speckle pattern (natural or painted) on the specimen surface. The deforming images are correlated spatially at each time step to calculate the sample deformations and strain fields. The DIC technique was attempted here, however, it was found that the strain in concrete before failure was too small to reliably measure displacement fields using this technique. The camera resolution and speckle size were the limiting factors for the strain sensitivity and the background noise. Consequently, it was not possible to measure the tensile strain and strain rate for the diametric compression tests reported here and the rate sensitivity is reported using the stress loading rate instead. The stress loading-rate divided by the quasi-static material elastic modulus has been used by other researchers [12,17] to estimate the tensile strain rate from diametric compression tests. However, this is avoided here due to apprehension in using quasi-static values for the modulus, which may potentially be different under dynamic loading conditions. High loading rate direct tension tests

18 The high rate direct tension tests were conducted using a tension SHPB with ¾” steel input and output bars instrumented with semiconductor strain gauges. The tension SHPB is shown schematically in figure 6. It uses a hollow striker bar which impacts a flange at the end of the input bar to produce the tensile pulse. The samples were glued to detachable platens, which screwed into the input and output bars. The glue used was Sikadur crack fix structural epoxy, which was found to have a high rate bond strength of 20-25MPa when used in this application. This was high enough to successfully test the mortar and SAM35 concrete, however not strong enough for the UHPC and granite. Several tests were attempted for the UHPC, however these resulted in failure at the glue line rather than in the material itself. Figure 7 shows some of the direct tension specimens for mortar, SAM35 and UHPC. Note the very large aggregate size for SAM35 concrete.

Figure 6: Split Hopkinson Pressure Bar schematic for high rate direct tension testing

Figure 7: High rate direct tension specimens

19 RESULTS SAM35 concrete The results for the SAM35 concrete are shown in figure 8. It was tested using the diametric compression technique at low, intermediate and high loading rates and also at high rate in direct tension. The results show significant scatter and this can be attributed to the material in-homogeneity (due to the aggregate components) for the small specimen size used here. The mean tensile strength at low rate (quasi-static loading conditions) was 2.5MPa. This is approximately 10% of the indicated quasi-static compressive strength of 24MPa (3500psi).

Figure 8: SAM35 concrete tensile strength versus loading stress rate

The SAM35 concrete exhibited a strong loading rate effect, with the tensile strength increasing with loading stress rate. This increase was fairly linear over the loading rate range considered here and the mean tensile strength of 5.5MPa at high rate was more than double the low rate (quasi-static) value of 2.5MPa. The data points for direct tension at high rate are also shown in Figure 8. The comparison between the diametric compression and direct tension test techniques at high rate was very good for this material.

20 Mortar The mortar was also tested at low, intermediate and high loading rates using the diametric compression technique and at high rate in direct tension. These results are shown figure 9. The mortar material is more homogeneous compared with the SAM35 concrete. It still contains sand particles within a cement matrix, however it has no large aggregate components. This resulted in less scatter of the data points for the diametric compression tests. The mean tensile strength at low and intermediate loading rates was approximately the same, 3MPa, and increased to 4MPa at the high loading rate. Thus, the loading rate effect on tensile strength for this material is only evident at the high rate and differs from that seen with the SAM35 concrete. The comparison between the two test techniques at high rate was poor for the mortar material. The tensile strength measured by the diametric compression technique was significantly lower than that measured by the direct tension tests (although there was significant scatter in the direct tension results). Thus, compared with the SAM35 concrete results, it seems that the agreement between the two test techniques may be material dependent. The direct tension test produces a nominally uniaxial tensile stress state, whereas the diametric compression test produces a more complex biaxial tension-compression stress state as discussed in [5]. Thus it is possible that the SAM35 concrete response is similar under both stress states, whereas the mortar response is different due to different macro and microstructural mechanisms.

Figure 9: Mortar tensile strength versus loading stress rate

The effect on specimen disc thickness for the diametric compression tests was investigated for the mortar. The diametric compression results shown in figure 9 were obtained using disc samples with nominally 20mm diameter (D) and ¼” (6.35mm) thickness (L), resulting in a cylinder with L/D of approximately 0.3. In comparison, the direct tension tests were performed with cylindrical specimens with an L/D=1 (20mm diameter, 20mm thickness/length). Consequently, the diametric compression tests at high rate were repeated with specimens with an L/D=1 (although in a different orientation to the direct tension tests) and these results are presented in table 1 and figure 10. The tensile strength determined by the diametric compression technique should not depend on the specimen thickness, however, the mean tensile strength increased by approximately 10% using the longer specimens (L/D=1). This is possibly due to inertial material confinement effects. However, despite the small increase, there was still significant disparity with the direct tension results.

21 Table 1: Mortar high rate tensile strength results – effect of specimen geometry

Mortar High Rate Tensile Strength (MPa) Direct Tension Diametric Compression L/D~0.3 L/D=1 L/D=1 4.1 4.7 8.2 3.6 4 8.2 4.1 4.5 5.7 3.7 4 7.3 4.1 4.7 5.7 Mean: 3.9 Mean: 4.4 Mean: 7.0

Figure 10: Tensile strength of mortar as a function of specimen L/D and experiment type

22 Granite The tensile strength of the granite samples was evaluated at the low, intermediate and high loading rates using the diametric compression technique. These results are presented in figure 11. At low rate the mean tensile strength was approximately 15MPa. This increased to approximately 21MPa for the intermediate loading rate and to 22.5MPa at high rate. These values are substantially higher than those for the SAM35 concrete and mortar samples and is due to the strong crystalline microstructure of the granite. The increase in tensile strength with loading rate was more pronounced over the low to intermediate rate range, which is in contrast to the mortar response where the increase was only seen over the intermediate to high rate range. Combining these two responses, we would expect a fairly linear increase over low to intermediate to high rate. This is exhibited in the behaviour of the SAM35 concrete which is a composite of these two representative components.

Figure 11: Granite tensile strength versus loading stress rate

The high strength glue used for the SAM35 concrete and mortar direct tension tests was not strong enough for the granite so no high rate direct tension tests could be performed for this material. Thus there is no comparison between the two test techniques for this material.

23 Alcatraz concrete The concrete sample obtained from the Alcatraz prison in San Francisco was tested at the three loading rates in diametric compression and its tensile strength results are presented in figure 12. The material response was similar to that of the SAM35 concrete. It showed an increase in tensile strength with loading rate, which was fairly linear over the range tested. The mean tensile strength at low rate was approximately 5.5MPa, increased to 6.6MPa at the intermediate loading rate and to approximately 8MPa at the high loading rate. The Alcatraz concrete was stronger than the SAM35 concrete, however, the SAM35 concrete is a relatively low strength concrete with only 3500psi or 24MPa quasi-static compressive strength.

Figure 12: Alcatraz concrete tensile strength versus loading stress rate

24 Ultra-high performance concrete The tensile strength results for the Ultra-High Performance (UHP) concrete, tested at the three loading rates in diametric compression, are shown in figure 13. The UHP concrete was substantially stronger than the other concretes tested, with tensile fracture strengths similar to that of the granite. The mean tensile strength at low rate was approximately 15.5MPa, this increased 21MPa at the intermediate loading rate and to 24.5MPa at the high rate. Thus, similar to the granite, the increase in tensile strength with loading rate was more pronounced for the low to intermediate loading rate.

Figure 13: Ultra high performance concrete tensile strength versus loading stress rate

As discussed earlier, the UHP concrete contains short steel fibers of length 12.7mm and diameter 0.2mm, which are randomly distributed throughout the concrete at 6.2% by weight. In work by Williams, et al [18], a similar steel fibre reinforced high strength concrete material, Cor-Tuf, was tested at low strain rate in direct tension both with and without the fibers. They observed a slightly lower peak stress in the material containing the fibers, which suggests that the steel fibers are not contributing to increase the peak tensile strength (rather the opposite). The lower strength is presumably because of the failure initiation at the weak interfaces introduced between the steel fibers and the concrete. However, the benefit of the steel fibres is assumably in their ability to increase the tensile post peak load bearing capacity (the ‘ductility’) of the material. This is evident by the condition of the samples after the tests reported here. Whereas the other materials investigated fractured into several pieces, typical of a brittle material response, the UHP concrete fragments stayed held together by the steel fibres, maintaining some residual strength in the sample. Isaacs, et al [1] also found a sustained load bearing post peak response for the UHP concrete, which they attributed to the steel reinforcing fibers. They conducted preliminary tests on the UHP concrete at high strain rate -1 (approximately 100s ) using a direct tension SHPB. They conducted two tests on 76.2mm diameter specimens which were notched to 57.2mm diameter. The samples were glued to the input/output bars using epoxy and the notch was required to reduce the concrete cross section area to prevent failure at the glue line (as the UHPC tensile strength is higher than available epoxies). They reported peak stresses of 16.6Mpa and 15.4MPa for the two tests. This is lower than the mean peak tensile strength of 24.7MPa at high rate reported here. This discrpeancy is due to the difference in test methodologies. The notched specimen used by Isaac, et al creates a stress concentration at the notch and so the peak stress is less than true tensile strength of the material. Whereas for the tests results reported here, the diametric compression test produces a biaxial stress state and so the induced tensile region is not pure uniaxial tension. The difference in specimen size, which affects the lateral

25 interia and confinement in the sample, may also play a role. Finally, the tensile strain rate for the tests reported here was probably lower, although it has been shown here that the strength decreases with decreasing strain rate. DISCUSSION A summary of the diametric compression results (mean values) for the five materials is shown in figure 14. This shows an increase in tensile strength with loading rate for all of the materials tested. For the mortar the increase only occurred from the intermediate to high loading rates, whereas for the granite the increase was most significant at the low to intermediate loading rates. For the SAM35 and Alcatraz concretes, which are both combinations of these two representative components, the response was a combination of these individual material responses, where the tensile strength increased fairly linearly over the total loading rate range investigated. The Alcatraz concrete was stronger than the SAM35 concrete, however, the rate of increase in tensile strength with loading rate was similar for both concretes, as shown in figure 14, where the two lines are parallel. The UHP concrete was substantially stronger than the other concretes tested, with a similar tensile strength to the granite. However, the failure response of the UHPC was much less brittle compared to the granite. The UHP concrete exhibited post fracture residual load bearing capacity due to the steel reinforcing fibres, which has also been reported by other researchers [1,18].

Figure 14: Summary of average tensile strength versus loading stress rate for the five materials tested

The material rate sensitivity for the five materials was reported using the loading stress rate. It is recognized that most rate dependent numerical model models require the material strength as a function of strain rate, not stress rate. However, it is difficult to determine the tensile strain and corresponding tensile strain rate using the diametric compression technique. The DIC technique was attempted, however, was not successful here due to the small strains in concrete before failure. The high-speed camera was setup to image the entire sample surface. It is possible that the DIC technique could be successful by zooming in on the sample center and using a smaller speckle size to improve resolution and provide greater strain measurement sensitivity. This was not pursued here. The DIC technique has been successfully used for measuring the tensile strain in diametric compression tests by [9], however, the material tested (polymer bonded sugar explosive stimulant) most likely produced larger strains than those seen in concrete.

26 It is also to possible to convert from stress rate to strain rate through the elastic modulus of the material; however, the modulus is usually obtained from quasi-static experiments and may also be rate dependent. Consequently, this conversion has not been performed here and the material tensile strength is maintained as a function of stress rate. The experimental results presented in figures 8, 9, 11-13 show significant scatter and this is a consequence of the heterogeneous nature of many of the materials tested. This results from the aggregate components in the concrete and the steel fibers in the UHP concrete. The specimen sizes used here (20mm & 25.4mm diameter) were relatively small; this was influenced by the available experimental apparatus, specifically, the diameter of the split Hopkinson pressure bars used for high rate testing. The Australian standard for quasi-static concrete testing [7] specifies specimen size requirements relative to aggregate size and recommends 150mm diameter samples where the aggregate is less than 40mm and 100mm diameter samples where the aggregate is less than 20mm.The Defense Science and Technology Organization (DSTO) in Australia are currently building a 100mm diameter split Hopkinson pressure bar system, capable of testing concrete in both direct tension and compression. Thus, high rate material characterization of larger sized concrete specimens will be the subject of continuing future work. The limited comparison between the diametric compression and direct tension test techniques performed here at high rate suggests that the agreement between the two techniques is material dependent. However, more materials need to be tested to confirm this. The different test techniques produce different stress states and so it is feasible that while one material might behave similarly under the two stress states, another material might have a different response for each stress state. The comparison for the SAM35 concrete was very good, whereas the results differed for the mortar. However, the diametric compression technique enables an estimation of the tensile response of high strength brittle materials, such as granite and the ultra-high strength concrete, which would be very difficult to test otherwise in direct tension. It also permits the use of more common compression test apparatus, particularly in the case of high rate testing using a split Hopkinson pressure bar. CONCLUSIONS The tensile strength for various concretes and their representative constituent components was determined at three loading rates using the diametric compression (Brazilian/ split-tension) technique. All five materials tested showed an increase in tensile strength with increasing loading rate. Direct tension tests at high loading rate were also performed for several of the materials. The diametric compression and direct tension test techniques produce different stress states and agreement between the two methods appears to be material dependent. However, this finding is preliminary and more tests are required to confirm this. The diametric compression test remains a useful technique. It enables tensile testing of high strength brittle materials, such as granite and ultra-high performance concrete, which are very difficult to test otherwise in direct tension. However the diametric compression technique does not provide any information on the post fracture response of the material, which is important for materials such as the ultra-high performance concrete. This requires direct tension testing. REFERENCES 1.

J. Isaacs, J. Magallanes, M. Rebentrost & G. Wight, Exploratory dynamic material characterization tests th on ultra-high performance fibre reinforced concrete, Proceedings of 8 International Conference on Shock and Impact Loads on Structures, Adelaide, Australia, 2009.

2.

H. Schuler, C. Mayrhofer & K. Thoma, Spall experiments for the measurement of the tensile strength and fracture energy of concrete at high strain rates, International Journal of Impact Engineering, Vol 32, Issue 10, pp1635-1650, 2006.

3.

E. M. Williams, S. A. Akers & P. A. Reed, Laboratory characterization of SAM-35 concrete, US Army Corps of Engineers, Engineer Research and Development Center, ERDC/GSL TR-06-15, 2006.

27 4.

B. A. Graybeal, Material Property Characterisation of Ultra-High Performance Concrete, US Department of Transportation, Federal Highway Administration, FHWA-HRT-06-103, 2006.

5.

H. J. Petroski & R. P. Ojdrovic, The concrete cylinder: stress analysis and failure modes, International Journal of Fracture, Vol 34, pp263-279, 1987.

6.

ASTM C 496/C 496M, Standard test method for splitting tensile strength of cylindrical concrete specimens, ASTM International, 2004.

7.

AS 1012.10 Methods of testing concrete, Method 10: Determination of indirect tensile strength of concrete cylinders (‘Brazil’ or splitting test), Australian Standards, 2000.

8.

Q.Z. Wang, W. Li & H. P. Xie, Dynamic split tensile test of flattened Brazilian disc of rock with SHPB setup, Journal of Mechanics of Materials, Vol 41, pp252-260, 2009.

9.

S. G. Grantham, C. R. Siviour, W. G. Proud & J. E. Field, High-strain rate Brazilian testing of an explosive stimulant using speckle metrology, Journal of Measurement Science and Technology, Vol 15, pp1867-1870, 2004.

10. G. T. Gray III, Classic Split-Hopkinson Pressure Bar Testing, ASM Handbook Volume 8 Mechanical Testing and Evaluation, 2000. 11. D. J. Frew, M. J. Forrestal & W. Chen, Pulse shaping techniques for testing brittle materials with a split Hopkinson pressure bar, Experimental Mechanics, Vol 42, No. 1, March 2002, pp93-106 12. M. L. Hughes, J. W. Tedesco & C. A. Ross, Numerical analysis of high strain rate splitting-tensile tests, Computers & Structures, Vol 47, No 4/5, pp653-671, 1993. 13. J. E. Field, S. M. Wally, W. G. Proud, H. T. Goldrein, C. R. Siviour, Review of experimental techniques for high rate deformation and shock studies, International Journal of Impact Engineering, Vol 30, pp725775, 2004. 14. T. C. Chen, W. Q. Yin, P. G. Ifju, Shrinkage measurement in concrete materials using cure reference method, Society for Experimental Mechanics, 2009. 15. C. R. Siviour, S. G. Grantham, D. M. Williamson, W. G. Proud, J. E. Field, Novel measurements of material properties at high rates of strain using speckle metrology, The Imaging Science Journal, Vol 57, pp326-332, 2009. 16. C. R. Siviour, A measurement of wave propagation in the split Hopkinson pressure bar, Measurement Science and Technology, IOP Publishing, Vol 20, 2009. 17. D. E. Lambert & C. A. Ross, Strain rate effects on dynamic fracture and strength, International Journal of Impact Engineering, Vol 24, pp985-998, 2000. 18. E. M. Williams, S. S. Graham, P. A. Reed, T. S. Rushing, Laboratory characterization of Cor-Tuf concrete with and without steel fibres, US Army Corps of Engineers, Engineer Research and Development Center, ERDC/GSL TR-09-22, 200

Influence of strain-rate and confining pressure on the shear strength of concrete Pascal FORQUIN1 1

Laboratoire d'Etude des Microstructures et de Mécanique des Matériaux (LEM3), Université Paul Verlaine - Metz, Ile du Saulcy, 57045 Metz Cedex 1, France [email protected]

ABSTRACT. The paper presents an experimental method used to investigate the shear behaviour of concrete and rock-like materials in quasi-static and dynamic loading. This method is based on the use of PunchThrough Shear (PTS) specimen and a passive confining cell. PTS sample is a short cylinder in which two cylindrical notches are performed. The displacement of the central zone beside the peripheral zone produces a shear fracture in the ligament. Metallic (steel or aluminium) confining ring allows inducing a confining pressure in the fractured zone due to the dilating behaviour of concrete under shear deformation. The experimental configuration has been designed through a series of numerical simulations in which the Drucker-Prager plasticity model is used for modelling the concrete behaviour. Computations showed the necessity to practice radial notches in the peripheral zone of the sample for deducing the radial stress in the ligament from data of strain gages glued on the outer surface of the confining ring. Finally this experimental method was employed to analyse the strain-rate and pressure sensitivity of dry and wet concrete under shear loading. 1. Introduction Concrete structures as bridges, nuclear power stations or bunkers can be exposed to intensive dynamic loadings such as earthquakes, industrial accidents or projectile-impacts. During such loading, tensile and shear damage modes as spalling, scabbing, cratering, shear fracturing can be observed [1, 2, 3, 4]. In a case of penetration of a rigid projectile in the core of a target triaxial compression and shear stresses are generated, and the inertia of the surrounding material creates a passive confinement in front of the projectile. To improve the understanding and modelling of concrete behaviour under such confined loading, quasi-static and dynamic shear tests have been developed in LEM3 laboratory. In the last decades, several experimental techniques have been used to characterise the confined or unconfined shear strength and mode II fracture toughness of concretes and rock like materials. Compression-Shear Cube tests were pioneered by Rao [5] to investigate the influence of confining pressure on the mode II fracture toughness of marble and granite. The author observed a linear increase of the stress intensity factor in mode II (KIIC) with the confining level. The Short Beam Compression test relies on the use of samples with 2 notches orientated perpendicular to the loading direction. This technique allowed investigating the quasi-static and dynamic unconfined shear strength of concretes [6, 7]. The authors noted low strain-rate sensitivity in comparison to results obtained in dynamic tension. The Punch Through Shear Test (Fig. 1a) was introduced by Watkins [8] and used more recently by Backers [9] to determine the stress intensity factor in mode II of six types of rock as function of confining pressure. The cylindrical specimen includes two cylindrical notches on top and bottom faces. In a first step, the specimen is subjected to a pure hydrostatic pressure. In a second step an axial load is added to create the shearing state. This technique was also used by Montenegro et al [10] to investigate the shear behaviour of concrete and fracture energy of T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_4, © The Society for Experimental Mechanics, Inc. 2011

29

30

concrete as function of the hydrostatic pressure (Fig. 1b). The authors noted a significant increase of strength and fracture energy with confining pressure. However, supposedly due to strong difficulties in performing dynamic tests with a hydraulic confinement, very few authors have investigated the confined shear strength of concrete and rock-like materials in a wide range of strain-rate. In the present work, in a similar way than in quasi-oedometric compression tests [11, 12] a passive confining cell has been applied to Punch-Through-Shear specimens to characterise the confined shear strength of concrete over a wide range of strain-rates (Fig. 1c). Inner diameter of the lower notch and outer diameter of upper notch coincide so a straight cylindrical fracture surface is obtained (Fig. 1c). Numerical simulations of shear testing, experimental method, data processing and some experimental results are detailed in the present paper. Compression crown Confining cell Shear surface Compression (c) plug (b) Fig. 1. Punch-Through-Shear tests (a) conducted by Backers [9], (b) by Montenegro et al. [10] (c) and in the present work.

(a)

2. Experimental method 2.1. Numerical simulation of PTS tests 2D-axisymetric (Fig. 2a) and 3D (Fig. 2b) numerical simulations of shear tests have been performed to set the dimensions of the confining cell and concrete sample. The Drucker-Prager model was used for the specimen to describe the pressure sensitivity and dilation behaviour of concrete. The length of the ligament (10 mm) allows ensuring an almost homogeneous shear stress field in the ligament. Moreover the confining ring is not centred toward the symmetry plane of the specimen to get a uniform pressure between the confining ring and the concrete sample (Fig. 2a). Finally, aluminium and steel confining rings 10 mm thick and 15 mm in height have been considered for quasi-static and dynamic experiments.

Specimen

Crown Crown

Confining ring

Confining ring

(a)

Radial notch

Fig. 2. (a) Mesh of 2D axisymetric computation, (b) Mesh of 3D computation.

(b)

31

Furthermore, 3D computations (Fig. 2b) have been performed to investigate the influence of radial notches on the stress fields in the sample. It was demonstrated that 4 radial notches at 90° through the outer part of the specimen allow avoiding a self confinement of the sample and any radial cracks triggered during the test. 2.2. Experimental procedure The experimental devices used in quasi-static and dynamic shear testing are presented on Fig. 3a and 3b. Prior to each test a bi-component resin (Chrysor®) has been used to fill the gap between the sample and the confining ring. Moreover, the mean radial stress in the ligament (sheared zone) is deduced from the contact force between the sample and the confining vessel knowing the contact surface between the specimen and the ring (Scontact) and the area of the fracture surface (Sligament):

σ radial =

S contact Pcontact , Sligament

(1)

Radial notches

Strain gauges placed on the confining cell

(a)

(b)

LVDT Concrete sample Damping system

Laser interferometer

(c)

(d)

Fig. 3. Shear tests performed on R30A7 standard concrete. (a) Concrete specimen with radial notches after shear testing, (b) Concrete sample without radial notches after shear testing, (c) Device used in quasi-static shear testing, (d) Device used in dynamic shear testing and high-speed hydraulic press (LEM3). In a similar way than in quasi-oedometric test [11, 12] the contact pressure between the sample and the confining ring is deduced from data of strain gages glued on the metallic cell (Fig. 3a and 3b). A high-speed hydraulic press has been used to perform shear tests for jack speed above 0.01 m/s. Above this loading rate, a damping system is included (Fig. 3d) to assure a correct balance of the specimen (equal input and output forces) and to avoid any shock waves. In dynamic tests laser extensometers have been used to deduce the axial displacement of the central part of the

32

specimen. The mean shear stress in the ligament is obtained from the axial load according to Equation (2):

σ shear =

Faxial Sligament

(2)

2.3. Tested concrete Basic mechanical properties and composition of the tested concrete are gathered in Table 1. R30A7 concrete is a standard concrete containing hard siliceous aggregates with a grain size from 2 to 8 mm, sand, cement and water. All the specimens have been cored out from large blocks and stored in water saturated by lime in order to avoid the dissolution of portlandite in water. Two sets of specimens have been considered for shear testing: saturated specimens have been kept in water until few minutes before testing. “Dry” specimens were oven-dried at 60ºC during several weeks until a constant weight has been achieved. Table 1. Composition and quasi-static compression strength of dry and wet R30A7 concretes Composition

R30A7 [13]

Aggregates [kg/m3]

1008

Sand [kg/m3]

838

Cement [kg/m3]

263

Water [kg/m3]

169

Water/Cement

0.64

Mechanical properties

Wet R30A7

Dry R30A7

2380

2290

Young modulus (GPa) [14]

40

30

Quasi-static compression strength (MPa) [13]

32

42

Density

3. Measurement results 3.1. Influence of lateral pressure A series of confined shear experiments has been conducted with both confining rings (aluminium or steel rings) and varying the concrete moisture and loading rate. A comparison of shear tests performed with dry samples is proposed on Fig. 4. After the stress peak, a strong softening behaviour is noted followed by a plateau. Similar trends have been reported in hydraulic confinement PTS experiments [10]. The plateau results supposedly from the friction on the fractured surface. Moreover the lateral pressure between the confining cell and the specimen deduced from data of strain gages is also plotted. The resulting radial stress in the shear zone is computed from Eq. 1. According to Fig. 4 the maximum radial stress reaches about 52 MPa (in absolute value) with the steel confining ring whereas it does not exceed 38 MPa with the aluminium cell. This difference of confining pressure may explain the difference in shear strength observed between both tests. A confirmation that pressure sensitivity of concrete under shear loading may be explored through PTS experiments with passive confining cells varying the stiffness of the ring.

33

(b) (a) Fig. 4. Results of quasi-static shear tests performed on dry specimens with (a) aluminium and (b) steel confining cells. Strain-rate: 2e-5/s, concrete samples with radial notches. 3.2. Influence of moisture content Experimental results obtained with dry and wet specimens in quasi-static loading with an aluminium alloy confining vessel are reported on Fig. 5. In wet specimen the maximum shear stress reached 41 MPa whereas a maximum value of 54 MPa was obtained in dry sample. Again a strong decrease of shear stress is observed after the stress peak. Again the change of lateral pressure between the confining cell and the specimen is reported on the same plot as well as the resulting radial stress in the shear zone. Oppositely to Fig. 4 a similar radial stress about 30 MPa (in absolute value) is observed in both tests (dry and wet specimens) in the shear zone when the peak shear stress is reached.

(a) (b) Fig. 5. Results of quasi-static shear tests performed on dry (a) and wet (b) specimens. Strain-rate: 2e-5/s, aluminium confining cell, concrete samples with radial notches. 3.3. Influence of strain-rate A series of shear tests has been performed with dry and wet samples and the aluminium confining ring varying the strain-rate from 2e-5/s to 4/s. Experimental results are gathered on Fig. 6. The shear strength of wet samples is markedly lower than that of dry specimens whatever the loading rate.

34

Furthermore, a very limited increase of strength is pointed out for both sets of specimens (dry and wet sets). This result significantly differs to the conclusion obtained for concrete under tensile loading. Indeed, as shown by several authors [4, 14, 15] the tensile strength of wet concrete is double in a similar strain-rate range (1e-6/s - 1/s) whereas a quasi-nil rate-effect is noted with dry specimens. Thus, the well-known Stephan effect supposed to explicate the strain-rate sensitivity of wet concrete in tension is supposedly inoperative under shear loading. 70 y = 64,36x0,017

Maximum shear stress (MPa)

60

y = 46,99x0,017

50 40 30 20 10

Dry R30A7 concrete Wet R30A7 concrete

0 1,0E-05

1,0E-04

1,0E-03

1,0E-02

1,0E-01

1,0E+00

1,0E+01

Strain-rate (1/s)

Fig. 6. Quasi-static and dynamic shear tests performed on dry and wet specimens (aluminium confining cell, concrete samples with radial notches). Summary An experimental method is proposed to investigate the shear behaviour of concrete under quasistatic and dynamic loading. The set-up based on PST (Punch Through Shear) testing technique relies on the use of a passive confining ring. Radial notches are also performed on each sample so the radial stress in the ligament may be deduced from the measurement of strain-gauges glued on the confining ring skirting around the effect of self confinement due to the peripheral part of the specimen. Geometries of the specimen and of the ring have been defined through a series of numerical simulations. Experiments have been conducted on dry and water saturated concrete samples over a large range of strain-rate with two confining cells (steel and aluminum rings). The obtained results show a higher strength with dry samples than in wet ones and with the steel ring than with the aluminum ring. Furthermore, oppositely to previous results obtained in the literature in tension, both sets of concrete (dry and wet sets) show very small strain-rate sensitivity in the considered range of strainrate. The so-called Stephan effect is though to be inoperative under shear loading. References [1] Li Q.M., Reid S.R., Wen H.M. (2005), Telford A.R., Local impact effects of hard missiles on concrete targets, Int J. Impact Eng. 32, 224-284. [2] Forquin P., Arias A., Zaera R. (2008), Role of porosity in controlling the mechanical and impact behaviours of cement-based materials, Int. J. Impact Eng., 35 (3), 133-146. [3] Forquin P., Hild F. (2008), Dynamic Fragmentation of an Ultra-High Strength Concrete during Edge-On Impact Tests, ASCE J Eng Mech, 134 (4), 302–315.

35

[4] Forquin P., Erzar B. (2010), Dynamic fragmentation process in concrete under impact and spalling tests, Int. J. Fracture, 163 : 193 – 215. [5] Rao Q. (1999), Pure shear fracture of brittle rock. Doctoral thesis, Division of rock Mechanics, Lulea University, Sweden. [6] Watkins J., Liu K.L.W. (1985), A finite element study of the short beam test specimen under mode II loading. Int. J. Cement Composites and Lightweight. 7, 39-47. [7] Ross C. A., Jerome D.M., Tedesco .J.W, et Hughes M. (1996), Moisture and Strain Rate on Concrete Strenght. ACI Material Journal 93. 293-299 [8] Watkins J. (1983), Fracture toughness test for soil-cement samples in mode II. Int. J. Fract. 23, 135-138. [9] Backers T. (2004) Fracture Toughness Determination and Micromechanics of Rock Under Mode I and Mode II Loading. Ph.D. thesis, University of Potsdam, Germany. [10] Montenegro O., Sfer, Carol I. (2007), Characterization of concrete in mixed mode fracture under confined conditions. ICEM13 conference, Alexandroupolis, Greece [11] Forquin P., A. Arias, R. Zaera. (2007), An experimental method of measuring the confined compression strength of geomaterials, Int. J. Solids Struct., 44 (13), pp. 4291-4317. [12] Forquin P., G. Gary, F. (2008b), Gatuingt. A testing technique for concrete under confinement at high rates of strain, Int. J. Impact Eng., 35 (6), 425-446. [13] Vu X.H., Malecot Y., Daudeville L., Buzaud E. (2009), Experimental analysis of concrete behavior under high confinement: effect of the saturation ratio, Int. J. Solids Struct., 46 1105-1120. [14] Erzar B., Forquin P. (2010), Experiments and mesoscopic modelling of dynamic testing of concrete, Mechanics of Materials (submitted for publication). [15] Cadoni E., Labibes K., Albertini C., Berr M., Giangrasso, M. (2001), Strain-rate effect on the tensile behaviour of concrete at different relative humidity levels, Materials Structures, 34, 21-26.

Dynamic Tensile Properties of Steel Fiber Reinforced Concrete

R. Chen1*, Y Liu2, X. Guo2, K. Xia3, and F. Lu1 1. College of Science, National University of Defense Technology, 410073 Changsha, P.R. China 2. National Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, 100081 Beijing, P.R. China 3. Department of Civil Engineering, University of Toronto, Toronto, Ontario, Canada M5S 1A4 ABSTRACT: This paper presents experimental results on three kinds of concretes, plain concrete (PC), 1.5% and 3% steel fiber reinforced concrete (SFRC), subjected to dynamic tensile loading. The cylinder splitting (Brazilian disc) specimens are loaded by a modified Split Hopkinson Pressure Bar (SHPB) with various loading rates (100~500 GPa/s). From the experiments it is found that there is a significant enhancement in tensile strength with increasing loading rates. Crack gauges mounted on the specimen showed that the average fracture velocity of 3% SFRC during the test is 730 m/s whereas that of PC is 790 m/s. Both the tensile stress history and the recovered specimen have demonstrated that SFRC has superior resistance to crack initiation and crack propagation as compared with PC. Keywords: Steel fiber reinforced concrete (SFRC); Brazilian Disc; SHPB; Fracture velocity INTRODUCTION Fiber reinforcement is one of the most important modification methods to alter the brittle nature of plain concrete (PC). The use of steel fiber reinforced concrete (SFRC) has been continuously increasing during the past decades because of its enhancement of material performance in toughness and crack control [1]. With the addition of steel fibers, SFRC shows an enormous increase in strength, toughness, and ductility from static mechanical tests [2]. As a result, SFRC has been widely used in many civil engineering structures. However due to its limitations in the flexibility and resistance to shrinkage cracking, it has been rarely used in bridge pavement. The success of SFRC in structural engineering has encouraged the development of newer high performance materials for critical infrastructures subjected to extreme loadings. Among these loading cases, intense dynamic loading is a unique one because of the worldwide increase of terrorist attacks against civilian targets. It is thus critical to better understand the impact resistance of SFRC and methods to enhance its performance under such loadings [3]. Over the past few years, significant progress has been made in the characterization of dynamic properties of SFRC [4-7]. However, the tensile strength has rarely been measured. It is thus the objective of this work to quantify the dynamics tensile behaviour of PC and SFRC. We use the Brazilian disc sample and apply the dynamic load with a modified split Hopkinson pressure bar (SHPB) system. The influence of the loading rate and fiber volume fraction on dynamic tensile strength is studied. To illustrate the tensile failure process, crack velocities are monitored. EXPERIMENT Sample preparation and geometry The concrete matrixes were designed for accommodating the volume fraction (Vf) of 0, 1.5% and 3% of steel fibers. The following materials were used in the fabrication of SFRC specimens: tap water, cement, steel fiber, standard sand, Silica fume, fly ash, water reducer and steel fiber. Tables 1 and 2 present the properties of raw material and the compositions of each component.

*

Corresponding author. Tel: +8673184573276; fax: +8673184573297. E-mail address:[email protected]

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_5, © The Society for Experimental Mechanics, Inc. 2011

37

38 Table1. The properties and manufactures of raw material Properties Cement Standard Sand Silica Fume Fly Ash Water Reducer Steel Fiber

Manufacture

Description

Density g/cm3

Grade 42.5 Dia.: 0.25-0.65mm 1250 Head Type I by JGJ28-86 Poly Ling salt water reducer Dia.: 0.15-0.2mm Average length: 15mm

2.59 2.15 2.2 1.06

Huxin Cement Co., Ltd. Xiamen ISO Standard Sand Co., Ltd. GaofengMiners Powder Co., Ltd. Chonghui Fly Ash Co., Ltd. JinSheng Tci. and Tech. Co., Ltd

7.8

Xintu Engineering Fiber Co., Ltd.

Table2 The compositions of SFRC/PC Water (W/B) 1# 2# 3#

0.2 0.2 0.2

Cement (C) 1 1 1

Binder (B) Silica Fume (SF/C) 0.3 0.3 0.3

Fly Ash (FA/C) 0.25 0.25 0.25

Sand (S/B)

Water Reducer (Solid Content)

Steel Fiber (Vf)

1.18 1.18 1.18

1% 1% 1%

0 1.5% 3.0%

The mixture was poured into a steel cubic mold (300 mm) for 2 hours. The block obtained was then cured in a standard condition of 20 ℃ and >95% relative humidity for 28 days. After curing, concrete cores with a nominal diameter of 50 mm were drilled from the block and then sliced to obtain discs with an average thickness of 24 mm. All the disc samples were polished afterwards resulting in a surface roughness variation of less than 0.5% of the sample thickness. The modified Split Hopkinson Pressure Bar system A 40 mm diameter SHPB system was employed as the loading apparatus in this study, as shown in Fig. 1. When the striker hit the incident bar, it generates an incident pulse in the incident bar. The incident wave travels through the flange without resistance because there is a gap between the flange and the rigid mass in the beginning. The incident pulse propagates along the incident bar to hit the sample, leading to reflected stress wave and transmitted stress wave. Denote the incident wave, reflected wave and transmitted wave by εi, εr, and εt. Based on the one-dimensional stress theory, and assuming the stress equilibrium prevails during dynamic loading (i.e., εi +εr =εt), we can determine the histories of the tensile stress σ(t) within the sample as [8]：

σ (t ) =

A0 E ε (t ) πRB 0 t

where E0 is the Young’s modulus of the bar, and A0 is the area of the bar; R is the radius of the sample and B is the thickness of the sample. Rigid mass Striker

Incident bar

Specimen

Transimitted bar

Absorbtion bar

Crack Gauge

Pulse shaper Flange

Fig. 1. Schematics of the preloaded spit Hopkinson pressure bar (SHPB) system with the Brazilian disc specimen. In order to ensure single pulse loading, the momentum trap technique is adopted in our Hopkinson bar setup. Detailed

39 explanation of the SHPB procedure and momentum trap can be found in the literature [9, 10]. EXPERIMENT RESULT Fig. 2 shows the original and recovered SFRC specimens. The recovered crack gauge showed that the crack travels along the crack gauge. It also can be observed as the steel fibers are pulled-out from the fragments.

Fig. 2 The virgin and recovered SFRC specimen. Dynamic force balance Fig. 3 shows the forces on both ends of the sample. The dynamic force on one side of the sample is the sum of the incident and reflected force waves, and the dynamic force on the other side of the sample is the transmitted force wave. It can be seen from Fig. 2 that the dynamic forces on both sides of the samples are almost identical during the whole dynamic loading period. We thus can use the equation mentioned above to determine the tensile strength. 300

Force (kN)

200 100 0 -100

In. Tr. Re. In.+Re.

-200 -300

0

40

80

120

160

Time (μs) Fig. 3. Dynamic force balance during a typical preloaded SHPB with the Brazilian disc specimen.

Effects of loading rate The tensile strength is the maximum value of the tensile stress history. There is a approximately linear region in σ(t), and its slope is taken as the fracture loading rate, with which σ varies. Fig. 4 shows some examples of stress versus time curves for different loading rates and two materials. One can observe how the maximum strength is reached for the three loading rates in different time due to the imposed loading rate. In the case of PC, the duration of the fracture process is about 120, 90 and 60 μs for 142, 253 and 409 GPa/s, respectively; while for the 1.5% FRC, it has been obtained 120, 100, 80 and 68 μs for 192, 233, 340 and 0.449 GPa/s, respectively. Fig.5 shows stress histories of different types of specimens with similar loading rate of 330 GPa/s. The tensile strength of the plain concrete is 14.7 MPa, while the one of 1.5% SFRC and 3% SFRC are 16.9 MPa and 19.3 MPa respectively. The postpeak ductility in the stress-time curve for the test of SFRC specimen shows the effect of reinforced fibers, as the steel fibers are being pulled-out from the materials. In addition, the post-peak ductility increases with increasing fiber volume fractions.

40 Fig.6 shows the conclusion of our tests. The tensile strength increase with increasing loading rates and fiber volume fractions. The slops of different materials are similar to each other.

20

25

Vf=0

142

253

409

12 8

Loading Rate (GPa/s)

192

232

340

449

15 10 5

4 0

Vf=1.5%

20

Stress (Mpa)

16

Stress (Mpa)

Loading Rate (GPa/s)

0

0

40

80

120

160

200

0

50

100

150

200

250

Time (μs) Time (μs) (a) (b) Fig. 4. Stress versus time curves for different loading rates: (a) PC; (b) 1.5% SFRC

25 Loading Rate: ~330GPa/s Vf : 0 1.5%

Stress (Mpa)

20

3%

15 10 5 0

0

50

100

150

200

250

Time (μs)

Fig. 5. Dynamic tensile histories of SFRC/PC specimens with various volume fractions. 22.5 0 1.5% 3.0%

Strength (Mpa)

20.0 17.5 15.0 12.5 10.0

0

100

200

300

400

500

600

Loading Rate (GPa/s)

Fig. 6. Tensile strength increase with increasing loading rates and fiber volume fractions. Fracture velocity A crack gauge was mounted on the specimen during the test. The typical signal of the crack gauge and corresponding loading history are shown in Fig. 7. There is a significant post-peak ductility after the crack travels through the specimen, which indicates the pull-out of the fiber from the concrete matrix. The details of the crack gauge can be seen from the insert of Fig.

41 7. The distance between each wile in the crack gauge is 1.1 mm, and the time step can be measured by the jump of the crack gauge signal. From which, the crack velocity can be measured for each test, as shown in Fig. 8(a). The average crack velocities of three types of specimens are shown in Fig. 8(b). The average crack velocity of PC is 790 m/s where as the ones of 3% SFRC is 730 m/s. The phenomenon demonstrates that one of the important properties of SFRC is its superior resistance to cracking and crack propagation, which is also proved by both the tensile stress history and the recovered specimen. 5

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(a) (b) Fig. 8. The crack velocities of different conditions: (a) the crack velocity of each test, and (b) the effects of fibers’ volume fraction. CONCLUSION We present the dynamic tensile properties of PC, 1.5% SFRC and 3% SFRC with the loading rates from 100 to 500 GPa/s. The experiments were conducted on the modified SHPB system with the Brazilian disc specimens. The results show that there is a significant enhancement in tensile strength with increasing loading rates and the volume of steel fiber. Crack gauges mounted on the specimen show that the average fracture velocity of SFRC during the test is 730 m/s whereas that of PC is 790 m/s. The result demonstrates the fibers do substantially increase the post-cracking ductility, or energy absorption of the material. ACKNOWLEDGMENTS This work was supported by the open fund of National Key Laboratory of Explosion Science and Technology through Grant No. KFJJ08-1, and the Natural Science Foundation of China (NSFC) through Grant No. 10872215 & 10902100. K.X. acknowledges the support by Natural Sciences and Engineering Research Council of Canada (NSERC) through Discovery Grant No. 72031326.

42 REFERENCE [1] Daniel J.I., Gopalaratnam V.S., Galinat M.A., et al., "Report on Fiber Reinforced Concrete", (2002). [2] Cadoni E., Meda A., and Plizzari G.A., "Tensile behaviour of FRC under high strain-rate", Materials and Structures. 42(9): 1283-1294 (2009). [3] Bindiganavile V. and Banthia N., "Generating dynamic crack growth resistance curves for fiber reinforced concrete", Experimental Mechanics. 45(2): 112-122 (2005). [4] Wang Z.L., Wu L.P., and Wang J.G., "A study of constitutive relation and dynamic failure for SFRC in compression", Construction and Building Materials. 24(8): 1358-1363 (2010). [5] Wang Z.L., Liu Y.S., and Shen R.F., "Stress-strain relationship of steel fiber-reinforced concrete under dynamic compression", Construction and Building Materials. 22(5): 811-819 (2008). [6] Lok T.S. and Zhao P.J., "Impact response of steel fiber-reinforced concrete using a split Hopkinson pressure bar", Journal of Materials in Civil Engineering. 16(1): 54-59 (2004). [7] Liu Y.S. and Chen M.C., "Study on mechanical properties of ultra-short steel fiber reinforced concrete under dynamic compression". in: Vol. Innovation & Sustainability of Modern Railway Proceedings of Ismr' 2008. 240-247 (2008). [8] Berenbaum R. and Brodie I., "Measurement of the tensile strength of brittle materials", British Journal of Applied Physics. 10(6): 281-286 (1959). [9] Xia K., Nasseri M.H.B., Mohanty B., et al., "Effects of microstructures on dynamic compression of barre granite", International Journal of Rock Mechanics and Mining Sciences. 45(6): 879-887 (2008). [10] Chen W.W. and Song B., "Split Hopkinson (Kolsky) Bar Design, Testing and Applications", Mechanical Engineering Series. 388 (2010).

Effect of Liquid Environment on Dynamic Constitutive Response of Reinforced Gels Sashank Padamati, Vijaya B. Chalivendra*, Animesh Agarwal, Paul D. Calvert Dynamic Material Testing Laboratory University of Massachusetts, Dartmouth, MA, 02747 Corresponding author: [email protected] ABSTRACT Quasi-static and dynamic compressive behavior of three different types of hydrogels used for soft tissue applications are tested using a modified split Hopkinson pressure bar. Three kinds of hydrogels: (a) Epoxy hydrogels, (b) Epoxy hydrogels reinforced with definite orientation of three-dimensional polyurethane fibers and (c) fumed silica nano particles reinforced hydrogels with different weight fractions are considered in this study. Swellability of the all the hydrogels considered are studied and controlled by mixing different ratios of jeffamines and epoxides. The three dimensional pattern of the fibers are generated by a rapid robo-casting technique. Split Hopkinson pressure bar (SHPB) was used for dynamic loading and a pulse shaping technique was used to increase the rising time of the incident pulse to obtain dynamic stress equilibrium. A novel liquid environment technique was implemented to observe the dynamic behavior of hydrogels when immersed in water. Experiments were carried out at dynamic loading conditions for different strain rates with and without water environment. Results show that the hydrogels are rate sensitive. Also the yield strength of hydrogels decreased and elongation percent increased when they were immersed in water. INTRODUCTION Mimicking human tissues with artificial materials which are bio-compatible and bio-degradable are of vast interest. Hydrogels are extensively used in biomedical applications [1] especially in tissue engineering. They are used as scaffolds to guide the growth of new tissues. Hydrogels are highly corrosion-resistant and their swelling nature provides an aqueous environment comparable to soft tissues. Hydrogels are water swollen polymer networks, which have a tendency of absorbing water when placed in aqueous environment. As the hydrogels tend to swell when fully saturated, these are suitable for biological conditions and are ideal in use of drug delivery. These are hydrophilic polymers that can absorb up to 1000 times their dry weight in water. This high water content makes it a resemblance to living tissues. When hydrogels are completely saturated with water they tend to have poor mechanical properties. Usage of hydrogels as biological material in different parts of human body is increasing at a rapid pace. With a huge advantage of varying the properties, hydrogels are being used as a replacement for many damaged parts of the body. One example of such a part is human cartilage, which is subjected to high amounts of loads. Recent studies show many researchers have tested and published the mechanical properties of the hydrogels in Quasi-static compression state. Testing soft materials for the dynamic behavior has been a challenging task. Many researchers have worked on soft materials and have achieved success in testing them. Mechanical properties of soft materials like lungs, stomach, heart and liver have been found out at dynamic loading conditions [2]. Dynamic compression tests using polymer split Hopkinson pressure bar (SHPB) apparatus were performed on bovine tissues [3]. PAG gels, an alternative for tissue stimulant has been tested for the dynamic response [4]. Subhash et al., has found the compressive strain rate behavior of ballistic gelatin. All the above testing was performed under open to air environment. Most of the tissues in human body are surrounded with some type of aqueous environment. This environment change in the body will possess the tissue to behave with different properties. Hence testing the hydrogels in open air environment might vary their properties as compared to aqueous environment. In this paper a novel technique for testing the hydrogels under aqueous environment in both static and dynamic loading conditions were designed, developed and implemented. Attempts were made to increase the mechanical properties of hydrogels by adding glass fibers and were partially successful [6]. In this study three-dimensional polyurethane fibers and fumed silica nano particles are reinforced into different hydrogels to improve low mechanical properties. A novel liquid environment chamber built with acrylic glass is developed for testing hydrogels in water. The need of testing materials at high strain rates developed a method which produced mechanical properties of materials at elevated strain rates which in recent time very well known as split Hopkinson pressure bar. This method was widely used for

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_6, © The Society for Experimental Mechanics, Inc. 2011

43

44 testing materials such as metals as they have high impedance values. However, soft materials like foams and hydrogels have very low impedance values which resist the use of SHPB. The main disadvantage for dynamic testing of soft materials using SHPB arises in capturing the transmitted pulse. Soft materials such as hydrogels have extremely low impedance values, which transmits a very low magnitude pulse into the transmission bar. This makes it very hard to capture the transmitted pulse. In literature, few approaches were adopted to magnify the transmitted pulse, (i) Using a hollow transmission bar Song and Chen, 2004, this technique magnifies the magnitude of the transmitted pulse up to 10 times. However, the magnitude of the pulse in case of hydrogels is so low when compared to incident pulse increasing it 10 times still makes it barely visible, (ii) Using polymeric bars Liu and Subhash, 2006, these bars can provide decent transmitted pulse but it requires lots of assumptions. The wave attenuation and dispersion in the bars requires idealized assumptions about the bar material properties and extensive mathematical treatment of pulses should be obtained in the experiments. To overcome this issue, foil gages were replaced with semiconductor gages on input and output bars. Semiconductor gages have a very high gage factor and these gages have 50-75 times more sensitivity than normal foil gages [4]. This technique is vastly used for characterizing ultra soft and very low impedance materials. Biological materials such as tissues, soft bones and hydrogels are some materials which have been using this technique in recent times. In this paper semiconductor gages are used for data acquisition. It can be noticed from the above studies that characterization of hydrogels was successfully studied in open to air environment. It was identified from the above studies that there was no detailed study conducted to understand the effect of liquid environment on hydrogels under static and dynamic loading conditions. Hence this paper mainly focused on studying the quasi-static and dynamic constitutive behavior of reinforced hydrogels inside liquid environment. EXPERIMENTAL DETAILS Materials and preparation Three types of hydrogels were considered in this study and are epoxy-amine based. Three kinds of hydrogels are (a) Epoxy hydrogels, (b) Three dimensional Polyurethane fiber reinforced hydrogels and (c) Nano particle reinforced hydrogels. Polyethyleneglycol diglycidylether(PEGDGE) was purchased from Sigma Aldrich. Jeffamine® ED 600 is donated from Huntsman Chemical Company. 15 Minutes Degassing

PEGDGE

ED 600

Mould Water

3 hours

24 hours

48 hours

Oven at 700C

Curing at room temperature

Curing in Deionized water

Fully swelled Hydrogel

Figure 1 Material preparation procedure Amines were cross linked to epoxides in aqueous solution to form hydrogels. An equal molar ratio of PEGDGE and ED 600 are mixed together with 20-30% weight fraction of H2O. The entire mixture is stirred at a regular pace for 15 min. Thus formed mixture is poured in to acrylic molds and placed in oven at 65-700C for 3 hours. These molds are then cured at room temperature for 24 hours. These samples are placed in de-ionized water for 48 hours to completely swell to get fully saturated epoxy amine hydrogels. Figure 1 shows the pictorial view of the material preparation

Figure 2 Asymtek dispensing system

Figure 3 Three dimensional optical image of PU fiber construct

45 Figure 2 shows a dispensing system machine were 3D fibers are extruded. This machine is produced from Asymteck. As Figure 2 shows the machine consists of a disposable syringe, a needle and a gas pressure apparatus to pump the solution on to the surface. Required patterns of fibers can be extruded from the machine through a computer program. The machine moves in all the three directions which make it more flexible to generate complex design patterns.10-20% polyurethane solution were prepared in dimethylformamide(DMF). The fibers are extruded through 100µ EFD needle and deposited into 3D structure by pressure driven Asymteck Automove 402 dispensing system as shown in Figure 3. The fiber formation process is based on wet spinning process and deposition technique is similar to robocasting technique. These 3D structures are placed in to hydrogel matrix and the solution is ultrasonicated to form fiber reinforced hydrogel. Fumed silica nano particles were purchased from Sigma Aldrich. Nano particles are measured for weight and are mixed with epoxy amine hydrogel. This mixture is ultrasonicated in a water bath up to 10 min allowing the particles to disperse. In this paper 2%, 4% and 6% are three different weight fractions of particles used.

Figure 4 Split Hopkinson pressure bar The split Hopkinson pressure bar was used for dynamic testing of hydrogels. Traditionally SHPB consists of an incident bar and a transmission bar and are all made of Aluminum 7075-T651. The striker bar used in these experiments has a diameter of 12.7mm and length 304.79mm. Incident and transmission bars have the diameter of 12.7 mm. Incident bar is 1828.80mm long and transmission bar is up to a length of 1220mm. A Copper C11000 pulse shaper was used in the testing to obtain dynamic stress equilibrium and is placed using KY jelly at the impact end of the incident bar as shown in Figure 4. The specimen is sandwiched between incident bar and transmission bar. Specimen has the thickness of 2.36mm and diameter of 6.35mm. Molybdenum disulfide lubricant is applied between specimen and the contacting surfaces of bars to minimize the friction. When the striker bar impacts the incident bar, an elastic compressive pulse is generated. This pulse deforms the pulse shaper mounted at the incident bar and creates a ramp in the pulse. This pulse propagates through the incident bar and reaches the specimen bar interface. When it reaches the interface due to impedance mismatch between the bar and specimen some part of the pulse (reflected pulse) is reflected back in to the incident bar and some transmits through the specimen (transmission pulse) to the transmission bar. Strain gages are mounted on both the bars which provide time-resolved measures of the elastic strain in the pulses. For softer materials strain gages were replaced with semi conductor gages as the gage factor of semiconductor gage are very high when compared to foil gages which gives a very high sensitivity to semiconductor gages. Typical pulses obtained from SHPB using semiconductor gages are shown in Figure 5.

Figure 5 Typical pulses obtained from SHPB

Figure 6 Liquid environment specimen fixture

46 A novel environmental chamber was designed for SHPB at the interface of incident and transmission bar as shown in Figure 6. This fixture ensures that specimen is placed in between bars and an aqueous environment is surrounded around the specimen during the testing time. The fixture is made of acrylic glass has rubber washers fixed in both the bars ensuring no leak of aqueous solution. Sample tests were ran with the liquid environment chamber and ensured that the setup doesn’t restrict the movement of bars. The specimen undergoes uniform homogenous deformation and the analysis based on one-dimensional wave theory. In order to make the experiment valid when testing soft materials the specimen should be in dynamic equilibrium and should have a near constant strain rate. In this paper, using SHPB we obtained a dynamic stress-strain behavior of hydrogels at a constant strain rate range of 3600-4000/s.

(a)

(b)

Figure 7 Specimen holders in Quasi-static compression (a) without water (b) with water conditions The quasi-static compression tests are performed on Instron Materials Testing Machine 5585. Experiments were performed at slow loading rates of 1mm/min. The tests were continued until the specimen is completely crushed at specified loading rate. A setup was also built for testing the hydrogels under water in quasi-static loading conditions. Figure 7 shows the liquid environmental setup built for quasi-static compression experiments RESULTS AND DISCUSSION

5

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As discussed above a novel environmental chamber was implemented in this study, consistency experiments were ran and the results are showed in Figure 8. Results show that the methods implemented and the material behavior in all tested cases remained same.

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47 As the hydrogels are soft materials dynamic testing of these materials is a challenging task. The data that we obtain from the tests should be trustable. So, experiments for consistency were performed on all the hydrogels. Currently 2% nano particle hydrogels were randomly considered and presented in this paper. 1.2

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Figure 9 Comparison of True Stress vs. True Strain curves under quasi-static compression (a) epoxy and nano composite hydrogel (b) fiber reinforced hydrogel Figure 9(a) shows the comparison of the data obtained from epoxy gels and nano particle gels. Figure 9 clearly indicates the increase of strength of material of pure epoxy gels as the nano particles are added to it. We can also observe that the rate of increase of stress of the material has a significant change when tested in with and without water conditions. When tested in water the material has a low rate of increase and higher elongation percentages. These results conforms that the materials such as hydrogels with a water content of 95% as their volume subjects to a different behavior when tested in different environmental conditions. Figure 9(b) shows the comparison of fiber reinforced hydrogels in with and without water environment condition. The curves shows that the material behavior is non linear. The compressive stress of the material increased with the increase in strain percent. But the flow stress in this case has a highly nonlinear effect when compared to epoxy and nano particle gels. The non linearity in the material behavior is due to the fibrous structure reinforcement in the hydrogels matrix. The fibers reinforced in the hydrogel when subjected to compression show more stability towards the material allowing it to take higher loads. The material when tested in normal atmospheric conditions has yield strength of 4MPa and has percentage elongation of 70%. Some resistance was observed around the specimen as a result yield strength of the material decreased increasing the percentage elongation of the specimen. 1.8

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Figure 10 Comparison of True Stress vs. True Strain curves of hydrogels considered under fatigue loading

48 From the quasi-static experimental data we observed that the data obtained for epoxy gels and nano particle reinforced gels has a linear increase in the stress in both the experimental conditions. And the fiber reinforced hydrogels had a highly non linear increase. So, to confirm the results and to see if there is any hysteresis effect in these specimens fatigue loading experiments was conducted. Fatigue loading was done at a rate of 1 mm/min and up to 10 cycles. The specimens were subjected to an elongation of 20%. Figure 10 shows the results of fatigue loading on all the hydrogels considered in this study. From the figure we can see that there is no significant hysteresis observed in epoxy and nano hydrogels. But the fiber reinforced gels have been subjected to hysteresis confirming the non linear behavior observed in the quasi-static experiments. 6

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Figure 11 Comparison of True Stress vs. True Strain curves under dynamic compression on epoxy and nano composite hydrogel (a) without water (b) with water conditions Figure 11(a) shows the dynamic true stress- strain curves for a constant strain rate plotted against the quasi-static true stressstrain curves for each hydrogel specimen without water conditions. It can be seen from figures that all the hydrogels are rate sensitive and show significant rate sensitivity. The constitutive response in all the above cases is non linear. The rate of increase of yield stress in case of without water experiments is very high. The yield strength of the material is reached with lesser percentage elongation. It can be observed from the graphs that the flow stress has a significant change when compared to the quasi-static conditions. Figure 11(b) shows the dynamic true stress- strain curves for a constant strain rate plotted against the quasi-static true stressstrain curves for each hydrogel specimen with water conditions. It can be seen from figure that the materials even when immersed in water are rate sensitive and show significant rate sensitivity. The constitutive response of the materials when immersed in water shows a high non linearity. The pulses when compared to that of regular without water experiments showed a significant change in the strength, elongation and signature of the pulses. The strength of the materials decreased, percentage elongation of the materials increased when tested in water. The reason for this effect in the material as mentioned in the above sections, a plasticizing effect is observed by the material when they are immersed in water and tested. The rate of increase of yield stress in case of with water experiments is less when compared to without water. As mentioned the water experiments produce plasticizing effect, as a result the material experiences high percent elongations. Even in the case of water experiments it can be observed from the graphs that the flow stress has a significant change when compared to the quasi-static conditions. Figure 12 shows the true stress vs. true strain curve for fiber reinforced hydrogel for both the experimental conditions. Fiber reinforced hydrogels also experiences same behavior as that of the other gels. They are rate sensitive and have the highest yield strength of all the hydrogels tested. Fiber reinforced hydrogel have the highest dynamic yield strength of 12MPa followed with 6% nano particle hydrogels with 6MPa and epoxy gel with 4MPa in without water conditions.

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Figure 12 Comparison of True Stress vs. True Strain curves of fiber hydrogel under dynamic compression CONCLUSIONS In this paper, a detailed experimental study was conducted to investigate the effect of water environment on three different hydrogel. The change in their material properties was showed significantly. Following are the major outcomes of this study: Quasi-static characterization 

A linear true stress vs. true strain response was observed with a large deformation in both epoxy hydrogels and nano particle reinforced hydrogels.



A non linear true stress vs. true strain response was observed for fiber reinforced hydrogels. These hydrogels also possess large percentage elongations.



Fatigue loading experiments showed no hysteresis in both epoxy hydrogel and nano particle reinforced hydrogel. This also confirms the hydrogel under low loading rates possess linearity.



In case of PU fiber reinforced hydrogels, fatigue loading on the specimens showed a significant change in the hysteresis. This confirms the fiber hydrogels even under low loading rates show non linearity.



In water the material is experiencing softening effects which increase the percentage elongation at break.

Dynamic characterization 

Epoxy hydrogel showed significant rate sensitivity at different strain rates. Considerable difference in yield stress from quasi-static conditions to dynamic conditions was observed and the difference ranges between 0.5MPa and 3.5MPa.



In water epoxy hydrogel showed almost similar characteristics as that of the regular experiment. The yield stress of the material is decreased by increasing the percentage elongation. Considerable difference in yield stress from quasistatic conditions to dynamic conditions was observed and the difference ranges between 0.5MPa and 2.0MPa.



Nano particle reinforced hydrogels also showed a significant change in their properties when tested under water by reducing the yield stress. Considerable difference in yield stress from quasi-static conditions to dynamic conditions was observed and the difference ranges between 0.5MPa – 0.7MPa and 4.0MPa – 5.5MPa. Under water the yield stress values show 0.5MPa – 0.7MPa and 1.5MPa – 3.0MPa.



Fiber reinforced hydrogels also show a decrease in yield stress and increase in elongation when tested under water environment.

50 

In conclusion the rate of increase of flow stress is less in water environment and constitutive response is non linear in all the hydrogels considered.

REFERENCES 1.

Metters AT, and Lin CC, Biodegradable Hydro gels: Tailoring Properties and Function through Chemistry and Structure, Biomaterials, Wong JY, and Bronzino JD, Eds. NY: CRC Press, 5.1-5.44, 2007

2.

Saraf.H Ramesh, K.T., Lennon, A.M., Merkle, A.C., Roberts, J.C., Measurement of the dynamic bulk and shear response of soft human tissues, Experimental Mechanics 47, 439-449, 2007.

3.

Sligtenhorst, C.V., Cronin, D.S., Broadland, G.W., High strain rate compressive properties of bovine muscle tissue determined using a split Hopkinson bar apparatus, Journal of Biomechanics 39, 1852-1858, 2007.

4.

Paul Moy, Tusit Weerasooriya, Thomas F. Juliano, Mark R. VanLandingham, and Wayne Chen, Dynamic Response of an Alternative Tissue Simulant, Physically Associating Gels (PAG), Army research laboratary, 2006.

5.

J. Kwon, G. Subhash, Compressive strain rate sensitivity of ballistic gelatin, Journal of Biomechanics 43, 420-425, 2010.

6.

Yang, Shukui L, Lili Y, Dongmei H and Fuchi W: Dynamic compressive properties and failure mechanism of glass fiber reinforced silica hydrogel, Material Science and Engineering., 824-827, 2010

7.

Song, B., Chen, W.W., Dynamic stress equilibration in split Hopkinson pressure bar tests on soft materials, Experimental Mechanics 44,300–312, 2004.

8.

Liu, Q., Subhash, G., Characterization of viscoelastic properties of polymer bar using iterative deconvolution in the time domain. Mechanics of Materials 38, 1105–1117, 2006.

Ballistic Gelatin Characterization and Constitutive Modeling D. S. Cronin Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario, Canada ABSTRACT Ballistic gelatin is widely used as a soft tissue simulant for non-penetrating and penetrating, and the mechanical properties of gelatin are known to be highly sensitive to strain rate and temperature. Mechanical compression testing was undertaken across a range of strain rates at constant temperature to evaluate the material response. The material strength and stiffness increased with increasing strain rate, while the strain to failure was relatively constant across a wide range strain rates. The mechanical test data was implemented in two constitutive models: a quasi-linear viscoelastic model, commonly available in explicit finite element codes, and a tabulated hyperelasticity model. The implementations were verified using simulations of the experimental tests and it was found that the quasi-linear viscoelasticity model did not adequately capture the low and high strain rate response across the range of data. The tabulated hyperelasticity model was found to provide accurate representation of the material across the range of strain rates considered, and included a damage function to predict material failure. INTRODUCTION Ballistic gelatin powder is produced from biological materials (skin, bone and tendons) through extraction with hot water in an acidic environment for Type A gelatin (Sellier 1994). This powder is then combined with water, heated and mixed, and conditioned at 4°C for a period of 2-3 days [Jussila 2004]. Type A, 250 Bloom is the most common gelatin formulation used in ballistic testing. The two commonly used mixtures are the Fackler formulation (10%, 4⁰C) (Fackler 1988) and the NATO formulation (20%, 10⁰C). The 10% formulation was considered for this study since it has been shown to better represent the properties of muscle tissue compared to the 20% gelatin 3 (Van Sligtenhorst 2004). Gelatin is known to have a density close to that of soft tissue (approximately 1060 kg/m ) (Sellier 1994) and a an elastic wave speed of approximately 1540 to 1550 m/s Van Bree (1996, 1998, 1999). Previous studies have investigated the characterization of ballistic gelatin, and evaluation as a tissue simulant (Van Sligtenhorst 2004, Cronin 2006, Caillou 1994, Van Bree 1998, Sellier 2004, Salisbury 2009) based on mechanical properties and penetration studies. A recent study on the mechanical properties of 10% gelatin -1 -1 -1 provided mechanical data at strain rates of 0.01 s , 0.1 s , and 1.0 s (Figure 1) (Cronin 2010). Intermediate -1 strain rate data, ~100 s has previously been reported (Cronin 2009) and was augmented for this study with -1 additional test data. High strain rate data on the order of 1000 s (Salisbury 2009) was not considered in this study.

Figure 1: 10% Gelatin compressive mechanical properties at low and intermediate strain rates

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52 It should be noted that all test data considered corresponds to 10% Type A, Bloom 250 Gelatin at a temperature of 4°C, which had been conditioned (aged) for approximately 72 hours. The actual conditioning time varied, and the gelatin was determined acceptable using the standard BB impact test (Jussila 2004). It has been shown that insufficient conditioning time or temperatures exceeding 4°C can significantly change the measured mechanical properties (Cronin 2010). METHODS The material data was recorded in terms of applied compressive force and deformation and was converted to engineering stress and strain based on the initial sample dimensions. The true stress and strain, required for constitutive modeling purposes, were calculated from equations 1 and 2 assuming constancy of volume of the samples. This is a reasonable assumption up to the initiation of material damage. σTrue= σEng(1+e)

Eqn.1

ε=ln(1+e)

Eqn.2

Based on the test data (Figure 1) it was apparent that the mechanical properties of ballistic gelatin were sensitive to deformation rate and this should be considered in any constitutive model. Available constitutive models for viscoelastic materials can be generally classified as linear, quasi-linear, and non-linear viscoelastic. The selection of a model was guided by the material response, and by commonly available models in numerical codes since the goal of this work was to use the properties and models to evaluate gelatin response to impact conditions. As a frame of reference, unpublished relaxation test data on 10% gelatin has shown that under relatively low strains (~20% engineering strain), 10% gelatin samples continue to relax for durations exceeding 15 hours. This -1 suggests that the stress strain curve at 0.01 s (Figure 1) does not represent the instantaneous elastic response or fully relaxed behavior of gelatin. However, this rate is relatively low for impact phenomena and was used to represent the fully relaxed behavior in this study. It should be noted that most constitutive models require complete definition of the material response across the range of strain rates and strains considered, and in the case of a inputting a discrete curve in a model, care must be taken to ensure the curve is defined across the entire range of anticipated strains. For this study, the instantaneous response curve was extended beyond the test data using a hyperelastic curve fit, and the tensile properties were determined by assuming symmetric behavior in terms of true stress and strain. This curve is shown in Figure 2, along with the original compression test data. The quasi-linear viscoelastic model as proposed by Fung (1993) for soft tissues and similar materials is widely implemented in numerical modeling codes and was evaluated for this study. Specifically, the implementation in the explicit finite element code LS-Dyna (LSTC, 2009) was investigated. This model requires a representation of the instantaneous elastic response of the material, typically expressed in terms of a hyperelastic stress (σH(ε)), and predicts strain rates through the addition of a viscoelastic stress component (σV(ε,t)) based on the convolution integral as shown in equations 3a, b and c. σ(ε, t) = σH(ε) + 𝜎𝑣 (ε, t) 𝑡

Eqn. 3a

𝜕𝜀

Eqn. 3b

𝐺(𝑡) = ∑𝑛𝑖=1 𝐺𝑖 𝑒 −𝐵𝑡

Eqn. 3c

𝜎v = ∫0 𝐺(𝑡 − 𝜏) 𝑑𝜏 𝜕𝜏 -1

The mechanical properties at a strain rate of 0.01 s were selected to provide the instantaneous elastic response of the gelatin material. It should be emphasized that this selection is somewhat arbitrary; typically a strain rate below which response will not be evaluated is selected to provide this response. It was found that commonly -1 2 available hyperelastic models (Mooney, Ogden) could represent the stress-strain curve at 0.01 s with r values exceeding 0.999. The specific implementation of the quasi-linear viscoelastic model allowed for an effective stress versus strain expressed as a polynomial, or actual test data to be implemented directly. For the purposes of this study, the properties were implemented directly using the tension and compression response shown in Figure 2.

53

Figure 2: Instantaneous elastic response data plotted with original test data A second constitutive model using tabulated stress-strain and strain rate data (Kolling et al. 2007, LSTC 2009) was investigated. This model uses the same concept of an instantaneous hyperelastic response (shown in Figure 2) and incorporates rate effects through the inclusion of actual stress-strain and strain rate data. Similar to the quasi-linear viscoelastic model, the compression test data was extended to strains beyond the material failure strain, and assumed symmetric in tension based on the true stress-strain data. This data was included in the material model, and material failure was addressed through a damage approach, described below. Following model fitting, the data was implemented in an explicit finite element code (LS-Dyna, LSTC 2009) and evaluated using single element test cases, and multiple element test cases. The element size for the simulations was on the order of 1mm, a typical value used to simulate ballistic gelatin in impact scenarios (Cronin, 2009). Both constitutive models considered incorporate a damage-based failure model that could be used to predict the onset of damage and final failure of the material. In this study, the form of the damage model used was based on the first invariant of stretch (equation 4). The full damage model allows for dependence on the square of the first invariant and the second invariant, but only the first term could be justified based on the available test data. The variable K corresponds to 100% damage (D). A second parameter (h) determines the initiation of damage in the material (equation 5). 𝑓(𝐼1 ) = (𝐼1 − 3) 1

𝐷 = � �1 + 2

𝑐𝑜𝑠�𝜋(𝑓−𝐾) �� ℎ𝐾

0.0 𝑓 ≤ (1 − ℎ)𝐾 (1 − ℎ)𝐾 < 𝑓 < 𝐾 1.0 𝑓 ≥ 𝐾

Eqn. 4

Eqn. 5

Figure 3: Finite element models – 1mm single element [L] and 25 mm diameter cylinder [R] (not to scale)

54 RESULTS AND DISCUSSION The quasi-linear viscoelastic model was evaluated relative to the data presented in Figure 1. Although an -1 2 adequate fit could be achieved with the quasi-linear viscoelastic model for strain rates of 0.1 s (r = 0.993) and -1 2 -1 2 1.0 s (r = 0.980), the fit for the intermediate rate data (105 s , r =0.885) was not satisfactory. Also, the general shape of the curve was not in good agreement with the experimental data. This was verified using single element simulations, but was not pursued further. It is anticipated that this modeling approach would not be able to accurately represent data at higher strain rates. The tabulated hyperelasticity model was evaluated using single element test cases in uniaxial compression, and found to reproduce the material data accurately across the range of strains and strain rates considered. Simulations on a multi-element cylindrical sample (Figure 3) were also undertaken and found to produce consistent results, although some oscillation occurred early in the simulation as the sample achieved equilibrium, and a small radial inertial effect was noted. The damage model was fit to the experimental test data presented in Figure 2 using a spreadsheet calculation -1 (K=5.7 and h=0.2) and incorporated in the material model. The compression test at a strain rate of 0.01 s was simulated and found to be in good agreement with the experimental data (Figure 5).

Figure 4: Tabulated hyperelasticity model predictions

Figure 5: Damage model predictions compared to experimental test data

55 CONCLUSIONS The mechanical properties for 10% Type A, 250 Bloom ballistic gelatin were investigated using two different constitutive models. A quasi-linear viscoelastic model, commonly available in finite element codes, was found to provide reasonable predictions for the low strain rate data, but did not accurately predict the intermediate strain rate data in terms of stress magnitude or the shape of the curve. A tabulated hyperelasticity model, based on the actual stress-strain and strain rate curves from the experimental tests, was found to accurately predict the material response across the range of strains and strain rates considered. In addition, the available material damage model was capable of predicting material damage and ultimate failure. Future work will investigate alternate hyper-viscoelastic models. The benefit of this class of material model is the capability to predict material response without the need or bias for specific test data across the range of strains and strain rates. ACKNOWLEDGEMENTS Natural Sciences and Engineering Research Council of Canada REFERENCES

Caillou J.P., Dannawi M., Dubar L., Wielgosz C. (1994) Dynamic behaviour of a gelatine 20% material numerical simulation. Personal Armour System Symposium, pp. 325-331. Cronin, D.S., Salisbury, C.P., Horst, C., “High Rate Characterization of Low Impedance Materials Using a Polymeric Split Hopkinson Pressure Bar”, SEM 2006, Society for Experimental Mechanics, St. Louis, 2006. Cronin, D.S. and Falzon, C., "“Dynamic Characterization and Simulation of Ballistic Gelatin”, Society for Experimental Mechanics, Albuquerque New Mexico, June 2, 2009. Cronin, D.S., Falzon, C.*, “Characterization Of 10% Ballistic Gelatin To Evaluate Temperature, Aging And Strain Rate Effects”, Journal of Experimental Mechanics, Online November 25, 2010, In press. Fackler, M. L. and Malinowski, J. A. “Ordnance Gelatin for Ballistic Studies,” The American Journal of Foresnic Medicine and Pathology, vol. 9 pp. 218-219, 1988. Fung, Y.C., Biomechanics: Mechanical Properties of Living Tissues, 2nd Edition, Springer-Verlag, 1993. Jussila J. (2004) Preparing ballistic gelatine - review and proposal for a standard method. Forensic Science International 141:91-98. Kolling, S., Du Bois, P., Benson, D., Feng, W., “A tabulated formulation of hyperelasticity with rate effects and damage”, Computational Mechanics, Volume 40, 2007. Kwon and Subhash, "Compressivestrainratesensitivityofballisticgelatin", Journal of Biomechanics, 43 (2010) pp 420-425. LSTC, 'LS-Dyna Theory Manual", LSTC, 2009 LSTC, 'LS-Dyna User's Manual", LSTC, 2009 Salisbury, C.P. and Cronin, D.S., "Mechanical Properties of Ballistic Gelatin at High Deformation Rates", Experimental Mechanics, Volume 49, Number 6 (2009), pp 829–840. Sellier, K.G. and Kneubuehl, B.P., Wound Ballistics and the Scientific Background; Elsevier, 1994, ISBN 0-444-81511-2. van Bree, J. and van der Heiden, N., "Behind armour pressure profiles in tissue simulant", Personal Armour Systems Symposium 96, September, 1996. van Bree, J. and van der Heiden, N., "Behind armour blunt trauma analysis of compression waves", Personal Armour Systems Symposium 98, Colchester, U.K., September, 1998. van Bree, J. and Fairlie, G., "Compression wave experimental and numerical studies in gelatine behind armour", 18th International Symposium on Ballistics, San Antonio Texas, November 15-19, 1999. VanSligentorst, C. "High Strain Rate Compressive Properties of Bovine Muscle Tissue", MASc Thesis, Department of Mechanical Engineering, University of Waterloo, 2004.

Strain Rate Response of Cross-Linked Polymer Epoxies under Uni-Axial Compression Stephen Whittie [email protected] Paul Moy [email protected] Andrew Schoch [email protected] Joseph Lenhart [email protected] Tusit Weerasooriya [email protected] Army Research Laboratory Weapons and Materials Research Directorate Bldg 4600 Deer Creek Loop Aberdeen Proving Ground, MD 21005-5069 ABSTRACT The strain rate responses of several cross-linked polymer epoxy materials were investigated under uni-axial compression at low to high strain rates. The properties of the epoxy were tailored through a variety of monomer choices including aromatic, which provide stiff, high glass transition structural materials, and aliphatic, which can form elastomers. The molecular weight and molecular weight distribution as well as the chemical functionality of the monomers can be varied to provide further control over the mechanical response. High rate experiments (greater than 1/sec rates) were conducted using a modified split-Hopkinson Pressure bar (SHPB) with pulseshaping to ensure that the compressive loading of the specimen was at constant strain rate under dynamic stress equilibrium. In this paper, moduli and yield strengths as a function of strain-rate of the epoxies are presented and compared in an effort to understand the effect of the different tailoring to their mechanical response. INTRODUCTION In general, there are two types of epoxies, the glycidyl and non-glycidyl epoxy resins. Each differs in the way that the epoxies are prepared. Glycidyl epoxies use a condensation reaction of dihydroxy compound, dibasic acid, and or a diamine and epichlorohydrin. Whereas, non-glycidyl epoxies are created using peroxidation of olefinic double bond. Each type of epoxy can be tailored to have a rubbery to a brittle toughness by changing the molecular weight, cross-link density, or adding a dispersed toughener into the cured epoxy. These chemically cross-linked thermosetting polymer networks are used in a wide range of applications including composite laminates, anticorrosive coatings, polymer membranes, and electronics. Epoxies can range from glassy structures to flexible gels [1]. In order for epoxies to be used for composite laminates for armor applications, the material response under a variety of strain rates needs to be fully understood to develop complete material models. As discussed in Rittel [2], as the epoxies undergo deformation majority of the mechanical energy is transformed into heat. The heat causes the specimens to undergo intrinsic strain softening followed by strain hardening. This phenomenon is described by Govaert et al. [3]. In 1995 Arruda et al. used an infrared detector to analyze the temperature effects of PMMA during quasi-static to intermediate strain rate compression testing. Arruda then reported that softening of the material occurring after the yield is a result from the combination of strain hardening/softening and thermal softening [4]. T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_8, © The Society for Experimental Mechanics, Inc. 2011

57

58 In this work, uni-axial compression experiments conducted on two epoxies are used to understand the mechanical responses from low to high strain rates. Quasi-static experiments were conducted using a servohydraulic Instron. High strain rate compression testing was carried out on a Kolsky bar which is also referred to as a Split-Hopkinson Pressure Bar (SHPB). The SHPB technique allows for the study of materials under a dynamic loading with high strain-rate deformation. -3 Jordan et al. [5] investigated the compressive behaviors of Epon 826/DEA epoxy from strain rates of 10 up to 4 -1 10 s . Similarly, Chen et al. [6] used a modified SHPB from high strength aluminum to study the dynamic response of Epon 828/T-403. Moy et al. [7] used the same testing techniques to study the mechanical behavior of PMMA at various strain rates. All showed results describing the effects of increasing yield strength and modulus as a function of strain rate. MATERIAL The epoxies used in this experimental investigation are Di-Glycidyl of Bisphenol A cross-linked with Jeffamine Diamine D400 (DGEBA D400), and Di-Glycidyl of Bisphenol F cross-linked with Jeffamine Diamine D400 (DGEBF D400). The polymers bisphenol (resin) and Jeffamine (curing agent) were acquired from Aldrich Chemical Company. The epoxy resin and curing agent were mixed and cured at the Army Research Laboratory Each epoxy mixture were poured into a custom fabricated stainless steel mold designed to provide a specimen diameter of 6.35 mm. The molded epoxy, right cylinder is about 152.4 mm long. Figure 1 shows a picture of each half of the mold. The bottom of the fixture is completely enclosed and the top has a threaded opening. Prior to the casting, mold release was used to ensure the epoxy can be easily removed from the steel mold. Otherwise forcible removal can damage the relatively thin epoxy rod. All compression experiments were conducted at room temperature. Once the epoxy has completely cured, right cylinder 3.18 mm gage length compression test specimens were then fabricated. Diameter to length ratio of these compression specimens is 2:1. Then, the specimen faces at both ends were machined to be flat and parallel to each other with a smooth surface finish. After the final machining, the test samples for DGEBA D400 and DGEBF D400 were then annealed in an oven at 76°C and 66°C respectively for 4 hours to eliminate any residual stresses caused from the cutting tool during machining. This annealing temperature is 20°C above the glass transition temperature, T g for DGEBA D400 and DGEBF D400 which is 56°C and 46°C, respectively.

Figure 1: Epoxy Molds for Compression Specimens LOW RATE EXPERIMENTS -3 -1 The quasi-static (10 /sec), (10 /sec), and intermediate (1/sec) strain rate experiments were conducted on an Instron 1331 servo-hydraulic test frame. A LabView-based program was used to generate an exponentially decaying waveform through a WaveTek function generator to command the Instron controller. This process

59 allowed for a constant true-strain rate compression experiments to be achieved. The load and displacement data were acquired using a separate LabView data acquisition program. Mineral oil was used as lubrication on the specimen ends to minimize friction during the compression. Hardened steel compression platens with a swivel base were used to compensate for any minor misalignment in the loading train. In order to correct the displacement data for the effect of machine compliance, the compliance of the Instron test machine was measured and the displacement due to machine compliance was subtracted from the measured machine displacement.. HIGH RATE EXPERIMENTS High rate experiments were conducted on a modified SHPB. A SHPB consists of a striker, an incident bar, and a transmission bar as shown in figure 2. The working principle of such setup is well documented [8, 9]. The bars used for the incident bar and transmit bar in the test setup were made from high strength Al 7075 that were specified to be centerless grounded to a diameter of 19 mm. Various annealed copper disks with different diameters and thicknesses were used for pulse shaping.

Transmission bar Strain Gage

Incident bar

Specimen

Striker

Strain Gage Pulse shaper

Power Supply

Power Supply

Digital Oscilloscope

Figure 2: Schematic of pulsed-shaped SHPB Set-up Assumptions were made during the SHPB testing that homogeneous deformation in the specimen occurs, identical incident and transmitted bars, and lastly analysis was based on one-dimensional wave theory [10]. The nominal strain rate described by Kolsky in the specimen is ,

(1)

Where is the elastic bar-wave velocity of the bar material, L is the original specimen gage length, and is the time-resolved strain from the reflective pulse from the incident bar. Integration of equation 1 with respect to time yields the time-resolved axial strain of the specimen. The axial stress, σ, of the specimen is determined using the equation ,

(2)

Where As is the cross-sectional area of the specimen, At is the cross-sectional area, Et is the Young’s modulus, and is the time-resolved axial strain from the transmission bar.

RESULTS AND DISCUSSION Figure 3 shows a typical set of stress pulses from the input and output bars from the SHPB with pulse shaping. The stress waves (incident, reflected, and transmitted) are identified on the graph.

60 0.03 Reflected

0.02

Volt (V)

0.01 Transmitted

0 -0.01

input bar (v)

-0.02

output bar (v)

Incident

-0.03 -0.04 -200

0

200

400

600

800

1000

1200

Time (sec) Figure 3: Typical input and output stress pulses Plotting the true strain as a function of time (Figure 4) for the high rate experiment shows that the epoxy sample is undergoing a near constant strain rate. Since a majority of the curve is linear, a valid constant strain rate compression test has been achieved. 0.5

True Strain

0.4

0.3

0.2

0.1

0

0

50

100

150

200

250

300

Time (sec) Figure 4: Compressive True Strain as a function of Time Stresses in the specimen/bar interface at each ends are shown in Figure 5. This indicates that dynamic stress equilibrium in the high rate experiments is achieved through pulse-shaping of the incident wave. This ensures that uniform loading is accomplished throughout the test without cyclic loading pulses going through the material from the incident stress wave.

61 200 specimen/input bar interface

True Stress (MPa)

150

specimen/output bar interface

100

50

0

0

20

40

60

80

100

Time (sec) -1

Figure 5: Stresses at Specimen/Bar Interfaces for DGEBF D400 at 4200s Strain Rate with Pulse-Shaping Figure 6 summarizes the mechanical response for DGEBF D400 from low to high strain rates. The individual plots with either an “X” or “O” states whether the specimen has failed or not failed, respectively. The “failed” compression test specimens are deemed by evidence of visible cracks. As a result, the epoxy specimens tested at 2000/sec strain rate and higher have various cracks in the direction of loading. In this paper, no high rate experiments were completed for the DGEBA D400. Specimen failure did not occur for the test at 1500/sec. Just like the specimens at the lower rate, this specimen did not have any visible signs of cracks. The specimen’s initial gage length was 3.19 mm. The total measured deformation is about 20%. Eight hours after testing, the specimen recovered to a near 100% of the initial gage length. It is quite evident that adiabatic heating effects are present at this strain rate. When the temperature for this epoxy material reaches above the T g the behavior would change to behaving similarly to an elastomer. Thus, the value for the yield strength at 1500/sec cannot be confirmed.

62 200

X X X True Stress (MPa)

150

X

4200/s

Failed X Did not Fail O

3500/s 2600/s 2000/s

O

100

1500/s 1/s 0.001/s

50

oo 0

0

0.1

0.2

0.3

0.4

0.5

0.6

True Strain

Figure 6.True Stress as a Function of True Strain for DGEBF D400 Figure 7 shows the stress-strain behavior for the epoxies DGEBA D400 and DGEBF D400 at the lower strain rates. The DGEBA D400 has a significant lower flow stress for a given strain rate than the DGEBF. For all quasi and intermediate experiments, strain hardening was evident at strains approximately 0.04. Interestingly, the stress-strain behavior for DGEBF at 0.001/sec is similar to DGEBA at 1/sec.

50 1/s DGEBF D400 0.001/s DGEBF D400

True Stress (MPa)

40

1/s DGEBA D400 0.100/s DGEBA D400

30

oo

0.001/s DGEBA D400

oo

20

10

0

o

0

0.1

0.2

0.3

0.4

0.5

0.6

True Strain Figure 7: True Stress as a Function of True Strain for DGEBA D400 and DGEBF D400 at Quasi-Static through Intermediate Strain Rates To illustrate the effects of strain rate on the different epoxies, the yield strength and modulus as a function of strain rate are shown in Figure 8 and Figure 9, respectively. From both figures, it demonstrates that the DGEBF

63 epoxy is rate sensitive. As stated earlier, high rate experiments were not conducted for DBEBA. Furthermore, these results show a bi-linear behavior for the DGEBF in both plots. 200

Yield Strength (MPa)

DGEBF D400 150 DGEBA D400 100

50

0 0.001

0.01

0.1

1

10

100

1000

4

10

Strain Rate Figure 8: Yield Strength as a Function of Strain Rate for DGEBA D400 and DGEBF D400

5 DGEBF D400

4

Modulus (GPa)

DGEBA D400 3

2

1

0 0.001

0.01

0.1

1

10

100

1000

4

10

Strain Rate Figure 9: Modulus as a Function of Strain Rate for DGEBA D400 and DGEBF D400 Figure 10 shows the pictures of both (a) DGEBA D400 and (b) DGEBF D400 epoxy compression specimens tested at the1/sec intermediate strain rate. In both cases, no signs of cracks were present.

64

(a)

(b)

Figure 10: Photo of the Tested Compression Experiments for (a) DGEBA and (b) DGEBF at 1/sec

(a)

(b)

(c)

Figure 11: Photo of the Tested Compression Epoxy DGEBF at (a) 1500/sec, (b) 2000/sec, and (c) 2600/sec Figure 11a, 11b, and 11c shows the pictures of the compression tested DGEBF D400 specimens tested at 1500/sec, 2000/sec, and 2600/sec, respectively. Since the specimen tested at 1500/sec did not have any cracks, the final gage length was measured at 3.18 mm which is only 0.24% of the initial gage length. Yet, the maximum strain from the SHPB calculations is about 20%. This proves that the epoxy transitioned from a glassy state to a rubbery state. The specimen DGEBF D400 at strain rate of 2000/sec showed cracking in the center of the specimen along with cracks radiating outward. Although the specimen failed, the specimen continued to remain intact. This specimen was tested to strain of 8%. Figure 11c shows DGEBF D400 at a strain rate of 2600/sec. This specimen was tested to strain of 9%. At this strain rate, the specimen showed many more cracks then the previous specimen tested at a strain rate of 2000/sec. The specimen exhibit cracking located in the center of the specimen which is in a circular pattern. The specimen also has radial cracks moving outward towards the specimen edges where the edges start to split apart from the body of the specimen. Although the damage is severe the specimen remains intact. Specimens tested at strain rates of 3500/sec and 4200/sec were pulverized during testing. In fact, at these higher strain rates, some of the recovered pieces show evidence of melting and fusing together of the debris. The specimens tested at the strain rate of 3500/sec had larger fragments compared to the specimens tested at 4200/sec. SUMMARY Compression testing on two epoxies, DGEBA D400 and DGEBF D400, was conducted at quasi-static, intermediate, and high strain rates. A near constant strain rate was achieved for the SHPB experiments by stress pulse-shaping the incident wave. Summary of the stress-strain responses for DGEBF show that the epoxy is rate

65 sensitive. Both the yield strength and modulus increase with the increase in the strain rate. Furthermore, results indicate a bi-linear mode from the yield strength and modulus as the function of strain rate graphs. Adiabatic heating effects are quite evident. In particularly to the compression experiment conducted at 1500/sec. The specimen at this rate recovered to a near 100%. Future efforts will investigate the adiabatic heating effects for the high rate experiments as well as other epoxy groups. References [1] Knox, C.K., Andzelm, J., Lenhart, J.L., High Strain Rate Mechanical Behavior of Epoxy Networks from th Molecular Dynamics Simulations. 27 Army Science Conference (2010). [2] Rittel, D., On the Conversion of Plastic Work to Heat during High Strain Rate Deformation of Glassy Polymers. Mechanics of Materials 31 (1999), pp. 131-139. [3] Govaert, L.E., van Melick, H.G.H., Meijer, H.E.H., Temporary Toughening of Polystyrene through Mechanical Pre-conditioning. Polymer 42 (2001), pp. 1271-1274. [4] Arruda, E.M., Boyce M. C., Jayachandran, R., Effects of strain rate, temperature and thermomechanical coupling on the finite strain deformation of glassy polymers. Mechanics of Materials. Volume 19, Issues 2-3, 1995, pp. 193-212. [5] Jordan, J.L., Foley. J.R., Siviour, C.R., Mechanical Properties of Epon 826/DEA Epoxy. Mechanics of TimeDepend Materials. (2008), pp. 249-272. [6] Chen, W., Zhou, B., Constitutive Behavior of Epon 828/T-403 at Various Strain Rates. Mechanics of TimeDepend Materials. (1998), pp. 103-111. [7] Moy, P., Weerasooriya, T., Chen, W., Hsieh, A., Dynamic Stress-Strain Response and Failure Behavior of PMMA. Conference Proceedings of 2003 ASME IMECE. Washington DC. [8] Davies, E.D.H., Hunter, S.C., The dynamic compression testing of solids by the method of the split Hopkinson pressure bar. Journal of the Mechanics and Physics of Solids. Volume 11, Issue 3 (1963), pp. 155-179. [9] Wu, X.J., Gorham, D.A., Stress Equilibrium in the Split Hopkinson Pressure Bar Test. Journal de Physique IV (1997), 7(C3), pp. 91-96. [10] Kolsky, H., An Investigation of the Mechanical Properties of Materials at very High Rates of Loading. Proc. Roy. Soc. London, B62, (1949), pp. 676-700.

Strength and Failure Energy for Adhesive Interfaces as a Function of Loading Rate

Tusit Weerasooriya1 [email protected] C. Allan Gunnarsson1 [email protected] Robert Jensen1 [email protected] 1

Army Research Laboratory Weapons and Materials Research Directorate Bldg 4600 Deer Creek Loop Aberdeen Proving Ground, MD 21005-5069 Weinong Chen2 [email protected] 2

Department of Aerospace and Mechanical Engineering Purdue University

ABSTRACT Adhesives are used to bond different materials to resist impact and penetration. This adhesive bond layer is a frequent source of failure when subjected to impact loadings. Therefore, it is necessary to measure the cohesive strength and failure behavior, especially at high loading rates. Experimental methods are limited in characterizing the mechanical behavior of adhesive bonds at high loading rate, so a unique experimental method and a specimen geometry was developed to determine the effect of loading rate on the failure (Mode I) of a commercially available adhesive (EPON 828). Four-point bend specimens consisting of two aluminum “wings” bonded together with the adhesive were tested at different loading rates, from quasi-static to high-rate. The high rate loading was performed using a unique modified Split-Hopkinson Pressure Bar (SHPB) setup. Embedded quartz transducers at the loading interfaces were used to measure the applied loads at both sides of the bar-specimen interfaces, which helped to optimize the input stress waves using wave-shapers to conduct valid dynamic experiments. Digital image correlation (DIC) method was used as an optical crack opening displacement (COD) gage to measure the onset of crack opening velocity (COV) to determine the failure initiation point on the loaddisplacement trace. The experimental results are used to obtain the energy required to initiate failure at different rates of loading. In this paper, the experimental methodology is presented, along with the results from these experiments. These data are being used to develop computational simulation methodologies to investigate the failure of bonded structures during dynamic impact loading. They will also be used to compare potential adhesives in determining the most qualified adhesive for various applications under different environmental conditions as well as different surface morphologies. KEYWORDS High rate loading, dynamic properties, bond/adhesive strength and energy, ultra high-speed digital image correlation, EPON 828

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_9, © The Society for Experimental Mechanics, Inc. 2011

67

68 INTRODUCTION Adhesives are used to bond dissimilar materials to form protective systems. This adhesive bond layer is a frequent source of failure when subjected to impact loadings. To develop simulation methodology to predict the behavior of systems that contain adhesives during impact and penetration, it is important to understand the failure behavior of the adhesive interface layer. Therefore, it is necessary to develop experimental methods to measure the cohesive strength and failure behavior, especially at the high loading rates experienced during impact. Experimental methods do not exist to characterize the fracture behavior of adhesive bonds at high loading rates. By developing this experimental technique, the cohesive strengths can be determined, and these material properties can be used in computer models to predict the fracture process. For fiber-resin based composites, understanding the failure behavior at the micro-structural level is important to develop accurate simulation methodologies. This ensures correct prediction of deformation and failure of these composites during high rate impact loading. During impact, fracture at the interface between fibers and resin is the dominant failure mechanism. Currently, there are no valid high rate experimental methods to investigate this type of interfacial failure. The high rate experimental techniques presented in this paper can be used to study such interfacial failure. The failure mechanisms of the adhesive layer depend on many variables, including the surface roughness of the substrate materials being adhered together and environmental conditions. The interface properties can be controlled by varying the surface treatment at the interface [1-2]. Similarly, the bond strength may depend on the temperature and surface morphology in addition to the mechanical behavior (strength) of the adhesive material. Researchers develop various adhesives to bond different materials with high adhesive strength and toughness, and to withstand different environmental conditions. Currently, there does not exist a way to compare different adhesives in terms of bond strength at high loading rates, or to determine how the bond strength of these different adhesives change due to changing environmental conditions or surface morphology. This research documents a methodology that can be used to determine adhesive bond strengths at high loading rates, and which can be extended in future work to investigate the effect of surface morphology and environmental conditions on bond strength at high rate. Many different materials are used to develop protective systems, all with different material properties. Frequently, disparate materials are bonded together, such as ceramic bonded to polymer composite. During impact loading, a frequent failure mechanism involves adhesive failing at the material interface. Therefore it is important to understand the failure along the interface between adhesive and material. Interface failure of the adhesive occurs one of three ways: mode A, where the crack propagates through the adhesive; mode B, where the crack travels between adhesive and bonded material; and mixed mode, where the crack alternates between propagating through the adhesive and at the interface (combination of mode A and B). Higher levels of adhesion usually cause a higher likelihood of mode A failure, where the crack propagates through the adhesive. These modes are not to be confused with traditional fracture modes (mode I, mode II, mixed mode). In this paper, we propose a novel experimental method to study and quantify failure of adhesives at high loading rate under mode I fracture. This information will be used for the development of cohesive zone based simulation methodologies as well as providing data to be used as an evaluation tool for adhesives that are being designed for Army applications. Only recently have experimental methods been developed to adequately investigate the fracture behavior of materials at high rate loading. Weerasooriya et al [3] investigated the fracture behavior of a SiC-N ceramic as a function of loading rate, including at high loading rates. They used a modified split Hopkinson pressure bar (SHPB) setup to perform four point bending experiments on notched and unnotched SiC-N beam specimens to determine fracture toughness (notched) and flexure strength (unnotched) at dynamic loading rates. They found that both properties increased with increased loading rate. They used the same techniques to characterize the dynamic fracture toughness of PMMA as a function of loading rate [4]. They found that the fracture toughness of PMMA is also directly dependent on the loading rate. A comprehensive review of recent progress in dynamic fracture toughness experimental techniques was performed by Jiang et al [5].

69 The effects of surface morphology on the failure of interfaces have been studied at low loading rates. For a glass alumina interface, Cazzato and Faber [6] determined that the crack path was not restricted to the inter-material interface. The failure properties are thus nearly independent of the alumina surface roughness. Zhang et al. [7] used a bi-layer double cantilever beam specimen to study the fracture behavior of an epoxy-aluminum interface as a function of substrate surface morphology. They showed that the roughness index at a microscopic level is more important that the nano-level features of the aluminum surface. Extending the work of Zhang et al [7], Syn and Chen [8] studied the surface morphology effects on an aluminum-epoxy interface when subjected to high rate loading. They developed a novel butterfly like 4point bend specimen that consisted of aluminum on one side and epoxy on the other. This new specimen design was subjected to high rate loading in a modified compressive split Hopkinson pressure bar (SHPB) apparatus, using the 4-point bend technique similar to those detailed in [3-4]. They found that both fracture toughness and energy dissipated by the fracture process increased with increasing surface roughness for specimens loaded in approximately 10 N/µs range. In this paper, novel adhesive specimens were created by using a thin amount of the adhesive to bond two aluminum “wings” together, creating similar geometry specimens as Syn and Chen [8]. The specimens were then subjected to four point bending experiments as in [3-4], which caused failure of the adhesive. These experiments were performed to obtain the fracture properties of the adhesive bond/interface, including failure load, maximum load, failure energy (energy required to initiate crack propagation), and specimen crack opening velocity. These experiments were conducted over a range of loading rates, from quasi-static to dynamic. For dynamic loading experiments, a modified SHPB setup was used with embedded quartz force transducers to measure load history on the specimens, similar to [3-4,8]. High-speed digital image correlation (DIC) was used during the experiments to determine the onset of fracture and to allow quantification of the failure energy absorbed by the adhesive. This was obtained by measuring the velocity at which the two aluminum wings traveled away from each other after the onset of crack growth; hereafter this is referred to as the crack opening velocity (COV). The COV begins to increase just before maximum load; this indicates the initiation of failure; by determining when the failure began, it is possible to measure the energy needed to cause the initiation of failure of the adhesive. This method is more accurate than the assumption that the failure begins at the maximum load point. MATERIAL The adhesive used in this was diglycidyl ether of bisphenol-A (DGEBA), obtained from Shell under the trademarked name EPON 828. DGEBA is a typical commercial epoxy resin; it was used as a “baseline”, against which, later, high performance adhesives can be compared to. The DGEBA was cured with diethylenetriamine (DETA). This adhesive is widely accepted as a strong, versatile adhesive, used in a wide variety of applications. The behavior of this adhesive has been widely investigated. Chen et al [9] and Chen et al [10] investigated the constitutive compressive response of a similar DGEBA, EPON 828 cured with T-403. They found that the compressive strength increased with increasing strain rate, as did the strain at maximum strength. In addition, Chen et al [11] investigated the tensile behavior of EPON 828 as a function of strain rate. EXPERIMENTS Specimens The “butterfly” specimens were created by bonding two aluminum (7075) wings together with the adhesive. The wing geometry is shown in figure 1(a) (all units are in mm). The specimens are assembled using custom molds, the detail of which is shown in figure 1(b); the wings (blue) are held in place with set screws, and the two halves come together guided by alignment pins (purple). Custom made gauge blocks (green) are used to set the adhesive thickness (red); bolts (tan) are used to tighten the two halves together, which squeezes out excess adhesive, leaving only a layer as thick as desired, set by the gauge block thickness. A small pre-crack is created in the adhesive by embedding a thin layer of Teflon at the

70 loading edge (not shown in figure). This ensures that over the narrow length of the Teflon, the adhesive is bonded to only one of the aluminum wings.

(a) (b) Figure 1 – (a) Aluminum wing geometry and (b) specimen fabrication schematic

The completed specimens are then observed with an optical microscope to measure adhesive thickness and uniformity, and initial crack length. Figure 2 shows a complete, untested specimen and a magnified detail of the cracked-end of the adhesive demonstrating the measurements being obtained for the specimen using an optical microscope.

Figure 2 – Complete specimen and microscopic view of adhesive layer and pre-crack

Loading Fixtures To allow for integration into a compressive SHPB setup, cylindrical aluminum pieces of diameter 31.8 mm (1.25 in) were fabricated to mount a pair of hardened steel loading pins of diameter 2.4 mm (0.094 in) parallel to each other and normal to the cylinder axis. These pieces were bonded to a quartz disc of same diameter and 3.175 mm (0.125 in) thickness using a conducting metallic based epoxy. The other side of the quartz disc was bonded to a 25 mm (1.0 in) long aluminum cylinder. The quartz disc is the key instrumentation of the measurement system; as it is piezo-electric, it develops a small charge as it is subjected to mechanical stress. Two small pinholes in the aluminum pieces allow for electrical connectors to measure the charge developed by the quartz disc during loading. Two of these fixtures were created, one for each side of the specimen. The center-to-center distance for the loading side fixture was 12.20 mm and the support side fixture was 3.00 mm. Shown in Figure 3 are the complete fixtures, with the aluminum pieces bonded to each side of the quartz disc. Masking tape on the end of the fixtures ensures a snug fit with the plastic sleeve which is used to hold the fixture against the Hopkinson bar.

71

Figure 3 – Loading and measurement fixtures

Digital Image Correlation Digital image correlation (DIC) is an optical measurement that allows the user to perform displacement measurements of an object from a series of digital images recorded during the experiment. The images are post processed using specialized software. The DIC software allows measurement of the relative displacement of the two speckled wings. Figure 4 shows an example of an adhesive specimen with a speckle pattern that was used for displacement measurement by the DIC software.

Figure 4 – DIC extensometer and speckle pattern

Figure 5 shows the loading and crack opening displacement (COD) history of a typical specimen obtained from DIC method; the COD starts to increase just prior to the peak load, showing the failure initiation (crack growth start) point in the load history. By computing the relative displacement between wings using the high speed camera images of the speckled wings and DIC, a COV can be calculated from the slope of the COD-time curve. DIC measurements were obtained for experiments at all three loading rates, including ultra high-speed DIC (1 million frames per second) on the dynamic loading rate experiments using the modified Hopkinson bar 4-point fracture set-up.

72

Figure 5 – Load history and COD history during typical adhesive experiment

Quasi-static and Intermediate Loading Rate Experiments Two loading rates, which were several orders of magnitude below the dynamic loading rate, were used to characterize the adhesive at slower rates of loading. The experiments were conducted using an Instron 8871 servo-hydraulic test machine, with a load cell of capacity 5 kN (1124 lb). The same high rate loading fixtures discussed previously were used here for the purpose of using the same four-point loading geometry, thus eliminating any variability that may come from using different loading fixtures. However, the quartz cells were not used for the load measurements. The Instron load cell provided the load history for the lower rate experiments. Figure 6 shows the experimental setup for these two lower rate loading experiments. An alignment fixture, not shown, is used to ensure that the upper and lower fixtures are concentric and that the loading pins are parallel. The upper fixture is allowed to pivot to correct for any slight misalignments that may occur and make certain that the full length of the pins are engaged on the specimen during loading.

Figure 6 – Experimental setup for quasi-static and intermediate loading rates

73 During the experiments, the Instron test machine was controlled in displacement mode. The actuator traveled at constant velocity; due to the linear elastic behavior of the adhesive, this resulted in a constant loading rate. The quasi-static loading rate was 0.0116 kN/s (2.61 lb/s), which corresponded to a actuator velocity of 0.51 µ m/s (20.08 µin/s). The intermediate loading rate was 13.92 kN/s (3.13 kip/s), which corresponded to an actuator velocity of 0.64 mm/s (.0252 in/s). A high-speed camera recorded images of the specimen during testing; at failure, a break load detector initiated by the Instron was used to trigger the camera. The camera used was a Photron APX-RS, recording at 60k fps (16.7 µs/pic) and at a resolution of 128 by 128 pixels. The images were then post-processed using DIC software to determine the rigid body motion of the aluminum wings and to calculate the COV. Dynamic Loading Rate Experiments A modified compressive SHPB setup was used to perform the dynamic loading rate experiments. The incident and transmission bars were both aluminum 7075 with a diameter of 31.8 mm (1.25 in) and length of 3.66 m (12 ft). The striker used was 0.41 m (16 in). Wave shaping was employed on the incident bar to ensure the specimens were in a state of dynamic equilibrium. Semi-conductor strain gauges were mounted at midpoint on both bars, and connected to a high-speed digital oscilloscope to record the strain histories in the bars. The specimen-bar interfaces were modified to include the loading fixtures; the fixtures were held concentric with the bars and in place with a plastic sleeve. A detailed view of the experimental setup can be seen in Figure 7.

Figure 7 – Detail of high loading rate experimental setup

The loading fixtures were connected to a Kistler charge amplifier, and the amplifier output was recorded on the oscilloscope. The quartz gauges were calibrated to the known strain (and therefore stress) history recorded by the strain gauges. The calibration was performed by sending a compressive pulse through the system without a specimen. Mating fixtures were used in place with the loading fixtures to allow for smooth stress pulse transfer between bars and fixtures. Figure 8 (a) shows a typical calibration curve for the quartz gauges. Figure 8 (b) shows the loading history for one typical adhesive experiment. The loads are approximately equal at both incident and transmitted bars for nearly the entire experiment, and the loading rate is near constant. For the dynamic loading experiments, the average specimen gap closure velocity was 888.7 mm/s (35.0 in/s) and the average loading rate was 29,648 kN/s (6670 lb/s).

74

(a) (b) Figure 8 – (a) Calibration of quartz gauge using strain gauge and (b) typical load histories at incident and transmitted bars during adhesive experiment

RESULTS A summary of the adhesive’s mechanical properties that were measured during the experiments at the three different loading rates is tabulated in Table 1. The failure energy is the energy absorbed by the adhesive up to the point of failure (crack growth initiation); the failure initiation time is determined using the DIC technique (an optical COD gage) to measure when the COD begins to increase. Energy to failure is calculated by integrating load with respect to displacement. For the high rate tests, load was measured from the transmission quartz gauge, and displacement was determined from bar end motions measured by the strain gauges. At the quasi-static and intermediate rates, the displacement and load were measured using the test machine’s instrumentation. Failure energy rate is simply the failure energy per unit area of adhesive. Table 1 – Summary of adhesive experimental properties

Adhesive Averages Rate

Disp Rate (mm/s)

Loading Rate (kN/s)

Max Load (kN)

Dynamic Intermediate Quasi-static

888.7 0.64 0.00051

29648.20 13.92 0.01158

3.01 2.18 1.55

COD Velocity Failure Energy Failure Energy Rate (m/s) (mJ) (J/m^2) 9.89 4.38 1.70

206.60 97.11 81.79

1525 717 384

Figure 9 shows the relationship between loading rate and (a) failure load, (b) COV, (c) energy rate. The black markers indicate the average for that loading rate group. There is a strong and direct correlation between loading rate and all three properties. This demonstrates the absolutely necessity to obtain the high loading rate behavior of adhesives that may be subjected to these rates.

75

(a)

(b)

(c) Figure 9 – Effect of loading rate on (a) failure load, (b) crack opening velocity, and (c) energy rate

CONCLUSIONS A methodology for performing high-rate four-point bend strength (mode I fracture) experiments on adhesives has been developed at the Army Research Laboratory’s experimental facilities. A unique butterfly-specimen is used to study the adhesive and interface fracture. A modified compressive SHPB setup dedicated for dynamic fracture experimentation was used with this specimen to study the fracture behavior of a DGEBA epoxy cured with DETA. Digital image correlation was used as an optical COD gauge to measure exactly when the crack opening displacement begins to increase, which is shortly before the maximum load, and provided a precise measurement of the initiation of crack growth. These capabilities were used to characterize the loading rate effects on the strength and energy required to initiate fracture of a typical adhesive (EPON 828) used in Army applications. It was found that the failure strength and energy required for fracture initiation are both directly correlated to loading rate. This technique will be used to characterize other adhesives in the future, and allow for direct comparison of high loading rate properties for different adhesives under different environmental conditions as well under different surface morphologies. These experimental failure criteria will be used in developing simulation methodologies to represent fracture along the adhesive interfaces. Furthermore, these simulation methods will be used to explore other possible failure criteria, such as stress and strain fields around the crack-tip region. Also, these simulation methods can be used to explore the mode mixity, where both mode I and II are present at the crack tip region.

76 Determining the mechanism of failure for the adhesive is an important future step for this work. Investigating whether the failure occurs at the bonded material (here, aluminum) adhesive interface (mode B), or within the adhesive (mode A), or some mixture of the two will help researchers better understand the failure behavior. Additionally, this research is focused on mode I failure of the specimens. Development of experimental techniques to understand the mode II failure of adhesives will be needed. ACKNOWLEDGEMENTS The authors would like to acknowledge and thank Jared Gardner for his dedication and development of specimen fabrication techniques. REFERENCES 1. Jennings, C.W. Surface Roughness and Bond Strength of Adhesives. J. Adhesion, 4, pp. 25-38. 1972. 2. Thouless, M. Fracture Resistance of an Adhesive Interface. Scripta Mater. 26, pp. 949-951. 1992. 3. Weerasooriya, T., Moy, P., Casem, D., Cheng, M., and Chen, W. A Four-Point Bend Technique to Determine Dynamic Fracture Toughness of Ceramics. Journal of American Ceramics Society, 89 [3], pp 990-995, 2006. 4. Weerasooriya, T., Moy, P., Cheng, M., and Chen, W. Dynamic Fracture of PMMA as a Function of Loading Rate. Proceedings of the Society of Experimental Mechanics Annual Conference. 2006. 5. Jiang, F., and Vecchio, K. Hopkinson Bar Loaded Fracture Experimental Technique: A Critical Review of Dynamic Fracture Toughness Tests. Applied Mechanics Reviews, Transactions of the ASME, 62. 2009. 6. Cazzato, A. and Faber, K. Fracture Energy of Glass-Alumina Interfaces via the Biomaterial Bend Test. Journal of American Ceramics Society, 80, pp 181-188. 1997. 7. Zhang, Y. and Spinks, G. An Atomic Force Microscopy Study of the Effect of Surface Roughness on the Fracture Energy of Adhesively Bonded Aluminum. Journal of Adhesion Science and Technology, 11, pp 207-223. 1997. 8. Syn, C., and Chen, W. Surface Morphology Effects on High-Rate Fracture of an Aluminum/Epoxy Interface. Journal of Composite Materials, 42, pp 1639-1658. 2008. 9. Chen, W., and Zhang, X. Dynamic response of Epon 828/t-403 under multiaxial loading at various temperatures. Transactions of the ASME, Journal of Engineering Materials and Technology, 45, pp 1303-1328. 1997. 10. Chen, W., and Zhou, B. Constitutive Behavior of Epon 828/T-403 at Various Strain Rates. Mechanics of Time-Dependent Materials, 2, pp 103-111. 1998 11. Chen, W., Lu, F., and Cheng, M. Tension and Compression Tests of Two Polymers under Quasistatic and Dynamic Loading. Polymer Testing, 21 [2], pp 113-121. 2002.

Fracture in Layered Plates having Property Mismatch across the Crack Front

Umesh H. Bankar, Anil Rajesh and P. Venkitanarayanan* Department of Mechanical Engineering Indian Institute of Technology Kanpur, Kanpur, India 208016 (* Corresponding Author: [email protected])

ABSTRACT Layered structures are used in protection systems such as personal and heavy armor, windshields and also in thermal barriers. Such materials have mismatch in the properties, both elastic and fracture, from layer to layer. The focus of this study is to understand the behavior of cracks in such systems, especially when the crack orientation is such that there are property changes along the crack front. Plates comprising of layers of epoxy and PMMA were prepared by bonding together sheets of 6 mm nominal thickness with an epoxy adhesive. Single edge notched (SEN) specimens were loaded in bending. The thickness averaged stress intensity factor (SIF) was obtained through photoelasticity. Subsequently the behavior of crackpropagation in these materials was also investigated by loading SEN specimen dynamically. The crack tip fields were recorded using high-speed imaging coupled with dynamic photoelasticity, from which the thickness averaged fracture parameters are extracted. 1.0

Introduction

Layered structures are used in many applications including, protection systems, thermal barriers, windshields and heavy armor. A layered architecture offers the scope for choosing the layer material and properties in order to optimize the overall performance of the structure. However this advantage comes with added complexities in terms of material characterization and analysis. Particularly one can expect the fracture oriented failure of such structures to be sensitive to the layer architecture, change of elastic and fracture properties from one layer to the other and also to the type of loading experienced and crack orientation. Without a thorough understanding on these aspects, one cannot successfully garner the full potential of layered architecture for critical applications. Fracture of layered materials has received considerable attention over the years. There have been several investigations attempting to explore the fracture of bi-material systems. A bi-material system consists of two material layers joined together along an interface across which there are property jumps. The behavior of stationary cracks both oriented along the interface and oriented normal to the interface have been investigated by several researchers [1-5]. The behavior of propagating cracks along the interface of a bi-material has also received extensive attention [6-10]. Crack propagation cross the interface in a multilayer system has also been studied [11-12]. In all these investigations, the crack orientation is such that there is no property change across the crack front and therefore a two dimensional approach is applicable. A through thickness edge crack in a layered plate can have both elastic properties and the fracture toughness varying along the crack front. In this situation it is logical to anticipate that the stress intensity factor (SIF) will vary along the crack front and this variation will be sensitive to the variation of the elastic properties along the crack front. Recent studies [13, 14] indicate that if the elastic property variation along the thickness (crack front) in a cracked plate is continuous, then under in plane bending, the SIF variation is coterminous with the elastic modulus. This implies that the stiffer layer will have a higher SIF compared to the compliant layer. Therefore, the critical condition for a crack to become unstable may not be reached simultaneously at all points across the thickness (crack front). This can either lead to an overall increase of the critical load or its decrease depending on the relative variation of the SIF and the fracture toughness across the thickness. The purpose of this study is to bring out the fracture behavior of layered plates having cracks with property gradients along the crack front. To this extent, thin plates made of two different polymers, Poly Methyl Methacrylate (PMMA) and Epoxy (LY556) having both

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_10, © The Society for Experimental Mechanics, Inc. 2011

77

78 elastic mismatch and fracture toughness mismatch were bonded together using an Epoxy adhesive. An edge crack in this plate was loaded in quasi-static and dynamic three point bending. The thickness averaged SIF is obtained through photoelasticity. 2.0

Experimental details

2.1

Specimen preparation and characterization

PMMA sheet used in this study was a commercial grade sheet of nominal thickness 5.5 mm. The epoxy sheets were cast in house and were of nominal thickness 5.8 mm. The elastic modulus and Poisson’s ratio of the materials were determined though tensile tests performed as per ASTM D638 using a 20kN UTM. The longitudinal and lateral strains were measured using a pair of strain gages. Fracture tests were also conducted using the three point bend specimen geometry following ASTM D5045-99. The fracture specimens had a natural crack which was initiated from a machined notch by tapping with a sharp razor blade. Five samples were tested for obtaining the fracture toughness. The elastic and fracture properties of the sheets are listed in table 1. One can easily observe from table 1 the variation of elastic modulus and fracture toughness from one layer to the other. The fringe constant of the materials was also determined by loading circular discs of the materials under diametrical compression and recording the fringes through a circular polariscope equipped with Tardy compensation. The two materials have different sensitivities, with the Epoxy almost 13 times more sensitive than the PMMA.

Material Epoxy PMMA

Table 1. Properties of materials used Elastic Modulus Poisson’s Fringe constant (GPa) ratio (MPa-m/fr) 3.44 0.34 0.018 2.67 0.34 0.239

Fracture toughness (MPa√m) 0.53 ± 0.04 0.95 ± 0.11

The specimen for fracture testing was prepared by bonding together a PMMA sheet to an Epoxy sheet using the two part epoxy adhesive, Araldite®, which has similar characteristics to the Epoxy sheet. The bonding surfaces were first abraded with fine grit paper and then cleaned with methanol. Then a thin layer of the premixed adhesive was applied with a serrated tool. The two sheets were then assembled and placed in a fixture under slight pressure for overnight curing. After curing the sheets were sized to a length of 200 mm and width of 50 mm. From the measured thickness of the specimen after bonding and the thickness of the individual sheets, it was estimated that the adhesive layer had an average thickness of around 170 micrometers. A notch of the required length was made in the specimen using a band saw and a natural crack was subsequently extended from the notch tip using a sharp razor blade. 2.2

Static testing Flash lamp

Single edged notch (SEN) specimens were subjected to both four-point bending and three-point bending in a UTM. The specimen was placed in a light field circular polariscope during loading and the isochromatic fringes were recorded using a CCD camera for further analysis. 2.3

Dynamic testing

Striker bar

V

Incident bar Strain Gages Specimen

Make trigger Trigger in

Circular Polarizer

SEN specimens were subjected to dynamic three-point Camera loading using a Hopkinson bar. A hollow polymeric bar Trigger out of 3 meter length was used for this purpose. Tests were performed on individual Epoxy and PMMA sheets as Trigger in SG Amp well as the combined PMMA-Epoxy specimens also. A SIM02-16 ultra high speed camera coupled with a circular polarizer was used to capture the isochromatic fringes during the fracture process. Sixteen images were captured at framing rates in the range of 100,000 to 150,000 frames per second. Figure 1 shows the schematic of the experimental setup. A make trigger circuit attached on the impact face of the bar was used Fig. 1 Schematic of the experimental setup for dynamic loading

79 to trigger the camera. The flash lamps and the strain gage data acquisition system was triggered by the camera itself. The captured images were analyzed to determine the crack speed and also the SIF as explained in the next section. 3.0

Analysis of isochromatics

The analysis of the crack-tip isochromatics to extract the fracture parameters (SIF) is a well established procedure for a homogeneous plate [15]. However, in the present study, the sample consists of two materials having different elastic and optical properties. Because of this, the stress field may not be constant through the sample thickness. The optical retardation is a function of the principal stress difference (σ1-σ2) and the fringe constant of the materials; both vary along the optical path (z-axis) in this case. The net relative retardation, ∆, can be written as h

1 (σ − σ 2 ) ∆ dz + =∫ 1 2π 0 fσ 1

h1 + h2

∫

h1

(σ 1 − σ 2 ) dz fσ 2

(1)

where, fσ1 and fσ2 denote respectively the optical fringe constant of material 1 and material 2 and h1 and h2 the respective thicknesses. For a cracked plate with elastic modulus varying along the crack front subjected to in plane bending, it has been shown that the variation in the stresses along the plate thickness is same as the elastic modulus variation, implying more or less an iso-strain type of situation [14]. Using this assumption, we can write the above equation as h h2  ∆ (σ
1 − σ
2 )  1 E E dz + ∫ dz  = ∫ Ee f 2π  0 fσ 1  h1 σ 2

(2)

In equation 2, Ee is an equivalent elastic modulus and (σ
1 − σ
2 ) is the principal stress difference in a homogeneous plate h

having an elastic modulus of Ee, where, Ee =

1 E ( z )dz and h=h1+h2 is the total thickness. In the present case, the elastic h ∫0

modulus and the fringe constant do not vary within a layer, hence we can define an equivalent fringe constant, fσe, for the whole plate as follows.

1 1  E1h1 E2 h2  = +   fσ e Ee h  fσ 1 fσ 2 

(3)

The optical fringe constant calculated using the above equation was used for analyzing the isochromatic fringes in this study, however, it should be pointed out that the usage is valid only for an iso-strain situation. A three-dimensional finite element analysis was carried out to investigate the variation of the stresses along the thickness. The contours of (σ1-σ2)/E for a homogeneous specimen and a layered specimen are shown in figure 2. The crack occupies the negative y axis. Figure 2(a) shows the variation of (σ1-σ2)/E at four locations across the thickness, two near surface (s) and two internal planes (i) for a homogeneous material having elastic modulus same as Ee. One can see that there is some variation of the stresses across the thickness even in a homogeneous plate. Figure 2(b) shows the contours of (σ1-σ2)/E for an Epoxy-PMMA layered plate. Interestingly, the variation of (σ1-σ2)/E across the thickness in the PMMA-Epoxy plate is almost identical to that in a homogeneous material indicating that once the stresses are normalized with the local elastic modulus, their variation is identical to that of a homogeneous plate. The method for extracting fracture parameters, particularly SIF from isochromatics is detailed in [15]. The procedure essentially involves fitting the near-tip stress field expressions to the experimentally observed fringe order using the stress-optic law. Recent investigations report that, continuous variation of elastic properties along the crack front, do not affect the structure of the first three terms, corresponding to r(-1/2), r(0) and r(1/2) in the stress field [13]. In the PMMA-Epoxy plate, the elastic properties are constant within each layer and vary only across the interface. Therefore we will assume the near-tip stress field to be identical to that in a homogeneous material. The SIF was calculated from the isochromatics following the over deterministic non-linear least square method using the equivalent optical fringe constant, fσε, in the stress optic law. The SIF thus obtained will be the thickness averaged SIF, Ke, and the SIF in the individual layers can be calculated as [14]

KP =

EP E Ke , K E = E Ke Ee Ee

(4)

80 where, subscripts P and E refer respectively to PMMA and Epoxy.

(a) (b) Fig. 2 Contours of (σ1-σ2)/E at four different planes along the thickness for (a) homogeneous plate and (b) PMMAEpoxy plate with an edge crack subjected to in plane bending. (s-close to surface, i-internal) 4.0

Results and discussion

4.1

Static tests

K (MP a-m ½ )

The results of the three-point and four-point bend tests will be discussed in this section. As mentioned earlier SENB specimens having a span of 200 mm and width of 50 mm were loaded in four-point bending and three-point bending. The thickness averaged SIF, Ke as a function of the applied moment in the four-point test is shown in figure 3 for two crack lengths. The solid line in the figure is the SIF calculated using 0.6 the analytical solution for an edge crack in a homogeneous plate a= 16.4 mm subjected to four point bending. It can be observed that Ke values in figure 3 are in good agreement with the theoretical values 0.5 a= 12.7 mm validating the use of fσe in the stress optic law. Edge cracks were also subjected to quasi-static three point bend test until failure of 0.4 the specimen. Figure 4 shows the load-displacement curve till failure of the sample. The load increased and close to a load of 0.3 300 N the crack in the epoxy layer jumped with a small drop in the load. The load further increased with stable crack growth in 0.2 the epoxy layer. Simultaneously the crack in the PMMA layer also grew stably; however, the crack tip in PMMA lagged the crack tip in epoxy leading to crack tunneling. Figure 5 shows the 0.1 SIF as a function of applied load. In figure the solid symbols represent the equivalent SIF, Ke, before crack jump and the open 0 symbols are the SIF calculated from the fringes assuming that 0 2 4 6 8 10 the fringes are caused by only the epoxy layer crack. The solid line is the SIF calculated using the theoretical expressions M (N.m ) available for an edge crack subjected to three-point bending in a Fig. 3 Equivalent stress intensity factor, Ke, as a homogeneous plate. Once again close agreement between the function of applied moment theoretical SIF and the experimentally determined SIF can be observed. The isochromatic fringes just before first crack jump, at the end of first crack jump and just before the final failure are shown in figure 6. One can see two crack tips in figure 6(c), one in the epoxy layer and one in the PMMA layer, suggesting crack tunneling. The equivalent SIF at the onset of first crack jump is 0.61 MPa-m1/2 which is about 15% larger than the fracture toughness of epoxy (see table 1). This indicates that the presence of the tougher PMMA layer helps in

81 increasing the fracture resistance of the pate and also delays the onset of final unstable failure. There was no sign of delimitation in the specimen. 0.7

500

0.6 K (MP a-m ½ )

Load (N)

400 300

200

0.5 0.4 0.3 0.2

100

0.1 0

0 0

0.1

0.2

0.3

0.4

0.5

0.6

Displacement (mm)

Fig. 4 Load displacement record for three point bend test a

b

0.7

0

100

200 300 L oa d (N)

400

500

Fig. 5 Stress intensity factor as a function of load in three point bending

c Epoxy crack tip

PMMA crack tip

Fig. 6 Stable crack growth in a PMMA-Epoxy plate under three point bending (a) just before first crack jump (load 310 N) (b) after first crack jump (load 300 N) and (c) just before final unstable failure (load 388 N) 4.2

Dynamic loading

SEN specimens were subjected to dynamic three-point bending as explained in section 2.3. The isochromatics recorded during the experiment were analyzed to determine the crack speed and also the value of the equivalent SIF at the point of crack propagation. Figure 7 shows the isochromatics in a PMMA-Epoxy specimen loaded dynamically. The pictures are separated in time by 10 microseconds. The first three pictures show the development of the opening mode stress field around the crack tip. In the fourth picture the crack has already started moving. From the seventh picture onwards, another butterfly shaped fringe (indicated by arrow in the picture) can be seen behind the crack tip (indicated by the vertical line). This fringe continues to follow the crack tip. From this observation, we can deduce that the crack propagation started only in the epoxy layer in picture four and the crack tip in PMMA starts propagating only in the seventh picture. The crack-tip location history obtained from the photographs of two nominally identical experiments is shown in figure 8. The initiation of the crack in epoxy layer could be recorded only in one case (test 02). In both experiments, the crack propagated with a constant velocity of 240 m/sec. From the plot of the crack-tip location versus time (test 02 in figure 8), the exact crack initiation time was determined as 26 microseconds after the first picture was taken. The crack tip in PMMA initiated about 30 microseconds after the crack tip in epoxy layer initiated. The equivalent SIF as a function of time was extracted from the isochromatic fringes and the variation of Ke with time for PMMA-Epoxy specimen is shown in figure 9. The SIF corresponding to a time of 26 microseconds was estimated as 0.6 MPa-√m, which is close to the dynamic initiation toughness of epoxy layer.

82

Fig. 7 Crack propagation in a PMMA-Epoxy layered plate subjected to dynamic three-point bending. The vertical line indicates the crack tip in Epoxy layer and the arrow indicates the crack tip in PMMA. The time interval between two pictures is 10 microseconds. 35

0.65

0.55

25

K (Mpa-m ½ )

Crack-tip location (mm)

30

20 15 PMMA Epoxy

10

0.35

Test-01

5

Test-02

0

0.25 0

20

40

60

80

100

120

Time (µ µ s)

Fig. 8 Crack-tip location history for dynamic fracture of PMMA-Epoxy layered plates under dynamic loading 5.0

0.45

0

5

10

15

20

25

T im e (µs)

Fig. 9 Stress intensity factor history up to crack initiation in PMMA-Epoxy layered plates under dynamic loading

Conclusions

The fracture behavior of layered plates having an edge crack subjected to quasi-static and dynamic three-point loading is investigated. The layered plate was prepared by bonding together PMMA and Epoxy sheets which have 30% mismatch in elastic modulus and 80% mismatch in the fracture toughness. Particularly the effect of sudden change in elastic modulus and fracture toughness across the crack front on the fracture behavior is investigated. The thickness averaged SIF was determined from the photoelastic fringes recorded during the fracture phenomena. The results of the study indicate that the presence of

83 the relatively tough PMMA layer can increase the fracture resistance of the material and also delay the onset of unstable fracture under quasi-static bending. When subjected to dynamic loading, the crack tips in Epoxy and PMMA propagated at different time instants with the crack tip in the relatively brittle epoxy layer initiating earlier than that in PMMA. Further studies are in progress to understand the beneficial effects of layered structure on their fracture tolerance under dynamic loading. 6.0

Acknowledgements

The authors would like to acknowledge the financial support under the FIST program by Department of Science and Technology, Government of India for the Ultra-high speed camera used in this study through grant number SR/FST/ETII003/2006. 7.0

References

1.

Rice J. R. and Sih G. C., Plane problems of cracks in dissimilar media, Journal of Applied Mechanics, Vol. 32, pp. 418423, 1965.

2.

Xu L. and Tippur H. V., Fracture Parameters for Interfacial Cracks: an Experimental-Finite Element Study of Crack Tip Fields and Crack initiation Toughness, International Journal of Fracture, vol. 71, pp. 345-363, 1995.

3.

Ricci V., Shukla A. and Singh R. P., Evaluation of Fracture Mechanics Parameters in Bimaterial Systems using Strain Gauges, Engineering Fracture Mechanics, Vol. 58, pp. 273-283, 1997.

4.

Erdogan F. and Biricikoglu V., Two Bonded Half Planes with a Crack Going through the interface, International Journal of Engineering Sciences, Vol. 11, pp. 745-766, 1973.

5.

Tippur, H. V. and Rosakis, A. J., Quasi-static and dynamic crack growth along bimaterial interfaces: A note on crack-tip field measurements using coherent gradient sensing, Experimental Mechanics, Vol. 31, pp. 243-251, 1991.

6.

Yang, W., Suo, Z. and Shih, C. H., Mechanics of dynamic debonding, Proceedings of the Royal Society (London), Vol. A433, pp. 679-697, 1991.

7.

Liu, C., Lambros, J. and Rosakis, A. J., Highly transient elastodynamic crack growth in a bimaterial interface: higher order asymptotic analysis and optical experiments, Journal of the Mechanics and Physics of Solids, Vol. 41, No. 12, pp. 1857-1954, 1993.

8.

Singh, R. P. and Shukla, A., Subsonic and intersonic crack growth along a bimaterial interface, Journal of Applied Mechancis, Vol. 63, pp. 919-924, 1996.

9.

Singh, R. P., Kavaturu, M. and Shukla, A. Initiation, propagation and arrest of an interface crack subjected to controlled stress wave loading, International Journal of Fracture, Vol. 83, pp. 291-304, 1997.

10. Shukla, A. and Kavaturu, M. Opening-mode dominated crack growth along inclined interfaces: Experimental observations. International Journal of Solids and Structures, Vol. 35, No. 30, pp. 3961-3975, 1998. 11. Singh, R. P. and Parameswaran, V., An Experimental Investigation of Dynamic Crack Propagation in a Brittle Material Reinforced with a Ductile Layer, Optics and Lasers in Engineering, Vol. 40, No. 4, pp.289-306, 2003. 12. Parameswaran, V. and Shukla, A., Dynamic Fracture of a Functionally Gradient Material Having Discrete Property Variation, Journal of Material Science, Vol. 33, No. 13, pp. 3303-3311, 1998.

13. Wadgaonkar S. C. and Parameswaran V., Structure of near tip stress field and variation of stress intensity factor for a crack in a transversely graded material, Journal of Applied Mechanics, Vol. 76, No. 1, 011014, 2009.

14. Kommana R., Parameswaran V., Experimental and numerical investigation of a cracked transversely graded plate subjected to in plane bending, International Journal of Solids and Structures, Vol. 46, No. 11-12, pp. 2420-2428, 2009. 15. Sanford, R. J. and Dally, J.W, A general method for determining mixed mode stress intensity factors from isochromatic fringe patterns. Engineering Fracture Mechanics, Vol. 11, pp. 621-633, 1979.

Stress Variations and Particle Movements during Penetration into Granular Materials Hwun Park, Weinong W. Chen Schools of Aeronautics/Astronautics and Materials Engineering, Purdue University 701 West Stadium Avenue, West Lafayette, IN 47907-2045 Email: [email protected]

ABSTRACT Granular materials such as sand are crushed and compacted locally when subjected to projectile penetration. It is necessary to obtain more diagnostic information during the experiment to develop a thorough understanding of the deformation mechanisms inside the target. In this study, we embedded piezoelectric film pressure gages at strategically distributed locations inside cylindrical sand targets. The gages are designed to measure only the pressure normal to the gage plane. The pressure gages are calibrated using a Kolsky bar before being embedded into the targets. The instrumented target is then subject to penetration by a cylindrical steel projectile with a semi-spherical nose. The distributed gages measure the pressure histories at the local gage positions. Flash X-rays installed around the target record the instant images of moving projectile and the gages inside the target. It was found that the penetration-induced pressure concentrates locally around the projectile. INTRODUCTION Sand has been widely used as a defensive material in military applications to defeat kinetic energy penetrators. Sand is a typical granular material and has complicated behaviors because of particle interactions [1]. Under dynamic loading condition, sand can be locally compacted and allow stress to propagate through specific chains, which may not be completely modeled by continuum mechanics [1-6]. Penetration into sand has been investigated in many studies [4-8]. However, predictive capabilities remain poor, indicating that improvements in the understanding of mechanisms are necessary. Most experimental investigations on penetration into sand have been focused on the relation between projectile conditions and maximum depth of penetration. In terms of numerical approaches, sand has been modeled using hydrodynamic assumptions that many theories of penetration have been based on [6-9]. These models give detail information on the status of targets such as stress profiles and medium displacements as well as the depth of penetration. However, the predicted quantities, such as stress field around a penetrator, have rarely verified by experiments. To develop more accurate models of sand penetration, obtaining diagnostic information beyond traditional experimental methods from experiments is necessary. The pressure measurement in sand and soil has been conducted with diaphragm gages or pressure transducers with static loading conditions in large scale samples [10]. For small scale samples, the measurement can vary with the ratio of gage size to particle size. Under dynamic loading conditions, sand can be locally deformed and inelastically packed, which makes the measurements more challenging [1,2]. Under projectile penetration conditions, stress propagates in three dimensions with various stress components in the medium. To capture the distribution of pressure inside the target, we use special piezoelectric gages designed to measure only the stress component normal to the gage planes. To validate their response and quantify their sensitivity, the in-house-assembled pressure gages are calibrated with a Kolsky bar before being embedded into the targets. To observe trajectory of projectiles during penetration and to evaluate the motion of sand in the target, we use flash X-rays that pass through the sand but are blocked by high density metals. EXPERIMENTS

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_11, © The Society for Experimental Mechanics, Inc. 2011

85

86 As a projectile move ahead, a cavity expands in the radial direction, generates stress propagation and transfers media in the radial directions [8,9]. The stress propagation and pressure caused by transferring media are generally three dimensional. We developed a special gage responding to only normal pressure as shown in Fig. 1.

Fig. 1 Design of piezoelectric gage to measure normal pressure The pressure gage consists of a piece of 0.0254 mm-thick piezoelectric film, PVDF, between two thin brass plates serving as electrodes. The gage is covered with Teflon tapes to avoid taking in-plane shear stresses. The brass plates and PVDF film are not bonded for the same purpose. To protect the gage from damage by sharp sand particles, steel or aluminum plates are attached at both sides. Even though the manufacturer provides electro-mechanical conversion factors, the gage needs to be calibrated because the specific arrangement of the gage inside the sand medium may affect the gage sensitivity. If there is no electric field around the gage, the response of piezoelectric film is calculated by [12].

Q   DdA  d15 31A1  d15 23 A2  d31 11  d32 22  d33 33 A3

(1)

where dij is a piezoelectric strain constant for each direction. (See Fig. 1 for the directions). The thickness of PVDF film is very thin, only 0.0254 mm, so the area A1 and A2 are negligible. Even though the PVDF film is not bonded with the brass plates, the film can have σ11 and σ22 because the edges of film are loosely confined with Teflon tapes and in-plane deformation is caused by Poisson effect. The low friction and relatively heavy protection cause little transfer of shear stress to the film. To calibrate the gage, a Kolsky bar was employed. The gages were sandwiched between the incident bar and transmission bar to let uni-axial stress be applied in the orthogonal direction. The sensitivity is obtained by adjusting the charge amplifier until the outputs from gages match the difference between the incident and reflected pulses. After the sensitivity of the gage is determined, a polycarbonate tube filled with sand was inserted between two bars. The gages were embedded in the sand tube to evaluate the gage response when the sand is loaded (Fig. 2).

Fig. 2 Configuration of a piezoelectric gage embedded in a sand column in a Kolsky bar The amount of sand in front and behind the gage affects the response because compacted sand reduces the stress transfer [2]. To simulate the environment inside the target, various amounts of sand were inserted between the gage and the incident bar, whereas the sand amount between the gage and the transmission bar remains constant.

87 Fig. 3 illustrates the experimental setup. A light gas gun was employed to launch projectiles at high velocities. The couples of lasers and sensors detect the passage of the projectile, which gives the flight velocity by the time interval. Projectiles are 50.8-mm long, 12.7-mm diameter cylinder and have semi-spherical heads. They are made with 4340 steel and weighs 46.5 g each.

Fig. 3 Configuration of experimental setup The cross-section of the sand target is 10.2 cm by 10.2 cm, which is the maximum depth of sand that X-ray penetrates and records images with reasonable quality. For consistency, Ottawa 20/30 sand conforming to ASTM C778 was used to fill the target. Density was measured as 1.61 kg/m3. Two types of sand targets were constructed; one having distributed iron balls for observation of particle motion by X-ray and the other having pressure gages for pressure measurement. Four flash X-ray tubes located in serial to record instance image of projectile in the target. The gage positions were decided by preliminary tests. The position of each gage is tabulated in Table 1. Fig. 3 showed the coordinate system and the center of target surface is the origin. The speed of projectile is approximately 180 m/s. Table 1 Lists of test with positions of gages Gage 1 Gage 2 Gage 3 X (cm) Z (cm) X (cm) Z (cm) X (cm) Z (cm) -1.3 5.1 -1.3 12.7 -1.3 25.4

Gage 4 X (cm) Z (cm) -1.3 38.1

1

Velocity (m/s) 151

2

184

-1.3

5.1

-1.3

12.7

-1.3

25.4

-1.3

38.1

3

184

1.3

5.1

1.3

12.7

1.3

25.4

1.3

38.1

No

EXPERIMENTAL RESULTS Fig. 4 shows a set of results from gage calibration tests with the Kolsky bar. The gage was inserted between the incident bar and transmission bar without sand. A small part of incident pulse reflects from the gage because the impedance of gage package containing polymer and aluminum is lower than the steel bar. The sensitivity of the charge amplifier was adjusted to make the plateau of gage output closed to the incident pulse. The sensitivity was found to be -3.4 pC/N. This calibration procedure was repeated with incident pulses of higher amplitudes. All the factors obtained from other tests agree well with 3.4 pC/N.

88

Fig. 4 Comparison of incident, transmitted pulses and adjusted gage output To simulate gage behavior in the sand target, the gage was embedded into a sand sample placed between the bars. To examine the effect of distance from a projectile to gages, the length of sand between the gage and the incident bar was set as 6.4 mm, 19 mm and 25 mm. The gas chamber pressure to drive the striker was set at 207 kPa, 345 kPa and 483 kPa. The corresponding outputs of gage for each case are shown in Fig. 5. (A) shows the transmitted pulses from the incident bar to the sand and was calculated by subtracting the reflected pulse from the incident pulse. (B) and (C) are the outputs from gage, which show the trends according to the impulse strength and the thickness of sand in front of the gage, respectively.

Fig. 5 Transmitted impulse to sand and corresponding outputs from gages Fig. 6 shows X-ray images recording the instants of particle motion in the sand target. Three flash X-rays were emitted on the film. Each number on the figure represents the time of each frame was taken after the projectile struck the surface of the target. The region marked by two black lines is affected by the projectile and narrows as the projectile proceeds and slows down. The maximum diameter of this region is approximately 36 mm, which is three times the projectile diameter. The small diameter of head of the projectile locating at the deepest position in the target indicates that the projectile tuned to left side (in the negative y direction in Fig. 3).

89

Fig. 6 X-ray image of target embedding iron balls (striking velocity: 178 m/s) Fig. 7 shows X-ray images capturing the moments of penetration. All gages were installed at the bottom (in the negative x direction in Fig. 3). The projectile turned to the top side away from gages during penetration. Asymmetric positions of gage may cause the curved trajectory. The projectile changed the direction again to the center as shown in the last shadow image. This may be caused by reflection from the aluminum wall of the target.

Fig. 7 X-ray images of sand target embedded with pressure gages (Test 2) Fig. 8 shows X-ray images capturing the moments of penetration. In this case, all the pressure gages were installed near the top (in the positive x direction) of the target box, opposite to previous set. Consequently, the projectile turned to the bottom side, again away from the gages. This indicates that the gages may indeed interfere with the motion of the projectile inside the target. After learning this lesson, we will install pressure gages at symmetric locations inside the sand targets in the future studies.

90

Fig. 8 X-ray image of sand target embedded pressure gages (Test 3) Fig. 9 shows stress variations at each gage location in Test 1. The output from Gage 1 has a sharp peak initially and then slowly decays. The high speed impact by the projectile induces the rapidly increasing pulse for a short duration. As shown in the X-ray images in Fig. 6, Gage 1 was supposed to be exposed at the cavity completely because the radius of cavity at the mouth of target is expected to exceed the distance from the centerline to Gage 1. It is believed that the slow decay and large pressure were caused by the cavity. Gages 2 and 3 recorded peaks when the projectile passed by the gage locations. Gage 3 has a lower peak value than Gage 2.

Fig. 9 Pressure induced by projectile in Test 1 Fig.10 shows the penetration-induced pressure variation recorded in Test 2. Even though the output from Gage 1 has rough fluctuations, it has a sharp peak and slow decaying, which is similar to the Gage 1 records in Test 1. Gages 2, 3 and 4 also had peaks as the projectile passed by the gage locations in sequence. The peaks value of each gage decreased as projectile traveled further because it moved farther away from the gages, as well as slowing down.

91

Fig.10 Penetration induced pressure variations in Test 2

CONCLUSIONS Piezoelectric gages were designed and fabricated to measure pressure variations in sand targets during projectile penetration. The sensitivity of the pressure gages was determined in the calibration tests with a Kolsky bar. The size of cavity during penetration was imaged with flash X-rays. The maximum size of cavity is three times the diameter of the projectile. The embedded pressure gages measured the pressure variations at different locations in the sand target as the projectile passed by. ACKNOWLEDGEMENT This research was supported by Basic Research Programs of Defense Threat Reduction Agency. REFERENCES Jaeger H. M, Nagle, S. R., Behringer R. P., The Physics of Granular Materials, Physics Today, 49(4), 32-38, 1996 Song B., Chen W., Luk V., Impact Compressive Response of Dry Sand, Mech. Materials, 41, 777-785, 2009. Martin B. E., Chen W., Song B., Akers, S. A., Moisture Effects on the High Strain-Rate Behavior of Sand, Mech. Materials, 41, 786-798, 2009. 4. Allen W. A., Mayfield E. B., Morrison H.L., Dynamics of a Projectile Penetrating Sand, J. App. Phy., 28, 370-374, 1957. 5. Borg J.P., Vogler T. J., An Experimental Investigation of the High Velocity Projectile Penetrating Sand, Proc. 11 th Int. Cong. Exp. Soc. Exp. Mech., 2008. 6. Borg J.P., Vogler T. J., Mesoscale Simulation of a Dart Penetrating Sand, Int. J. Impact Eng., 35, 1435-1440, 2008. 7. Zukas J. A., Impact Dynamics, Krieger, 1992. 8. Backman M. E., Goldsmith W., Mechanics of Penetration of Projectiles into Targets, Int. J. Eng. Sci., 16, 1-99, 1978. 9. Forrestal M. J., Luk V. K, Dynamic Spherical Cavity-Expansion in a Compressible Elastic-Plastic Solid, Trans. ASME, 55, 275-279. 1988. 10. Nichols. T. A., Bailey A. C., Johnson C. E., Grisso R. D., A Stress State Transducer for Soil, Trans. ASAE, 30(5), 1237-1241, 1987. 1. 2. 3.

92 11. Gran J. K., Frew D. J., In-Target Radial Stress Measurements from Penetration Experiments into Concrete by Ogive-Nose Steel Projectiles, Int. J. Impact Eng, 19, 715-726, 1997. 12. Moheimani S. O. R., Fleming A. J., Piezoelectric Transducers for Vibration Control and Damping (Advances in Industrial Control), Springer, 2006.

Sand Particle Breakage under High-Pressure and High-Rate Loading Md. E. Kabir, Weinong Chen Schools of Aeronautics/Astronautics and Materials Engineering, Purdue University. 701 W. Stadium Ave. West Lafayette, IN 47907-2045 Email: [email protected] ABSTRACT Under intensive loading, either hydrostatic or dynamic shear, sand particles are broke up to smaller particles. In this study, we randomly embedded color-coated sand grains of five different initial sizes inside cylindrical sand specimens under dynamic tri-axial loading. The quasi-static pressure was varied from 25 to 150 MPa at 25 MPa intervals and then unloaded. The embedded particles were retrieved and found fractured particles increased with increasing pressure. However, many of the colored particles remained intact. Under 100-MPa -1

hydrostatic pressure, additional axial load was added at three different strain rates, 0.01, 500, and 1000 s . It was found that, under additional dynamic loading nearly all the colored particles were fractured. In addition, to the attention on the embedded colored particles, the overall size change of the sand from the loaded specimens were also quantified. INTRODUCTION Particle fracture plays a major role in the mechanical behavior of sands. When sand particle breaks into a number of pieces, the total surface area of sand increases whereas the number of contacts per particle decreases. Particle size distribution of sand is, therefore, a key characteristic, which influences its properties, handling and domain of application. Loading rate on sand may vary significantly in applications. The quantity and

type of particle breakage at high rates of loading may differ from that at low rates. However, there are very limited experimental results found in the high rate range, late alone particle breakage analysis. In our previous work we investigated the shear behavior of sand at high strain rates [1]. In this study, we investigate the effect of strain rate on sand particle breakage behavior. In addition, the effect of hydrostatic pressure was also investigated. All other factors were kept constant. EXPERIMENTS AND RESULTS In order to study the particle breakage in a sand specimen, five different size sand particles (particles retained on US sieve No. 30-50) were first color-coated with different colors. Twenty sand particles corresponding to each of the five different sizes were randomly dispersed in the specimen. The specimen was then subjected to either isotropic consolidation only or isotropic consolidation followed by triaxial shear. Figure 1 shows all the experimental conditions in terms of void ratio and mean pressure for this study. The isotropically consolidated specimens were taken to 6 different confining pressure levels ranging (as shown by different symbols in Figure 1) from 25 MPa to150 MPa at an interval of 25 MPa. On the other hand, triaxial shear experiments were conducted by first applying a hydrostatic pressure of 100 MPa, followed by axial loads that were applied at -1

constant strain rates of 0.01, 500 and 1000 s (as shown by the arrow in Figure 1). After the loading, the specimens were carefully dismantled and spread on a bench from where the colored particles were removed. Presence and type of failure as well as presence of significant fracture surfaces were recorded for each of the colored particles using an electron microscope. Based on the damage observed, the T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_12, © The Society for Experimental Mechanics, Inc. 2011

93

94 colored sand particles were categorized into three groups. These are ‘Type 1’ if there was no visible damage, ‘Type 2’ if there was only surface abrasion of the particles leaving the parent particle largely intact but usually more rounded, and ‘Type 3’ if there was a major fracture of a particle into two or more pieces. Table 1 tabulates the results.

Figure 1: State paths for compression and shear experiments Table 1: Particle Failure Observed for Isotropic Consolidation Experiments Pressure

Grain size

Level (MPa)

(sieve#)

50

100

150

Grain damage Type 1

Type 2

Type 3

(No damage)

(Attrition)

(Fracture)

30

17

1

2

35

10

7

3

40

9

7

4

45

7

9

4

50

7

8

5

30

8

9

3

35

7

12

3

40

7

7

5

45

6

7

7

50

5

6

8

30

7

8

5

35

6

7

7

40

4

6

8

45

3

8

9

50

2

4

11

Lost

1 2

2 3

REFERENCE: [1] Kabir, Md. E., and Chen, Weinong W. “High Rate Triaxial Experiment on Dry Sand”, Proceedings of SEM Conference, IN (2010).

Experimental and Numerical Study of Wave Propagation in Granular Media

T. On1, K.J. Smith1, P.H. Geubelle1, J. Lambros1, A. Spadoni2, C. Daraio2 1

2

Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801

Department of Aeronautics and Applied Physics, California Institute of Technology, Pasadena, CA 91125

ABSTRACT One dimensional stress waves travelling in granular chains exhibit interesting characteristics such as filtering, tunability and wave mitigation because on the formation of solitary waves, and solitary wave trains, within them. An idealized one dimensional granular medium, consisting of a linear array of contacting spherical brass beads, was loaded dynamically in a modified split Hopkinson pressure bar, with loading pulses that span a variety of rates and profiles using pulse shaping techniques. Different chain lengths were studied to determine how solitary waves form in a varying length granular medium, as well as the speed of wave propagation. It is found that the wave speed propagates faster for longer chains of brass beads. The high loading rates of the Hopkinson bar also allowed us to investigate plastic dissipation effects in the granular chain when composed of different types of metals. To further our understanding of wave propagation in ductile ordered granular media, the experimental results are compared with companion numerical simulations based on a particle contact law that accounts for plastic dissipation. Knowing the behavior of a stress wave propagating through such materials can lead to arrangements that can produce desired stress wave mitigation characteristics as the waves travel through the granular chain. INTRODUCTION A granular chain can be characterized as a group of particles which can displace independently of one another and interact only when in contact with an adjacent particle. In such a chain, each element can only transmit information to its neighbors via compression. Contact between two spherical surfaces, even elastic, is a nonlinear process, the best known case of which is the Hertz contact solution developed in the late 1890s (Timoshenko and Goodier [1]). Subsequent efforts pertain to the plastic deformation of contacting spheres, which neglect volume conservation of the plastically deformed sphere, were based on the model of Abbott and Firestone [2]. In addition to contact models, considerable amount of work has been done towards the study of the static response in granular materials by Drescher [3]. Dynamic contact studies have been conducted by Iida [4], Hughes and Kelly [5], Shukla [6, 7] and Xu [8] who investigated the effects of particle size and wave velocity in one and two dimensional granular chains. The current paper focuses on the wave velocity traversing through a one dimensional granular medium consisting of brass beads with varying lengths. EXPERIMENTAL PROCEDURE AND RESULTS This set of experiments was conducted using a split Hopkinson pressure bar (SHPB) for dynamic loading at high strain rates (102– 104 s-1). The SHPB consists of an incident bar and a transmitted bar, with the specimen sandwiched between the two. The bars are long enough to be considered uniaxial and are hardened to remain elastic during the loading process. The loading is generally done via a striker bar, usually made from the same material, impacting the

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_13, © The Society for Experimental Mechanics, Inc. 2011

95

96 incident end, sending a compressive stress wave to the specimen. A momentum trap, developed by Nasser [9], is added to the bar to provide a single loading wave, followed immediately by an unloading wave. This ensures the specimen undergoes a single impact for the duration of the test. Strain gages are placed along the incident and transmitted bar to record the stress waves traversing along the bar as shown in Figure 1.

Figure 1. Schematic of a split Hopkinson pressure bar with momentum trap. Two strain gages are placed in the middle of each bar, and on opposite ends to cancel strain readings caused by possible bending of the bars. Strain gage (EA-06-250BK-10C) was used for all tests. The strain gages were connected through a signal conditioner which amplifies the strain and sent to an Agilent Technologies Digital Oscilloscope. The specimens used were brass beads (Alloy 260) of 9.525 mm diameter obtained from McMaster-Carr. In order to maintain a one dimensional chain of spheres, a holder, adapted from Spadoni and Daraio [10], was fabricated which consists of a hollow metal tube with threaded end caps which can be placed onto the bar. The holders are sized such that a hemisphere extrudes from the tube and comes in contact with the bar, allowing a point-load contact for the stress wave to travel. The edges of the tube are threaded by an end cap of larger inner diameter to ensure the caps will not interfere with the sphere specimens. A holder setup with one end cap mounted is shown in Figure 2. The holder was tested and showed no alterations to the experimental data. Chains of brass ranging from one sphere to twelve spheres are tested, with different length holder tubes for each length of brass chains.

Figure 2. Image of the one dimensional sphere holder with one end cap in place. Figure 3 shows a typical incident reflected and transmitted signal form the strain gauges (compensated for bending). Among other things, data from the experiment is used to calculate the wave velocity through the brass chain using the equation (1) where V is the wave velocity, N is the number of beads in the chain, d is the diameter of the bead, and is the time duration starting when the incident strain gage receives a signal and ending when the transmitted strain gage receives a signal. The travelling time of the wave within the chain can be calculated by knowing the positions of the strain gages. Since the material of the bar is known, both the wave velocity of the bar and the distance from the

97 strain gage to the specimen can be calculated. This results in the travelling time of the wave within the incident bar and transmitted bar, which are denoted as and , respectively. Thus, the denominator in equation (1) denotes travelling time of the wave within the chain.

Figure 3. Raw incident and transmitted data acquired from an experiment. The wave speeds were calculated for tests of various chain lengths and are plotted in Figure 4.

Figure 4. Wave velocity through one dimensional brass granular medium for varying chain lengths. Note that the force in each case, which is known to control the wave speed of such nonlinear waves, is about the same. Aside from the single bead experiments, the wave speed appears to be similar, although possibly somewhat increasing, for increasing chain lengths. The wave propagation in granular medium is governed by the contact mechanism and the wave speed calculated is much smaller than the dilatational wave or shear wave velocity for the same material, although it does depend on loading amplitude. The wave speed for a single bead has significant variations, more than the experimental error. The reason for this is not clear, but we believe it is associated with progressive yielding of the two contact points of the bead.

98 CONCLUSIONS The experimental data obtained by the strain gages show an increasing wave speed for an increasing distance for the wave to traverse. The wave speed also appears to approach a constant velocity value around 25% of the wave speed for a solid brass bar. This result is similar to that observed by Iida [4], and Xu and Shukla [8]. REFERENCES [1] Timoshenko, S. P. and Goodier, J. N., Theory of Elasticity, 3rd Ed., McGraw-Hill, New York, 1970. [2] Abbott, E.J. and Firestone, F.A., Specifying Surface Quality - A Method Based on Accurate Measurement and Comparison, Mech. Eng. Am. Soc. Mech. Eng., 55, p. 569, 1933. [3] Drescher, A., Application of Photoelasticity to Investigation of Constitutive Laws for Granular Materials, Proc. IUTAM-Symposium on Optical Methods in Solid Mechanics, Poities, France, 1979. [4] Iida, K., Velocity of Elastic Waves in a Granular Substance, Bulletin Earthquake Research Institute, Vol. 17, p. 783-808, 1939. [5] Hughes, D.S. and Kelly, J.L., Variation of Elastic Wave Velocity with Saturation in Sandstone, Geomechanics, Vol. 17, p. 739-752, 1952. [6] Shukla, A. and Zhu, C.Y., Influence of the Microstructure of Granular Media on Wave Propagation and Dynamic Load Transfer, J. of Wave Material Interaction, Vol. 3, No. 3, 1988. [7] Shukla, A. and Damaia, C., Experimental Investigation on Wave Velocity and Dynamic Contact Stresses in an Assembly of Disks, Exp. Mech., Vol 27, No. 3, 1988. [8] Xu, Y. and Shukla, A., Stress Wave Velocity in Granular Medium, Mech. Research Communications, Vol. 17, p. 383-391, 1990. [9] Nasser, S.N., Issacs, J.B, and Starrett, J.E., Hopkinson Techniques for Dynamic Recovery Experiments, Math. And Phy. Sciences, Vol. 435, p. 371-391, 1991. [10] Spadoni A. and Daraio C., Private Communication, 2010.

Communication of Stresses by Chains of Grains in High-Speed Particulate Media Impacts William L. Cooper, Mesoscale Diagnostics Engineer, Air Force Research Laboratory, AFRL/RWMW, 101 W. Eglin Blvd Suite 135, Eglin AFB, FL 32542 ABSTRACT Right-circular (φ 15 mm x 26 mm) projectiles were fired vertically-downward (150-720 m/s) into acrylic containers (φ 80-190 mm) containing quartz Eglin sand. Decreasing container size increased projectile drag and decreased total penetration depth. Thus, the container is within the projectile’s event horizon for at least a portion of penetration path length and some mechanism(s) exists for communication between projectile and container. The particulate media fractured near the projectile nose and created a rigid, conical false nose on the front face of the projectile, but the fractured media domain does not extend beyond 1.5 projectile diameters of the shot line. Jammed grains (i.e. mechanically-compacted, but not fractured) can be found adjacent to the fractured media; surrounded by nominally initial-density grains. It is theorized that the projectile communicates with the container via stress chains in the un-fractured grains which span the distance between the projectile/crushed media and container wall. The stress chain event horizon may be limited by either mechanical decay of forces along the chains or limited stress wave speeds in the particulate media. This paper focuses upon the mechanical force decay limits. Multiple analytical models are presented to illustrate how stress chain curvature and friction can limit the maximum stress chain length and thereby the ability of projectiles to communicate through the particulate media with the container wall. INTRODUCTION Researchers at Osaka University [Watanabe 2010] conducted experiments to observe the high-speed impact of right-circular cylinder projectiles (12.3 g, steel front face, polycarbonate body) with quartz Eglin sand (φ 75-1,400 µm grains, d50=400 µm, as-poured ρ=1.53 g/cc) [Cooper 2010]. Projectiles were launched vertically downward and impacted the sand surface normally (flat face parallel to sand) at 150-720 m/s. Projectile velocities were measured using induction loops and impact attitude was validated with high-speed photography. Two containers were used: φ 80 mm & φ 190 mm internal diameter, 300 mm length. Total penetration depth did not vary appreciably with impact velocity, but was strongly affected by the container size. Decreasing the container size from 190 mm to 80 mm cut the penetration depth in half as shown in Fig. 1. Thus, it can be inferred that the projectile communicates with the container--the stresses at the projectile surface and the drag on the projectile are affected by the container size. The goal of this analysis is to examine the mechanics that enable the projectile to communicate with the container. 200 180

φ 190 mm Container

Penetration depth [mm]

160 140 120 100 80 60 40

φ 80 mm Container

20 0 0

100

200

300

400

500

600

700

800

Impact velocity [m/s]

Figure 1 Projectile penetration depth as function of impact velocity

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_14, © The Society for Experimental Mechanics, Inc. 2011

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100 High pressure directly ahead of and beside the projectile nose comminute and fracture the sand media, forming a rigid, conical false nose on the front of the projectile. They also comminute and fracture sand up to 1.5 projectile radii from the projectile centerline. The increased density of the comminuted sand can easily account for the sand volume displaced by the projectile. However, this region spans only a fraction of the sand container diameter and does not fully explain the communication between the projectile and container unless the container is on the order of 3 projectile diameters or less. Outside the comminuted sand region the sand can be mechanically-compacted, but does not fracture; suggesting a maximum density of 1.73 g.cc (the maximum mechanical compaction density achieved previously [Cooper 2010]) which decays with distance to the initial density of 1.53 g/cc. 2-D experimental work [Howell et al. 1999] and 2-D discrete element modeling (DEM) [Peters et al. 2005, Tordesillas et al. 2005] indicate that grain contact mechanics allow formation of stress chains which can propagate stress information over considerable distances. Howell et al. measured stresses in a 2-D Couette flow (inner radius 10.34 cm, outer radius 19.14 cm) using birefringent discs and showed that chains have a mean length of 2 to 5 particles, depending upon the particle packing fraction. Most chains are quite short with only a few percent of the chains exceeding 8 or 10 discs in length. However, visual inspection of experimental images shows chains which bridge the gap between inner and outer radius (roughly 100 discs in width) and much longer chains might be visually constructed because the chains form at angles to oppose rotation of the inner drive wheel. The authors note the difficulty in defining the length of chains. In this case the authors defined the chains “to be any set of nearly co-linear discs carrying stress larger than the mean.” The chains end at branches (which are common) or boundaries. This definition is excellent for assessing the force structure in the particulate material fabric. However, it does not address the primary question of interest to the present effort; How far can chains of grains propagate stresses to facilitate communication between points or surfaces in a particulate media (PM)? This question requires assemblies of chains (as defined by Howell et al.) such that the stress at one end affects the stress at the other. Depending upon interactions with surrounding grains, chains can satisfy this question while loaded with stresses more or less than the mean. Such chains can split or re-combine to produce chain segments with more or less stress than the mean. Peters et al. considers a quasi-static case more analogous to the present projectile impact—the impinging of a square punch upon the upper surface of a 2-D PM bed—and reports average chain lengths of approximately 5 grains with an exponential decay of the chain length PDF. In this case the definition of a chain is more specific than above—a “linear string of at least three rigid particles in point contact that can support loads along its axis, with only small amounts of rotation involved.” [Peters 2005, Cates et al. 1998, 1999] Chains are allowed some maximum degree of curvature. Exceeding this value (e.g. ±45º) breaks the chains into shorter segments in a method similar to Howell’s branching criteria. The stress chains organize to oppose the motion of the punch and roughly 40% of the particles are more highly stressed than the average (the “strong network”). Of these stressed particles only half are involved in chains longer than 2 particles. Again, this chain length definition is effective for assessing the stress-carrying structure in the PM fabric, but does not specifically address the communication question. Consider the question of chain lengths if an international cell phone call is made. Each specific segment (cell phone to tower, fiber optic cable, satellite up/down-link, etc.) may have a specific length, but if connected end to end allows the termination points to communicate--affect each other. Fig. 10 in Peters et al. serves to both illustrate this point and to emphasize the far-from-equilibrium behavior that typifies particulate materials [NRC 2007]. Standalone force chains (greater than average stress) are observed in the middle of grain ensembles with less than average stress. This suggests that stress information is communicated through the region of low stress by multiple low-stress force chains which can converge to load a chain more highly than average, before branching again. Thus a communications-focused chain length definition is needed: Communication chains are any ensemble of grains in contact for which stress changes at one location (chain end) affect the stress at another location (other end of chain). Such chains can incorporate multiple segments at higher/lower-than-average stress levels according to the definitions above. It is also possible, even likely, that stress changes at one grain may affect the stress levels in a variety of locations throughout the particulate material. This introduces the idea of ad hoc communication networks in particulate materials. The present effort focuses on a very narrow subset of the theory required to explain ad hoc communication networks in PM. Several analytical models are developed which help explain the stability of communication chains and to set bounds on their lengths based upon consideration of both the chains’ radius of curvature and friction (inner-chain and inter-chain). ANALYTICAL MODELS Given the communication chain definition, a model particulate chain can be constructed as shown in Fig. 2. The chain has both a beginning & end and all grains are in contact (point-wise for spherical grains considered here). Geometrically, any two neighboring grains have a characteristic radius (rc) of curvature. In practice the origin of each rc can vary wildly in three dimensions, but flattening this to 2-D provides a simpler illustration. Following the example of the projectile impacting the PM, the projectile applies a force to one end of the chain and it is theorized that communication with a confining wall can affect this force. In practice the experimental results indicate that the closer the wall, the larger the applied forces by/on the projectile, the higher the drag, and the shorter the total penetration length.

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Figure 2 Theoretical particulate material communication chain model Kinematically, the grains must be confined by some forces in order to maintain contact along the full chain length. These confining forces are provided by other grains in the PM as shown in Figs. 2 & 3. Fr,i is the confining force for the ith grain. Fi-1 and Fi are the inter-grain forces. Frs,i, Fs,i-1 and Fs,i are the shearing (frictional) forces between grains. The angles φ and β are defined as follows: φ=2asin(r/rc) and β=(π−φ)/2. Γ is the angle of the confining force; which is limited by the presence of the i-1 and i+1 grains. Note that if Γ is negative then the confining force can amplify the Fi-1 loading force such that the force can increase along the length of the chain. For all calculations r, rc, Γ and friction coefficient values (µ) are assumed constant along the entire chain length, although they would vary in practice from grain to grain.

Figure 3 Communication chain inter-grain forces FRICTIONLESS CASE Inter-grain friction requires both that surface roughness and/or asperities exist and that these are activated by sufficient macroscopic deformation (resulting in relative grain-grain lateral sliding) and/or deformation at the grain contact (due to grain loading forces) [Cole 2008]. There are a number of materials and conditions which fail to satisfy these criteria and are represented reasonably by a frictionless model, e.g. as-poured smooth glass beads. Such contacts are dominated by normal contact forces and result in a conservative system of grains which lacks a dissipative mechanism. Ignoring all interfacial friction forces, it can be shown that the inter-grain forces are governed by:

102

tan β −1 Fi tan Γ =Χ= tan β Fi −1 +1 tan Γ Fr ,i

 sin β 1 =  − 1 Fi  Χ  sin Γ

Eq. 1

Eq. 2

If F0 is the applied force on the initial (i=0) grain then the force propagates as follows:

Fi = Χi F0 Fr ,i F0

(

= Χ i −1 − Χ i

Eq. 3

β ) sin sin Γ

Eq. 4

Plotting X as a function of non-dimensionalized Γ & rc shows that the forces increase along the chain for Γ < 0 and decrease for Γ > 0 (see Fig. 4). As noted previously this effect occurs due to the alignment of the confining force. The radius of curvature accentuates this effect for very small values (rc/r < 10). Small radii of curvature transition more of the chain load to the confining force instead of the adjacent grain; effectively branching the chain for very small radii of curvature.

Figure 4 X (ratio of forces for adjacent grains) plotted as function of Γ/Γmax & rc/r. Plotting the number of grains required to decay the initial force (F0) to 5% of its initial value provides a more intuitive view of the same data. These values are capped at 300 grains both for practical readability reasons and because 300 grains easily span the container radii of interest (~40 – 100 mm for 0.4 mm diameter grains). As expected, Γ=0 is the dividing line between chains whose inter-grain forces decay and those that do not. The total chain length rapidly diminishes for Γ>0. This result helps explain how inter-grain forces in PM can be amplified locally (as noted by Peters 2005) by confining forces with Γ<0. The convergence of two chains is equivalent to Γ<0. Chains in materials with no friction have no theoretical length limit according to this analysis for Γ<0. Recall that this is a conservative system with no dissipative mechanism. As a result forces loads can be added to the chain by advantageous confining forces (Γ<0) or removed from the chain by chain branching (Γ>0 and small rc/r).

103

Figure 5 Number of grains required to decay inter-grain forces to 5% of initial load plotted as function of Γ/Γmax & rc/r. FRICTION AT CONFINING FORCE, NO INTER-GRAIN FRICTION The previous model can be generalized slightly by including friction at the confining force interface (Frs,i). This is a reasonable approximation for loosely-poured particles which do not experience enough macroscopic deformation to activate inter-grain friction (between grains in the chain), but for which small motion along the chain can activate friction at the radial constraint. The inter-grain force is again an exponential function of the number of grains as follows:

 cos Γ − µ i sin Γ    sin β − cos β sin Γ + µ i cos Γ  Fi  =Ψ= Fi −1  cos Γ − µ i sin Γ    sin β + cos β  sin Γ + µ i cos Γ  Fr ,i

sin β  1 =  − 1 Fi  Ψ  (sin Γ + µ i cos Γ )

Eq. 5

Eq. 6

If F0 is the applied force on the initial (i=0) grain then the force propagates as follows:

Fi = Ψi F0 Fr ,i F0

=

(Ψ

Eq. 7

)

− Ψ i sin β sin Γ + µ i cos Γ i −1

Eq. 8

A variety of friction coefficient values can be chose to represent various materials. Choosing µ=0.5 yields Figs. 6 & 7. Increasing the friction shifts the Ψ=1 values. To maintain the force level in the chain, substantially negative Γ values are required to compensate for friction at the confining force. The system is no longer conservative and forces in the chain can degrade due to friction at the confining force interface (i.e. with grains adjacent to the chain).

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Figure 6 X (ratio of forces for adjacent grains) plotted as function of Γ/Γmax & rc/r. The friction reduces the chain length for neutral Γ=0 values from ~ 300 grains to 75 or less. For 400 micron grains this cuts the chain length from 120mm to 30 mm and eliminates the ability of a single communication chain to communicate from projectile to container wall for containers smaller than φ 75mm (for a φ 15 mm projectile).

Figure 7 Number of grains required to decay inter-grain forces to 5% of initial load plotted as function of Γ/Γmax & rc/r. The friction coefficient at the confining force has a strong effect upon the chain length as summarized in the following figure. For these calculations Γ/Γmax = 0.2. The longest chain lengths occur for the frictionless case and have roughly a parabolic dependence upon both the friction coefficient and the Γ/Γmax ratio (near the X or Ψ = 1plateau). The chain lengths have a nearly linear dependence upon the radius of curvature.

105

Figure 8 Chain length summary for various coefficient of friction values at the confining force INTER-GRAIN FRICTION, NO FRICTION AT CONFINING FORCE The multi-grain model shown in Fig. 3 is ill-posed if friction occurs at multiple interfaces. Additional information must be supplied to enable the system to properly utilize the friction coefficients. These coefficients are in fact maximum values which limit the frictional force up to (and potentially including) the onset of dynamic sliding. In practice the actual friction values might assume any value between zero and the limiting value. Simply choosing a single value for the inter-grain and confining force coefficients of friction results in highly non-intuitive and physically impossible scenarios. However, there exists a very practical observation for chains with inter-grain friction (i.e. friction between particles in the communication chain). If the friction coefficient is sufficiently large (and grain rotation is assumed constrained) then no confining force is required. This is a reasonable approximation for mechanically-compacted particles where sufficient macroscopic deformation has occurred along the inter-grain interfaces, surface roughness exists, or grains deform enough (or some combination thereof) to activate inter-grain friction. Mechanical compaction forces grains into openings between adjacent grains, e.g. compaction may force the center grain in Fig. 3 down into the opening between the left & right-hand grains. If the inter-grain friction coefficient is large enough (assumed constant over the length of the chain for this analysis) then the center grain requires no confining force. The inter-grain frictional forces will constrain it. This action is responsible for both the residual normal & frictional forces between grains resulting from mechanical compaction; these forces produce a much stiffer particulate material skeleton and substantially increased shear strength. Assuming the friction coefficient at all inter-grain interfaces is constant, there is no confining force, and the r/rc ratio is fixed along the chain length, it can be shown that the minimum inter-grain coefficient of friction is given by:

µ≥

1 = tan β

1 π −φ  tan    2 

=

1 π r tan  − sin −1   rc 2

    

= µ min

Eq. 9

Plotting the geometric angle β and µmin as a function of the r/rc ratio (the inverse of the rc/r ratio used previously—to simplify the figure) results in Fig 9. β is a linear function of r/rc and the minimum friction coefficient is nearly a linear function. As one might expect, when β=90º then the chain is straight and the minimum coefficient of friction is zero. As β decreases the chain becomes more curved and larger coefficients of friction are required. At the smallest curvature ratio, a minimum friction coefficient of nearly 0.6 is required to confine the grains in the chain without an external confining force. This result helps explain the mechanical compactability of various particulate materials. For example, Eglin sand (used in previous experiments) has many surface artifacts which help activate friction between grains. Such sand mechanically compacts easily (roughly 10% by density) and produces a very stiff “crunchy” sand mass. Angles of repose greater than 90º are easily achieved and the sand must be mechanically sheared to enable subsequent flow. Compacted sand will not drain out of a bucket with a hole in the bottom. Clearly this sand has a coefficient of friction high enough to activate and utilize intergrain friction to lock in normal and frictional stresses during mechanical compaction.

106 Glass beads, by contrast, are very smooth and difficult to mechanically compact. They retain little interfacial stresses and require little or no shearing to produce flow. Their coefficient of interfacial friction is so low that residual stresses can only be reatined in rather straight chains. Mechanically compacted glass beads can drain (with little or no stimulation) from a bucket with a hole in it.

Figure 9 Minimum inter-grain friction coefficient to eliminate confining force It can be shown that if the minimum inter-grain friction coefficient is satisfied, the inter-grain forces do not decay along the length of the chain. Again, there is no mechanism for decay (no dissipative friction) and this system is again conservative. Thus, chains in a mechanically-compacted system could theoretically bridge long distances. The curvature of the chains would be limited by the inter-grain friction coefficients as noted in Eq. 9. Of course, distances are more easily bridged by straight chains and these are satisfied by the lowest coefficients of inter-grain friction. This could be a factor in the projectile impacts noted in the introduction. Compacted sand in and around the crushed sand region would easily propagate stresses to the perimeter of the compacted region. If the particulate material were compacted prior to the experiment then this region could extend to the container wall. If the material was not pre-compacted then we might expect chains outside this compacted region as described by the “Friction at Confining Force” model. SUMMARY It was observed experimentally that projectile penetration depth is a function of container size when projectiles impact PM in cylindrical containers. Increasing the container size increases the penetration depth, suggesting a feedback mechanism between the container and projectile. Prior research suggests that the PM between the projectile and the container wall is composed of crushed PM (ahead of & adjacent to high-speed projectiles), compacted PM (either due to mechanical compaction prior to the experiment or due to dynamic compaction during the experiment), and as-poured PM. The highlydensified crushed PM is limited to a region within 1.5 projectile diameters of the shot line. Thus, some mechanism must be responsible for communicating stresses between the projectile (and crushed media) and larger containers. It is proposed, in concurrence with previous researchers, that such stresses are communicated by stress chains. Previous stress chain definitions have been offered as a means to diagnose and understand the stress-carrying structure of the PM fabric. A further definition is therefore proposed to account for “communication chains” composed of multiple chain segments. Such communication chains are distinguished by the ability of the stress at one point or grain to influence the stress level at a remote point along the same communication chain. Three analytical models are presented which indicate how stresses might be propagated along chains in frictionless PM, PM with friction at the confining forces, and PM with friction at inter-grain interfaces (i.e. between grains in the chain of interest). Each of these cases corresponds with a particular set of PMs. In the frictionless case the inter-grain force growth or decay is primarily influenced by the angle of the confining forces. For Γ<0 it is possible for the confining force to amplify forces in the chain. For chains with a fixed radius of curvature the force grows or decays exponentially with the number of grains. In the case with friction at the confining force, chain lengths can be cut by an order of magnitude (compared with the frictionless case) for the same Γ. The coefficient of friction and Γ strongly influence the chain length. In the case of inter-

107 grain friction, it is shown that the chain radius of curvature suggests a minimum coefficient of friction which eliminates the need for a confining force to maintain the chain. When this coefficient of friction criteria is met then the force does not decay along the chain length. In practice, each communication chains is likely to encounter a combination of these idealized conditions along their length, but this analysis helps decompose the problem and bound the limiting conditions for various PM types. ACKNOWLEDGEMENTS This research was funded by a grant from the Air Force Office of Scientific Research. The author is indebted to Professor K. Watanabe (Osaka University) for providing the projectile impact data and for analysis suggestions from Professor K. Tanaka (Chubu University) and Professor K. Takayama (Tohoku U.). REFERENCES M. E. Cates, J. P. Wittmer, J.-P. Bouchaud, and P. Claudin, Jamming and Stress Propagation in Particulate Matter, Physica A 263, 354 1999. M. E. Cates, J. P. Wittmer, J.-P. Bouchaud, and P. Claudin, Jamming, Force Chains, and Fragile Matter, Phys. Rev. Lett. 81, 1841 1998. National Research Council (NRC) Committee on CMMP 2010, Condensed-Matter and Materials Physics: The Science of the World Around Us, ISBN: 978-0-309-10965-9, 224 pages, 7 x 10, 2007. Cole, D.M., Laboratory observations of stiffness and friction of normal and sliding contacts, Earth and Space Conference 2008: Proceedings of the 11th Aerospace Division International Conference on Engineering, Science, Construction, and Operations in Challenging Environments, v 323, ISBN-13: 9780784409886, 2008. Cooper W.L., Breaux B.A., Grain Fracture in Rapid Particulate Media Deformation and a Particulate Media Research Roadmap from the PMEE Workshops, International Journal of Fracture, Special IUTAM Fracture & Fragmentation Issue 162, 1-2, p 137-150, March 2010. Howell D., Behringer R.P., Stress Fluctuations in a 2D Granular Couette Experiment: A Continuous Transition, Phys Rev Letters, 82 no 26, pp. 5241-5244, 28 Jun 1999. Peters, J.F., Muthuswamy, M., Wibowo, J., Tordesillas, A., Characterization of force chains in granular material, Phys Rev E 72, 4, p 41307-1-8, Oct. 2005. Watanabe, K., Personal communication, July 2010.

Effects of thermal treated on the dynamic facture properties using a semi-circular bend technique

T.B. Yin1, 2, X.B. Li2, K.W. Xia1,∗ , S. Huang1 1 Department of Civil Engineering and Lassonde Institute, University of Toronto, Toronto, ON M5S 1A4, Canada 2 The School of Resources and Safety Engineering, Central South University, Changsha, Hunan 410083, PR China

ABSTRACT: Dynamic fracture toughness of Laurentian granite (LG) subjected to heat treatment was tested by means of a notched core-based semi-circular bend (SCB) specimen with a modified split Hopkinson pressure bar (SHPB) apparatus. The samples for testing the fracture toughness were manufactured and heat treatment up to 850℃ according to the requirements of experiment. The relationship between the fracture toughness and temperature were investigated of rocks subjected to thermal treatment. The experimental results showed that fracture toughness decreases with increasing temperature. Furthermore, the fracture surfaces of rocks in thermal treated were observed by means of the scanning electronic microscope (SEM). These results provide the theoretic direction for engineering application in thermal treatment. Keywords: Dynamic fracture toughness; thermal treated; NSCB; SHPB INTRODUCTION Temperature plays an important role in many engineering practices, such as rock drilling, ore crushing, deep petroleum boring, geothermal energy extraction, and deep burial of nuclear wastes [1]. So far many researchers have investigated the effect of temperature on mechanical properties of rocks under static loading [1-3]. Nasseri, et al. reported that with the increasing of the temperature of thermal treatment (250 °C to 850 °C), the number of fractures of rock sample becomes more and the opening of cracks becomes wider; and then the values of P-wave velocity and fracture toughness of Westerly Granite decrease [2]. Yavuz, et al. measured the bulk density, P-wave velocity, and effective porosity of carbonate rocks suffered thermal damages introduced by various temperatures from 100 °C to 500 °C [3]. They found that the P-wave velocity of measured rocks is higher than initial values at 100 °C, than the velocity decreases sharply with the rising of the temperature. Based on above researches, a conclusion can be formed that with the rising of the temperature, the mechanical properties of rocks will be diminished due to the thermal induced fractures. Some researchers also considered the effects of the temperature on dynamic mechanical properties of rocks [4, 5]. Lindholm et al. utilized the split Hopkinson pressure bar to measure the uniaxial compressive strength of Dresser basalt with up to 527 °C [4]. They obtained a relationship between temperature, strain rate, and rock flow stress for Dresser basalt. Zhang et al. measured the dynamic fracture toughness of Fangshan gabbro and Fangshan marble under high temperature, by employing a short-rod (SR) specimen performed on the SHPB system [5]. Zhang et al. considered the axial pressure on the rock samples during tests in order to eliminate the difference between the measured fracture toughness and the true toughness. But it will cost lots of time to prepare the SR specimen for dynamic fracture toughness due to the complex sharp of samples. Dai et al. employed a notched semi-circular bend (NSCB) method to measure the dynamic fracture of toughness [6]. This NSCB sample, compared with SR, chevron bending (CB) [7], and cracked chevron notched Brazilian disc (CCNBD) [8], has little time consuming and can works for small segment of cores. It has an attraction for the weaker core pieces, which are too thin or fragmented to be used in short rod or uniaxial compressive strength [9]. The objective of this study is to examine the correlation between dynamic fracture toughness and various temperatures for heat treatment under dynamic loading for Laurentian granite. The notched semi-circular bend (NSCB) specimen with a split Hopkinson pressure bar (SHPB) apparatus is introduced in this work [6]. The recovered fracture surfaces of rocks were ∗

Corresponding author. Tel.:+14169785942; fax:+14169786813. E-mail address:[email protected]

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_15, © The Society for Experimental Mechanics, Inc. 2011

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examined using the scanning electronic microscope (SEM). DYNAMIC NOTCHED SEMI-CIRCULAR BEND FRACTURE TEST Sample preparation In this study, Laurentian granite (LG) is adopted to perform dynamic fracture test, which is taken from the Laurentian region of Grenville province of the Precambrian Canadian Shield, north of St. Lawrence and north-west of Quebec City, Canada. The mineralogical composition is obtained using X-ray diffraction technique: feldspar 60%, quartz 33%, biotite3-5%. Rock cores with a nominal diameter of 40mm were drilled from the same block of Laurentian granite, and then sliced to obtain discs with an average thickness of 18 mm. The SCB samples were subsequently made from the discs by diametrical cutting. A notch with approximately 1 mm thickness is then machined using a rotary diamond- impregnated saw from the center of the disc perpendicular to the diametrical cut. Then a 1 mm wide, 5 mm length of notch was cut in the semicircular rock disc and the tip was sharpened with a diamond wire saw to achieve a tip diameter of 0.5 mm shown in Fig. 1b. The average grain size is about 0.59 mm [10, 11], so the diameter of the crack tip is similar to the thickness of naturally formed cracks. This will ensure accurate measurements of fracture toughness.

Fig.1 Schematics of the straight through notched semi-circular bend fracture test in (a) the material testing machine and (b) the SHPB system Here, Groups of ten samples were heat-treated at 100℃, 250℃, 450℃, 600℃ and 850℃, respectively, with ten samples remained untreated at room temperature as a contrast. The heat treatment was carried out in an electrical furnace with a 2 ℃/min heating/cooling speed, which was sufficiently slow to avoid crack due to thermal shock [2]. Split Hopkinson pressure bar system The dynamic test was carried out on a 25 mm diameter split Hopkinson pressure bar (SHPB) system (Fig. 2). A typical SHPB system consists of a striker bar, an incident bar, a transmitted bar, and specimen is located between the incident and transmitted bars[12]. The lengths of the striker bar, incident bar and transmitted bar are 200mm, 1500mm and 1200mm, respectively, which are made from maraging steel, with high yield strength of 2.5GPa. Strain gauges are mounted at 733 and 655 mm away from the bar-specimen interfaces on the incident bar and transmitted bar respectively. The data processing unit used an eight-channel Sigma digital oscilloscope by Nicolet, to record strain gauge signals collected from the Wheatstone bridge circuits after amplification, and another channel to collect the signal from the strain gauge cemented near the crack tip on the sample surface. The incident wave, reflected wave and transmitted wave were denoted with subscripts i, r and t, respectively. The loading forces P1 and P2 on the two ends of the specimen induced by the SHPB are[13]: P1 = AE (ε i + ε r ) P2 = AEε t (1) Where E is Young’s modulus of the bar material, A is the cross-sectional area of the bar and ε denotes strain.

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Fig.2. Schematics of the notched semi-circular bend (NSCB) specimen in the split Hopkinson pressure bar (SHPB) system [6]. Static test is performed by an MTS hydraulic servo-control testing system (Fig. 1a). Determination of Mode-I fracture toughness Compared with the static test, it is hard to achieve the balance of the dynamic loading forces, thus the inertial effect will induce during dynamic test and complex calculation is required to measure the fracture toughness of rocks [14]. To achieve the stress equilibrium of the samples, pulse shaper technique is utilized [14, 15]. In this work a composite pulse shaper (a combination of a C11000 copper and a thin rubber shim) is utilized to shape the loading pulse. In a test the dynamic forces on both ends of the NSCB sample have been achieved, the inertial effect in the sample can be effectively minimized, and the quasi-static data reduction scheme can be utilized to determine the fracture toughness of rocks. We thus only need guarantee the balance of the time-resolved dynamic force on both ends of the NSCB sample. To do so, pulse shaping technique is employed for all the dynamic tests and the dynamic force balance on the two loading ends of the sample has been compared before data processing. Fig. 3 shows the forces on both ends of the sample in a typical test. From Eq. 1, the dynamic force P1 is proportional to the sum of the incident (In) and reflected (Re) stress waves, and the dynamic force on the other side P2 is proportional to the transmitted (Tr). It can be seen that the balance of dynamic forces on both end of the sample is clearly achieved.

Fig. 3. Dynamic force balance check for a typical dynamic punch test with pulse shaping Based on the ASTM standard E399-06e2 for rectangular three-point bending sample, a equation for calculating the stress intensity factor (SIF) for mode-I fracture is proposed[13]:

KI =

PS BR

3

2

a Y( ) R

(2)

Where KI is the quasi-static Mode-I SIF, P is the time-varying loading force, and Y (a/R) is a function of the dimensionless crack length a/R, which can be calculated using the finite element software ANSYS [6]. Then the maximum value of KI is the

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fracture toughness KIc of tested sample. RESULTS Effect of thermal treated on ultrasonic velocity The P-wave velocity of rock samples was measured in a quality detector of rock and soil engineering .Ten sample sets were prepared from each rock and each set was subjected to one of temperature levels of 25, 100, 250, 450, 600, 850 ℃, respectively. The variation of P-wave velocity (Vp) with increasing temperature as shown in Fig.3. Experiments showed that P-wave velocity measured through dry rocks was very sensitive to thermal treated. The ultrasonic velocity of all rocks markedly decreased with increasing temperature (Fig. 4). This phenomenon can be explained as the thermally induced fractures increase with the temperature [2, 3]. At 100 ℃, the P-wave velocity is similar with that of rock sample at 25 ℃. Similar results also have been reported by many researchers [3]. The main reason is the new thermally induced microcracks are hardly found with this temperature.

Fig.4 Variation of P-wave velocity (Vp) with increasing temperature

Fig.5 Sample recovered was separated into two equal parts from the NSCB in a SHPB

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Fracture toughness Fig. 5 gives out a typical NSCB LG sample. After the test, the sample was separated into two equal parts through its main crack, as shown Fig. 5. The relationship between fracture toughness and loading rate as a function temperature is shown in Fig. 6. The results indicate that the fracture toughness increase linearly with increasing loading rates at different temperature. In addition, the results show that, at lower loading rate (less than 60 GPam1/2/s) the fracture toughness values of rocks are close to each other at different temperature; then at higher loading rate, the slope of fracture toughness and loading rates decreased, that means fracture toughness decreased with increasing temperature. Further investigation is required to explain this phenomenon.

Fig. 6.The effect of loading rate on the fracture toughness as a function temperature Study of micro-cracks This research was carried out by scanning electron microscope（SEM） ，which works on the basis of impingement of electron beam. Six measured samples at different temperatures were prepared for microcrack study. The surface of samples was polished and sprayed gold to observe microcracks. The microphotos of the samples with various temperatures are shown in Fig. 7, all of them are magnified 400 times. For temperature ranged started from 25 ℃ to 250 ℃, it is hardly to observe microcracks in Fig. 7. When temperature rises to 450 ℃, the thermal induced microcracks are clearly observed, and the amount and the width of microcracks generally increase with the temperature. It can be referred to the fracture toughness testing results that the fracture toughness decreases with the rising of the temperature due to the induced microcracks. The result indicate that the pre-existing microcracks of samples fracture surfaces less than widening and development of new microcracks, thus the temperature effect is a fatal factor influence the fracture toughness of rocks. CONCLUSION In this study, the notched semi-circular bend method with split Hopkinson pressure bar system is utilized to measure the

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dynamic fracture toughness of thermally treated Laurentian granite up to 850 ℃. A dynamic force equilibrium technique is introduced to eliminate the inertia effect during dynamic test, thus the static analysis can be used to calculate the dynamic fracture toughness of rock. The fracture toughness increases with increasing loading rates in various temperatures. The investigation of microcracks was performed by scanning electron microscope. The thermal induced microcracks are observed when the temperature increases to 450 ℃, which is the main factor reducing the fracture toughness of rock in this study. Higher temperature, more and wider of microcracks can be observed. All above mentioned conclusions is an improved understanding of the effect of temperature on fracture properties of rocks which improves engineering practices.

25℃

100℃

250℃

450℃ 600℃ 850℃ Fig. 7. SEM photos of the section of Laurentian granite (LG) specimens as a function temperature REFERENCE [1]. [2]. [3]. [4]. [5]. [6]. [7]. [8]. [9]. [10]. [11].

Heuze F. E., "High-Temperature Mechanical, Physical and Thermal-Properties of Granitic-Rocks - a Review", International Journal of Rock Mechanics and Mining Sciences: 20, 3-10 (1983). Nasseri M. H. B., Tatone B. S. A., Grasselli G., and Young R. P., "Fracture Toughness and Fracture Roughness Interrelationship in Thermally treated Westerly Granite", pure and applied geophysics: 166, 801-822 (2009). Yavuz H., Demirdag S., and Caran S., "Thermal effect on the physical properties of carbonate rocks", International Journal of Rock Mechanics and Mining Sciences: 47, 94-103 (2010). Lindholm U. S., Yeakley L. M., and Nagy A., "Dynamic Strength and Fracture Properties of Dresser Basalt", International Journal of Rock Mechanics and Mining Sciences: 11, 181-191 (1974). Zhang Z. X., Yu J., Kou S. Q., and Lindqvist P. A., "Effects of high temperatures on dynamic rock fracture", International Journal of Rock Mechanics and Mining Sciences: 38, 211-225 (2001). Dai F., Chen R., and Xia K., "A Semi-Circular Bend Technique for Determining Dynamic Fracture Toughness", Experimental Mechanics: 50, 783-791 (2009). Ouchterlony F., "On the background to the formulas and accuracy of rock fracture toughness measurements using ISRM standard core specimens", International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts: 26, 13-23 (1989). Fowell R. J., Hudson J. A., Xu C., and Chen J. F., "Suggested method for determining mode-I fracture toughness using cracked chevron-notched Brazilian disc (CCNBD) specimens", International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts: 32, 57-64 (1995). Ulusay R., and Hudson J. A., The complete ISRM suggested methods for rock characterization, testing and monitoring:1974-2006. The ISRM Turkish National Group, Ankara, Turkey, (2007). Iqbal M. J., and Mohanty B., "Experimental calibration of ISRM suggested fracture toughness measurement techniques in selected brittle rocks", Rock Mechanics and Rock Engineering: 40, 453-475 (2007). Nasseri M. H. B., and Mohanty B., "Fracture toughness anisotropy in granitic rocks", International Journal of Rock

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[12]. [13]. [14]. [15].

Mechanics and Mining Sciences: 45, 167-193 (2008). Frew D. J., Forrestal M. J., and Chen W., "A split Hopkinson pressure bar technique to determine compressive stress-strain data for rock materials", Experimental Mechanics: 41, 40-46 (2001). Chen R., Xia K., Dai F., Lu F., and Luo S. N., "Determination of dynamic fracture parameters using a semi-circular bend technique in split Hopkinson pressure bar testing", Engineering Fracture Mechanics: 76, 1268-1276 (2009). Frew D. J., Forrestal M. J., and Chen W., "Pulse shaping techniques for testing brittle materials with a split Hopkinson pressure bar", Experimental Mechanics: 42, 93-106 (2002). Dai F., Xia K., and Luo S. N., "Semicircular bend testing with split Hopkinson pressure bar for measuring dynamic tensile strength of brittle solids", Review of Scientific Instruments: 79, - (2008).

Development and Characterization of a PU-PMMA Transparent Interpenetrating Polymer Networks (t-IPNs)

K.C. Jajama, S.A. Birdb, M.L. Auadb, H.V. Tippura,* a

b

Department of Mechanical Engineering, Auburn University, AL 36849, USA Department of Polymer and Fiber Engineering, Auburn University, AL 36849, USA

ABSTRACT This paper presents our on-going efforts on development and characterization of transparent Interpenetrating Polymer Networks (t-IPNs) with polyurethane (PU) and poly(methyl methacrylate) (PMMA) as constituents. The resulting molecular composite is a fully cross-linked IPN with PU as the tough phase and PMMA as the stiff phase. By varying the volume fraction of the stiff and tough phases, t-IPNs with PMMA/PU content in the range of 90%/10% to 70%/30% have been prepared. Reaction kinetics and process parameters are carefully controlled to avoid phase separation to achieve optical transparency in the resulting IPN. Tensile tests show that a significant failure strain enhancement can be achieved but with a loss of stiffness relative to PMMA. Preliminary fracture tests show that an optimum PMMA/PU ratio in the IPN can produce enhancement in fracture toughness relative to PMMA. Keywords: Transparent interpenetrating polymer networks (t-IPNs); Polyurethane; Poly(methyl methacrylate); Tensile strength; Quasi-static fracture; Fracture toughness

Introduction Tough, lightweight and optically transparent polymers are desirable in many civil and military applications as hurricane resistant windows, protective goggles, aircraft canopies, automotive windows, sound and vibration damping materials to name a few. Interpenetrating polymer networks (IPNs) are a relatively new class of materials suitable for these kinds of applications. IPNs are molecular composites where one polymer is synthesized and/or crosslinked in the immediate presence of the other(s) [1]. Over the years several investigators have attempted to develop such materials [2-10]. However, most have resulted in opaque and/or translucent IPNs and very limited work exists from the perspective of transparent IPNs for the aforementioned applications. Furthermore, the reported ones mostly do not address IPN material characterization in terms of mechanical and fracture behavior. The present work aims at development and characterization of a novel lightweight, tough and transparent interpenetrating polymer network (t-IPN) materials by combining stiff and compliant polymer phases where each polymer forms its own network and the networks interpenetrate other during polymerization. In this work, poly(methyl methacrylate) (PMMA) is used as stiff/hard phase whereas polyurethane (PU) is employed as compliant/soft phase. Depending upon the ratio of soft and hard phases, the mechanical and fracture properties of the resultant IPNs can be tailored for a given application. The hard phase obtained from the PMMA network imparts higher strength, while the soft phase formed by PU network provides ductility and crack growth as well as impact resistance to resultant t-IPN systems. In this study, the mechanical and fracture responses of t-IPNs with varying PMMA/PU ratios ranging from 90/10 to 70/30 were studied by performing a series of tensile and fracture tests under quasi-static loading conditions. *

Corresponding author. Tel.: +1 334 844 3327; fax: +1 334 844 3307. E-mail address: [email protected] (H.V. Tippur).

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_16, © The Society for Experimental Mechanics, Inc. 2011

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Material Preparation The reagents used for the PMMA system were: methyl methacrylate (MMA, 99%, ACROS Organics), trimethylolpropane trimethacrylate (TRIM, Sigma-Aldrich) and 2,2´-Azobisisobutyronitrile (AIBN, 98%, Sigma-Aldrich), and the reagents used for the PU system were: poly(tetramethylene ether)glycol (PTMG), 2-Ethyl-2-(hydroxymethyl)-1,3-propanediol (TRIOL, 98%, ACROS Organics), 1,6 diisocyanatohexane (DCH, 99+%, ACROS Organics) and dibutyltin dilaurate (DBTDL, 98%, Pfaltz and Bauer, Inc.). The five different compositions (PMMA/PU ratio) of t-IPNs were prepared, namely 90/10, 85/15, 80/20, 75/25 and 70/30. A typical experimental protocol was determined to prepare t-IPNs of desired PMMA/PU ratios. In this protocol, a homogeneous mixture was prepared at room temperature by dissolving the required amounts of PTMG, TRIOL and DCH in the methacrylic monomer MMA and crosslinker TRIM under vigorous stirring for about 10 minutes. Next, the freeradical initiator, AIBN was dissolved during mixing and finally the desired amount of DBTDL (catalyst) was added followed by further stirring of the mixture for another 5 minutes. This homogeneous mixture was then poured into a closed mold made of poly(tetrafluro ethylene) (Teflon) sheet. Enough care was exercised to avoid any possible Fig. 1. A t-IPN (80/20, PMMA/PU) sheet (Dimensions: 170 x 80 x 8 mm3). evaporation of PMMA from the mixture by sealing the mold interfaces with a thin layer of caulk. The mold containing the PMMA/PU mixture was kept in an oven at 60ºC for 24 hours followed by further curing at 80ºC for another 24 hours. After curing, the mold was left in the oven at room temperature for another 12 hours for complete cooling of the casting. It should be noted that, slow cooling from 80ºC to room temperature in the oven prevents warpage of the cured sheet and minimizes residual stresses. A completely transparent 80/20 (PMMA/PU) t-IPN sheet of dimensions 170 mm x 80 mm x 8 mm is shown in Fig. 1.

Experimental and Testing Procedure The uniaxial quasi-static tensile tests were carried out in an Instron universal testing machine to measure properties such as elastic modulus, tensile strength, percent elongation at break. Dogbone-shaped specimens were prepared from t-IPN sheets of thickness 4 mm according to ASTM D638 test method [11]. All the experiments were conducted at room temperature under displacement controlled conditions at a constant crosshead speed of 2 mm/min. Typically 3 specimens were tested at the same crosshead speed for each t-IPN category. In order to characterize toughness of t-IPNs in terms of the P critical-stress-intensity factor, KIc, quasi-static fracture tests B were performed in accordance with ASTM D5045 test method [12]. For quasi-static fracture tests, the cured t-IPN sheets were S = 80 mm W = 20 mm machined into rectangular coupons of dimensions 80 mm x 20 W B = 8 mm mm x 8 mm. An edge notch of 6 mm in length was cut into the a /W = 0.3 a samples and the notch tip was sharpened using a razor blade. The three-point bend specimen geometry and loading configuration can be seen in Fig. 2. The single edge notched S bend, SENB, specimens were loaded in displacement control Fig. 2. Three-point bend specimen (SENB) geometry and mode with testing speed of 0.25 mm/min. The load vs. loading configuration for quasi-static fracture tests. deflection data was recorded up to crack initiation and during stable crack growth, if any, and the crack initiation toughness, KIc was calculated using the load (P) at crack initiation. Again, for each t-IPN category at least three sets of experiments were performed at laboratory conditions. The mode-I stress intensity factor for a single edge notched bend (SENB) specimen loaded in three-point bending using the linear elastic fracture mechanics is given by [13],

119

⎞ ⎟ ⎠ , where f ⎛ a K Ic = ⎜ ⎝W B W Pf

⎛a ⎜ ⎝W

3

S

a

⎞ W W ⎟= 3/ 2 ⎠ a ⎞⎛ a ⎞ ⎛ 2 ⎜1 + 2 ⎟ ⎜1 − ⎟ W ⎠⎝ W ⎠ ⎝

⎡ a ⎢1.99 − W ⎣

⎛ a ⎜1 − ⎝ W

⎞⎧ ⎛a ⎟ ⎨ 2.15 − 3.93 ⎜ ⎠⎩ ⎝W

2 ⎞ + 2.7 ⎛ a ⎞ ⎫ ⎤ ⎟ ⎜ ⎟ ⎬⎥ ⎠ ⎝ W ⎠ ⎭⎦

Experimental Results The mechanical properties of t-IPNs obtained from quasi-static tensile tests are shown in Figs. 3 and 4. A set of representative stress-strain curves for neat PMMA and various t-IPN specimens can be seen in Fig. 3(a). The initial response in each case depicts linear elastic region with only a modest nonlinearity in the case of neat PMMA and 90/10 t-IPN before failure compared to the other cases showing large nonlinear responses. The t-IPN specimens 85/15 and 80/20 show yielding at ~30 and ~26 MPa respectively followed by some strain hardening before failure. In the case of 75/25 and 70/30 cases, a yield plateau can be seen between ~4 to 20% strain followed by appreciable strain hardening region until ultimate failure. The elastic modulus for each specimen was deterimined by constructing a tangent to the initial part of the stress-strain curves and the variation of elastic modulus as a function of relative pecentanges of PMMA and PU is shown in Fig. 3(b). It can be seen that there is a reduction in Young’s modulus as the PU content increases in the t-IPNs. 3 .5

80 PMMA IPN(90/10) IPN(85/15) IPN(80/20) IPN(75/25) IPN(70/30)

Stress (MPa)

60

3 .0

E la stic m o d u lu s (G P a )

70

50 40 30

2 .5 2 .0 1 .5 1 .0

20

0 .5

10

0 .0

0 0

10

20

30

40

50

60

Strain (%)

(a)

(b)

Fig. 3. (a) Typical stress-strain response for t-IPNs, (b) Elastic modulus variation as a function of PMMA-PU percenatge. The ultimate stresses and failure strains for each material are plotted against the relative percentages of PMMA and PU in Figs. 4(a) and (b) respectively. Similar to elastic modulus, a decreasing trend in ultimate stress can be seen in Fig. 4(a) as the percentage of PU increases in the t-IPN. Fig. 4(b) depicts variation of percent elongation at break and as expected the failure strain increases with increasing proportions of PU in the PMMA network. The quasi-static fracture tests results for t-IPNs are shown in Fig. 5. The normalized load-deflection curves for each specimen are shown in Fig. 5(a). It can be seen that the curves are generally linear in the initial stage of deformation for each specimen. The neat PMMA and 90/10 t-IPN show complete linear region upto peak load followed by a sudden drop in the load, signalling crack initiation. However, in the case of other specimens, as the applied load increases, the specimens start to respond nonlinealry before reaching the peak load, and again the extent of nonlinearity increases with increasing PU content. This nonlinearity in the pre-peak load region is attributed to crack-tip blunting and crazing before crack initiation. Moreover, propagation of the macro-crack is arrested unless the applied load is increased. Because of this observed mechanism, the crack may deviates from its original propagation plane requiring additional load and energy for further propagation.

120 60

80 70

50

40

Failure strain (%)

Failure stress (MPa)

60 50 40 30

30

20

20 10

10

0

0

PMMA (%) 100 PU (%)

0

95

90

85

80

75

70

65

5

10

15

20

25

30

35

(a)

(b)

Fig. 4. (a) Variation of failure stress for t-IPNs with changes in PMMA-PU ratios, (b) Variation of strain at failure for t-IPNs with changes in PMMA-PU ratios. From Fig. 5(a), it can also be seen that at peak load the deflections are more as the PU content increases in t-IPNs. Furthermore, the overall deflection also increases with increasing PU content and the area under these curves represents the strain energy absorbed. Therefore, it is worth noting that the strain energy increases as the amount of PU increases in t-IPNs. 50

2.0

PMMA IPN(90/10) IPN(85/15) IPN(75/25) IPN(70/30)

1.5 1/2

KIc (MPa m )

P/B (kN/m)

40

30

20

1.0

0.5

10

0.0

0 0

1

2

3

4

5

6

δ (mm)

(a)

(b)

Fig. 5. (a) Normalized load-deflection response for t-IPNs, (b) Variation of quasi-static fracture toughness for t-IPNs. The quasi-static crack initiation toughness, KIc was calculated using the load at fracture in each case and the variation of KIc with changes in PMMA/PU ratios can be seen in Fig. 5(b). A ~60% improvement in static crack initiation toughness is quite evident for t-IPNs when compared to the neat PMMA. Furhter, the trends in KIc values with increasing PU content suggest that there is an optimum PMMA/PU ratio for which the quasi-static fracture toughness is highest. In this work, 90/10 and 85/15 t-IPNs provide maximum fracture toughness among all cases.

Summary In this study, transparent Interpenetrating Polymer Networks (t-IPNs) with polyurethane (PU) and poly(methyl methacrylate) (PMMA) as constituents were prepared by varying the volume fraction of stiff and tough phases with PMMA/PU content in the range of 90%/10% to 70%/30%. The t-IPNs were mechanically characterized in terms of tensile

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strength, elastic modulus, elongation at break and crack initiation toughness by performing a series of quasi-static tensile and fracture tests. The effects of variation of the PMMA/PU ratios on the aforementioned properties were examined. Tensile tests show that there was gradual reduction in elastic modulus and tensile strength with consequent increase in elongation at break relative to neat PMMA. Preliminary quasi-static fracture tests show that an optimum PMMA/PU ratio in the IPN can produce greatly enhanced fracture toughness with respect to neat PMMA.

Acknowledgments The authors would like to thank Defense Threat Reduction Agency for supporting this research through a grant HDTRA109-1-0023. References [1] Sperling, L.H., Interpenetrating polymer networks and related materials. 1981: Plenum Press, New York [2] Djomo, H., et al., Polyurethane-poly(methyl methacrylate) interpenetrating polymer networks: 1. Early steps and kinetics of network formation; intersystem grafting. Polymer, 1983. 24(1): p. 65-71. [3] Akay, M. and S.N. Rollins, Polyurethane-poly(methyl methacrylate) interpenetrating polymer networks. Polymer, 1993. 34(9): p. 1865-1873. [4] Sperling, L.H. and V. Mishra, The current status of interpenetrating polymer networks. Polymers for Advanced Technologies, 1996. 7(4): p. 197-208. [5] Chou, Y.C. and L.J. Lee, Mechanical properties of polyurethane-unsaturated polyester interpenetrating polymer networks. Polymer Engineering & Science, 1995. 35(12): p. 976-988. [6] Chakrabarty, D., B. Das, and S. Roy, Epoxy resin–poly(ethyl methacrylate) interpenetrating polymer networks: Morphology, mechanical, and thermal properties. Journal of Applied Polymer Science, 1998. 67(6): p. 1051-1059. [7] Chakrabarty, D., Interpenetrating polymer networks: Engineering properties and morphology. Polymer Gels and Networks, 1998. 6(3-4): p. 191-204. [8] Harismendy, I., et al., Morphology and thermal behavior of dicyanate ester-polyetherimide semi-IPNS cured at different conditions. Journal of Applied Polymer Science, 2000. 76(7): p. 1037-1047. [9] Bonilla, G., et al., Ternary interpenetrating networks of polyurethane-poly(methyl methacrylate)-silica: Preparation by the sol-gel process and characterization of films. European Polymer Journal, 2006. 42(11): p. 2977-2986. [10] Widmaier, J.-M. and G. Bonilla, In situ synthesis of optically transparent interpenetrating organic/inorganic networks. Polymers for Advanced Technologies, 2006. 17(9-10): p. 634-640. [11] ASTM D638-01: Standard test method for tensile properties of plastics. [12] ASTM D5045-96: Standard test method for plane strain fracture toughness and strain energy release rate of plastic materials. [13] Anderson, T.L., Fracture mechanics: Fundamentals and applications, 3rd edition, CRC press, New York.

Dynamic Ring-on-Ring Equibiaxial Flexural Strength of Borosilicate Glass Xu Nie1*, Weinong Chen1 1

*

AAE&MSE schools, Purdue University Corresponding author: Xu Nie, 701 W. Stadium Ave. West Lafayette, IN 47907-2045 Email: [email protected]

ABSTRACT A novel dynamic ring-on-ring equibiaxial flexural testing technique with single pulse loading capability is established on a modified Kolsky bar. This technique is then utilized to investigate the loading-rate and surface-condition effects on the flexural strength of a borosilicate glass. Quasi-static and dynamic experiments are performed at loading rates ranging from 5x10-1 to 5x106 MPa/s. It is found that the flexural strength of the borosilicate glass strongly depends on the applied loading rates. HF acid corrosion on the surface promotes the flexural strength to above 1.3 GPa. Fractographic analysis shows that surface modification has changed the type of flaws that govern the flexural strength of glass samples. INTRODUCTION Over the past several decades, vast amount of research has been carried out to explore the biaxial flexural strength of glass and ceramic materials [1]. However, it was only until recently that these studies have been expanded to dynamic loading conditions, for which the data are of great desire to the high speed impact applications. Cheng et. al [2] tested the dynamic biaxial flexural strength of a thin ceramic substrate with a modified piston-on-three-ball testing configuration. In these experiments, dynamic loading is applied on the center of the specimen through a thin incident bar driven by a force hammer. Except under extremely high pressure or temperature, the failure of brittle materials under impact is controlled by flaw nucleation, propagation, and coalescence. Flexural tests on glass materials suggested that the shape and severity of surface flaws are the key factors in strength determination. Consequently, the dynamic biaxial bending behavior of glass materials should be determined together with the loading-rate and surface-flaw effects. In this paper we studied the surface-flaw and loading-rate effects on the biaxial flexural strength of a borosilicate glass utilizing a modified Kolsky bar, with its testing section customized into a ring-on-ring equibiaxial bending configuration. Pulse shaping technique is applied on the Kolsky bar to ensure both force equilibrium and constant loading rate in the specimen. It is found that the flexural strength of the borosilicate glass increases with increasing loading rates under all surface conditions. The HF acid etching promotes glass surface tensile strength by a factor of 4 under equibaxial bending, while sandpaper grinding compromises strength for over 50% due to the severe surface flaws introduced by abrasive particles. EXPERIMENTS AND RESULTS The calculated strength values for borosilicate glass samples at different loading rates and surface conditions are summarized in Table 1. The results indicate that the surface modifications significantly affect the flexural strength of the glass material. The sandpaper grinding degrades the strength by 60-70% from the as-polished surface condition. However, HF acid etching on as-polished specimens promotes the surface tensile strength T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_17, © The Society for Experimental Mechanics, Inc. 2011

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by 200-400%, depending on the applied loading rates. The experimental results also indicate that the loading rate has remarkable effects on the flexural strength. Under all surface conditions tested, the strength universally increases with loading rates. But the rate of strength increase levels out at the loading rate of ~3,500 MPa/s. The observed strength variations under different surface conditions stimulated further fractography investigations to better understand the fracture mechanisms of borosilicate glass under equibiaxial flexural loading conditions. In this research, fracture surface images were taken by scanning electron microscopy (SEM) and are shown in Fig. 1. For the sandpaper-ground samples, sharp cracks that penetrate into the sub-surface are visible at the center of the fracture zone. Further polishing on the as-ground surfaces resulted in reduction of critical crack size, and thus an increase in flexural strength. The strength governing flaw size on HF acid etched surface is similar to that on the as-polished surface, whereas the flexural strength of the etched sample is four times higher than that of as-polished samples. The fracture surface of an etched sample reveals that failure was initiated from a severe surface pit. However, no sharp front of a pre-crack was identified. This indicates that the fracture of an etched specimen is originated from a blunt surface flaw, unlike the case of as-polished specimens where a sharp surface crack is located in the failure origin. As a result, although acid etching developed similar flaw depths as is possessed by as-polished samples, the blunt nature of etching pits offers much less stress intensity compared to the sharp crack front in polished samples, and thus raises the flexural strength significantly. Table 1. Equibiaxial flexural strength of borosilicate glass under different loading rates and surface conditions. (Strength data are in MPa)

Loading rate

0.52 MPa/s

42 MPa/s

3500 MPa/s

5x106 MPa/s

HF etched

352  35.7 146  11 46  3

744  96 180  13 52  4

1267  124 245  15 77  8

1383  137 255  19 83  7

As-polished Sandpaper ground

(a)

(b)

(c)

Figure 1. SEM images showing the fracture origins of glass samples receiving different surface modifications. (a) Ground by sandpaper, (b) As-polished, and (c) Polished and etched by HF acid. Fracture surfaces for acid-etched samples are taken from 4-point bending experiment for comparison purpose. REFERENCE: [1] D. K. Shetty, A. R. Rosenfield, G. K. Bansal and W. H. Duckworth, “Biaxial fracture studies of a glass-ceramic”, Journal of the American Ceramic Society, 64[1], 1-4, 1981 [2] M. Cheng, W. Chen and K. R. Sridhar, “Experimental method for a dynamic biaxial flexural strength test of thin ceramic substrates”, Journal of the American Ceramic Society, 85[5], 1203-1209, 2002

Stress-Strain Response of PMMA as a Function of Strain-Rate and Temperature Paul Moy [email protected] C. Allan Gunnarsson [email protected] Tusit Weerasooriya [email protected] Army Research Laboratory Weapons and Materials Research Directorate Bldg 4600 Deer Creek Loop Aberdeen Proving Ground, MD 21005-5069 Wayne Chen [email protected] Purdue University School of Aeronautics and Astronautics and School of Materials Engineering 315 N. Grant Street West Lafayette, IN 47907 ABSTRACT The strain rate response of PMMA was investigated under uniaxial compression at room temperature at strain-rates ranging from 0.0001/sec to about 4300/sec. In addition, the temperature response of PMMA was investigated at strain-rates of 1/sec and 0.001/sec at temperatures ranging from 0°C to 115°C (below Tg). High rate experiments at room temperature (greater than 1/sec rates) were conducted using a split-Hopkinson Pressure bar (SHPB) with pulse-shaping. This is necessary to induce a compressive loading on the specimen at a constant strain rate to achieve dynamic stress equilibrium. Results conducted at room temperature show that PMMA is strain rate sensitive from quasi-static to dynamic loading. Additionally, the stress-strain response exhibits a decrease in the flow stress with an increase in temperature. These experimental data are being used to develop constitutive behavior models of PMMA. INTRODUCTION Polymethyl methacrylate (PMMA) is a thermoplastic polymer that is widely used in many applications in the automotive, medical, industrial, and consumer markets [1] due to its exceptional clarity and lightweight (density of about 1.181.19g/cm3). In addition, this polymer has exceptional high-impact strength characteristic. This makes the material appealing to use for windows for protection against bullets and blast. Thus, it is essential to understand the dynamic mechanical response of PMMA in transparent armor applications for the Army. PMMA has been extensively investigated over the past decades for its toughness, tensile, and compressive strengths. However, the mechanical response and failure behavior at high strain rates are still not completely understood. The mechanical properties of polymers are dependent upon two key factors, the rate of deformation and temperature. Observation of polymers, tested at high rates of strain, typically has an increase of the yield strength and the modulus and a decrease in strain to failure when compared to quasi-static results [2, 3]. Another study by Hall [4] reports the temperature of PMMA increases during deformation at high strain rate, whereas no temperature change at lower rates. Work by Walley et al [5] has shown that the strain rate and temperature affects the strain hardening behavior of glassy polymers. Three-dimensional material models by Arruda et al [6] and Boyce et al [7] for both PC and PMMA have been proposed to predict the trend for differing strain rates. Observation in their experiments has shown that adiabatic heating occurs over a range of strain rates (10-3/s to 10-1/s) for axial and in-plane compressive loading. Since polymers in nature having poor thermal diffusivity, adiabatic conditions are expected to prevail during high deformation rates. One common technique employed to study materials under dynamic loading and high strain-rate deformation is the Kolsky [8] bar method or prevalently known as split-Hopkinson pressure bar (SHPB). For example, Weerasooriya [9] and Green et T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_18, © The Society for Experimental Mechanics, Inc. 2011

125

126 al [10], each studied the compression behavior affected by a low-to-high strain rate response for Tungsten heavy alloys on a Maraging steel split-Hopkinson bar. Maiden and Green [11] used the SHPB to determine the dynamic compressive stressstrain behavior of Lucite and Micarta. Chou et al [12] also measured the compressive behavior of plastics using similar methods and Briscoe and Nosker [13] determined the rate effects on the flow stress of high-density polyethylene. Both works have also observed that the yield strengths increase with an increase in the strain rate. More recent work, Chen et al [14, 15] used a modified aluminum SHPB to study an epoxy, Epon 828/T-403, under high strain rate uni-axial tension and compression conditions. However, the testing of polymeric materials at high rates using the split Hopkinson bar has many challenges. These are 1) during high rate of loading, the specimen should be in dynamic stress equilibrium and deform uniformly for nearly the entire duration of the experiment, 2) during deformation, the strain rate should be nearly constant, and 3) the transmitted signals from strain gages are minute, and therefore there is a need to improve methods to measure these signals for valid experiments. These issues are discussed later in this paper. MATERIAL The polymethyl methacrylate (PMMA), used in this study is Plexiglas G™ manufactured by Atofina Chemicals and acquired from a local distributor. The Plexiglas G is made from a cell-casting process, which offers exceptional optical-clarity, high design stress, weatherability, and ease of fabrication from acrylic sheets [16]. The sheets were fabricated into a cylindricalshaped specimen from stock material of 3 mm thickness of the as-received sheet stock material, resulting in gauge length of the same dimension. A 6 mm diameter test specimen yields a diameter to length ratio of 2:1 and permitted an improved signal-to-noise ratio resolution from load cell in the low rate experiments. Also, the relatively smaller cross sectional area with the small gage length allowed higher strain rates to be attained on the aluminum Hopkinson bar beyond 2000/s without exceeding the bars’ elastic limit. Considerations were taken into account for the geometry of the compression samples, since it has been known that the thickness, or gage length in this case, affects the results in Hopkinson bar experiments [17]. A small batch of PMMA specimens was heat treated to relieve any possible residual stresses initiated from the lathe-machining process. The annealing temperatures were set for 2 hours at the materials’ glass transition temperature, 105 oC in a vacuum oven. Experimental results would reveal there was no significant deviation between the annealed samples and the unannealed. LOW RATE EXPERIMENTS Uni-axial quasi-static (10-3/s) and intermediate (1/s) strain rate experiments were conducted on an Instron, model 1331, servo-hydraulic test frame. A (TBASIC) computer program was written to command an exponentially decaying signal to achieve the desired constant strain rate. Load and displacement data were acquired from the Instron using a Nicolet digital oscilloscope. Petroleum jelly was used as lubricant on the specimen ends to minimize friction effects that possible cause barreling during loading. For each test, the specimen was placed on the center of the hardened steel platen, in-line with the actuator for conformity to ensure uni-axial compressive loading. To correct the measured displacement data, the compliance of the machine was also obtained without any specimens. For the low and high temperature experiments, an Instron model 3119-007 environmental chamber was used. The chamber has the capability to operate up to 350°C and with the use of liquid nitrogen, cryogenic temperatures to -40°C. All compression samples were temperature soaked to a minimum of 1 hour prior to testing. SPLIT-HOPKINSON BAR EXPERIMENTS A conventional split Hopkinson pressure bar (SHPB) consists of a striker, an incident bar, and a transmission bar as shown in Figure 1. The working principle of such a setup is well documented [18, 19]. Assuming a homogeneous deformation in the specimen and identical incident and transmission bars, analysis based on one-dimensional wave theory [8] shows that the specimen’s nominal strain rate,  (t), as

2 c0  r (t ) , (1) L where L is the original gage length of the specimen, r (t) is the time-resolved strain associated with the reflected pulse in the

 (t )  

incident bar, and C0 is the elastic bar-wave velocity of the bar material. Integration of equation (1) with respect to time gives the time-resolved axial strain of the specimen. The nominal axial stress, , in the specimen is determined using the equation

 (t ) 

At Et  t (t ) , As

(2)

127 where As is the cross-sectional area of the specimen, r(t) is the time-resolved axial strain in the transmission bar of crosssectional area At and Young’s modulus Et. Traditionally, the use of the Hopkinson bar apparatus to obtain large deformation in low-impedance materials such as polymers was very limited due to a high noise to signal ratio in the transmission bar. This is due to the high impedance mismatch from the metal incident bar to the polymer specimen, the sign is reduced several fold, when compared to metal alloy samples. Transmission bar Strain Gage

Incident bar

Specimen

Striker

Strain Gage Pulse shaper

Power Supply

Power Supply

Digital Oscilloscope

Figure 1. Schematic of pulsed-shaped SHPB Set-up

To mitigate this problem, Chen at al [14, 15, 20] used aluminum bars with a pulse-shaping technique. This ensures that the noise to signal ratio is low and that the equilibrium stress state and homogenous deformation in the polymer had been reached before failure/yielding occurred. A version of an aluminum Hopkinson pressure bar at the US Army’s Rodman Materials Research Laboratory was used to carry out the PMMA high rate experiments incorporating similar methods. High strength aluminum bars were acquired and specified to be centerless grounded to a diameter of 19 mm. In this experimental setup, strikers of varying lengths from 50 mm to 254 mm, comprised of the same aluminum material for the bars, provided the necessary amplitude and duration of the incident pulse. The pulse length is a function of the pulse shaper and striker length. Annealed copper disks of different thickness and diameters were employed for pulse shaping and placed at the incident bar end where the striker impacts. In some cases, different diameter and thickness copper disks were stacked to obtain the correct pulse shape. For the work presented in this paper, only room temperature experiments were conducted at the high strain rate. RESULTS AND DISCUSSION High Rate Effects A typical set of stress pulses in the input and output bars is shown in Figure 2. The reflected pulse shows that a plateau has been reached after an initial rise indicating the specimen is deforming under a constant strain rate, which is essential for testing of polymeric materials.

128 0.04 reflected 0.02

Volts

0 transmitted

incident

-0.02 -0.04 -0.06

Input Bar

-0.08

Output Bar

-0.1 -100

0

100

200

300

400

500

600

Time (sec) Figure 2. Typical input and output bar stress waves.

Polymeric materials are much more rate sensitive compared to traditional metals even during elastic deformation and thus it is important to reach a constant strain rate during testing. To further indicate that a constant strain rate was achieved, Figure 3 shows the true strain as a function of time. Since the strain increases almost linearly over a majority of the time, this confirms that a true strain rate controlled test has been achieved on the Hopkinson bar with pulse-shaping. 0.1

True Strain

0.08

0.06

0.04

0.02

0

0

20

40

60

80

100

Time (sec) Figure 3. True Strain as a Function of Time for PMMA at about 1300/s.

For a typical SHPB experiment, the stresses at the specimen/input-bar interface and specimen/output-bar interface are shown in Figure 4. The stress profiles, at both interfaces are approximately equal. This shows that the specimen is in a state of dynamic equilibrium.

129 350 300

Stress (MPa)

250 200 150

Specimen / Output Bar Interface

100

Input Bar / Specimen Interface

50 0

0

20

40

60

80

100

Time (sec) Figure 4. Stresses at Specimen / Bar Interfaces for PMMA at 1300/s Strain Rate with Pulse-Shaping

400

~4300/sec ~3400/sec ~2000/sec ~1300/sec 1/sec 0.001/sec 0.0001/sec

True Stress (MPa)

300

200

100

0

0

0.2

0.4

0.6

0.8

1

True Strain Figure 5. Stress-Strain Behavior for PMMA at Various Strain Rates

Figure 5 summarizes the stress-strain behavior for PMMA over a strain rate range from 10 -4/s to 4300/s. The low rate results reveal a typical behavior for thermoplastics under compression. First the material undergoes intrinsic softening followed by increasing strain hardening. Intrinsic softening appears to be the governing factor in the initiation of plastic deformation like necking or crazing in these glassy polymers [21]. Between these two low strain rates, the observed flow stress increases with the increase of strain rate. However, in the case for 1/s, the intrinsic softening is completely dominant beyond yield. The maximum strain achieved was consistently about 0.9 the for low rate experiments. At higher strain rates conducted on the Hopkinson bar, PMMA fractures at lower total strain and at a higher flow stress. The decrease in strain to failure was evident with the increase in the strain rate. The apparent modulus from the flow stress data increases from 1.78 GPa for 10 -4/s to 123.55 GPa at 4300/s. Table 1 shows the apparent modulus, yield strength, and strain to failure for the PMMA at the strain rates carried out at room temperature. From these values, both the moduli and yield strengths increase with increase in strain rate. Yield strength for 3400/sec and 4300/sec were unobtainable since the PMMA failed prior to yielding.

130 Table 1. Apparent Modulus and Apparent Yield Strength of PMMA at Strain Rates Tested Strain Rate

Modulus (GPa)

Yield Strength (MPa)

Strains to Failure

-4

10 /s

1.78

93.91

N/A

-3

10 /s

1.75

112.81

N/A

1/s

2.38

192.95

N/A

1300/s

5.14

335.55

0.0810

2000/s

11.42

350.00

0.0730

3400/s

58.71

N/A

0.0250

4300/s

123.35

N/A

0.0059

400

Yield Strength (MPa)

350

y = 221.19 + 35.434log(x) R= 0.98994

300 250 200 150 100 50 -5 10

0.001

0.1

10

1000

-1

Strain Rate (s )

Figure 6. Yield Strength as a Function of Strain Rate for PMMA at Room Temperature Figure 6 shows the plot of the apparent yield strength as a function of strain rate. This plot clearly indicates that PMMA is rate sensitive in which the yield strength increases with the increase of strain rate. Although the test conducted at 1/s reveals some thermal softening, it is not well understood how much of the intrinsic adiabatic heating affects the yield strengths. Temperature and Quasi/Intermediate Rate Effects Figures 7 and 8 are the stress-strain responses for PMMA conducted over a range of temperatures at strain rates 10-3/sec (quasi-static) and 1/s (intermediate), respectively. For the quasi-static experiments at the lower temperatures of 0°C and 15°C, PMMA behaves similar to the room temperature test. There is an intrinsic softening followed by strain hardening. The only subtle difference is that the flow stresses gradually increases from room temperature to the low temperature. All three plots are nearly parallel to each other. However, at the higher temperatures there is little strain hardening effects. In fact, this strain hardening effect decreases with the increase in temperature, in particular to the test at 115°C since this is about 10°C above the material’s Tg. Also, the hardening effects at 75°C and 115°C appears to develop at much higher strains. For the intermediate strain rate experiments under similar temperature conditions as the quasi-static, the mechanical response of PMMA at the elevated temperatures are almost the same as the quasi-static experiments. The main deviation is that the strain softening is much more dominant at the room and lower temperatures due to some adiabatic heating. There is no strain hardening effects at 0°C and 15°C. Interestingly is that strain hardening is present for room temperature result and thus the stress-strain crosses over the low temperature experiments. It is not like the quasi-static where the plots are parallel. Although the flow stresses have increased in comparison to the higher temperature test, the expected response for most glassy polymer would be a brittle like behavior. This shows that there are more adiabatic effects due to an increase in the loading rate even

131 for these polymers tested at the lower temperatures. Furthermore, relative to the corresponding temperatures, there is an increase in the flow stresses between the quasi-static and intermediate experiments. It is anticipated that there is greater adiabatic effects at higher loading rates. The lowest strain rate in the SHPB experiments for this effort is about 1300/s. For all of these high rate experiments, the specimen has either developed visible cracks (3400/s) or complete failure into smaller debris that appears to have melted and fused back together (4300/s). Yet for all the low rate experiments, no specimen failure occurred. Further study requires experiments to be conducted at other strain rates (i.e. 102/s and 100/s) to determine a failure threshold and measuring the adiabatic temperature rises during these loading rates.

250 0°C 15°C 20°C 75°C 115°C

True Stress (MPa)

200

150

100

50

0

0

0.2

0.4

0.6

0.8

1

True Strain

Figure 7. Stress-Strain Responses of PMMA at High and Low Temperature Conducted at 0.001/s Strain Rate

250 0°C 15°C

True Stress (MPa)

200

20°C 75°C 115°C

150

100

50

0

0

0.2

0.4

0.6

0.8

1

True Strain

Figure 8. Stress-Strain Responses of PMMA at High and Low Temperature Conducted at 1/s Strain Rate

132

250

Yield Strength (MPa)

200

-3

10 /s 1/s

150

100

50

y = 234.65 - 1.6844x R= 0.99517 y = 140.38 - 1.1192x R= 0.99724

0 -20

0

20

40

60

80

100

120

o

Temp ( C)

Figure 9. Yield Strengths as a Function of Temperature and Strain Rate A plot of the yield strength as a function of test temperature for PMMA compression experiments conducted at 0.001/s and 1/s is shown in Figure 9. As stated earlier, none of the compressed PMMA samples showed any cracks. The result of this plot emphasizes that adiabatic heating is quite dominant at the intermediate strain rate in comparison to the quasi-static. There is a significant decrease of the yield strengths for the corresponding temperatures. The linear fit between the two strain rates diverges as the lower temperatures. However, it is unclear if this trend continues below 0°C. SUMMARY The mechanical response of PMMA was determined for strain rate and temperature effects. By stress pulse-shaping the incident pulse, a near constant strain rate controlled experiments can be achieved resulting in valid results from high rate Hopkinson experiments. The outcome of the compression Hopkinson bar experiments of the PMMA material resulted in lower strains to failure with different fracture modes as the strain rate is increased. In addition, the apparent modulus and yield strength increase with increasing strain rate. Finally, adiabatic thermal softening is evident at strain rate of 1/s in comparison to rates at 10-4/s and 10-3/s even at the lower temperatures. For the higher temperature experiments, PMMA strain hardens at much higher strains than for room and low temperature conditions. Further study requires experiments to be conducted at the 102/s to determine a failure threshold and the ability to accurately measure the adiabatic temperatures during these loading rates. Experimental data are being used to develop constitutive model for the PMMA. REFERENCES 1. Cassidy, Robert. T. Acrylics. Engineered Materials Handbook: Engineering Plastics. ASM International, Vol. 2, pp. 103-108. 1988. 2. Lo, Y. C., Halldin, G. W. The Effect of Strain Rate and Degree of Crystallinity on the Solid-Phase Flow Behavior of Thermoplastic. ANTEC ‘84, pp. 488-491. 1984. 3. Kaufman, H. S. Introduction to Polymer Science and Technology. John Wiley & Sons Press, New York. 1977. 4. Hall, I. H. Journal of Applied Polymer Science, 12, pp 739. 1968. 5. Walley, S. M., Field, J. E., Pope, P. H., and Stafford, N. A. A Study of the Rapid Deformation Behavior of a Range of Polymers. Philos. Trans. Soc. London, A, 328, pp.783-811. 1989. 6. Arruda, E. M., Boyce, M. C., Jayachandran, R. Effects of Strain Rate, Temperature, and Thermomechanical Coupling on the Finite Strain Deformation of Glassy Polymers. Mechanics of Materials, 19, pp. 193-212. 1995.

133 7. Boyce, M. C., Arruda, E. M., Jayachandran, R. The Large Strain Compression, Tension, and Simple Shear of Polycarbonate. Polymer Engineering and Science, Vol. 34, No. 9, pp. 716-725. 1994. 8. Kolsky, H. An Investigation of the Mechanical Properties of Materials at Very High Rates of Loading. Proc. Roy. Soc. London, B62, pp. 676-700. 1949. 9. Weearsooriya, T. Deformation Behavior of 93W-5Ni-2Fe at Different Rates of Compression Loading and Temperatures. ARL-TR-1719. July 1998. 10. Green, J., and Moy, P. Large Strain Compression of Two Tungsten Alloys at Various Strain Rates. MTL-TR92-66. September 1992. 11. Maiden, C. J., Green, S. J. Compressive Strain-Rate Tests on Six Selected Materials at Strain Rates from 10 3 4 to 10 in/in/sec. Transactions of ASME, Journal of Applied Mechanics, pp. 496-504. 1966.

-

12. Chou, S. C., Robertson, K. D., Rainey, J. H. The Effect of Strain Rate and Heat Developed During Deformation on the Stress-Strain Curve of Plastics. Experimental Mechanics, Vol. 13, pp. 422-432. 1973. 13. Briscoe, B. J., Nosker, R. W. The Flow Stress of High Density Polyethylene at High Rates of Strain. Polymer Communications, 26, pp. 307-308. 1985. 14. Chen, W., Zhang, B., Forrestal, M. J. A Split Hopkinson Bar Technique for Low-Impedance Materials. Experimental Mechanics, 39, pp. 81-85. 1999. 15. Chen, W., Lu, F., Cheng, M. Tension and Compression Tests of Two Polymers under Quasi-Static and Dynamic Loading. Polymer Testing, 21, pp. 113-121. 2002. 16. http://www.atofinachemicals.com/atoglas/p2_allproducts_plex.cfm#1 17. Dioh, N. N., Leevers, P. S., Williams, J. G. Thickness Effects in Split Hopkinson Pressure Bar Tests. Polymer 34, pp 4230-4234. 1993. 18. Meyers, M. A. Dynamic Behavior of Materials. J. Wiley. 1994. 19. Gray, G. T. Classic Split-Hopkinson Pressure Bar Testing: In Mechanical Testing and Evaluation. American Society for Metals, Metals Handbook, Vol. 8, pp. 462-476. 2000. 20. Chen, W., Lu, F., Zhou, B. A Quartz-Crystal Embedded Split Hopkinson Pressure Bar for Soft Materials. Experimental Mechanics, 40, pp. 1-6. 2000. 21. Govert, L. E., van Melick, H. G. H., Meijer, H. E. H. Temporary Toughening of Polystyrene Through Mechanical Pre-Conditioning. Polymer, 42, pp. 1271-1274. 2001.

Dynamic Behavior of Three PBXs with Different Temperatures J. L. LIa,b, F. Y. LU 1,a, R. CHENa, J. G. QINa, P. D. ZHAOa, L. G. LANc, S. M. JINGc a. National University of Defense and Technology; b. Institute of Fluid Physics, Chinese Academy of Engineering and Physics; c. Institute of Chemical materials, Chinese Academy of Engineering and Physics. ABSTRACT Polymer-bonded explosives (PBXs) came into being and are widely used as energetic fillings in many systems as they are more chemically and mechanically stable than traditional ones. There are three kinds of materials named by PBX1, PBX2 and PBX3, the last two of which are newly invented. To investigate the mechanical properties of newly manufactured PBXs, we carried out the dynamic compression tests as a function of strain rate and temperature with Split Hopkinson Pressure Bar (SHPB). The experiment temperatures were set to be 25°C, 50°C and 65°C. The constant strain rate is assured by using brass as a pulse shaper, and as for brittle materials, the constant loading strain rate is limited at about 550s-1. The results show that the compressive strengths and moduli of each material decrease with increasing temperature and increasing with increasing strain rate. Failure is by brittle fracture for each condition and the cracks’ major axis makes an angle of 30° with the direction of the applied stress. Keywords: PBX, dynamic behavior, temperature, strain rate, failure Introduction As many unexpected explosions happened, many researchers commit themselves to develop more secure but more violent explosives [1,3,5]. At present, the polymer binder explosives (PBX in short) are widely used as its attributes of low sensitivity and high detonation energy. PBX is alike particle reinforced composite containing main energetic crystal granite and matrix including polymer binders，plasticizer and other materials. With different kind of energetic crystals, polymer binders or their distribution, the mechanical behavior would change a lot [2-5]. PBXs (Polymer bonded explosives) behave quite differently at high loading strain rates compared with static cases The strength increases as strain rate increases, and decreases as temperature increases

[6,7]

.

[6,8]

. Explosives would detonate

earlier than expected as the changes of temperature or strike by shock waves. Besides, the materials in outer space would change its properties for the discrepancy of the temperature. Therefore, it is crucial to study the dynamic behavior of temperature-dependent materials, and it is also important in many areas, such as penetration, protection from explosion, explosive weld, and so on. SHPB (Split Hopkinson Pressure bar) has become a widely used technique to measure the uniaxial compressive stress-strain relation of various materials in the strain rate range of 102~104 s-1. The stress, strain and strain rate history of the specimen can be calculated by the records of on the incident and transmitter bars based on one-dimensional wave propagation theory. In an effort to understand newly manufactured PBXs, we carried out uniaxial compressive experiments with different strainrates (50~550s-1)and different temperatures(25℃、50℃、65℃) by the use of SHPB.

1

Science College, National University of Defense Technology, Changsha, Hunan, 410073, China. E-mail: [email protected]; [email protected].

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_19, © The Society for Experimental Mechanics, Inc. 2011

135

136

Experimental design On purpose of finding out the most insensitive but high detonation behavior explosives, two PBXs (PBX2 and PBX3) are newly engineered, with HMX and TATB as energetic crystals, and F2311 and F2314 as polymer binders. As a comparison, PBX1, containing TATB and F2311, was also investigated next. Three experimental temperatures(25℃、 50℃ and 65℃) were adopted to study the temperature effects on the three energetic materials. We can realize constant strain rate with proper diameter and length of the brass pulse shaper in SHPB. Here are three original curves records obtained by oscilloscoper in Fig.1, in which three strain rate loadings are presented. It’s widely accepted as constant strain rate when the reflected pulse has a flat before the curve uprises again, which reflects the beginning of the specimen failure. The failure point has been noted by the arrows in Fig.1. The constant strain rate requirement is satisfied very well in our experiments as shown in Fig.1. But the materials seem to be brittle, for the failure is controlled by its strain. So the strain rate is limited to a critical value for some size combination of the specimen and striker.

Fig.1 Constant strain rate loads

Experimental results We know constant strain rate can minimize the inertia effect of SHPB system and simplified the constitutive model. Now we discuss the results of those experiments with brass wave shaper. 1) Under condition of normal temperature, PBX1 seems strain-rate-independent (Fig. 2). But as the temperature rises, the strain rate effect becomes more obvious. We can learn about from he figures directly that the maximum stress decreases as the temperature rising. 2) PBX2 behaves strain-rate-dependently at room temperature (Fig.3). But at 50℃, the difference among three strain rates is very small. Maybe the transition temperature of this material is around 50℃, which brings some uncertainty of strength (symbolized by the maximum stress). At 65℃, the discrepancy becomes larger as strain rate varies, and the highest strength is only a little less than the case of 50℃. It means the apparent change of material’s mechanical behavior is close to 50℃or below. 3) The maximum stress of PBX3 changes not so much as the other two materials while temperature and the strain rate varies (Fig.4). There is a sudden reduction of ascending slope at 65℃ with the strain rate of 414s-1. PBX3 behaves much brittle on the whole. The obtained stresses decrease rapidly after the peak value as the strain increases in most experiments. It means that once specimen gets the critical state of fracture, the inner cracks would spread rapidly to cause the specimen crushing.

137

As the strain rate increases, all the three materials’ strengths increase and the specimen would break into pieces because of the much strain energy released by fracture. At high temperature, specimen would have shear areas to prevent from fracture，which leads to higher failure strain. PBX1 is very brittle at normal temperature. But at 50℃ and 65℃, the stress after peak value decreases slowly and the unloading process likes an undamaged material as the load ends. PBX2 has similar behaviors at 65℃. PBX3 seems the most brittle.

Fig. 2 Stress-strain curves of PBX1

Fig. 3 Stress-strain curves of PBX2

Fig. 4 Stress-strain curves of PBX3

Temperature and strain rate effects We now discuss about temperature and strain rate effect on such mechanical parameters such as modulus, failure strength and failure strain.

1) Dynamic elastic modulus The dynamic elastic modulus is believed to be the slope of the 5%-20% of the linear part of the stress-strain curve. So called dynamic elastic modulus here is to differ it from the usual called Young’s modulus because they don’t numerically correspond to the static tests. Although these values are discrete in some extent, we could also see that dynamic moduli of the three materials decrease apparently while the temperature rises, but changes little while strain rate rises (Table.1-3). PBX1’s dynamic modulus is only 2~4Mpa. This can be explained by the main energetic crystal (TATB)’s stiffness, which is much smaller than HMX. The dynamic modulus of PBX3 is relatively higher. In general, PBX1 has the smallest stiffness, and thus dynamic modulus is the least.

138

2) Failure strength It has been defined before that the maximum stress is the failure stress or strength. At the same temperature and strain rate, as for strength, we have PBX1
25°C

50°C

65°C

Strain Rate

Dynamic Modulus

Failure Stress σ

（1/s）

E（GPa）

（Mpa）

100

3.64

43.4

1.86%

220

3.81

43.06

2.14%

360

3.10

42.66

2.14%

480

3.20

42.16

2.14%

Failure Strain ε

90

3.52

37.45

1.7%

300

3.15

35.3

1.42%

550

2.92

39.31

1.96%

100

2.32

29.5

1.83%

200

2.47

31.98

2.14%

430

2.48

33.89

2.14%

Tabel 2 Experimental Results of PBX2 Temperature

25°C 50°C

Strain Rate

Dynamic Modulus

Failure Stress σ

（1/s）

E（GPa）

（Mpa）

140

5.20

45.45

1.58%

240

5.98

48.48

1.58%

400

4.58

50.5

1.76%

80

4.38

38.67

1.54%

210

3.43

38.92

2.03%

Failure Strain ε

139

65°C

450

3.61

40.33

2.56%

70

3.87

32.45

1.16%

180

3.95

38.17

1.66%

550

3.70

39.9

2.1%

Tabel 3 Experimental Results of PBX3 Temperature

25°C

50°C

65°C

Strain Rate

Dynamic Modulus

Failure Stress σ

（1/s）

E（GPa）

（Mpa）

140

6.10

53.18

1.59%

230

7.96

54.05

1.33%

460

6.66

57.12

1.7%

130

4.95

46.82

1.44%

260

6.22

50.0

1.67%

450

7.88

50.63

1.73%

50

4.96

33.92

0.92%

70

4.48

38.26

1.35%

230

4.27

44.58

1.67%

450

2.88

49.03

2.09%

Failure Strain ε

Failure mode In our tests, all the cracks, either on the surfaces or inside the specimen, develop in such directions that have an angle of 30° with the direction of the applied stress(Fig.5). This is similar to the work of Griffith on glass several decades ago (represented by Wiegand[8]). One part of the failed specimen in Fig.9a is a cone, which is strict axial symmetry. It should be noted that only PBX2 have such standard failure mode in our tests. The angles are almost the same although the strain rate variations are different, which may indicate such angle reflects inner property of material.

30°

(b)

（a） -1

Fig.5 Failure modes of specimen (a) PBX2-240s -25℃, (b) PBX2-210s-1-50℃ Conclusion Dynamic compression tests have been carried on three PBXs with different temperatures and strain rates. The compressive strengths and dynamic moduli of each material decrease with increasing temperature and increase with increasing strain rate. And it should be noted that the strain rate sensitivity is very small, especially in the range of strain rate from 50s-1 to 550s-1. Failure strains of each material are nearly insensitive to temperature and strain rate and can be adopted as the criterion for failure. The cracks develop along the direction that that have an angle of 30° with the direction

140

of the applied stress. Acknowledgements This research is supported by National Basic Research Program of China under grant No. 61383. We also would like to acknowledge the support of National Natural Science Foundation of China under grant No.10872215, No.10672177, and No. 10902100. References [1]. D. R. Drodge, D. M. Williamson, S. J. P. Palmer, W. G. Proud and R. K. Govier. The mechanical response of a PBX and binder: combining results across the strain-rate and frequency domains. J. Phys. D: Appl. Phys. 43: 335-403, 2010. [2]. N.K. Bourne, G.T. Gray lll. Dymanic response of binders; teflon, estaneTM, and Kel-F-800TM. Journal of Applied Physics. 98,123503, 2005. [3]. Siviour C R, Laity P R, Proud W G，et al. High strain rate properties of a polymer-bonded sugar: their dependence on applied and internal constraints[J]. Proceedings of the Royal Society. A464:1229-1255, 2008. [4]. D. A Wiegand, S. Nicolaides, J. Pinto, Mechanics and thermomechanical properties of NC based propellants, J. of Energetic Materials, Vol. 8: 442-461, 1990. [5]. Daniel R. Drodge and William G. Proud. THE EFFECTS OF PARTICLE SIZE AND SEPARATION ON PBX DEFORMATION, shock compression of condensed matter, 2009 [6]. G.T. Gray lll, D.J. IDAR, W.R. Blimenthal, et al. High-and-low-strain rate, compression of several energetic material composites as a function of strain rate and temperature. 11th international detonation symposium. AUG. 31-SEP 4, 1998 [7]. Daniel R. Drodge, John W. Addiss, David M. Williamson and William G. Proud. HOPKINSON BAR STUDIES OF A PBX SIMULANT. Shock compression of condensed matter, 2007. [8]. D. A. Wiegand, J. Pinto, The mechanical response of TNT and a composite, composition B, of TNT and RDX to compressive stress: І Uniaxial stress and yield, J. of Energetic Materials, Vol. 9: 19-80,1991. [9]. X. F. Ma, MD simulation of the structure and behavior of polymer bonded explosives, Nanjing, China, 2006.

DYNAMIC COMPRESSIVE PROPERTIES OF A PBX ANALOG AS A FUNCTION OF TEMPERATURE AND STRAIN RATE J. Qin1*, Y. Lin1, F. Lu1, Zh. Zhou2, R. Chen1 and J. Li1 1. College of Science, National University of Defense Technology, 410073 Changsha, Hunan, P.R. China 2. National Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, 100081 Beijing, P.R. China ABSTRACT: The compressive strength behavior of a Polymer Bonded Explosives (PBX) analog was measured as a function of temperature (25°C to 95°C) and strain rate (100 to 700 s-1) using the split Hopkinson pressure bar (SHPB).The result exhibits the flow stress of the PBX analog is strong dependency on temperature and strain rate. According to the result, an improved Sargin constitutive model was used to describe the dynamic compressive behavior of the material, and the modeling curves fit well with the experimental. Key words: PBX analog; Compressive strength; SHPB; Constitutive model INTRODUCTION Physically-based constitutive models are needed to predict the mechanical behavior, damage evolution, and performance of modern energetics for their safe application. Understanding and modeling the mechanical response of polymers and polymerbased composites is of great interest for defense and commercial applications related to (1) the need for predictive constitutive model descriptions for use in large-scale finite-element simulations of damage and deformation, and (2) focused emphasis on understanding the dynamics of localization phenomena and mechanical failure of polymeric composites. New continuum models, based on actual physical and chemical mechanisms, to describe complex loading processes must account for complex phenomenology, including temperature, strain rate, orientation effects like crosslinking and chain stretching, or texture (created by extrusion or directional formation), and aging effects on mechanical performance, if a predictive capability is to be achieved. Conventional methods have been used to measure the mechanical properties of polymers and polymer composites at low strain rates. However, high strain rate methods using the split Hopkinson pressure bar (SHPB) must be modified to achieve stress equilibrium with these low sound speed materials; to obtain adequate pressure bar signal output at very low stress levels; and to minimize undesirable stress triaxiality caused by friction on specimen interfaces. A number of previous studies have probed the constitutive response of a wide variety of plastic bonded explosives (PBX).[1-10] Some literatures investigated PBX at low strain rate.[2,5,7] PBX behave quite differently at high strain rates compared to low strain rate. Although many literatures gave the curves and trends about the mechanical properties of PBXs as function of temperature and strain rate,[6,8-10] there are few literatures showed the expressions to quantify them. In the present investigation, a PBX analog instead of a true PBX was studied because of safety. Uniaxial compression tests were performed at strain rates from 100 to 700 s-1 and for temperatures from 25°C to 95°C on the PBX analog. From the data of experiments, a constitutive model was used to describe the dynamic compressive behavior of the PBX analog. The model curves can not only predict the raise part but also the decline part of experimental. EXPERIMENTAL The SHPB used for this study was equipped with 20mm diameter 7075A All-alloy, to improve the signal-to-noise output *

Corresponding author. Tel.:+8673184573276; fax: +8673184573297. E-mail address: [email protected]

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_20, © The Society for Experimental Mechanics, Inc. 2011

141

142 associated with low strength materials. The length of incident bar, transmission bar and striker bar were 2000mm, 1200mm and 200mm, respectively. The specimens geometry were 10mm diameter and 6mm long. Specimens were tested at temperatures of 25°C, 50°C, 65°C, 80°C and 95°C. Controlled temperature variations between 25°C to 95°C were achieved putting the specimen in a small box which was heated using hot water. The water temperature was controlled to the tested temperature, and specimen was equilibrated at temperature for ~15 min prior to testing. After heated, the specimen was quickly taken out of the box and then installed on the bars. The gas gun was fired once the installation was finished. The whole time from take the specimen out of the box to finish loading was 2~3seconds. The temperature variety of the specimen during installation is show in figure 1. Because of the low constant of heat exchange of the specimen, it could been seen that the temperature variety of the specimen was less than 1°C in 3 seconds although at 95°C. The error of temperature less than ~2 percent proved the heated method is valid. The surfaces of bars which connected the specimen during loading were lubricated with either a thin spray coating of boron nitride or a thin layer of molybdenum disulfide grease before the specimen installed, which could minimize undesirable stress triaxiality caused by friction on specimen interfaces. A small cylinder made of brass as a shaper was installed on the impacted-surface of incident bar, to change the shape of incident wave. The existence of shaper enlarged the time of incident wave rose to maximum, which made it is more easier to achieve stress equilibrium on the both surfaces of specimen and constant strain rate loading. The typical wave of loading process is showed in figure 2. As is shown in figure, the stress equilibrium is clearly achieved between the two sides of the specimen since the transmitted stress wave is nearly identical to the sum of the incident and reflected waves during the whole loading period. 96 94 93

Stress(MPa)

o

Temperature( C)

in tr re in+re

20

95

92 91 90 89 88

10

0

-10

87 86

0

1

2

3

4

5

6

7

8

-20

9

0

100

Time(sec)

200

300

Time(μs)

Fig. 1 Variety of temperature of specimen during testing

Fig. 2 The stress histories in the bars. in: input; tr: transmitted; re: reflected

RESULTS

40

40

35

35

Engineering stress(MPa)

Engineering stress(MPa)

The stress-strain curves of specimens under different temperatures and strain rates are shown in figure 3.

30 25 20 15 strain rate 152/s 367/s 689/s

10 5 0 0.00

0.02

0.04

Engineering strain

(a)25℃

0.06

30 25 20 15 strain rate 142/s 297/s 551/s

10 5 0 0.00

0.02

0.04

Engineering strain

(b)50℃

0.06

Engineering stress(MPa)

Engineering stress(MPa)

143

20 15 10 strain rate 150/s 336/s 482/s

5 0 0.00

0.02

0.04

20 15 10 strain rate 157/s 327/s 529/s

5 0 0.00

0.06

0.02

0.04

0.06

Engineering strain

Engineering strain

(c)65℃

(d)80℃

Engineering stress(MPa)

20

15

10 strain rate 202/s 313/s 570/s

5

0 0.00

0.02

0.04

0.06

Engineering strain

(e)95℃ Fig. 3 Curves of engineering stress-strain at different temperatures and strain rates These data show that the flow stress peaks consistently between 2% and 3.5% strain before slowly decaying with further strain.

1.6

40

1.4

35

Failure strength(MPa)

Dynamic modulus(GPa)

The temperature dependence of elastic modulus, flow stress and failure strain at high strain rate is summarized in Figure 4. The points in the Figure 4 are the experiment results. These plots show the elastic modulus, flow stress increase with increasing strain rate, and the higher strain rate is, the higher increasing speed becomes. The elastic modulus are approximately independent of temperature, because at the nearly same strain rate, elastic modulus at different temperatures present confusion. The flow stress decreases with increasing temperature except 50°C, which performs nearly the same value of 25°C. The result could be analyzed as the stronger force connect crystal and bonder at ~50°C, so the flow stress maintained. As the temperature went on increasing, the effect of the temperature-soften presented and the flow stress decreased. Failure strain of the specimen is independent of strain rate and temperature, and presents between 2% and 3.5% (Figure 4 (c)).

1.2 1.0 0.8 0.6

O

25 C O 50 C O 65 C O 80 C O 95 C

0.4 0.2 0.0

0

100

200

300

400

500

Strainrate(1/s)

(a) elastic modulus

600

700

800

30 25 20 15

O

25 C O 50 C O 65 C O 80 C O 95 C

10 5 0

0

100

200 300

400 500 600

Strainrate(1/s)

(b) flow stress

700 800

144 4.0

Failure strain(%)

3.5 3.0 2.5 O

25 C O 50 C O 65 C O 80 C O 95 C

2.0 1.5 1.0

0

100

200 300

400 500 600

700 800

Strainrate(1/s)

(c) failure strain Fig. 4 Varieties of mechanic behaviour at different temperatures and strain rates Constitutive models of PBX now available usually can only describe the climb part of the stress-stain curve depend on temperature and strain rate, because of the complex structure of PBX. And the other part of curve that the flow stress coming down with further strain takes an important part in understanding the remain-strength of material. Because of the similar shape of stress-stain curve between the specimen and concrete, the Sargin model which described concrete behavior originally is modified to describe the behavior of the specimen.[11] The Sargin model describe as:

ε ε + ( β − 1)( ) 2 ) εf εf σ= ε ε 1 − (α − 2) + β ( ) 2 εf εf −σ f (−α

(1)

Where α=E/Ef, (Ef=σf/εf), Ef is the secant modulus at peak compressive strength, parameter β affects decline part of stressstrain curve. From analysis above, the mechanical behavior affected by temperature and strain rate were explained as the following equations:

E ε& T − Tr ε& T − Tr = AE + BE exp( ) + CE + DE exp( ) ε&0 ε&0 Tr E0 Tr

(2)

σf T − Tr ε& ε& T − Tr = Aσ + Bσ exp( ) + Cσ + Dσ exp( ) Tr σ0 ε&0 ε&0 Tr εf ε& T − Tr ε& T − Tr = Aε + Bε exp( ) + Cε + Dε exp( ) ε0 ε&0 Tr ε&0 Tr

(3) (4)

Where A, B, C and D are the parameters controlled by experimental results, B, C and D mean strain rate effect, temperature effect, both strain rate and temperature effect, respectively. From the experiment results, take

=138s-1, room temperature Tr=25°C. σ0=19.29Mpa, ε0=2.98% and E0=0.802Gpa are flow

stress, failure strain and elastic modulus at strain rate

and temperature Tr respectively. β=0.5.

Constitutive parameters fitting by experimental results used the inheritance arithmetic are shown in table 1. Table 1 Modify Sargin model parameters A E 1.173 σ 1.104 ε 0.934

B 2.83E-03 4.70E-03 1.91E-03

C -0.1163 -0.1573 -0.0202

D 5.41E-04 0 -8.77E-04

145 The solid lines in figure 4 (a), (b), and (c) are the fitting results by using equations (2), (3), and (4) and the parameters in table 1. And the model curves fitting by equation (1) are compared to experimental result in figure 5. 40

Engineering stress(MPa)

Engineering stress(MPa)

40 35 30 25 20 15

experimental of 152/s experimental of 367/s experimental of 689/s modeling of 152/s modeling of 367/s modeling of 689/s

10 5 0 0.00

0.02

0.04

35 30 25 20 15

experimental of 142/s experimental of 297/s experimental of 551/s modeling of 142/s modeling of 297/s modeling of 551/s

10 5 0 0.00

0.06

Engineering strain

0.02

Engineering stress(MPa)

Engineering stress(MPa)

(b)50℃

20 15

5 0 0.00

experimental of 150/s experimental of 336/s experimental of 482/s modeling of 150/s modeling of 336/s modeling of 482/s

0.02

0.06

Engineering strain

(a)25℃

10

0.04

0.04

0.06

20 15 10 5 0 0.00

Engineering strain

experimental of 157/s experimental of 327/s experimental of 529/s modeling of 157/s modeling of 327/s modeling of 529/s

0.02

0.04

0.06

Engineering strain

(c)65℃

(d)80℃

Engineering stress(MPa)

20

15

10

5

0 0.00

experimental of 202/s experimental of 313/s experimental of 570/s modeling of 202/s modeling of 313/s modeling of 570/s

0.02

0.04

0.06

Engineering strain

(e)95℃ Fig. 5 Comparison of stress-strain curves with modeling to experimental CONCLUSION Split Hopkinson pressure bar tests were made on a PBX analog at different temperatures. The following conclusions can be drawn: 1) the compressive flow strength of specimen is strongly dependent on both strain rate and temperature. The elastic modulus is only mildly dependent on strain rates. The failure strain is not evidently temperature and strain rates dependent. 2) the specimen has an invariant strain to failure of approximately 2~3 percent. 3) The modify Sargin constitutive models could describe and predict the stress-strain curve of the specimen as a function of temperature and strain rate not only the climbing part, but also the declining part.

146 ACKNOWLEDGMENTS This work was supported by National Basic Research Program (973 program) under grant No. 61383. And we also would like to acknowledge the support of National Natural Science Foundation of China under grant No.10872215, No.10672177, and No. 10902100. REFERENCES [1] Hoge, K. G．The Behavior of Plasticbonded Explosives under Dynamic Compressive Loads．Appl. Polym. Symp., vol. 5, 1967, pp. 19-40. [2] Peeters, R. L．Characterization of Plastic Bonded Explosives．J. Reinf. Plast. Compos., vol. 1, 1982, pp. 131-140. [3] Field, J. E., Palmer, S. J. P., Pope, P. H., Sundararajan, R. and Swallowe, G. M．Mechanical Properties of PBX’s and their Behaviour during Drop-weight Impact．Proc. 8th Int. Detonation Symposium, White Oak, Maryland, USA, 1985, pp. 635-644. [4] Palmer, S. J. P., Field, J. E. and Huntley, J. M．Deformation, Strengths and Strains to Failure of Polymer Bonded Explosives．Proc. R. Soc. Lond. A, vol. 440, 1993, pp. 399-419. [5] Funk, D. J., Laabs, G. W., Peterson, P. D. and Asay, B. W．Measurement of the Stress-Strain Response of Energetic Materials as a Function of Strain Rate and Temperature: PBX 9501 and mock 9501．In Shock Compression of Condensed Matter-1995; AIP: Woodbury, New York, 1996, pp. 145-148. [6] Gray III, G. T., Blumenthal, W. R., Idar, D. J. and Cady, C. M．Influence of Temperature on the High-Strain-Rate Mechanical Behavior of PBX 9501．In Shock Compression of Condensed Matter-1997; AIP: Woodbury, NY, 1998, pp. 583-586. [7] Idar, D. J., Peterson, P. D., Scott, P. D. and Funk, D. J．Low Strain Rate Compression Measurements of PBXN-9, PBX 9501, AND Mock 9501．In Shock Compression of Condensed Matter-1997; AIP: Woodbury, NY, 1998, pp. 587-590. [8] Gray III, G.T., Idar, D. J., Blumenthal, W.R., Cady, C.M., and Peterson, P. D ． High- and Low-Strain Rate Compression Properties of Several Energetic Material Composites as a Function of Strain Rate and Temperature．Proc. 11th Int. Detonation Symposium, Snowmass Village, CO, USA, 1998, pp. 76-83. [9] Blumenthal, W.R., Gray III, G.T., Idar, D. J., Holmes, M.D., Scott, P.D., Cady, C.M., and Cannon, D.D．Influence of Temperature and Strain Rate on the Mechanical Behavior of PBX 9502 and Kel-F 800TM．In Shock Compression of Condensed Matter-1999; AIP: Woodbury, NY, 2000, pp. 671-674. [10] Idar, D. J., Thompson, D. G., Gray III, G. T., Blumenthal, W. R., Cady, C.D., Peterson, P. D., Jacquez, B. J., Roemer, E. L., Wright, W. J．Influence of Polymer Molecular Weight, Temperature, and Strain Rate on the Mechanical Properties of PBX 9501．In Shock Compression of Condensed Matter-2001; AIP: Woodbury, NY, 2002, in press. [11] Sargin M．Stress-strain Relationships for Concrete and the Analysis of Structural Concrete Sections．University of Waterloo, Solid Mechanics Division, SM Study. 1971, (4):23-46.

Dynamic Response of Shock Loaded Architectural Glass Panels

Puneet Kumar, and Arun Shukla*. Dynamic Photomechanics Laboratory, Department of Mechanical, Industrial and Systems Engineering, University of Rhode Island, Kingston, RI 02881, USA. *Corresponding author email: [email protected]

Abstract A controlled study has been performed to understand fracture and damage mechanisms in glass panels subjected to air blast. A shock tube apparatus has been utilized to obtain the controlled blast loading. Five different panels, namely plain glass, sandwiched glass, wired glass, tempered glass and sandwiched glass with film on both the faces are used in the experiments. Fully clamped boundary conditions are applied to replicate the actual loading conditions in windows. Real-time measurements of the pressure pulses affecting the panels are recorded. A post-mortem study of the specimens was also performed to evaluate the effectiveness of the materials to withstand these shock loads. The real time full-field in-plane strain and out-of-plane deformation data on the back face of the glass panel is obtained using 3D Digital Image Correlation (DIC) technique. The experimental results show that the sandwich glass with two layers of glass joined with a PVB interlayer and protective film on both the front and back face out performs the other four types of glass tested. 1. INTRODUCTION Accidental explosions or bomb blasts cause extreme loading on glass structures. This results in the shattering of glass panels into small pieces which have sharp edges and move at very high velocities. These high velocity glass fragments are the major cause of injuries to people. Apart from this, the blast pressure entering the building through the shattered window panels can also cause additional injuries to the occupants. Five different types of glass panels are subjected to blast loading using a shock tube to study their dynamic response. Post-mortem analysis has been conducted on the blast loaded panels to evaluate the effectiveness of the material to mitigate blast loading. Previously, the main focus of research in this area has been on the numerical/theoretical analysis of glass panels subjected to an explosion. Recently, experimental studies have been done on glass panels to analyze their blast mitigation properties. However, these experiments used either an indenter or an impactor to simulate the blast condition. The aim of this study is to analyze the damaged area, midpoint transient deflection, and other characteristics of the dynamic response of glass panels subjected to a controlled blast loading. Saito et. al [1] modeled the blast process on glass using the indenting method. They discussed the mechanism of formation of residual stress in the indenting process, both analytically and experimentally, in order to optimize the processing conditions to produce the desired residual stress in a blast loading. Gogotsi et al. [2] used different shapes of indenters to analyze the fracture in rectangular shaped optical and technical glasses and showed that the fracture resistance of float glass was higher than that of fused silica and other optical glasses Bouzid et al. [3] studied glass material under impact conditions where stress waves and their interactions are dominant. They proposed a damage model characterized by the damage volume to evaluate the damage development and fragmentation. It was found that damage volume is a function of impact duration and critical stress. Wei et al. [4] formulated a failure criterion based on the energy balance approach for a laminated glass panel subjected to a blast loading. They developed a damage factor to assess the failure of the laminated glass panel. According to them, the negative phase of the blast load will cause the breakage of the laminated glass if the positive phase of the blast load is not violent enough to cause failure. They also predicted the size of the glass shards using the surface energy based failure model. Wei at al. [5] developed a 3-D nonlinear dynamic finite element model to characterize the stress distribution in a laminated architectural glazing subjected to blast loading. They considered the viscoelastic parameter of PVB interlayer on the dynamic response of the glass panel. The parametric study showed that the panel exhibited a non-linear response to the blast overpressure. At the same time they found that the through thickness stress and displacement distribution are nearly linear. Karuthammer et al. [6] analyzed the effect of the negative phase of blast waves on glass panels. They developed an approximate numerical model for the dynamic response simulation of glass panels subjected to blast loading. This also included the stochastic considerations of the glass flaw characteristics. They also conducted a parametric study showing that the glass panels exhibit different responses at different scaled ranges, and for different charge sizes. In one of the other publications, Wei et al. [7] studied the response of a rectangular laminated glass panel based on the classical small deflection and large deflection theory. Their main conclusion was that the mid-span deflection and tensile stress due to the negative pressure is almost double of that in the case of positive pressure. They also showed that the tensile stress develops on the T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_21, © The Society for Experimental Mechanics, Inc. 2011

147

148 back face of the laminate panel where as the compressive stress develops on the front face or the face which experiences the blast loading. Glasses have higher compressive strength, which is about ten times that of their tensile strength [8]. The present paper focuses on the response of five different types of glass panels subjected to a controlled blast loading applied by a shock tube. Real-time measurements of the pressure pulses affecting the panels are recorded. Post-mortem study is used to evaluate the effectiveness of the panels to withstand these shock loads. The real time deformation mapping is done using the Digital Image Correlation (DIC) technique Tiwari et al. [9] successfully combined the use of the high-speed stereo vision with 3D Digital Image Correlation to quantify the dynamic specimen response during a buried blast loading with an inter-frame time ranging from 16 to 40 μs. They confirmed that dynamic stereo-vision systems are capable of accurately measuring surface deformation data at high rates, such as in the case of impact, penetration and blast loading. In the following sections, the methods used to carry out these experiments are presented, and the experimental results are discussed in detail. 2. Experimental Procedure 2.1 Material Details The five different panels used during these experiments include a clear glass panel, tempered glass panel, wired glass panel, sandwiched glass panel and laminated sandwiched glass panel with a protective film on both of its faces (three specimens are shown in fig. 1). Each experiment is repeated three times. The specimens are 305 mm long x 305 mm wide x 6.5 mm thick. Laminated sandwiched glass panel has a thickness of 7.5 mm because of the protective film on both front and back face of the sandwiched panel.

Plane Glass Panel

Sandwiched Glass Panel

Wired Glass Panel

Figure 1: Specimens The panels are made out of soda-lime-silica glass which has a tensile strength in the range of 20-100 MPa and a compressive strength of approximately 10 times of that. The clear glass panel is the regular glass panel on which no additional treatment is performed. The tempered glass panel is made from the clear glass panel. Specimens of a specific size are cut out from the clear glass panel which is then heat treated to release the pre-stress and induce beneficial residual stresses. The wired glass panel is manufactured by building the whole panel on a wire frame such that the wire frame is imbedded within it. The sandwiched glass panel consists of two clear glass panels which are bonded by a polyvinyl butyral (PVB) interlayer. This bonding process takes place utilizing heat and pressure treatment. The PVB layer has a good bonding strength, is optically clear and does not diminish the optical properties of the glass panel. The laminated sandwiched glass panel is made by adhering a protective film from XO ARMOR® on both of the outer faces of the sandwiched glass panel. The XO® protective film is 0.5 mm thick and a special adhesive XO® bond was used to adhere the protective film onto both of the face of the sandwiched glass panel. According to the manufacturers, XO® bond penetrates the glass surface and forms a chemical bond between the glass and XO® film at nano level. 2.2 Shock loading apparatus The shock tube apparatus used in this study to obtain the controlled dynamic loading is shown in fig. 2. A complete description of the shock tube and its calibration can be found in [10]. In principle, the shock tube consists of a long rigid cylinder, divided into a high-pressure driver section and a low pressure driven section, which are separated by a diaphragm. By pressurizing the high-pressure section a pressure difference across the diaphragm is created. When this pressure differential reaches a critical value, the diaphragm ruptures. The following rapid release of gas creates a shock wave, which travels down the tube to impart dynamic loading on the specimen. The specimen is held in a fixture that ensures the proper specified boundary conditions. When the shock wave impacts the test panel at the end of the tube, the gas is superheated and the wave is reflected at a higher pressure than that of the initial shock front. An extensive derivation of the theoretical equations for shock tubes has been

149 previously established in literature and is briefly discussed in the following section [11]. In basic shock wave theoretical formulations the following assumptions are generally used to describe the gas flow: 1. The gas flow is one-dimensional. 2. The gas is ideal and has constant specific heats. 3. Heat transfer and viscosity effects are neglected. 4. Diaphragm rupture is instantaneous and does not disturb the subsequent gas flow. Using conservation of energy, mass, and momentum as described by Wright [16], the following relationships for pressure, temperature and density across a shock front can be derived:

P2 2 M 12  (  1)  P1  1 T2 {2 M 12  (  1)}{(  1) M 12  2}  T1 (  1) 2 M 12

2 M 12 (  1)  1 (  1) M 12  2 P1 , T1 , 1 are pressure, temperature and density ahead of the shock front and P2 , T2 , 2 are the pressure, temperature and density behind the shock front,  is the adiabatic gas constant and M 1 is the mach number of the shock

where

wave relative to the driven gas.

Figure 2: The URI shock tube facility. The shock tube utilized in the present study has an overall length of 8 m, consisting of a driver, driven, converging and muzzle sections. The diameter of the driver and driven section is 0.15 m. The final muzzle diameter is 0.07 m. Two pressure transducers (fig. 3), mounted at the end of the muzzle section measure the incident shock pressure and the reflected shock pressure during the experiment. All of the glass specimens are subjected to the same level of incident pressure in this experiment. A typical pressure profile obtained at the transducer location closer to the specimen is shown in fig. 4. The reflected velocity for the plane glass panel is 450 m/s, for tempered glass is 330 m/s, for the wired glass panel is 400 m/s, for sandwich glass panel is 310 m/s and for laminated sandwich glass panel is 300 m/s. 2.3 Loading conditions The square flat plate specimens utilized in this experimental study are held under fully clamped boundary conditions prior to blast loading. The size of the specimens is 305 mm × 305 mm x 6.5/7.5 mm. The dynamic loading is applied over a central circular area of 76.2 mm in diameter.

150

Figure 3: Schematics of the muzzle of the shock tube and fixture

Reflected Pressure

Incident Pressure

Figure 4: A typical pressure profile 2.4 Digital Image Correlation (DIC) Technique The digital image correlation technique is one of the most recent non-contact methodologies for analyzing full-field shape and deformation. It involves the capture and storage of high speed images in digital form and subsequent post-processing of these images to get the full-field shape and deformation measurements. The DIC system involves the mapping of predefined points on the specimen to measure the full-field shape and deformation. Capturing the three dimensional response of the plates requires that 2 cameras be used in a stereo configuration and they must be calibrated and have synchronized image recording throughout the event. The calibration of the cameras is performed by placing a grid containing a known pattern of dots in the test space where the glass sample is located during the test. This grid is then translated and rotated in and out of plane while manually recording a series of images. As this grid pattern is predetermined, the coordinates of the center of each dot is extracted from each image. The coordinate locations of each dot extracted uniquely for each camera allows for a correspondence of the coordinate system of each camera. The DIC is then performed on the image pairs that are recorded during the shock event. Prior to testing the back face of the sample is painted white and then coated with a randomized speckle pattern (Figure 5). The post processing is performed with the VIC-3D software package which matches common pixel subsets of the random speckle pattern between the deformed and un-deformed images. The matching of pixel subsets is used to calculate the three dimensional location of distinct points on the face of the panel throughout time. A speckle pattern is placed on the back face of the glass panel (as seen in fig. 5). Two high speed digital cameras, Photron SA1s, are positioned behind the shock tube apparatus to capture the real time deformation and displacement of the glass panel, along with the speckle pattern. The high speed cameras are set to capture images at 20,000 frames per second (inter frame time of 50 µs). During the blast loading event, as the panel responds, the cameras track the individual speckles on the

151 back face sheet. Once the event is over, the high speed images are analyzed using DIC software to correlate the images from the two cameras and generate real time in-plane strain and out-of-plane deflection histories. A schematic of the set-up is shown in fig. 5.

Figure 5: Schematics of DIC system There are two key assumptions which are used in converting images to experimental measurements of objects shape, deflection and strain. Firstly, it is assumed that there is a direct correspondence between the motion of the points in the image and that in the object. This will ensure that the displacement of points on the image have a correlation with the displacement of points on the object. Secondly, it is assumed that each sub-region has adequate contrast so that accurate matching can be preformed to define local image motion. 3. Experimental results 3.1 DIC Analysis The DIC technique (as discussed in section 2.4) is used to obtain the out-of-plane deflections and the in-plane strains on the back surface for all the five panels. The speckle pattern is applied onto the back face of the panels (fig. 5) which are subjected to shock loading. The high speed images captured using two Photron SA1 cameras are analyzed to get the back face deflections from the DIC as shown in fig. 6. Experiments have already been done to compare the back face deflection from the real time transient image and DIC to verify the accuracy of the DIC results. The error between the maximum deflection from DIC and real-time transient images is 4% [12]. The DIC results are within the acceptable error limits and so the DIC results can be used to better understand the failure and damage mechanism in the panel. The full-field DIC analysis for the five different glass panels is shown in fig. 6. The figure shows the real-time deflection of the different panels for the first 600 µs. The lamination of the sandwiched glass panel improved the blast mitigation property of the laminate and also resulted in delayed deflection and damage propagation. For a better understanding, the center-point of this full-field analysis was chosen and out of plane deflection and in-plane strain data were extracted at this point. The center point deflections in all the five panels are shown in fig. 7. The sandwich glass panel has a maximum deflection of 18 mm prior to complete fracture, whereas at the same time, the laminated sandwich glass panel shows a deflection of 9mm and no through hole formation. The tempered glass panel has a maximum deflection of 8 mm prior to fracture, the wired glass panel has 6 mm and the plane glass panel shows a deflection of only 2 mm. The deflection-time history for the laminated sandwich glass panel is shown in fig. 8. This shows that the laminated sandwich panel has a maximum deflection of 28 mm and recovers back to a final deflection of 16 mm. The other important point is that it experiences fragmentation and cracking in glass panel, but the protective film is able to contain the shattered glass pieces from flying off. Also, the in-plane strains on the back face of the five different glass panels tested are shown in fig. 9. The sandwich glass panel has a strain of 5% before fracture initiates and at the same time the laminated sandwich glass panel only has a 1.7% strain (there was no through hole formation at this time), whereas in the case of the tempered glass panel it is 2%, 1% for the wired glass panel and 0.01% for the plane glass panel before fracture. The in-plane history for the laminated sandwich glass panel is shown in fig. 10. It shows that the laminated sandwich panel has a maximum in-plane strain of 6% after which it recovers to 3%. This time-deflection and in-plane history shows that the laminated sandwiched glass panel behaves in a more ductile manner as compared to the other glass panels.

152

(a)

(b)

(c)

(d)

(e)

0 ms

200 ms

400 ms

600 ms

Figure 6: Time-deflection history of the back face for: (a) Plane Glass, (b) Tempered Glass, (c) Wired Glass, (d) Sandwiched Glass, & (e) Laminated Sandwiched Glass Panels Plain Glass Sandwich Glass Laminated Glass

20

Deflection (mm)

18 16

Tempered Glass Wired Glass

14 12

10 8

6 4 2

0 0

200

400

600

800

Time (ms) Figure 7: Time-deflection history of the back face for five glass panels

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Figure 8: Time-deflection history of the back face for laminated sandwich glass panel 0.07

Plain Glass Sandwich Glass Laminated Glass

In plane Strain, exx

0.06

Tempered Glass Wired Glass

0.05

0.04 0.03 0.02

0.01 0

0

200

400

600

800

Time (ms) Figure 9: Time-in-plane strain history of the back face for five glass panels

Figure 10: Time-in-plane strain history of the back face for laminated sandwich glass panel 3.2 Macroscopic post-mortem analysis The result of post-mortem evaluation of the shock loaded glass panels is shown in Fig. 11-13. The post-mortem analysis of the clear glass and tempered glass panels have not been shown as they lost the structural integrity and shattered into pieces. The post-mortem analysis of a wired glass panel is shown in fig. 11. The panel shows a large amount of fragmentation but in comparison to the clear and tempered glass panel, which shattered completely, it retained structural integrity. The postmortem image of the sandwiched glass panel is shown in fig. 12. There is heavy fragmentation on both the front and back

154 face as seen in figs. 12(a)-12(b). The PVB interlayer is able to withhold a substantial amount of these fragments from flying off. The post-mortem images of the laminated sandwich glass panel are shown in Fig. 13. It is evident from the post-mortem images that there is substantial fragmentation in the case of the laminated sandwich glass panel. However, the protective film is able to capture these pieces and prevent them from flying off. Also, there is no cracking in either of the layers (both on the front and back face of the glass panel) of the protective film. Overall, it can be concluded that the laminated sandwich glass panel has better blast mitigation properties than the other four panels. The clear glass panel and tempered glass panel have the worst blast mitigation properties and are shattered into pieces when subjected to the shock loading. The sandwiched glass performs better than the wired glass panel, but it still has fragmentation and shattered glass pieces flying around. The fragmentation in the case of the sandwich glass panel is lower as compared to that in the wired glass panel. Also, the diameter of the through hole formed in the wired glass panel is larger as compared to that in the sandwich glass panel. This improvement in the blast response of the sandwich glass panel can be attributed to the PVB interlayer which helps in withholding some of the shattered glass pieces from flying off.

(a) (b) Figure 11: Post-mortem evaluation of Wired Glass Panel (a) Front view; (b) Back view

(a) (b) Figure 12: Post-mortem evaluation of Sandwich Glass Panel (a) Front view; (b) Back view

(a) (b) Figure 13: Post-mortem evaluation of Laminated Sandwich Glass Panel (a) Front view; (b) Back view 4. Conclusions Five different panels are subjected to a controlled shock loading using a shock tube. The high speed photography and DIC analysis is applied to obtain the out-of-plane deflection and in-plane strain on the back face of all the five panels. 1. The macroscopic post-mortem analysis and DIC deflection analysis shows that the sandwich glass panel has less damage due to blast loading as compared to the wired, tempered and clear glass panels. The PVB interlayer increases the flexural rigidity of the panels, and results in less damage when subjected to the shock loading.

155 2.

The area of the through hole formed in the case of the sandwich glass panel was smaller as compared to that in the case of the other three glass panels. This will minimize the blast overpressure entering in the buildings and thus lower the damage inflicted as compared to the wired, tempered and plain glass panels.

3.

The application of the protective film (XO-ARMOR®) on the front and back face of the sandwich panel further improves the blast mitigation property of the sandwich glass panel.

4.

The laminated sandwich glass panel has fragmentation and cracking in the glass panel but the protective film is able to withhold the shattered glass pieces from flying off. Also, there is no through hole formation in the case of the laminated sandwich glass panel. This prevents the blast overpressure from entering the building and thus restricting the damage because of the overpressure. Overall, the laminated sandwiched glass panels with PVB interlayer and protective film on both the faces has a better blast mitigation properties as compared to the other four panels. 5. Acknowledgement The authors acknowledge the financial support provided by the Department of Homeland Security (DHS) under Cooperative Agreement No. 2008-ST-061-ED0002. We also thank XO-Armor for providing the XO- Film ® for the preparation of laminated sandwich glass panel. References 1. Saito, H. and Masuda, M., Modeling of blast process using indenting method, Pre Eng, pp. 369-377, 2004. 2. Gogotsi, G. A. and Mudrik, S. P., Glasses: New approach to fracture behavior analysis, J Non-Crystalline solids, pp. 1021-1026, 2010. 3. Bouzid, S., Nyoungue, A., Azari, Z., Bouaouadja, N., and Pluvinage, G., Fracture criterion for glass under impact loading, Int J Impact Eng, pp. 831-845, 2001. 4. Wei, J. and Dharani, L. R., Fracture mechanics of laminated glass subjected to blast loading, The App Fracture Mec, pp. 157-167, 2005. 5. Wei, J., Shetty, M. S., Dharani, L. R., Stress characteristics of a laminated architectural glazing subjected to blast loading, Computers & Structures, pp. 699-707, 2006. 6. Krauthammer, T. and Altenberg, A., Negative phase blast effects on glass panels, Int J Imp Eng, pp. 1-17, 2000. 7. Wei, J., Dharani, L. R., Response of laminated architectural glazing subjected to blast loading, Int J Impact Eng, pp. 2032-2047, 2006. 8. Mencik, J., Strength and fracture of glass and ceramics, New York, Elsevier, 1992. 9. Tiwari, V., et al., Application of 3D image correlation for full-field transient plate deformation measurements during blast loading, I J Impact Eng, pp. 862-874, 2009. 10. LeBlanc, J., et al., Shock loading of three-dimensional woven composite materials, Compos Struct, pp. 344–355, 2007. 11. Wright, J., Shock Tubes, John Wiley and Sins Inc., New York, 1961. 12. Gardner, N. and Shukla, A., The blast response of sandwich composites with a functionally graded core and polyurea interlayer, 2010 SEM Annual Conference & Exposition on Experimental and Applied Mechanics, Indianapolis.

A Dynamic Punch Method to Quantify the Dynamic Shear Strength of Brittle Solids

S. Huang, K. Xia ∗), F. Dai Department of Civil Engineering and Lassonde Institute, University of Toronto, Ontario M5S 1A4, Canada ABSTRACT: Shear strength is a basic material parameter of rocks. It plays a vital role in the applications field of mining engineering and geotechnical engineering. Although static standards for measuring static shear strength of rocks are available, the shear behavior of rocks under the dynamic loading it is not well understood. This paper presents a punch device loaded by split Hopkinson pressure bar system (SHPB) to determine the dynamic shear strength of rocks. Thin disc samples are used to minimize bending stresses. An isotropic and fine-grained sandstone is used to demonstrate the measurement principle. It is observed that the shear strength of rocks increases with the loading rate. This device is applicable to fine-grained rocks with intermediate hardness. Keywords: Dynamic shear strength; Punch shear test; SHPB INTRODUCTION Shear strength is one of the most important material parameter for brittle solids. There are several suggested methods for the quantification of this parameter for brittle solids such as rocks [1, 2] and ceramics [3], and ductile materials such as polymers [4]. There are usually two methods proposed to measure the static shear strength of brittle solids: direct shear-box test and punch shear test. As compared to the direct shear-box test, the punch shear test owns the merits of applicability for high strength solids, minimization of the bending stresses on the samples, and facilitation of the sample preparation [5, 6]. The early works of punch shear tests have been performed to measure shear strength of rock using simple punch apparatus with thin disc samples [5, 7, 8]. As shown in Fig. 1a, Mazanti and Sowers introduced a simple punch to simulate the punching of a hard rock layer into a soft or compressible layer with concentrated load [7]. They concluded that the punch test results were comparable to the shear strength with zero confinement (i.e., the apparent cohesion in the Mohr-Coulomb theory). To minimize the bending stress on the rock specimen during shear tests, Stacey [5] employed a simple punch shear apparatus (Fig. 1b), which is adaptable for traditional compression testing frames. The compressive load produces two parallel shear failure surfaces in the test sample. Later, a Block Punch Index (BPI) test was developed [9]. The BPI test apparatus was designed to be fit into the frame of point load testing device and the index value was calculated by dividing the maximum load on the punch by the shear failure area. Schrier then established empirical relationships between the uniaxial compressive strength (UCS) and BPI, and between the Brazilian tensile strength and BPI. He suggested that the BPI can be used to predict the UCS of rocks [9]. Ulusay and Gokceoglu investigated the size effect of the test sample and a size-corrected BPI was provided and related to UCS (UCS = 5.5 BPI) [10]. They suggested a slightly different formula to calculate BPI and concluded that this index can be used as an alternative input parameter for intact rock strength in rock mass classification. This method became one of the suggested methods by the International Society of Rock Mechanics (ISRM) [11]. It is noted that in various engineering applications, brittle solids may be subjected to dynamic loadings due to impacts or blasting. It is thus desirable to develop a method to quantify the dynamic shear strength of brittle solids. In the laboratory, the common tool to carry out dynamic test is the split Hopkinson pressure bar (SHPB) developed by Kolsky [12]. Although the straightforward application of this apparatus is the dynamic compressive tests, SHPB can be use to measure other properties with proper modification of the sample geometry and assembly [13, 14]. Indeed, there were a few attempts to measure the dynamic shear properties of ductile materials and structures using SHPB. Li et al. utilized SHPB with punch system to conduct the dynamic fiber debonding and push-out experiment on model single fiber composite systems [15]. Dynamic ∗

Corresponding author. Tel.:+14169785942; fax:+14169786813. E-mail address:[email protected]

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punch tests were also used to investigate the mechanical properties of metals [16, 17]. Motivated by the needs from engineering applications and the existing studies, the objective of this paper is to develop a dynamic punch shear device to quantify the dynamic shear properties of brittles solids. Loading

a)

Punch head Sample

Holder

Loading

b)

Punch head Sample

Holder

Fig. 1. Schematics of punch shear devices used by: a) Mazanti and Sowers [7] and b) Stacey [5]. GENERIC TESTING PRINCIPLES A 25 mm diameter SHPB system is utilized to apply the dynamic load for punch shear tests (Fig. 2). The SHPB consists of a striker bar, an incident bar and a transmitted bar. The length of the striker bar is 200 mm. The incident bar is 1500 mm long and the strain gauge station is 787 mm from the specimen. The transmitted bar is 1000 mm long and the stain gauge station is 522 mm away from the specimen. The bars are made from Maraging steel, with a yielding strength of 2.5 GPa, density 8100 kg m-3, Young’s modulus 200 GPa and one dimensional stress wave velocity 4970 m/s. A gas gun lunches the striker bar to impact on the incident bar and generates an elastic compressive wave toward the sample. At the sample assembly, the incident wave will be separated into two waves: an elastic tensile wave reflected back into the incident bar and a compressive wave transmitted into the transmitted bar. The incident wave ε i , reflection wave ε r and transmitted wave ε t are measured by strain gauges mounted on the incident bar and the transmitter bar, respectively.

Strain gauge

Striker

Incident bar

Transmitted bar Sample

holder

Fig. 2. Schematics of Split Hopkinson Pressure Bar for dynamic punch shear tests. Using these three waves, the dynamic forces P1 and P2 on both ends (Fig. 3) of the sample assembly can be calculated [18]:

P1 (t ) = EA[ε i (t ) + ε r (t )] P2 (t ) = EAε t (t )

(1) (2)

159

where E and A are Young’s modulus and cross-sectional area of the bars, respectively. Front cover

Rear supporter

P1 Incident bar

P2 Transmitted bar

Fig. 3. Schematics of the sample holder in SHPB. When the test is under dynamic force equilibrium condition (i.e. P1 = P2), the inertial effect in the dynamic test can be ignored [14, 19]. In this case, the punch shear stress in the sample can be calculated using the following equation:

τ=

P πDB

(3)

where τ is the punch shear stress; P = P1 = P2 is the loading force; D and B are the diameter of incident bar and the thickness of the disc specimen, respectively. It is noted here that we divide the load by the total shear area to obtain the shear stress, in a similar way to most other static punch shear studies. The maximum value of τ is considered as the punch shear strength τ 0 of the sample tested. A special holder is designed to support and protect the sample during dynamic punch tests. Conventional punch shear systems for static tests usually have two kinds of punch heads: cylindrical punch head and block punch head (Fig. 1). For dynamic punch tests using SHPB, an annular holder is usually adopted [15-17]. In this paper, the stainless steel holder consists of a front cover and a rear supporter, which are jointed by screw to hold the sample as shown in Fig. 3. The purpose of the front cover is to reduce the bending force during tests and additional damage on samples during and after the tests. The inner diameter of the rear supporter is 25.4 mm, 0.4 mm larger than the diameter of the incident bar to accommodate shear deformation. The incident bar serves as the punch head and the rear supporter is attached to the transmitted bar. The outer diameter of the entire holder is 57 mm. APPLICATIONS TO LONGYOU SANDSTONE Sample preparation

Fig. 4. a) Typical virgin and tested samples; b) The rock ring and rock plug produced in a typical dynamic punch shear test (the unit in the picture is centimeter).

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In this study, Longyou sandstone (LS) from Zhejiang Province of China is chosen to demonstrate the feasibility of the proposed dynamic punch device. LS is a fine-grained homogeneous sandstone, with negligible clay content (Fig. 4a). The mineral composition of the rock has been reported by Huang et al. [20]. The porosity of LS is measured as 17%, the density 2150 kg/m3, and the P-wave velocity 1600 m/s. The sample for the punch test is first drilled into 44 mm in diameter cores. These cores are then sliced to discs with nominal thickness of 16 mm and further polished into 14 mm thick thin discs. Dynamic force balance In traditional SHPB tests, the mismatch between forces added on both end of sample leads to the so-called axial inertia effect [21]. This problem can be overcame by using the pulse shaping technique, which is especially useful for investigating the dynamic response of brittle materials [13, 22]. In this work, the C1100 copper disc is used as the shaper material. During tests, the striker impacts on the pulse shaper before the incident bar, generating a non-dispersive ramp pulse propagating into the incident bar and thus facilitating the dynamic force balance of the specimen.

80

In Re In+Re (P1)

60

Force (kN)

40

Tr (P2)

20 0 -20 -40 -60 -80 0

50

100

150

200

250

300

Time (μs)

Fig. 5. Dynamic force balance check for a typical dynamic punch test with pulse shaping Fig. 5 shows the forces on both ends of the sample in a typical test. From Eq. (1) and Eq. (2), the dynamic force P1 is proportional to the sum of the incident (In) and reflected (Re) stress waves, and the dynamic force on the other side P2 is proportional to the transmitted (Tr). It can be seen from Fig. 5 that the dynamic forces on both sides of the specimens are almost identical during the entire dynamic loading period. The inertial effects are thus eliminated because there is no global force difference in the specimen to induce inertial force. Thus the static shear strength formula Eq. (3) can be utilized to analyze the dynamic results. Determination of the loading rate The dynamic strength of brittle solids exhibits the rate dependence [13, 23]. The loading rate for punch shear is characterized by τ& obtained from the time evolution of the shear stress. Fig. 6 shows dynamic loading history for a typical shear punch test. There exits a regime of approximately linear variation of shear stress from 50 µs to 85 µs. The slope of this region is determined from a least squares fit, shown as a line in the figure and this slope is used as the shear loading rate for the dynamic test.

161 25

Stress (MPa)

20

15

10

5

.

τ=440 GPa/s

0 0

30

60

90

120

150

Time (μs)

Fig. 6. Typical shear stress-time curve for determining the loading rate. Experimental results The thin disc sample is punched into a rock ring and a rock plug as shown in Fig. 6b. Few visible radial cracks can be identified on the ring. To recover the plug, a momentum-trap technique is utilized in this study. The momentum-trap technique proposed by Song and Chen [24] is adopted in this work. The working principle of this technique was presented elsewhere [25]. The main idea of using this method here is to constrain the displacement of the incident bar and thus to protect the plug from multiple loading. The soft-recovery of the rock ring and rock plug is thus possible as shown in Fig. 6b. The shear stress – shear displacement curve for a typical test is shown in Fig. 7. It can be seen that the shear strength is achieve at the displacement of 0.12 mm. We used displacement here to be consistent with the suggested direct shear and torsional shear methods by ISRM [1]. 25

Stress (MPa)

20

15

10

5

0 0.00

0.05

0.10

0.15

0.20

0.25

Displacement (mm)

Fig. 7. Shear stress vs. displacement for a typical test. Dynamic punch shear experiments were conducted under different loading rates to investigate the rate effect for LS. The dynamic punch shear strengths were obtained at loading rates ranged from 566 GPa/s to 1800 GPa/s. The maximum dynamic strength is 36.8 MPa. For reference the static punch shear strength is measured as 11 MPa for this rock. The static test was performed on Material Test System with 0.001 mm/s loading speed. The variation in flexural tensile strength as a function of loading rate is illustrated in Fig 8. It is evident from Fig.8 that the strengths of LS with the loading rates in the loading. The punch shear strength of LS is thus strongly rate dependent.

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Punch Shear Strength (MPa)

40

30

20

10 0

500

1000

1500

2000

Loading Rate (GPa/s)

Fig. 8. Punch shear strength of LS and test loading rate. CONCLUSION A dynamic punch shear method has been developed in this work to quantify the dynamic shear strength of brittle solids. The dynamic load was applied by a split Hopkinson pressure bar (SHPB) system, and the pulse-shaping technique was adopted to achieve the dynamic force balance and thus eliminate the axial inertial effect. A special sample holder was designed to protect the sample and in combination with the momentum-trap technique, it enabled soft-recovery the ring and plug produced by the punch. This dynamic punch method was applied to Longyou sandstone (LS). This dynamic punch shear method can be readily applied to various brittle solids in addition to the sandstone used in this work. ACKNOWLEDGEMENTS This wok was financially supported by the CAS/SAFEA International Partnership Program for Creative Research Teams (No. KZCX2-YW-T12). S.H. and K.X. acknowledge the support by NSERC/Discovery Grant No.72031326. REFERENCE [1]. [2]. [3]. [4]. [5]. [6]. [7]. [8]. [9].

Franklin J. A., Suggested methods for determining shear strength. In: Ulusay R., and Hudson J. A., (Eds.), The complete ISRM suggested methods for rock characterization, testing and monitoring: 1974-2006. The ISRM Turkish National Group, Ankara, Turkey, pp. 165 (2007). ASTM, "D5607-08 Standard Test Method for Performing Laboratory Direct Shear Strength Tests of Rock Specimens Under Constant Normal Force", (2008). ASTM, "C1292-10 Standard Test Method for Shear Strength of Continuous Fiber-Reinforced Advanced Ceramics at Ambient Temperatures", (2005). ASTM, "D3846-08 Standard Test Method for In-Plane Shear Strength of Reinforced Plastics", (2008). Stacey T. R., "A Simple Device for the Direct Shear-Strength Testing of Intact Rock", Journal of the South African Institute of Mining and Metallurgy: 80, 129-130 (1980). Sulukcu S., and Ulusay R., "Evaluation of the block punch index test with particular reference to the size effect, failure mechanism and its effectiveness in predicting rock strength", International Journal of Rock Mechanics and Mining Sciences: 38, 1091-1111 (2001). Mazanti B. B., and Sowers G. F., Laboratory testing of rock strength. International symposium on testing techniques for rock mechanics, Seattle, Washington, United States, 1965, pp. 207-231. Vutukuri V. S., Lama R. D., and Saluja S. S., "Handbook on mechanical properties of rocks", Trans Tech Publication, Clausthal, Germany (1974). Schrier van der J. S., "The block punch index test", Bulletin of the international association of engineering geology:

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[10]. [11]. [12]. [13]. [14]. [15]. [16]. [17]. [18]. [19]. [20]. [21]. [22]. [23]. [24]. [25].

38, 121-126 (1988). Ulusay R., and Gokceoglu C., "The modified block punch index test", Canadian Geotechnical Journal: 34, 991-1001 (1997). Ulusay R., Gokceoglu C., and Sulukcu S., Suggested method for determining block punch strength index. In: Ulusay R., and Hudson J. A., (Eds.), The complete ISRM suggested methods for rock characterization, testing and monitoring: 1974-2006. The ISRM Turkish National Group, Ankara, Turkey, (2007). Kolsky H., "An investigation of the mechanical properties of materials at very high rates of loading", Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences: B62, 676-700 (1949). Dai F., Xia K., and Luo S. N., "Semicircular bend testing with split Hopkinson pressure bar for measuring dynamic tensile strength of brittle solids", Review of Scientific Instruments: 79, - (2008). Dai F., Chen R., and Xia K., "A Semi-Circular Bend Technique for Determining Dynamic Fracture Toughness", Experimental Mechanics: 50, 783-791 (2010). Li Z. H., Bi X. P., Lambros J., et al., "Dynamic fiber debonding and frictional push-out in model composite systems: Experimental observations", Experimental Mechanics: 42, 417-425 (2002). Qu J. B., Dabboussi W., Hassani F., et al., "Effect of microstructure on static and dynamic mechanical property of a dual phase steel studied by shear punch testing", Isij International: 45, 1741-1746 (2005). Dabboussi W., and Nemes J. A., "Modeling of ductile fracture using the dynamic punch test", International Journal of Mechanical Sciences: 47, 1282-1299 (2005). Kolsky H., "Stress waves in solids", Clarendon Press, Oxford (1953). Dai F., Huang S., Xia K. W., et al., "Some Fundamental Issues in Dynamic Compression and Tension Tests of Rocks Using Split Hopkinson Pressure Bar", Rock Mechanics and Rock Engineering: 43, 657-666 (2010). Huang S., Xia K., Yan F., et al., "An experimental study of the rate dependence of tensile strength softening of Longyou sandstone", Rock Mechanics and Rock Engineering: 43, 677-683 (2010). Frew D. J., Forrestal M. J., and Chen W., "A split Hopkinson pressure bar technique to determine compressive stress-strain data for rock materials", Experimental Mechanics: 41, 40-46 (2001). Frew D. J., Forrestal M. J., and Chen W., "Pulse shaping techniques for testing brittle materials with a split Hopkinson pressure bar", Experimental Mechanics: 42, 93-106 (2002). Chen R., Xia K., Dai F., et al., "Determination of dynamic fracture parameters using a semi-circular bend technique in split Hopkinson pressure bar testing", Engineering Fracture Mechanics: 76, 1268-1276 (2009). Song B., and Chen W., "Loading and unloading split Hopkinson pressure bar pulse-shaping techniques for dynamic hysteretic loops", Experimental Mechanics: 44, 622-627 (2004). Xia K., Nasseri M. H. B., Mohanty B., et al., "Effects of microstructures on dynamic compression of Barre granite", International Journal of Rock Mechanics and Mining Sciences: 45, 879-887 (2008).

A Sensored Projectile Impact on a Composite Sandwich Panel Matthew Mordasky1*, Weinong Chen1,2 1

School of Materials Engineering, Purdue University. School of Aeronautics and Astronautics, Purdue University. * Corresponding author: Matthew Mordasky, 701 W. Stadium Ave. West Lafayette, IN 47907-2045 Email: [email protected] 2

ABSTRACT Common impact events occurs between a single target and a single projectile, and traditionally the targeted is of most interest.

Various sensors, such as strain gauges and force transducers, are commonly applied to the

target in order to understand and record the impact event through the prospective of the target [1,2]. This approach leaves much to be desired due to the limited, and more so, qualitative data that is usually recorded. Witnessing the impact event through the prospective of the projectile not only provides more meaningful data regarding the impact, but records data through the entire impact event of the composite sandwich panel as the projectile travels through the face sheets and core.

Additionally, the material – target impact interaction can

be more closely evaluated. However, sensoring the projectile in conjunction with the inherent speed and confinement issues of a gas gun has presented many challenges for experimental implementation. INTRODUCTION With continued interest in the projectile-target interaction, experiments with an instrumented projectile are carried out. Force histories of impact events have been performed on laminate targets sensored with a force transducer [1] as well as composite sandwich targets [2].

By utilizing an instrumented projectile the force

history seen at the projectile-target interface is recorded rather than the average force seen by the back surface of the composite panel. A manganin gauge embedded into a projectile is used to record the force history. Embedding the gauge protects it from the impact interaction between the projectile and panel, the tensile waves that will occur in the projectile, as well as ensure the gauge stays intact during the experiments. The wires connecting the gauge to the data acquisition system travel from the gauge, out the back of the cured urethane projectile, through the base of the sabot, then down the barrel of the gas gun, and finally to the oscilloscope via an air tight connector. This connection, allows the oscilloscope to record the pressure seen by the gauge.

A schematic of this setup is shown in Figure 1.

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_23, © The Society for Experimental Mechanics, Inc. 2011

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Fig. 1 Schematic of the instrumented projectile exiting the low velocity gas gun EXPERIMENTS AND RESULTS A 25.4mm diameter cylinder made from Task®2 high strength pourable urethane by Smooth-On is used for the projectile. The cylindrical projectile is 7.6 cm in length. A manganin pressure gauge is embedded 3.2mm from the striking face of the projectile. The composite sandwich panel used in these experiments is an IM7 carbon fiber unidirectional prepreg with an 8552 resin system. A 12.7 mm thick NOMEX® core with a density of .064 g/cm3 and a cell size of 3.2 mm was used.

An FM300 film adhesive was used to aid in bonding the

face sheets to the NOMEX® core. The force history of the impact event is recorded, and through the use of a NAC K3 high speed camera, the force history and damage progression of the panel are explained. Figure 2 illustrates the impact event of the instrumented projectile onto the composite sandwich target.

Fig. 2 High speed image of an instrumented projectile striking the composite sandwich target. REFERENCE: [1] H. Kim, D. Welch and K. Kedward. “Experimental investigation of high velocity ice impacts on woven carbon/epoxy composite panels”, Composites: Part A, 34, 25-41, 2003 [2] L. Aktay, A. Johnson, M. Holzapfel. Prediction of impact damage on sandwich composite panels. Computational Materials Science, 32, 252-260, 2005

Cut Resistance and Fracture Toughness of High Perfomance Fibers Jessie B. Mayo, Jr.*1,2 and E. D. Wetzel2 1

Tuskgee Center for Advance Materials, Tuskegee University, Tuksegee, AL 36088 2 U.S. Army Research Laboratory, Aberdeen Proving Ground, MD 21005 *[email protected], 410-306-0699

A study to understand the fundamentals of fiber mechanics during cutting has been conducted. In the current study, high performance organic and inorganic fiber types are examined during different modes of cut testing. The scope of the current study is to explore the relationship between fiber type on cut resistance and to provide comprehension of how fiber structure influences cut behavior. All tests are conducted on single fibers. Fabric architecture parameters such as fiber number, yarn twist, and weave patterns are disregarded to reveal the individual strengths behind the cut resistance of the fibers alone. Organic and inorganic fiber types were held in a specially designed fiber clamping mechanism for cut resistance testing (Fig. 1). In order to examine the effects of orientation, the mechanism has a rotary platform that allows fiber cutting at different angles with a Celanese blade (longitudinal cut resistance testing). A number of commercially available blades were utilized for testing. The Celanese blade was chosen because of low error and data repeatability. For shear effects, blade angle was also varied (transverse cut resistance testing).

Figure 1. Photographs of (a) fiber clamp fixture showing normal incidence cut resistance testing, (b) longitudinal incidence and (c) transverse incidence.

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Figure 2. Blade stress as a function of (a) longitudinal fiber angle and (b) transverse blade angle after cut resistance testing. The results of the cut resistance testing are given with respect to fiber type and angle of incidence. Blade stress is defined as

σ=

F cos θ D ⋅ 2r

σ =

F cos α . D ⋅ 2r

and

Here F is the force of the blade on the fiber, θ is the fiber angle, α is the blade angle, D is the diameter of the fiber and r is the radius of the blade tip. Figure 2 shows the blade stress as a function of longitudinal and transverse angles. Within accuracy of results, all of the organic fibers had similar behavior. Inorganic fibers showed higher cut resistances than organic fibers. Longitudinal cut resistance testing showed very little angle dependence while organic fibers showed decreasing failure stress with increasing longitudinal angle. Increased transverse angle showed decreasing failure stress for all fiber types during transverse testing. Microscale observations (Figure 3) show traditional brittle fracture zones in the cut tested Sglass fiber. The fracture zones indicate that S-glass failed due to isotropic fracture mechanics. Mixed failure mechanisms (cut failure, longitudinal tensile failure, transverse tensile failure, transverse shear) are seen in the Kevlar fibers. Kevlar’s anisotropy was the main contributor to transverse failure during cut resistance testing.

S-glass

5 µm

Figure 3. Scanning electron micrographs of S

The results show that cut resistance of these materials are governed by fiber structure angle of incidence of cutting instrument. Trends of higher failure stresses for inorganic fibers indicate the potential for using these materials for flexible stab resistant armor. The belief here is also that in order to examine materials for ballistic resistan material for its fracture toughness. Fracture toughness may govern the way that a material behaves ballistically. Kevlar, one of the most widely used materials in body armor, has several uses but its fracture toughness has yet to be determined because this parameter may have direct influence on ballistic resistance. current study are to implement a process for determining the fracture toughness of single filaments and to determine the fracture toughness of single Kevlar fibers. After putting a precise notch into a Kevlar filament with a focused 600), tensile testing of the notched fibers was completed at a constant strain rate of 0.098 mm/min until failure with a NanoUTM universal testing system. of the notched specimens are in Fig. 4. The microgr matrix for the fracture toughness study. terms in this document.

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Figure 4. Focused ion beam notched Kevlar fibers of 0.2 µm width/6.5 µm length flaws. Figure 5 shows the failure stress as a function of crack length for 0.2 µm and 1.0 µm crack width fibers. As the notch length increased for both for the 0.2 µm tip diameter fiber the failure stress remained constant within error bars. This means that there was no significant response to notch introduction. However, with 1.0 µm tip diameter fiber, the failure stress decreased as notch length increased. The difference in notch tip diameter also made a difference. The 1.0 µm notch tip diameters fiber had a lower failure stress than the 0.2 µm notch tip diameter fibers. Figure 6 shows the strain to failure for 0.2 and 1.0 µm notch tip diameter fibers. The strain to failure decreases as the notch length increases for both the 0.2 and 1.0 µm fibers. The 1.3 µm and 6.5 µm notch lengths have a higher strain to failure for the 0.2 µm fiber than the 1.0 µm fiber. The 2.6 µm notch lengths for both of the notch tip diameters are close in magnitude.

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Scanning electron micrographs (Fig. 7) show that Kevlar’s anisotropy caused mixed modes of failure during testing. Although the notched fiber was pulled in tension, shear failure (Mode II) was also evident. When considering only Mode I, the fracture toughness of Kevlar fiber was found to be 2.99 MPa√m using linear elastic fracture mechanics. The fracture toughness was found to be 3.13 MPa√m when using fracture toughness derived from the Arrhenius Life Equation (Oh, 1996). Experiments are under way to examine the Mode II aspect of Kevlar fiber failure. 4000 HDPE

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Figure 8 contains specific fracture toughness as a function of specific modulus for some common materials. The belief here, again, is that the combination of fracture toughness and modulus may govern the ballistic response of the material. The further that a material is to the upper right corner of the plot, the better the material will perform against ballistic threats. Kevlar has a slope that is close to +1 while carbon fiber has a much lower slope. Kevlar has good ballistic resistance, while carbon fiber, which has a higher modulus than Kevlar, has poor ballistic resistance. The only materials that are fibers included in the plot are the ductile Kevlar 129 (PPTA) fiber and the brittle carbon fiber. Polycarbonate, which is used in bulletproof glass applications, has high fracture toughness but low modulus. This points to the fact that fibers are governed by different micro-properties when failure is the concern. More detailed treatment is needed for Kevlar because of fiber anisotropy that leads to mixed failure modes. This includes analysis through in-situ experimentation.

Reference Oh, Hung-Kuk. Determination of Fracture Toughness by Uniaxial Tensile Test. Eng. Fract. Mech., v55 no. 5, pp. 865-868, 1996

Kolsky Tension Bar Techniques for Dynamic Characterization of Alloys

Bo Song, Helena Jin, Bonnie R. Antoun Sandia National Laboratories, Livermore, CA 94551-0969, USA Dynamic stress-strain response associated with damage and failure mechanisms of alloys is desired to be quantitatively determined. Dynamic tensile characterization has been demonstrated an efficient method for such an investigation. However, reliable dynamic tensile characterization relies on appropriate experimental instruments and procedure. In this study, we employed a newly developed Kolsky tension bar to characterize the tensile properties of alloys [1, 2]. Challenges and remedies in the dynamic tensile characterization with the Kolsky tension bar are presented. At 2010 SEM Annual Conference, we presented the Kolsky tension bar developed at Sandia National Laboratories, California. This new Kolsky tension bar facilitates reliable loading and easy pulse shaping. An additional laser-beam measurement system is employed to directly measure the displacement of the incident bar end such that the specimen deformation is able to be accurately measured. However, for a dumbbell specimen threaded into the bar ends, the specimen strain is measured in terms of average strain over the specimen length inclusive of the transition portion from the threads to the gage section. The actual strain over the specimen gage length needs to be corrected. In addition, it is desirable for the specimen to deform uniformly such that the average strain can represent any point wise strain over the specimen gage section. The uniform deformation, which is usually associated with stress equilibration, needs to be verified. An abnormal stress peak was also observed in the stress response of 4340-V steel, as presented at 2010 SEM Annual Conference [1]. The stress peak needs to be properly addressed in order to obtain reliable tensile stress-strain response of alloys.

Fig. 1. Stress in the specimens with different gage lengths

Fig. 2. Comparison of stress in the specimens with and without Taflon tape applied

We firstly address the abnormal stress peak. An early stress peak, which occurs around plastic yielding, has been commonly observed in dynamic tensile characterization of alloys with Kolsky tension bars. However, different interpretations have been presented in regard to the mechanism of the stress peak [3, 4]. In this study, we used a 4330-V steel as an example to determine the effects of specimen length and installation on the

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176 amplitude of the stress peak in Kolsky tension bar experiments. Figure 1 shows the stress histories in the specimens with two different gage lengths (6.35 and 9.53-mm-long) at an identical strain rate of approximately -1 1100 s . It is observed that the amplitude of the peak stress was reduced by 12% when the specimen gage length was reduced from 9.53 mm to 6.35 mm. In order to facilitate the same strain rate on the specimens, the longer specimen requires higher impact energy by means of higher impact speed of striker. When the contact between the specimen and the bar ends is not perfect, additional impact, i.e., between the thread teeth on the specimen and bar ends, may be generated. The higher impact speed amplifies such an effect of imperfect contact. To confirm this, Teflon tape was applied on the threads of both the specimen and the adaptors. Figure 2 clearly shows that the amplitude of the peak stress was significantly reduced after the Teflon tape was applied. This also confirms that the stress peak observed in the Kolsky tension bar experiments is pseudo, which should be eliminated. Applying Teflon tape has been demonstrated an efficient method to minimize the amplitude of the peak stress. After the pseudo stress peak is minimized, the stress histories measured with the strain gages on the transmission bar becomes a reliable representative of actual stress response in the specimen. However, the stress over the entire specimen gage length should be equilibrated. In other words, the specimen should deform uniformly. Due to the complication of the tension bar system,

Fig. 3. DIC pattern on the specimen surface

T=0

T=12 µs

T=24 µs

T=60 µs

T=120 µs

T=180 µs

Fig. 4. Specimen axial deformation (Exx) from DIC analysis the reflected pulse may not be reliable to calculate the stress at the front end of the specimen with classic “2wave” method. In this study, we employed digital image correction (DIC) to the specimen to measure the deformation field over the specimen gage length. Figure 3 shows the DIC pattern on the specimen surface. The DIC pattern was photographed with a Phantom V12.1 digital camera at the speed of approximately 83,000 frames per second. Figure 4 shows the DIC results during the tensile loading. Figure 4 clearly indicates that the specimen has already achieved uniform deformation within the first 12 µs. A significant strain localization was not observed until t=120 µs (Fig. 4). This strain localization became more and more severe until macroscopic visible necking is observed. Based on the DIC results, the specimen deformed uniformly before necking occurred. Therefore, the signal from the strain gages on the transmission bar can be used for calculating the stress history

177 in the specimen. The DIC results can be used for the strain history in the specimen. However, the limited number of data points from the DIC results may not yield accurate stress-strain response of the specimen under investigation. In this study, we incorporated laserbeam measurement and direct strain-gage measurement on the specimen surface, as presented in Ref. [5], to calculate the specimen strain history. Since the laserbeam measurement provides only displacement information, the effective specimen gage length needs to be determined for accurate calculation of specimen strain. Figure 5 shows the mechanical drawing of the steel specimen. When the specimen is subjected to mechanical loading, the total deformation includes the deformation on not only the gage section but also the transition (non-gage) section. The portion of the displacement over the non-gage section to the total specimen displacement is not negligible when the specimen is under elastic deformation. Direct strain-gage measurement on the specimen gage section compensates this obstacle. The strain gage on the specimen is able to measure the specimen strain up to 2%. The specimen strain over 2% can be calculated with the laser-beam system presented in [1, 2]. However, the displacement on the nongage section should be deducted from the total displacement measured with the laserbeam system. It is noted that, in this study, with increasing load on the specimen, the gage section will enter into plasticity while the non-gage section still remains in elasticity. In this case, the displacement on the non-gage section is approximated to a constant of 0.0178 mm (or 0.0007 inch) if the specimen, specified in Fig. 5, is assumed to have a perfectly plastic response with a yield strength of 1400 MPa. This non-gage-section displacement has been deducted from the laser-beam measurement for calculating the plastic strain of the specimen in this study.

Fig. 5. Mechanical drawing of the tensile specimen

Fig. 6. Dynamic tensile stress-strain curves of 4340-V steel

Based on above analysis, dynamic tensile stress-strain curves of 4330-V steel were obtained at two -1 different strain rates, 1340 and 680 s . Figure 6 shows the mean curves of three identical experiments at each strain rate. Again, it is noted that the specimen strain was obtained directly from specimen strain gage measurement when it is smaller than 2%; while the strain larger than 2% was obtained from laser-beam measurement after proper correction. The pseudo early stress peak was minimized but not fully eliminated, as -1 shown in Fig. 6. The 4330-V steel shows little difference in tensile stress-strain response at 680 and 1340 s . However, the flow stress at both dynamic strain rates significantly increases in comparison with that at quasistatic strain rates.

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ACKNOWLEDGEMENTS Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

REFERENCES 1. Song, B., Antoun, B. R., Connelly, K., Korellis, J., and Lu, W.-Y., 2010, “A Newly Developed Kolsky Tension Bar,” In: Proceedings of 2010 SEM Annual Conference and Exposition on Experimental and Applied Mechanics, Indianapolis, IN, June 7-10, 2010. 2. Song, B., Antoun, B. R., Connelly, K., Korellis, J., and Lu, W.-Y., 2011, “Improved Kolsky Tension Bar for High-rate Tensile Characterization of Materials,” Measurement Science and Technology, 22, 370304 (7pp), in press 3. Rusinek, A., and Klepaczko, J. R., 2003, “Impact Tension of Sheet Metals – Effect of Initial Specimen Length,” Journal de Physique IV France, 110, 329-334. 4. Kuroda, M., Uenishi, A., Yoshida, H., and Igarashi, A., 2006, “Ductility of Interstitial-Free Steel under High Strain Rate Tension: Experiments and Macroscopic Modeling with a Physically-Based Consideration,” International Journal of Solids and Structures, 43, 4465-4483. 5. Song, B., Antoun, B. R., and Connelly, K., 2010, “Dynamic Tensile Characterization of a 4330-V Steel with Kolsky Bar Techniques,” In: Proceedings of IMPLAST 2010, Providence, RI, October 12-14, 2010.

Prediction of Dynamic Forces in Fire Service Escape Scenarios M. Obstalecki, J. Chaussidon, P. Kurath, G.P. Horn Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign 1206 W. Green St. Urbana, IL 61801 ABSTRACT When firefighters become trapped on elevated floors in a burning structure, their only option for escape is often through an exterior window utilizing an escape rope system. The rope becomes a firefighter’s lifeline until he can either rappel to safety or be rescued by other means. Quantifying the dynamic loading experienced by the anchor, rope, and firefighter is critical in order to understand the risk of injury and rope failure for various systems and establishing appropriate criteria for system acceptance. An experimental setup was fabricated with instrumentation to measure the dynamic loading for realistic escape scenarios. The experiments illuminated the importance of dynamic loading and the significant effects of friction on the dynamic loads felt by the firefighter and the rope anchoring system. The load experienced by a firefighter can be 30% larger than the load measured at the anchor as a result of friction at the window sill. A numerical model based on conservation of energy mirrored the experimental results. Model results are improved by including friction, though further improvements will be made by characterizing energy absorption due to knot slippage. I. INTRODUCTION When firefighters enter a burning structure, they must carry with them all the tools and protective equipment required to perform their job safely. Firefighters operating on an above grade level may encounter conditions where they become trapped or overrun by fire due to sudden changes in ventilation, fire growth, or lack of situational awareness. In this case, there are often few options for safe egress from within the building. Often, the only option will be to exit out of a nearby window, in which case, the firefighter’s escape rope system becomes his lifeline until he can rappel down to safety or be rescued depending on the height of the exit point. The typical length of rope utilized in escape rope systems is about 15 meters, so a firefighter would not be able to rappel down if the rope does not reach ground level. The Fire Service has faced several lineof-duty fatalities in recent years where escape rope systems were not available or could not be deployed safely. As a response to these recent incidents, the Fire Service has invested resources into developing improved escape rope systems with the goal of reducing the risk of injury to firefighters during emergency escape scenarios. Escape rope systems are typically composed of three major components; an anchor, a polymeric rope, and a descent control device. During an escape, both the anchor and a portion of the rope will remain within the room. Since the anchor is made out of steel its risk of failure is minimal; however, the rope will degrade at elevated temperatures and most likely be the first component to fail. Escape systems typically utilize a 7.5 mm rope and are composed of two parts: a sheath and a core. The sheath’s primary purpose is to protect the core of the rope from abrasion; however, it does also have the ability to carry load. Ropes are typically constructed from pure nylon, pure Technora, or a combination of the two materials. Pure nylon ropes generally have the highest compliance and are best able to minimize dynamic loads by absorbing energy through deflection. On the other hand, pure Technora ropes offer the best resistance to degradation at elevated temperatures. The creation of a hybrid Technora-nylon rope (Technora sheath and nylon core) attempts to balance the benefits of both materials. In order to study the effects of different rope materials on escape scenarios, it is necessary to quantify and compare the effects of dynamic loading and elevated temperatures on each category of rope. Previous work has shown that increasing temperature tends to enhance rope compliance; however, it also reduces breaking strength of the rope [1]. This study focuses on the effects of dynamic loading on Technora-nylon rope. During an emergency escape, a firefighter’s first objective is to secure his escape system anchor onto a solid structure within the room. The firefighter can anchor onto many structurally sound objects including pipes, wall studs, door jambs, floor joists, and the window sill itself. Based on the variety of potential anchoring locations, multiple escape scenarios must be tested. Escape scenarios will be broken down into three main categories for experimental testing and comparison to model results. The first scenario (Scenario A) utilizes the window sill as the anchoring point. Scenario A minimizes the amount of

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180 rope exposed to elevated temperatures within the room; however, it also can result in large dynamic loads since the length of deployed rope is minimized. Assuming a constant modulus, rope stiffness can be computed using the following equation where is the rope modulus and is its length. (1) Since the modulus is a property of the rope, the structural stiffness of a particular rope – and thus maximum load transmitted to the firefighter – can only be decreased by increasing its length. In the second scenario (Scenario B), the rope is anchored near the ground to either a floor joist or pipe. Scenario B introduces more rope into the room, heightening the possibility of rope degradation due to high temperatures, but Simulated provides a system that is less structurally stiff and thus better able to absorb Ledge energy. In addition, Scenario B increases the amount of friction between the rope and window sill since contact area is increased. The anchoring point for Anchoring the final scenario (Scenario C) is across the room, on wall stud for example. Point Depending on room geometry, the distance between the window sill and the anchoring point could extend as much as ten to twelve feet. The peak load experienced during a Scenario C escape would be minimum when compared to the other scenarios for the same fall height; however, Scenario C exposes the greatest amount of rope to the conditions inside of the room. It is important to identify the best rope for a firefighter to use, which balances the rope’s resistance to degradation at elevated temperatures with its ability to absorb dynamic loads. It is also critical to identify a preferred scenario which Figure 1: Experimental testing frame for minimizes injury to a firefighter during an escape. simulation of escape scenarios A review of the current literature has found no published experiments focusing on fire service escape scenarios; however, dynamic testing has been conducted in association with characterizing rope properties and fall scenarios in relation to industrial fall protection and sport climbing. Sport climbing accidents have been analyzed using a model based on the conservation of energy [2].The model assumes a constant modulus and incorporates the effects of knot slippage but lacks validation through testing. Arbate utilized drop weight testing to characterize the nonlinear dynamic behavior of parachute static lines and obtained displacement data by looking at force-time history curves from the drops and considering the motion of the mass as a rigid body [3]. Baszczyński studied the dynamic strengths of lanyards and emphasized the importance of rope compliance on loads measured during dynamic loading [4]. Spierings characterized the difference in properties exhibited by yarns during quasistatic and dynamic testing and observed significant stiffening during dynamic loading when compared to quasistatic loading [5]. The current project aims to build upon this previous work in order to develop a comprehensive model capable of accurately predicting loads during emergency escape scenarios. This document presents ongoing work pertaining to experimental testing and numerical modeling of firefighter escape scenarios, focusing on identifying the effect of dynamic loading on maximum loads transferred to the firefighter and escape system anchor. The goal is to couple experimental testing with a numerical model in order to predict the dynamic loads anchor based on input parameters including firefighter weight, rope length, fall height, coefficient of friction at the window sill, and , in the future, energy absorbing capabilities of a descent control device and knot slippage. Such a model would reduce the number of required experimental tests, aid in the optimization of escape rope systems, and could assist in the development of improved emergency escape protocol.

181 II. EXPERIMENTAL SETUP AND PROCEDURES FOR ESCAPE SCENARIO TESTING

Corner Radius

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Dynamic rope testing was conducted using a modified experimental frame (Figure 1) at the Advanced Materials Testing and Evaluation Laboratory at the University of Illinois at UrbanaChampaign. The load frame is capable of simulating the three categories of escape scenarios described above. In the current configuration, an I-beam simulates a window ledge. The rope is bent around a radius which is approximately 3.2 centimeters in diameter, allowing for uniform friction (Figure 2). In

Figure 2: Detailed view of anchor side of experimental load addition, vertical drop tests can also be performed to frame measure dynamic rope properties up to failure of the rope. The current system utilizes a solid drop mass consisting of Olympic weightlifting plates as a substitute for the firefighter’s mass (Figure 3). The solid mass ignores the human body’s capability to absorb energy during a fall, focusing the data on rope properties rather than dynamic system properties. Eliminating the effects of the human body allows for more straightforward comparison with model results. Later tests will incorporate an anthropomorphic mannequin in order to capture energy absorption of the human body. The first round of testing was conducted using a mass of 84 kg. The experimental frame incorporates load cells located at the anchor point and falling mass (Omega, LCCD-10K, Stamford, CT). In addition to gathering load data during experiments, an instrument (Figure 4) was designed to gather dynamic deformation data. Previous experiments utilized accelerometers [6], differentiated load cell data [3], or calculated stiffness data from quasi-static tests using crosshead displacement [1,2]; however, such methods can be confounded by the effects of knot slippage in the measurements. The new instrument eliminates common sources of error found in traditional rope testing methods and enhances the goal of obtaining accurate rope properties. The instrument is composed of two crossheads which slide across ground rods on linear bearings. Attached to one crosshead is the body of a linear variable displacement transducer (LVDT, Schaevitz, MHR1000, Hampton,VA), while the core of the LVDT is attached to the second crosshead. Each crosshead is attached to the rope using a single pin through the rope diameter. Since earlier testing revealed that the rope rotates around its axial direction up to 60° during testing, the instrument allows the rope to undergo this rotation without introducing error. The measured deformations are then considered representative of all the rope between the anchor and ledge and utilized to extract strains, which are employed in subsequent calculations.

Quick Release

The rope is tied to eye bolts using figure-eight knots, which are mounted to each load cell. Figure-eight knots were utilized since they allow less slip when compared to other knots, and they simplify sample preparation when compared to sewn eyes. In the current configuration the rope is dropped Load over a ledge, such that the rope has a 90° Cell LVDT contact angle with the simulated ledge. Rope lengths before and after the simulated ledge Drop are recorded with the drop weight hanging Mass statically. The strain instrument crossheads are Figure 3: Drop pinned to the rope with the rope slightly mass setup tensioned. The drop weight is raised a prescribed distance above the static hanging position. After starting the digital data acquisition program, which gathers two data points every millisecond, a quick release is used to release the drop weight. Some ropes are subjected to multiple drops in order to study the effects of repeated dynamic loading on rope properties. In the case that a rope is tested more than once, the lengths are remeasured Figure 4: Dynamic rope deformation measurement before each test since the knots slip during the previous drop, and the instrument rope may have permanently elongated.

182 III. NUMERICAL MODEL OF ESCAPE SCENARIOS A numerical model was developed in MATLAB to parallel experiments in an effort to predict maximum loads experienced by both the system anchor and drop mass. Required inputs into the model include the anchor and drop mass side rope lengths, gauge length of strain gauge, drop height, mass of the drop weight, coefficient of friction, and contact angle with the corner radius. In addition to these inputs, a load-strain curve must also be supplied to the model. The model is based upon conservation of energy, where the change in potential energy is equated to the energy that has been lost due to work done by friction and stored in the rope in the form of strain energy. The current model does not take into account the effects of knot slip, heating of the rope, or a descent control device; however, these effects may still be acting on the system. The current model simulates a worst case scenario where the descent control device locks and causes an abrupt stop. Successful model development will allow for confident and efficient evaluation of dynamic system behavior over a broad range of escape scenarios. Where previous approaches have assumed a linear load-strain curve (constant rope modulus) [1], the present model incorporates nonlinear rope properties observed during both quasistatic and dynamic testing. Experiments have shown that the modulus varies significantly with strain and depends on rate of loading. The current model incorporates dynamic rope properties obtained during experiments. The total change in potential energy of the system is composed of three terms; the drop height, h, the rope deflection between the anchor and simulated window sill, δ1, and the rope deflection between the simulated sill and the drop weight, δ2. The equation for the total loss in potential energy follows. (2) Rope can be modeled as a nonlinear spring, so the strain energy within the rope is computed by integrating the force over the displacement. Since displacement of the rope on either side of the corner is influenced by both friction and initial lengths, the following equation gives the total strain energy in the system. (3) The rope deflection is a function of its initial length, so Equation 3 can be modified to be a function of strain using the simple relationship below. (4) Making this substitution, Equation 3 becomes the following. (5) Incorporating the strain instrument into the experimental setup allows for the generation of load-strain data. A third order polynomial is fit to the load-strain data to provide a mathematically simple constitutive relationship, and this curve is integrated to obtain the strain energy according to Equation 5. The work done by friction is defined as the coefficient of friction multiplied by the integral of the normal force times the deflection on the anchor side of the rope. The normal force varies between the anchor side and drop side forces, so the average of the two is taken and assumed constant over the radius. (6) Since the equation integrates the deflection on the anchor side, the drop side force term must be replaced with an equivalent relation, which is in terms of the anchor side force. The capstan friction equation given below relates the two forces through the coefficient of friction and the contact angle,β, which is 90° for the current experiment. (7) Substituting Equations 4 and 7 into Equation 6 and simplifying provides a relationship for the work done by friction as a function of the load and strain on the anchor side of the rope. (8) Since strain is only measured on the anchor side, a relation between the strain on either side of the corner is required to calculate the strain energy. In order to obtain this relation the capstan friction equation, Equation 7, is utilized along with the following equation relating displacement and force.

183 (9) Combining Equations 7 and 9 gives a relationship between the deflections on either side of the corner. (10) The difference in nonlinear modulus on either side of the simulated ledge is ignored, allowing for the modulus terms to drop out. This assumption introduces a small amount of error into the calculation; however, the difference in the deflections is relatively small for the current set of tests and would not introduce a significant error. Knowing the relation between deflections, the relationship between strains can be calculated by plugging Equation 4 into Equation 10. (11) The coefficient of friction for the current radius on the corner was unknown initially, so it can be estimated by solving for the coefficient of friction in Equation 6. (13) An important note to make is that the coefficient of friction is independent of the drop radius, and only depends on the angle of contact according to the capstan equation. Since the system of equations is nonlinear, a MATLAB solver finds the equivalent strains where the residual defined below approaches zero. (14) The calculated strains are plugged into the polynomial fit of the load-strain curve, resulting in predicted loads. IV. EXPERIMENTAL TESTING RESULTS AND DISCUSSION Initial tests are categorized as Scenario C, where the rope is anchored across the room from the exit window. The length of rope between the anchor and simulated sill is constant at 80 cm, while the length of rope deployed over the sill (slack length) is varied along with the drop height. Table 1 summarizes the drop parameters for the tests conducted. Table 1: Experimental test parameters for Technora-nylon rope

Rope # 1 1 2 2 3 3 4 4 5 5

Drop # 1 2 1 2 1 2 1 2 1 2

Slack Length (cm) 80 87 93 98 67 73 103 110 96 104

Drop Height (cm) 26 25 57 58 38 43 51 59 25 28

Figure 5 shows the evolution of load experienced by both load cells resulting from a drop on a typical Technora-nylon rope. The difference between the two recorded loads shows the significant effect of friction on the system. Differences of over 30% in peak load have been recorded, and emphasize the role of friction in dissipating energy from the system.

184

600

Load (kg)

500

Drop Weight Anchor

400 300 200 100 0 0

0.2

0.4 0.6 0.8 1 Time (sec) Figure 5: Anchor and drop mass load evolution for Technora-nylon rope By taking the average of the ratio of drop weight force to anchor force of the initial loading curve for each drop and plugging the value into Equation 13, the coefficient of friction during each drop was estimated. The calculated coefficients of friction are summarized in Table 2. The average coefficient of friction is 0.142 with a standard deviation of 0.019. Once calculated, the average value was used within the model. Table 2: Estimated coefficient of friction of 7.5 mm Technora-nylon rope

Rope # Drop # μ

1 1 -

1 2 0.107

2 1 0.147

2 2 0.138

3 1 0.171

3 2 0.150

4 1 0.153

4 2 0.156

5 1 0.130

5 2 0.125

Figure 6 shows the first loading and unloading curves of the anchor side of rope 5 for two consecutive drops. The difference between the slopes of the two curves shows a significant stiffening of the rope system after the first drop. The recorded measurements reinforce the current practice of discarding an escape rope after it has been used once. Small oscillations observed in the curves are most likely due to a combination of slip in the knots and a stick-slip condition due to friction around the corner radius. The second drop also reaches a higher maximum load with a larger area under the curve than the first drop. Knot slippage is significantly larger during the first drop and the reduction in energy absorption through the knot slippage in the second drop must be accommodated through elastic deformation of the rope. 900 750

Load (kg)

600 Drop 1 Drop 2

450 300 150 0 0.00

0.02

0.04

0.06

Strain (-/-) Figure 6: Initial loading and unloading curves for two drops on Technora-nylon rope (Rope 5)

185 Varying the type of knot tied also has a significant effect on measured peak loads for equivalent drop parameters. For example, tying a simple overhand knot results in a much lower measured load when compared to a figure-eight knot. The reason behind the difference is due to the amount of slip allowed through each knot. Many current escape rope systems utilize a sewn eye instead of a knot, which minimizes additional friction loss from slippage. Simply replacing the sewn eyes in these systems with a tied knot can significantly reduce peak loads experienced by a firefighter during an escape scenario. V. COMPARISON OF EXPERIMENTAL AND MODEL RESULTS Table 3 shows a comparison between measured experimental peak anchor loads and predicted values obtained from the numerical model with and without friction included. The load predicted by the numerical model exceeds the measured load for each drop, indicating that the magnitude of dissipative terms is too small and the model lacks the ability to describe certain important dissipative terms, such as knot slippage. The average error for the first drop on each rope with and without friction are 37% and 40% respectively. The corresponding values for the second set of drops are 25% and 27%. Observations revealed that about 2.5 cm of rope slips through the knot during the first drop. The exact amount of slip will depend on maximum load during a test as well as the tightness and dressing of the initial knot. The knots themselves dissipate energy, and the slip of the anchor side knot will also increase the amount of rope that slides across the ledge, increasing the energy loss due to work done by friction. One way to verify this hypothesis would be to utilize ropes with sewn eyes on either end in order to lower the effect of knot slippage from the experiments. In each case, the comparison between experiment and theory improves in the second drop, which can be attributed to tightening of the knots during the first drop. This model formulation is quite sensitive to the coefficient of friction and the value estimated here (0.14) is relatively low for polymer materials on aluminum (typically 0.20-0.35). By increasing the coefficient of friction to 0.26, the average error of the second set of drops can be reduced to below 7%. The method for estimating friction coefficient will be improved in future iterations of the model. Table 1: Comparison of experimental and predicted peak anchor loads

Rope #

Drop #

1 1 2 2 3 3 4 4 5 5

1 2 1 2 1 2 1 2 1 2

Experimental Load (kg) 513 694 355 469 332 415 444 558 604 812

Predicted Load w/o Friction (kg) 756 847 650 706 613 663 694 726 986 1079

Predicted Load w/ Friction (kg) 725 817 625 681 585 633 670 703 951 1045

VI. SUMMARY AND FUTURE WORK An experimental testing frame was developed to simulate firefighter escape scenarios. The apparatus incorporates an instrument capable of gathering dynamic deflection data directly from a rope. The foundation of a numerical model was developed which consistently overpredicts peak loads experienced by the anchor and firefighter. The overprediction is attributed to an incomplete accounting of dissipative terms in the model which will be incorporated in the future. Preliminary testing shows considerable difference between loads felt by the anchor and drop mass, stressing the importance of incorporating friction in the model. In addition, the Technora-nylon rope exhibits considerable stiffening during consecutive drops. Future tests will utilize ropes with sewn eyes in order to minimize the effects of knot slippage on peak loads experienced during dynamic drop testing. The energy absorbing capability ability of various knots will also be studied and incorporated into the numerical model. ACKNOWLDEGEMENTS The authors would like to thank Armand Beaudoin for his continued support throughout the project along with Rick Rottet for his assistance in preparing the experimental equipment for testing.

186 REFERENCES

1. Horn, G.P. and P. Kurath. Failure of firefighter escape rope under dynamic loading and elevated temperatures. 2010. 2. Schad, R., Analysis of climbing accidents. Accident Analysis & Prevention, 2000. 32(3): p. 391-396. 3. Abrate, S., Nonlinear dynamic behavior of parachute static lines. Composite structures, 2003. 61(1): p. 3-12. 4. Baszczyński, K., Dynamic strength tests for low elongation lanyards. International journal of occupational safety and ergonomics : JOSE, 2007. 13(1): p. 39-48. 5. Spierings, A.B., Methodology for the development of an energy absorber: Application to worker security ropes. International journal of impact engineering, 2006. 32(9): p. 1370-1383. 6. Hennessey, C.M., N.J. Pearson, and R.H. Plaut, Experimental snap loading of synthetic ropes. Shock and Vibration, 2005. 12(3): p. 163-175.

Tensile Behavior of Kevlar 49 Woven Fabrics over a Wide Range of Strain Rates

Jeremy D. Seidt1, Thomas A. Matrka1, Amos Gilat1,* and Gabriel B. McDonald1 1

The Ohio State University, Department of and Mechanical and Aerospace Engineering Corresponding Author: Scott Laboratory, 201 W 19th Ave, Columbus, OH, 43210, [email protected]

*

ABSTRACT Kevlar and other ballistic fabrics are frequently used in dynamic loading applications such as personnel protective equipment (armor) and soft engine containment systems for fan blade out events in aircraft engines. Numerical simulations of these events require constitutive models that are based on mechanical experiments that approximate the load conditions present in the application. Kevlar 49 fabric is tested in tension at strain rates ranging from 10-4 to 1500 s-1. A servohydraulic load frame is used for low strain rate experiments up to 1 s-1. -1 Experiments at strain rates above 400 s are conducted on a direct-tension split Hopkinson bar apparatus. 3D digital image correlation is used to measure specimen surface displacements and strains.

INTRODUCTION Aircraft engine fan blade out events can lead to catastrophic results. Because of this, the Federal Aviation Administration (FAA) requires full-scale tests to ensure that blade out events can be safely mitigated. Recent advances in numerical codes, such as LS-DYNA [1], have made it possible for simulations of blade out events to offset some of the costs associated with testing. Numerical simulations of these events require models that can accurately describe the material behavior under dynamic load conditions. Soft-wall aircraft engine blade containment systems consist of layers of Kevlar 49 fabric wrapped around a metal ring [2,3]. While there are many constitutive models that can adequately predict the dynamic behavior of metals [4-7], there are significantly fewer material models that can predict the dynamic behavior of dry fabrics. Recently, constitutive models have been developed to model dry fabrics under dynamic loads [8,9]. These models require dynamic mechanical experiments for calibration and validation. Several authors have conducted dynamic tensile tests on single fibers and fiber bundles. Wang and Xia studied Kevlar 49 fiber bundles at strain rates ranging from 10-4 to 1350 s-1 using a servohydraulic load frame and a tension split Hopkinson bar (SHB) apparatus [10]. The authors report significant strain rate effects on the fiber bundles; the dynamic stresses are roughly 35% higher than quasistatic stresses. Cheng, Chen and Weerasooriya tested Kevlar KM2 single fibers at strain rates up to 1496 s-1 using a modified SHB apparatus [11]. In this experiment, the specimen is connected between a quartz force transducer and the end of an incident Hopkinson bar. The loading wave is generated using a striker tube that is fired into a flange at the opposite end of the incident bar. Tan, Zeng and Shim tested Twaron CT716 yarns at strain rates ranging from 3.0x10-4 to 400 s-1 using a load frame and a tension SHB [12]. The authors report a 30% increase in dynamic stresses. Zhu, Mobasher and Rajan conducted tests on Kevlar 49 single yarns at strain rates ranging from 20 to 100 s-1 using a high speed servohydraulic load frame [13]. The authors report that the tensile strength of Kevlar 49 single yarn is dependent on the strain rate, however this sensitivity is heavily dependent on the initial gage length of the specimen. Researchers have also investigated the mechanical behavior of fabrics. Naik, et al., conducted low rate tension tests on Kevlar 49 and Zylon fabrics using a servohydraulic load frame [3]. Shim, Lim and Foo tested Twaron CT 716 fabric at strain rates ranging from 10-2 s-1 to 495 s-1 using both a load frame and a SHB apparatus [14]. The authors report dramatically different results from the low rate and dynamic setups. In the present research, tensile tests are conducted on 1420 Denier Kevlar 49 fabric in a plain weave at strain -4 -1 -1 -1 rates of 10 s , 500 s and 1500 s using both a hydraulic load frame and a tension SHB apparatus. A grip T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_27, © The Society for Experimental Mechanics, Inc. 2011

187

188

technique for dynamic experiments that does not damage the specimen with mechanical pressure and matches the impedance of the Hopkinson bars is presented. 3D digital image correlation is used to measure specimen surface deformations for both static and dynamic experiments.

EXPERIMENTAL Specimen Preparation Gripping fabric specimens for dynamic tensile (SHB) experiments is challenging since it is difficult to hold the fabric material using mechanical clamp pressure without a) damaging the clamped section of the fabric and b) creating impedance mismatches from bulky grip fixtures that cause premature wave reflections. The specimen assembly procedure for dynamic experiments is illustrated in Figure 1. First, a swath of fabric is cut from the fabric using ceramic scissors, Figure 1 (a). The swath is prepared by removing yarns, Figure 1 (b). Longitudinal yarns, or those oriented in the direction of loading, are removed until six remain. Transverse yarns are removed in the regions that are epoxied to the aluminum grip fixtures. Transverse yarn removal in the grip region enhances the ability of the epoxy to hold the specimen during loading. Aluminum clamp fixtures are attached with screws to a custom alignment fixture that sets the sample gage length at 0.3 in, Figure 1 (c). The prepared fabric swath is epoxied to the aluminum clamp fixtures with Emerson and Cuming TRA-BOND 2106T epoxy, Figure 1 (d). Aluminum adaptor fixtures are epoxied to the fabric swath-clamp assembly, Figure 1 (e). The specimen assembly is then epoxied between the incident and transmitter bars of a tension SHB apparatus, Figure 1 (f). Finally, the alignment fixture is removed and the specimen is ready to be tested, Figure 1 (g). The specimen preparation procedure is labor intensive but results in a grip method that both matches the impedance of the Hopkinson bars and does not damage the specimen with mechanical pressure and pinch points. The low rate specimen preparation procedure is nearly identical to that for the dynamic procedure. This ensures that the grip technique itself is not responsible for differences observed in the low and high rate data. In the low rate specimen assembly procedure, the specimen adaptors that are glued to the swath-clamp fixture assembly in the step illustrated in Figure 1 (e), are rods with a through hole at one end for pin attachment to the load frame. (a)

(b) (d) (c)

(f) (e)

(g)

Figure 1. Fabric specimen preparation procedure: (a) a swath is cut from fabric material, (b) longitudinal and transverse yarns are removed from the swath, (c) the specimen clamp fixtures are attached to a fixture that sets the specimen gage length, (d) prepared fabric swath is epoxied to the specimen clamp fixtures, (e) specimen adaptors are epoxied to the fabric swath-specimen clamp assembly, (f) Specimen assembly is epoxied to the SHB apparatus, (g) assembly fixture is removed.

Low Strain Rate Tests

Low strain rate tests ( ε& < 1.0 s-1) are conducted on an Instron 1321 load frame. The low rate specimen assembly is pinned to pull-rods that have upper and lower double universal joints to ensure that the tensile loading is uniform and uniaxial. Load is applied to the specimen by moving the load frame’s actuator at constant velocity -5 (3.0x10 in/s). Load is measured using an Interface 1216 CEW-2K load cell. Specimen surface deformations are measured using 3D digital image correlation (DIC) [15]. Point Grey Research digital cameras with 1600 pixel by 1200 pixel resolution are used to record the deformation sequence.

189

High Strain Rate Tests

High strain rate experiments ( ε& > 400.0 s-1) are conducted on a direct tension SHB apparatus [16]. A sketch of the apparatus is presented in Figure 2. The specimen is attached between the incident and transmitter bars. A section of the incident bar (between the clamp and the pulley is pre-loaded with a hydraulic cylinder. The clamp is released rapidly by fracturing a pin in the clamp which results in a tensile loading wave that travels from the clamp to the specimen. The strain record in the incident bar is measured at two locations (Gage A and Gage B in Figure 2). The strain in the transmitter bar is measured at Gage C. The relative displacement between the ends of the bars, u1(t)–u2(t), is determined from these strain gage records using traditional SHB data reduction techniques. Engineering strain is calculated by dividing this relative displacement by the initial gage length of the specimen (in this case 0.3 in). The transmitted record, measured by Gage C is proportional to the load sustained by the sample and thus the engineering stress. Specimen surface deformations are also measured using 3D DIC. Two Photron SA1.1 cameras, Figure 3, operating at typical frame rates of 120,000 fps (at 128 pixel by 208 pixel resolution) are used to record the deformation sequence.

Figure 2. Sketch of a direct-tension SHB apparatus.

Figure 3. Dynamic experimental setup.

190

Experimental Data Typical data from a low strain rate test are presented in Figure 4. Engineering stress and strain vs time data are presented in Figure 4 (a). Strain is calculated in two ways. First, the relative motion between the upper and lower fabric fixtures, measured with DIC, are divided by the specimen gage length (green trace). Second, the average Lagrangian surface strain on the fabric swath is calculated from the DIC measured displacement field (red trace). The two strain measures agree reasonably well for the first 400 seconds, however they diverge significantly afterward. The strain at fracture calculated from the grip motion is nearly 0.038, significantly higher than the average surface fracture strain (0.027). The divergence is evidence of compliance and potential pull-out in the epoxy grip bond. Strain calculated from the grip motion is overestimated, since it does not account for compliance or pull-out in the grips. The Lagrangian surface strains, measured with DIC, are more accurate since they are measured directly on the surface of the sample and are not influenced by grip compliance. Engineering stress vs strain data are shown in Figure 4 (b). This plot also highlights the difference between the two strain records. The curve constructed with surface strain (solid trace) is nearly linear elastic to failure after an initial crimping phase where the individual yarns engage the axial load. The grip motion strain curve (dashed trace) exhibits some curvature which is due to compliance in the grip region. 350

350

0.05 Eng. Stress (ksi) Strain, Grip Motion Strain, DIC Lagrange

300

(a)

250 0.03

200 150

0.02

Strain Stress (ksi)

250

Stress (ksi)

(b)

300 0.04

200 150 100

100

Strain, Grip Motion Strain, DIC Lagrange

0.01

50

50 0

0

200

400

600

Time (s)

800

0.00 1,000

0 0.000

0.010

0.020

0.030

0.040

0.050

0.060

Strain

Figure 4. Data from a low strain rate experiment: (a) history data, (b) stress vs strain data. Data from a SHB experiment are presented in Figure 5. The waves, recorded by Gages A, B and C (see Figure 2), are presented in Figure 5 (a). Engineering stress, strain rate and two measures of strain are shown in Figure -1 5 (b). This experiment has an average strain rate of 500 s . The green trace represents the strain calculated from SHB data reduction and the black triangles represent the average Lagrangian surface strain from DIC. The two measures of strain diverge at roughly 60 μsec. The DIC strain measurement is more accurate, since the deformation is measured directly on the surface of the sample. The SHB strain overestimates the strain in the sample, since it does not account for compliance and pull-out in the grips. Stress vs strain curves constructed using the two strain records are shown in Figure 5 (c). The fabric failure strain calculated from the Hopkinson bar waves is roughly 0.035, almost 30% higher than that measured on the surface of the sample (0.027). Stress vs strain curves at three strain rates are shown in Figure 6. Dashed traces are constructed with strains calculated from the relative motion of the grip fixtures, while solid traces are composed using average Lagrangian surface strains. The average surface strain curves from tests at three strain rates (10-4 s-1, 500 s-1 and 1500 s-1) coincide. Thus it is concluded that the modulus of Kevlar 49 fabric is insensitive to strain rate. The ultimate tensile stress of the fabric at 10-4 s-1 and 500 s-1 are nearly identical (~280 ksi). The ultimate tensile stress at 1500 s-1 is 310 ksi, however additional tests are required to determine if this increase is truly due to strain rate sensitivity or if it is due to data spread caused by inconsistent grip compliance.

191 1200

(a)

Force (lbf)

800

400

0 Gage A (Incident Bar) Gage B (Incident Bar) Gage C (Transmitter Bar)

-400 -200

0

200

400

Time (μs)

600

800

600

0.050

350

(b) 500

0.040

400 0.030

200 150

0.020

300 200

100

Eng. Stress Strain Rate Strain, SHB Strain, DIC Lagrange

50 0

Strain

Stress (ksi)

250

0

50

Time (μs)

100

0.010

0.000 150

Strain Rate (s-1)

300

100 0

350

(c)

300

Stress (ksi)

250 200 150 100 Strain, SHB Strain, DIC Lagrange

50 0 0.000

0.010

0.020

0.030

0.040

0.050

Strain Figure 5. Data from a high strain rate experiment: (a) wave records, (b) history data, (c) stress vs strain data.

192 350 300

Stress (ksi)

250 200 150 1E-4 1/s, Grip Motion 1E-4 1/s, DIC 500 1/s, Grip Motion 500 1/s, DIC 1500 1/s, Grip Motion 1500 1/s, DIC

100 50 0 0.000

0.010

0.020

0.030

0.040

0.050

Strain Figure 6. Stress vs strain curves at different strain rates.

SUMMARY AND CONCLUSIONS Quasistatic and dynamic tensile tests on 1420 Denier Kevlar 49 fabrics have been conducted. A grip method that matches the impedance of the Hopkinson bars and does not damage the sample with mechanical pressure is presented. 3D DIC has been used to measure specimen surface deformation. Results from static and dynamic tests show that the strains computed from the relative motion of the specimen fixtures diverge from those measured on the surface of the sample, indicating that there is compliance and potential yarn pull-out in the epoxy grip bond. The results indicate that the modulus of Kevlar 49 fabric is not sensitive to strain rate. -1 -4 -1 Preliminary results show that the peak stress is higher at 1500 s (320 ksi) than at 10 and 500 s (280 ksi), however, additional tests are required to determine if this is truly rate sensitivity in the material or if it is due to inconsistent compliance in the grips.

ACKNOWLEDGEMENTS This work was supported by the Federal Aviation Administration (FAA). Thanks to Don Altobelli, William Emmerling and Chip Queitzsch of the FAA. Thanks to Subby Rajan and Barzin Mobasher of Arizona State University. Also thanks to Gary Gardner and Josh Gueth of The Ohio State University for specimen fixture fabrication.

REFERENCES [1] LSTC, LS-DYNA Keyword User’s Manual, Volumes I and II, Version 971, Livermore Software Technology Center (LSTC), Livermore, CA, 2007. [2] Carney, K., Pereira, M., Revilock, D., Matheny, P., “Jet Engine Fan Blade Containment using Two Alternate Geometries”, Proceedings of the 4th European LS-DYNA Users Conference, Ulm, Germany, May, 2003. [3] Naik, D., Sankaran, S., Mobasher, B., Rajan, S.D., Pereira, J.M., “Development of Reliable Modeling Methodologies for Fan Blade Out Containment Analysis – Part I: Experimental Studies”, International Journal of Impact Engineering, Vol. 36, 2009, pp 1-11. [4] Johnson, G.R., Cook, W.H., “A Constitutive Model and Data for Metals Subjected to Large Strains, High Strain Rates and High Temperatures”, Proceedings of the 7th International Symposium on Ballistics, The Hague, The Netherlands, April 1983. [5] Johnson, G.R., Cook, W.H., “Fracture Characteristics of Three Metals Subjected to Various Strains, Strain Rates, Temperatures and Pressures”, Engineering Fracture Mechanics, Vol. 21, 1985, pp31-48. [6] Steinberg, D.J., Cochran, S.G., Guinan, M.W., “Constitutive Model for Metals Applicable at High Strain Rate”, Journal of Applied Physics, Vol. 51, 1980, pp1498-1504. [7] Zerilli, F.J.; Armstrong, R.W. “Dislocation-Mechanics-Based Constitutive Relations for Material Dynamics Calculations.” Journal of Applied Physics, Vol. 61, 1987, pp1816–1825. [8] Stahlecker, Z., Mobasher, B., Rajan, S.D., Pereira, J.M., “Development of Reliable Modeling Methodologies for Fan Blade Out Containment Analysis – Part II: Finite Element Analysis”, International Journal of Impact Engineering, Vol. 36, 2009, pp 447-459.

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[9] Bansal, S., Mobasher, B., Rajan, S.D., Vintilescu, I., “Development of Fabric Constitutive Behavior for Use in Modeling Engine Fan Blade-Out Events”, Journal of Aerospace Engineering, Vol. 22, 2009, pp249-259. [10] Wang. Y., Xia, Y., “The Effects of Strain Rate on the Mechanical Behaviour of Kevlar Fibre Bundles: an Experimental and Theoretical Study”, Composites: Part A, Vol. 29, 1998, pp1411-1415. [11] Cheng, M., Chen, W., Weerasooriya, T., “Mechanical Properties of Kevlar KM2 Single Fiber”, Journal of Engineering Materials and Technology, Vol. 127, 2005, pp197-203. [12] Tan, V.B.C., Zeng, X.S., Shim, V.P.W., “Characterization and Constitutive Modeling of Aramid Fibers at High Strain Rates”, International Journal of Impact Engineering, Vol. 35, 2008, pp1303-1313. [13] Zhu, D., Mobasher, B., Rajan, S.D., “Experimental Study of Dynamic Behavior of Kevlar 49 Single Yarn”, Proceedings of the SEM Annual Conference, Indianapolis, IN, 2010. [14] Shim, V.P.W., Lim, C.T., Foo, K.J., “Dynamic Mechanical Properties of Fabric Armour”, International Journal of Impact Engineering, Vol. 25, 2001, pp1-15. [15] Sutton, M., Orteu, J.-J., Schreier, H.W., Image Correlation for Shape, Motion and Deformation Measurements, Springer, New York, NY, 2009. [16] Staab, G. H., Gilat, A., “A Direct-Tension Split Hopkinson Bar for High Strain-Rate Testing”, Experimental Mechanics, Vol. 31, 1991, pp 232-235.

The Effect of Loading Rate on the Tensile Behavior of Single Zylon Fiber C. Allan Gunnarsson [email protected] Tusit Weerasooriya [email protected] Paul Moy [email protected] Army Research Laboratory Weapons and Materials Research Directorate Bldg 4600 Deer Creek Loop Aberdeen Proving Ground, MD 21005-5069 ABSTRACT The commercially available Zylon™ single fiber has been investigated to understand the effect of strain rate on the tensile response. Experiments were conducted on single Zylon fibers across a range of strain rates ranging from 0.001/s to ~1000/s. The low rate uniaxial tensile experiments were performed using a Bose TestBench™ system. The high rate dynamic tensile experiments were performed using a modified Kolsky Bar setup. This paper discusses the experimental setup and presents the effect of strain rate on the tensile deformation and failure behavior of this single fiber. These experimental data will be used to develop continuum and micro-mechanics based fabric models and computational simulation methodologies for impact of these fabrics by projectiles; this will allow engineers to better understand the impact behavior of composite structures created with fabrics made of this fiber. KEYWORDS High strain rate experimentation, dynamic loading, single fiber testing, ultra-high speed imaging INTRODUCTION There has been growing interest to fully understand the dynamic mechanical response of textile materials for use in impact protection applications (such as aramid fabrics in body armor). Traditionally, dynamic mechanical experiments are carried out on woven fabrics or laminated polymer-matrix composites, to obtain dynamic material response. In general, these continuum experiments do not provide valid experimental data at high rates of loading. In the case of fabrics and yarns, errors associated with gripping issues frequently dominate the dynamic experimental data. Therefore, new material models are being developed at the fiber level for computer simulations that provide increased accuracy and obtain enhanced details of how individual yarns interact with projectiles during impact events. Yet, there is a lack of material properties at the single fiber level, especially at high strain and loading rates. This is mainly due to the lack of experimental methods to characterize single fiber behavior at high loading rates. To understand how a fabric based protection system deforms during ballistic impact events, many variables, such as material properties, fabric structure, projectile velocity and geometry, layer interactions, friction, and global boundary conditions must be determined [1]. Paramount among these is material properties, particularly at the high strain rates which are relevant to ballistic and stabbing impacts. These high rate properties are vital to insure that these “fiber-level” material models are accurately capturing the constituent material behavior of the composites during impact and ballistic simulations. Researchers have recently begun to study the mechanical properties of single fibers. After experiencing the inability to avoid slipping of fabrics and yarns during gripping, Cheng et al [2] pioneered a unique experimental technique to obtain the dynamic tensile and transverse responses of single fibers. They T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_28, © The Society for Experimental Mechanics, Inc. 2011

195

196 demonstrated the experimental method for Kevlar KM2 single fibers and found that the tensile behavior was not strain rate sensitive. Mayo et al [3] determined the fracture toughness of single Kevlar 129 fiber and explored the relationship between fiber hardness and fracture toughness on cut resistance. Lim et al [4], using the experimental methods developed by Cheng et al [5], determined the mechanical behavior of A265 (“Termotex”) single fibers using axial tension, transverse compression, and torsion experiments. -1 This included experiments conducted at high (~1500 s ) strain rates. Additionally, they measured the effect of gauge length on the mechanical properties and discovered a “critical” gauge length above at which a strength-limiting defect influences failure. Increased gauge length beyond this critical value had a reduced effect on the A265 mechanical properties compared with varying the gauge length below the critical point. Other research has been performed at “near fiber” or fiber bundle level to the capture high rate behavior of yarns. High strain rate behavior of PPTA fiber bundles was recently determined by Languerand et al [6] using a conventional split Hopkinson tensile bar (SHTB). Shim et al [7] investigated the strain rate effect on Twaron fabrics and found that the tensile strength and modulus increase and failure strain decrease as the strain rate is elevated. Zhu et al [8] performed dynamic experiments on Kevlar 49 single yarn. The -1 yarns were tested using a high speed servo-hydraulic test machine over the 20 – 100 s strain rate range to determine rate effects on yarn modulus, tensile strength, strain to failure, and toughness. Their results indicate that the Kevlar yarns are highly strain rate dependent, with increasing strain rate causing -1 increased modulus, strength, and toughness over a small strain range (20 – 100 s ). They also found that increased strain rate caused an increase in strain to failure of the Kevlar yarns. Farsi et al [9] studied quasi-static and dynamic behavior of several different yarns, including three types of Kevlars (129, KM2, LT), Twaron, and Zylon. Their research indicated that Kevlar was not particularly strain rate sensitive, while the Twaron and Zylon were more rate dependent. Xia et al [10] investigated the strain rate effect on -1 Kevlar 49 fiber bundles, over strain rates between quasi-static and dynamic (~1400 s ). They found that modulus and tensile strength were directly proportional to strain rate. In this research, the mechanical behavior of Zylon single fiber is investigated at both quasi-static and dynamic strain rates. The effect of strain rate on tensile strength, modulus, and failure strain is reported in this paper. The gauge-length effect on Zylon single fiber is also investigated at quasi-static strain rate by varying the gauge length between 5 and 25 mm. A modified SHTB, similar to the one pioneered by -1 Cheng et al [2], is used to conduct tensile experiments at strain rates from 600 – 1700 s . This experimental methodology allows the determination of dynamic tensile properties for this single fiber. MATERIAL Zylon is the trademarked name of an organic polymer commercial fiber, produced by Kuraray (Osaka, Japan), and has been widely used in many applications, such as composites foruse in racecars and boats, where high strength and thermal stability is required. Zylon gained wide use as a body armor in 1998 when it started being used in police protective vests by a prominent body armor company. However, in 2003, it became controversial when two officers were mortally wounded while wearing Zylon based vests, leading to the decertification of Zylon for use in ballistic vests by the National Institute of Justice in 2005.[11] However, this does not mean that Zylon is unsuitable for protective applications. It is used as a safety component in tethering systems in both Formula One (2001) and IndyCar (2008) racing series. This fiber is also used in roping and bomb containment vessels. Zylon’s chemical composition is polybenzoxazole (PBO), which is an aromatic heterocyclic polymer. PBO fibers were developed as a result of an Air Force program whose goal was the development of lightweight, high strength fibers for composite applications [12]. They are spun from liquid crystalline solution, and then heat stretched for improvement of mechanical properties. PBO is thermally and thermo-oxidatively stable; it chars instead of burns when exposed to fire. It has higher tensile strength and modulus than most other high performance fibers (see table). Table 1 [13] summarizes Zylon’s mechanical properties and several other high performance fibers with data from their respective manufacturers.

197 Table 1 – Summary of mechanical properties of Zylon and other high performance fibers [13]

Tensile Density Modulus Strength 3 (g/cm ) (Gpa) (Gpa)

Trade Name

Chemical Composition

Denier

Diameter (µm)

Strain to failure (%)

Kevlar 129a

aramid

840

13.0

3.3

1.44

96

3.38

aramid

840

9.46

3.3

1.44

90

3.31

UHMWPE

400

18.8

3.6

0.98

88

2.62

PBO

500

12.3

3.5

1.54

180

5.80

aromatic polyester

1500

23.5

3.3

1.41

75

3.20

aramid

1500

12.3

4.6

1.39

73

3.44

glass

2980

9.29

5.5

2.48

85

4.80

graphite

800

8

2

1.8

234

4.83

b

Twaron

Dyneema SK-76 Zylon

d e

Vectran

Technora S2-glass

f

g

Carbon fiber

h

c

Data from:

a b c d e f g JPS Composites DSM Dyneema, Netherlands Toyobo, Inc. Kuraray, Inc. Teijin Aramid Owens Corning Grafil, Inc.

The manufacturer’s data included a nominal diameter for all Zylon fibers as 12.3 µm. However, to ensure accuracy, several fibers were measured using an optical microscope (figure 1). As can be seen in the figure, the fiber diameter is consistent and uniform along the fiber axis; this was the case for all fibers examined. The average nominal fiber diameter from the measurements was 12.26 µm.

Figure 1 – Optical microscope picture of Zylon single fiber at fifty times (50X) magnification

EXPERIMENTS Tensile experiments were performed on single Zylon fibers to measure the mechanical response of the material at both quasi-static and dynamic strain rates. Tensile experiments were performed on specimens with two gauge lengths at the quasi-static strain rate. This allowed for both strain rate dependence and gauge length effect to be quantified. Quasi-static Experiments To perform the quasi-static strain rate experiments, a Bose test bench system was utilized with a 2.45 N (250 g, 0.55 lb) load cell. This machine allowed for experiments at 0.001/s strain rate, and is also capable -1 of performing experiments at strain rates in the intermediate (1 s ) strain rate range. The tensile behavior of single fibers has been studied before at quasi-static rates and there exist test standards for this purpose (ASTM D3822-96 “Standard Test Method for Tensile Properties of Single Textile Fibers” and ASTM D3379-75 “Standard Test Method for Tensile Strength and Young’s Modulus for High-Modulus Single-Filament Materials”). The quasi-static tests in this study were performed closely following ASTM D3379-75; however, the standard specifies a gauge length of 10 mm, and in the experiments a gauge

198 length of 5 mm was used. This was done to ensure comparability with the dynamic experiments; the need for a short gauge length in dynamic experiments will be discussed later. The single fibers were extracted from a spool (not pulled from a woven fabric) and glued to a backing paper with an industrial structural adhesive. The backing paper has a cut out section that acts as the gauge length of the specimen. The fiber is nearly taught over the cut-out, but not quite, to ensure there is no pre-load. This “picture frame” technique allows for ease of handling and gripping of the single fiber for testing. The paper is then mounted in the Bose test machine and, just prior to testing, the paper is cut to allow the system to measure only the fiber response. The slack taken up prior to the commencement of loading is then added to the nominal gauge length to obtain the “actual” gauge length, and this adjusted gauge length is used for strain calculation. The slack is generally 0.10-0.15 mm. This same procedure was used to perform quasi-static experiments at 0.001/s strain rate for paper mounted single fibers with a nominal gauge length of 25 mm. Figure 2 shows (a) the two different gauge length specimens with paper backing and a Zylon single fiber adhered in place (although the fibers are difficult to see) and (b) the quasi-static test setup.

(a) (b) Figure 2 – (a) Quasi-static specimens with 5 mm and 25 mm gauge lengths (b) Bose test machine with specimen

High Strain Rate Experiments It is necessary to determine if there is a change in mechanical response of Zylon as the tensile strain rate increases into the dynamic strain rate region. To perform dynamic tensile tests on single fibers, a modified SHTB setup was created, similar to those discussed in the introduction, to allow for single fiber tensile -1 testing at strain rates of approximately 1000 s , many magnitudes higher than the quasi-static experiments. The SHTB experimental technique has been widely used to investigate the mechanical response of a wide variety of engineering materials, including metals, ceramics, polymers, soft materials, tissues, etc. In the setup used here, a traditional incident bar made of aluminum 7075, with diameter of 6.35 mm and length of 1.83 m has a threaded flange at one end. A hollow striker tube is accelerated by a series of springs along the incident bar and impacts the incident bar flange, which creates a tensile stress pulse that travels down the length of the incident bar. The stress pulse creates strain in the bar, which is measured by two pairs of strain gauges mounted approximately midpoint of the bar; one pair is a traditional metal foil type and the other pair is a semi-conductor type to allow for smaller strain measurements due to their greater signal-to-noise ratio. The stress pulse travels down the bar to the end, where the single fiber specimen is glued to a concentric flat machined into the end of the bar. A traditional transmission bar would not be sensitive enough to measure the small forces (below 1 N) transmitted through a single fiber, even with high sensitivity semi-conductor strain gauges, so the other end of the fiber is adhered to a small capacity, high sensitivity quartz force transducer, with capacity to 22 N (5 lb). Figure 3 (a) shows a schematic of the high rate experimental setup [4], and figure 3 (b) shows a close up of the setup.

199

Figure 3 (a) – High rate experimental setup schematic [4]

Figure 3 (b) – High rate single fiber experimental setup

The gauge length of the specimen was chosen at 5 mm to ensure that the fiber specimen is in dynamic force equilibrium during the experiment. This is a necessary criterion for valid Hopkinson bar experiments, as well as having a nearly constant strain rate during the experiment. The fiber is extracted, and then adhered to the load cell side mounting flat. The adhesive is applied at the ends of the bar flat and load cell flat; the gauge length is assumed to be the distance between the flat ends. Setting and measuring this distance prior to fiber mounting ensures proper gauge length. Care must be taken to ensure the fiber is running axially through the adhesive to avoid stress concentrations and non-uniaxiality. After adhesive on the load cell flat mount has hardened, the fiber is pulled to the flat on the incident bar side, and then glued it to the bar-flat. Again, the fiber is not completely taut, and the displacement due to the slack is used to account for the final gauge length. Once the adhesive on both sides is cured, the specimen is ready for the experiment. Figure 4 shows a close up of the adhesive joints at the fiber/flat connectors.

200

Figure 4 – Close up of the adhesive joints at the fiber/flat connections

Compression of the spring system and release of the striker is accomplished by hand; complete compression of all springs ensures the striker impact velocity remains constant throughout the experiments. Changing the number of springs in series and the spring stiffness allows for different striker impact velocities and, therefore, strain rates for a given gauge length. During the high rate experiments, a high-speed camera was used to record the specimen behavior and determine failure location and timing. A Photron APX-RS high-speed camera was used, recording at a rate of 100,000 fps (10 μs inter-frame time) and at a resolution of 384 x 48 pixels. Three high power lamps were used to illuminate the specimen. These were carefully energized just prior to testing, as the heat generated affects the response of the load transducer. A large majority of the high rate experiments failed in the gauge length; this was indicated by the high-speed video showing the presence of fiber continuing to protrude out of the adhesive. These observations help to confirm the validity of the mounting system. Figure 5 shows a close up of the fiber post-test where it is clearly visible that the specimen broke midway in the gauge length.

Figure 5 – Close up of failed specimen

The mechanical impedance of the load cell is much greater than the fiber, and is considered to be a rigid body. Therefore, the stress history (σS) of the specimen can be determined by measuring the load at the transducer (P) and dividing by the cross section area of the fiber (AS).

 S (t ) 

P (t ) AS

(1)

The displacement of the bar can be measured by recording the incident and reflected pulse time histories. Using one dimensional wave theory, the bar end velocity, u, is a function of the incident pulse (εI), reflected pulse (εR), and wave speed in the bar material (cB).

u (t )  CB [ I (t )   R (t )] The bar end velocity divided by gauge length (lS) provides specimen engineering strain rate.

(2)

201

 (t )  

u (t ) cB (t ) [ I (t )   R (t )]  ls ls

(3)

Integration of the specimen strain rate determines engineering strain in the specimen. t

 (t )    (t )dt  0

cB ls

t

 [ 0

I

(t )   R (t )]dt

(4)

Using these relationships, the dynamic stress-strain response of the single fiber can be obtained. Figure 6 shows typical signals for (a) bar strain and (b) load transducer history.

0.01

0.2

0.005

0.1 Load (v)

0.3

Strain (v)

0.015

0

0

-0.005

-0.1

-0.01

-0.2 -0.3 -0.6

-0.015 -0.6

-0.4

-0.2

0 0.2 Time (ms)

0.4

0.6

0.8

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Time (ms)

(a) (b) Figure 6 – (a) Typical strain gauge signal and (b) typical load transducer signal

RESULTS AND DISCUSSION The stress-strain response of the single fiber with 5 mm gauge length for both (a) high and (b) low rate is shown below in figure 7. As can be seen from figure 7(a), there is some scatter at high rate due to specimen variability. For both rates, only some of the experimental data is shown for clarity. Some of the specimens exhibited some minor slipping; these specimens were not used for the failure strain statistics. Specimens that slip, obviously, lead to exaggerated strain values; this is why proper specimen preparation is vital for valid experiments. 8000

8000

7000 Engineering Stress (MPa)

Engineering Stress (MPa)

7000 6000 5000 4000 3000 2000

6000 5000 4000 3000 2000 1000

1000

0

0 0

0.02

0.04 0.06 Engineering Strain

0.08

0.1

0

0.01

0.02

0.03

0.04

0.05

Engineering Strain

(a) (b) Figure 7 – (a) High rate and (b) Quasi-static response of Zylon single fiber at 5 mm gauge length

0.06

202 The statistics for all three experimental configurations are summarized below in Table 2. There is a large gauge length effect on tensile strength and failure strain; both are much lower for the 25 mm gauge length than the 5 mm gauge length. The tensile strength and modulus are both directly related to strain rate, while failure strain is inversely related. Table 2 – Mechanical properties for Zylon single fiber

Tensile Strength (GPa)

Failure Strain

Modulus (GPa)

Gauge Length (mm)

Strain Rate (/s)

Average

Standard Deviation

Average

Standard Deviation

Average

Standard Deviation

25

0.001

2.73

0.45

0.01411

0.0045

175.12

43.35

5

0.001

4.44

1.03

0.04322

0.0108

149.51

30.75

5

~ 1000

5.99

0.98

0.04095

0.0062

249.39

37.60

The relationship between the three mechanical properties investigated (tensile strength, failure strain, and modulus) and strain rate for 5 mm gauge lengths are shown below in Figures 8(a), 9(a), and 10(a), respectively. A trend line has been plotted to show the relationship between the mechanical property and strain rate. The relationship between the three mechanical properties and gauge length at 0.001/s strain rate are show below in Figures 8(b), 9(b), and 10(b). In all figures, the black dots represent the average for that strain rate group.

Figure 8 – (a) Tensile strength vs strain rate for 5mm gauge length and (b) tensile strength vs gauge length for 0.001/s

203

0.08

0.07

0.07

0.06

0.06

Failure Strain

Failure Strain

y = 0.042102 - 0.00035319log(x) R= 0.12002 0.08

0.05 0.04 0.03

0.05 0.04 0.03

0.02

0.02

0.01

0.01

0 0.0001 0.001 0.01

0.1 1 10 Strain Rate (/s)

100 1000

0

104

0

5

10 15 20 Gauge Length (mm)

25

30

(a) (b) Figure 9 – (a) Failure strain vs strain rate for 5mm gauge length and (b) failure strain vs gauge length for 0.001/s

350

350

300

300

250

250

Modulus (GPa)

Modulus (GPa)

y = 201.82 + 15.827log(x) R= 0.84549

200 150 100 50

200 150 100 50

0 0.0001 0.001 0.01

0.1 1 10 Strain Rate (/s)

100 1000

104

0

0

5

10 15 20 Gauge Length (mm)

25

30

(a) (b) Figure 10 – (a) Modulus vs strain rate for 5mm gauge length and (b) Modulus vs gauge length for 0.001/s

SUMMARY A methodology for performing single fiber tensile experiments at the Army Research Laboratory’s experimental facilities has been developed. This includes a modified Kolsky tensile bar dedicated to dynamic strain rate experimentation on micron-size single fibers. These capabilities were used to characterize the rate effects on the tensile mechanical properties of single fiber Zylon (PBO). It was found that tensile strength and modulus are both directly correlated to strain rate, while failure strain is inversely correlated. Additionally, the effect of gauge length on the mechanical properties of single fiber Zylon was investigated. Although the fiber diameter appears to be constant and uniform along the fiber axis, there is definitely a gauge length effect present. Longer gauge lengths increase the likelihood for a critical defect to be present, causing an earlier failure at lower strain and stress; however, this defect presence does not affect the tensile modulus. -1

In the future, additional experiments need to be performed at an intermediate strain rate (1 s ) for the short (5 mm) gauge length to understand the rate effect fully. Also, additional experiments must be performed at various other gauge lengths (2 mm, 10 mm, 15 mm, 50 mm, 100 mm) to fully develop the effect of gauge length at quasi-static strain rate on the mechanical response. This will demonstrate if there is some critical gauge length, past which, the gauge length effect becomes less pronounced as Lim et al concluded for A265 fiber [4]. Once that is complete, it would be necessary to investigate if the gauge length effect is the same at dynamic strain rates. Finally, it is hoped that optical strain measurement

204 techniques can be developed to increase the accuracy of strain measurements and decrease the importance of having to demonstrate no fiber slippage. ACKNOWLEDGEMENTS The authors would like to acknowledge the following people for their assistance in specimen preparation and experimentation: Jeffrey Gair, Mark Foster and Aristotle Gazonas. REFERENCES 1. Cheeseman, B., and Bogetti, T. Ballistic Impact Into Fabric and Compliant Composite Laminates. Composite Structures. Vol 61. pp 161-173. 2003. 2. Cheng, M., Chen, W., and Weerasooriya, T. Mechanical Properties of Kevlar KM2 Single Fiber. ASME Journal of Engineering Materials and Technology. Vol 127. pp 197-203. April 2005. 3. Mayo, J., Hosur, M., Jeelani, S., and Wetzel, E. Fracture Toughness of Single Kevlar 129 Fiber. Draft version, to be published 2011. 4. Lim, J., Zheng, J., Masters, K., and Chen, W. Mechanical Behavior of A265 Single Fibers. SEM Annual Conference Proceedings. June 2010. 5. Cheng, M., Chen, W., Modeling Transverse Behavior of Kevlar KM2 Single Fibers with Deformation Induced. International Journal of Damage Mechanics. Vol 15. pp 121-132. April 2006. 6. Languerand, D., Zhang, H., Murthy, N., Ramesh, K., and Sanoz, F. Inelastic behavior and fracture of high modulus polymeric fiber bundles at high strain rates. Material Science and Engineering A 500. pp. 216-224. 2009. 7. Shim, V., Lim, C., and Foo, K. Dynamic mechanical properties of fabric armor. International Journal of Impact Engineering. Vol 25. pp 1-15. 2001. 8. Zhu, D., Mobasher, B., and Rajan, S. Experimental Study of Dynamic Behavior of Kevlar 49 Single Yarn. SEM Annual Conference Proceedings. June 2010. 9. Farsi, D., Nemes, J., and Bolduc, M. Study of Parameters Affecting the Strength of Yarns. Journal of Physics IV. Vol 134. pp 1183-1188. 2006. 10. Xia, Y., and Wang, Y. The Effects of Strain Rate on the Mechanical Behavior of Kevlar Fibre Bundles: An experimental and Theoretical Study. Composites Part A. Vol 29A. pp 1411-1415. 1999. 11. http://en.wikipedia.org/wiki/Zylon 12. Clements, Linda L. Organic Fibers. Handbook of Composites. S. T. Peters. London, Chapman and Hall. pp 202 - 241. 1998. 13. Mayo, J. Studies on Cutting and Fracture Mechanics of High Performance Fibers. Doctoral thesis. Tuskegee Center for Advance Materials, Tuskegee University. 2010.

Statistical analysis of fiber gripping effects on Kolsky bar test1

J.H. Kim1, N.A. Heckert2, S.D. Leigh2, H. Kobayashi1, W.G. McDonough1, R.L. Rhorer3, K.D. Rice4 and G.A Holmes1* 1

Polymers Division (M/S 8541), 2Statistical Engineering Division (M/S 8980), 3Intelligent Systems 4 Division (M/S 8220), Law Enforcement Standards Office (M/S 8102) National Institute of Standards and Technology Gaithersburg, MD 20899 *

Corresponding author: [email protected]

ABSTRACT Preliminary data for testing fibers at high strain rates using the Kolsky bar test by Ming Cheng et al. [1] indicated minimal effect of strain rate on the tensile stress-strain behavior of PPTA [poly (p-phenylene terephathalamide)] fibers. In a different study, Lim et al. [2] reported that the tensile strengths of the copolymer aramid fibers are strain rate dependent. Usually with single fiber tests, a large sample size is needed to get statistically significant results. To date with high strain rate tests, technical issues associated with specimen preparation appear to limit the number of samples that can be tested within a reasonable time. In this study, the authors investigate the feasibility of a gripping design to establish a reliable, reproducible, and accurate Kolsky bar test methodology for single fiber tensile testing with high throughput. Preliminary results measured under a quasi-static strain rate with 5 mm and 60 mm gauge lengths showed that the PPTA fibers are gauge length dependent. This size effect must be considered in determining the true strain rate effect on fiber strength in conjunction with the gripping effect.

1. INTRODUCTION Although the material properties of high performance fibers are routinely characterized quasi-statically, two recent workshops have underscored the need to obtain accurate and reproducible high strain rate (HSR) test data on , these fibers to ensure their ballistic performance in lightweight soft body armor (SBA) applications [3 4]. In a previous publication [5], it was noted that the Kolsky bar technique, which requires test specimen gauge lengths to be less than 10 mm, offers promise as a measurement tool for providing these critical data. It was also highlighted that the increased percentage of perturbed gauge length in short fibers relative to long fibers and the wicking of the glue along the length of the fiber that is often used to attach the fiber specimen to the Kolsky bar can create measurement problems by rendering the fiber gauge length essentially unknown. In addition, the gluing approach has been found to be time consuming, often limiting testing to only a couple of samples per day.

1

Official contribution of the National Institute of Standards and Technology; not subject to copyright in the United States.

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_29, © The Society for Experimental Mechanics, Inc. 2011

205

206 In our initial publication [5], quasi-static tests on 2 mm and 60 mm long guage length samples were performed using two different gripping methods. In gripping method 1, the fiber is glued to a template of known gauge length, while in gripping method 2, the fibers are directly gripped between two tension-controlled polymethyl methacrylate 2 (PMMA) blocks (Textechno Herbert Stein GmbH & Co. KG. ). The data were initially analyzed by the Stoner et al. [6], and Newell et al. [7] grip effects model that account for failures caused by end effects. For both gripping techniques, this model only adequately predicted the survival behavior at the 60 mm gauge length and was found to be a poor predictor of the survival behavior at the 2 mm gauge length. Although the grip effects model was found to be of minimal value, the survival results indicate that the failure behavior of the fiber may be changing with shorter gauge lengths. In this study, a detailed statistical analysis of the failure data at 60 mm and 5 mm gauge lengths are evaluated using the direct gripping method under quasi-static loading conditions to provide a basis for assessing the fidelity of HSR data using the Kolsky bar. 2. EXPERIMENTAL PROCEDURE 2.1 Quasi-static loading For the tensile test under quasi-static loading, poly (p-phenylene terephthalamide) fibers (PPTA) and the Favimat testing machine were used. A fiber gripping technique (direct grip) used is shown in Figure 1. While the fiber is under a slight tensile load of 0.01 N to maintain the straightness and alignment to the grips, a single PPTA fiber is directly clamped between two blocks on both ends, and the clamping force of the blocks is controlled by tightening a spring. The gauge lengths of the fiber specimens were 5 mm and 60 mm. The lengths were chosen to investigate the length dependency of the fiber under the quasi-static loading condition. The tensile tests were -1 performed under the constant strain rate 0.00056 s with uncertainty in the load cell of 0.1 %. Fiber diameters were measured using the optical microscope at five locations and the average of the five values was used for the strength calculation.

Figure 1 Schematic and closed up of the direct grip for quasi-static loading. 3. RESULTS AND DISCUSSION The tensile test results are presented in Table 1. The average failure load measured at the 5 mm gauge length is 11 % higher than that for the 60 mm gauge length, while the average diameters of the two gauge lengths are Table 1. Summary of PPTA fiber properties Gauge length (mm)

Load (N)

Diameter (m)

Strength (GPa)

5

0.390.06

13.290.70

2.790.38

60

0.350.06

13.370.57

2.530.44

 refers the standard deviation of the fiber properties taken as an estimate of the standard uncertainty

2

Certain commercial materials and equipment are identified in this paper to specify adequately the experimental procedure. In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply necessarily that the product is the best available for the purpose.

207 relatively close. As a result, the average tensile strength for the 5 mm gauge length is 10 % higher, illustrating the gauge length dependency of the tensile strength. In order to do a valid comparison of tensile strengths between 5 mm and 60 mm fiber lengths, it is important to know if the overall distribution changes as the gauge length changes. A graphical approach can be used to examine the distribution computing nonparametric kernel density estimates [8] of tensile strength for both 5 mm and 60 mm gauge length data, and comparing the resulting density plots as shown in Figure 1. It is apparent from Figure 1 that the strength distribution of the 5 mm length data shows bigger values than the 60 mm gauge length data, with the 5 mm profile clearly shifted to the right of the 60 mm. As noted in the density plot, higher strength values for the 5 mm case show higher relative frequencies of occurrence than the 60 mm case above 2.75 GPa. The approximately 10 % increase in strength value for the 5 mm case is attributable to these more frequently occurring higher strength values.

Figure 1. Kernel density plots of the tensile strength for the 5 mm and 60 mm gauge length tests. An alternative approach to demonstrate the nonequivalence of the two distributions is to perform a QuantileQuantile (Q-Q) comparison [9] between the 5 mm and 60 mm tensile strengths. For the Q-Q plot, the strength sets for 5 mm and 60 mm were sorted in ascending order. Figure 2 shows the Q-Q plot for the 60 mm versus 5 mm. The empirical profile falls below the 45 distribution equivalence reference line and shows clearly that the strengths of the 5 mm case generally dominate those of the 60 mm case on a matching quantile by quantile (i.e., strength by strength) basis.

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Figure 2. Quantile-Quantile plot of the tensile strengths for the 5 mm and 60 mm gauge length tests As can be seen from the kernel density plot, the strength distributions for both 5 mm and 60 mm do not conform to a Gaussian distribution, so conventional comparison procedures to distinguish between the strength values (e.g., t-test, ANOVA (analysis of variance)) may not be appropriate. To demonstrate statistically the nonequivalence, nonparametric Kruskal-Wallis [10] and Kolmogorov-Smirnov tests [10] were carried out. The null hypothesis for the statistical analyses is that the strength distributions of the 5 mm and 60 mm are indistinguishable, so the strength distributions (e.g., the kernel density plots) must be similar if only fiber flaws influence to the tests. In this study, both the Kruskal-Wallis and Kolmogorov-Smirnov tests reject a null hypothesis of equality of distribution at >99 % confidence (K-W P-value 0.00084). 4. CONCLUSIONS Fiber tensile tests using a direct fiber grip technique have been carried out under quasi-static loading conditions to ascertain the feasibility of this fiber gripping method for possible use for the HSR testing with the Kolsky bar. A goal was to look at the trend of fiber failure (i.e., strength distribution) as the gauge lengths were decreased from 60 mm to 5 mm. This reflects on the question of whether this gripping method influences on experimental error. Based on the test results, there is a strong indication that the distribution of the fiber strengths for 5 mm is different from 60 mm. Under the quasi-static test, the average fiber tensile strength with the 5 mm gauge length was higher than that of the 60 mm. This suggests that the fiber tensile strengths in this study depend on the gauge length, and this length dependency of the fiber strength must be considered in determining the true strain rate effect of the fiber when tested under HSR condition. It may be that the discrepancies between both the strength distributions and the average strengths between 5 mm and 60 mm cases are an effect of the direct fiber grip used in this study.

209

References

1. Cheng,M., Chen,W.N., and Weerasooriya,T., "Mechanical properties of Kevlar (R) KM2 single fiber," Journal of Engineering Materials and Technology-Transactions of the ASME, 127 (2), 197-203 (2005). 2. Lim,J., Zheng,J.Q., Masters,K., and Chen,W.N.W., "Mechanical behavior of A265 single fibers," Journal of Materials Science, 45 (3), 652-661 (2010). 3.

NIST technical workshop on research and metrology of ballistic fibers (2007).

4. Multiscale materials behavior in ultra-high loading rate environments, US Army research laboratory workshop (2008). 5. Kim, J.H, Rhorer, R.L., Kobayashi, H., McDonough, M. G., and Holmes, G. A., Effects of fiber gripping methods on single fiber tensile test using Kolsky bar. Society of Experimental Mechanics, Indianapolis, IN (2010). 6. Stoner, E.G., Edie, D.D., and Durham, S.D., "An End-Effect Model for the Single-Filament Tensile Test," Journal of Materials Science, 29 (24), 6561-6574 (1994). 7. Newell, J.A. and Sagendorf, M.T., "Experimental verification of the end-effect Weibull model as a predictor of the tensile strength of Kevlar-29 (poly p-phenyleneterephthalamide) fibres at different gauge lengths," High Performance Polymers, 11 (3), 297-305 (1999). 8.

Silverman, B., Density estimation for statistics & data analysis,Chapman & Hall (1986).

9. Chambers, J.M., Cleveland, W.S., Kleiner, B. and Tukey, P.A. Graphical methods for data analysis, Duxbury Press (1983). 10.

Conover, W.J. Practical nonparametric statistics, John Wiley & Sons (1999).

Perpendicular Yarn Pull-out Behavior under Dynamic Loading Jihye Hong1, Jaeyoung Lim2, Weinong W Chen1* School of Aeronautics and Astronautics, Purdue University. 2 Central Advanced Research & Engineering Institute, Hyundai Motor Company. * Corresponding author: Weinong Chen, 701 W. Stadium Ave. West Lafayette, IN 47907-2045 Email: [email protected] 1

ABSTRACT Multi-layers of woven fabric, characterized by high strength, high flexibility and low density, have been widely used for protective systems, from bulletproof vests to turbine engine fragment barriers. Under ballistic impact, the kinetic energy of the projectile is absorbed by the fabrics deforming largely in out-of- plane direction, which retards the motion of the projectile. Interfacial friction between yarns is known as one of the factors that affect the energy absorption by the fabric. In this study, we experimentally investigated the mechanical response during the pulling out of a single yarn from a single layer of Kevlar® or Twaron® fabric at high rates. In order to perform the dynamic experiments, a pendulum impact was generated to pull out the yarn. Displacement was measured from a noncontact laser system and pull-out force signal was received by a quartz transducer connected to the yarn being pulled out. Quasi-static experiment was also performed to study the effects of loading rates on the mechanical response when a yarn is pulled out perpendicular to the fabric plane. INTRODUCTION Woven fabrics with aramid fibers have gained great attention as protective systems under the ballistic impact as providing unique characteristics of high strength, low density, and high flexibility in the woven structure. The multiple layers of the woven fabrics with these materials play an important role in absorbing the energy transferred from impact of the projectile. It is known that mechanism of energy absorption of the fabrics under the ballistic impact is determined by various factors, such as material property of single fiber, structure of the fabric, projectile geometry, impact velocity, multiple ply interaction, far field boundary condition and inter yarn friction. [1] As one of critical mechanisms affecting the ballistic impact energy absorption, friction between yarns has come into the spotlight for developing the appropriate model of the fabrics. It was emphasized that interfacial friction helped to develop more accurate fabric models [2] and contributed to increasing the fabric energy absorption below certain level of friction coefficient [3]. To quantitatively investigate the role of friction between yarns, pull out experiments have been performed by many researchers. Kirkwood et el.[4] performed quasi-static yarn pull-out to find the effects of fabric size, transverse tension, and multiple yarn pull-out and correlated the results to ballistic impact. Dong et al. [5] found that yarn pull out force is positively related to impact performance and developed a finite element model to estimate pull-out force for a single yarn. However, most studies were focused on yarn pull out at quasi-static or in plane loading although large yarn pull out occurs in the out of plane direction. In this study, a new pendulum setup was designed to perform the yarn pull out experiments at high rates. To study the rate dependency of the pull out behavior, quasi-static experiments were also executed. EXPERIMENTS AND RESULTS 1) Quasi-static Experiment Each specimen was prepared from a single ply woven fabric made of Kevlar® . The sample (10 cm by 10 cm) was placed over a rectangular plate and fixed by two separate plates over the fabric by tightening the screws. One of the yarns in the middle was carefully picked out and hanged onto a small hook attached to the upper grip on MTS 810. Cross head was moving at 1 mm/s to pull yarns from the fabric. The pull out load was measured by a load cell and displacement was measured by sensors built in MTS. The set up is shown in Fig1. The result shown in Fig 2 illustrates that pull-out force increases up to a maximum force during the yarn stretch and drops when the T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_30, © The Society for Experimental Mechanics, Inc. 2011

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212 yarn starts moving toward the center. Oscillation was caused by the friction between yarns as the yarn pulled out passed each crossover. Peak force in warp direction is higher than that in the weft direction. Warp Weft

Puu-out Force (N)

12

6

0 0

20

40

Yarn pull-out Displacement (mm)

Fig1. Yarn pull-out Set up at Quasi-static rate

Fig2. Yarn pull-out behavior at Quasi-static rate

2) High - rate Experiments To overcome limitations found in a Kolsky tension bar setup for yarn pull out, such as inertia effect of the quartz crystal at the end of the incident bar and short loading time, pendulum impact to the fabric fixture was utilized to generate high rate loading for yarn pull out. In this setup, the yarn initially fixed on the quartz transducer was pulled out as the fabric was driven to move backward at a certain velocity produced by the transferred impact energy from the pendulum motion. The fabric could be moved smoothly after the impact by placed on the linear bearing slide. The pendulum setup is shown in Fig3. Fig 4 shows the force history of pull out of weft yarn of Kevlar® when the fabric was moving at around 1 m/s. Since the history of displacement was not available at this time, only force data could be compared to the results from quasi-static results. It was noticed that the maximum pull out force was higher in high rate experiments. More accurate and complete comparison between quasi-static results and high rate ones will be obtained with accurate measurement of pull out displacement. 9

Force (N)

Force (N)

6

3

0

0.00

0.05

Time (s)

Fig3. Yarn pull-out set up at high - rate

Fig4. Force history during yarn pull-out at high - rate

REFERENCES [1] Cheeseman BA, Bogetti TA, “Ballistic impact into fabric and compliant composite laminates”, Composite Structures, Volume 61, Issue 1-2, Pages 161-173, July 2003 [2] Lim, Shim, Ng, “Finite-element modeling of the ballistic impact of fabric armor”, Int. J. Impact Engng, Volume 28, pp. 13-31, 2003 [3] Zeng, Tan, Shim, “Modelling inter-yarn friction in woven fabric armour”, Int. J. Numer. Engng, 66, pp. 13091330, 2006 [4] Kirkwood, Lee, Egres JR, Wagner, Wetzel, “Yarn Pull-Out as a Mechanism for Dissipating Ballistic Impact Energy in Kevlar® KM-2 Fabric, Part 1: Quasi-Static Characterization of Yarn Pull-Out”, Textile Research Journal, 2003 [5] Dong, Sun, “Tesing and modeling of yarn pull-out in plain woven Kevlar Fabrics”, Composites Part A:Applied science and Manufacturing, Volume 40, Issue 12, pp. 1863-1869, Dec 2009.

Dynamic Response of Homogeneous and Functionally Graded Foams When Subjected to Transient Loading by a Square Punch Chandru Periasamy and Hareesh Tippur* Department of Mechanical Engineering Auburn University, AL 36849 * [email protected]

Abstract In this work, failure response of structural syntactic foams subjected to dynamic square punch impact loading is studied. Experiments are being carried out on homogeneous and compositionally graded syntactic foam sheets. The former are made of hollow microballoons dispersed uniformly in an epoxy matrix at different volume fractions whereas in the latter the volume fraction of microballoons is spatially varied in the impact loading direction. An experimental set-up comprising of a long-bar apparatus is developed in conjunction with a gas-gun for subjecting unconstrained syntactic foam sheets to transient stress wave loading. Digital image correlation method and high-speed photography are used to measure time-resolved, inertia induced, in-plane deformations near the vicinity of the punch. The effects of volume fraction and spatial variation of microstructure on punch tip deformations are currently being studied. Feasibility of characterizing punch tip deformations using an analogous crack tip model is explored. Complementary finite element computations are also carried out using measured particle velocity in the long-bar as input boundary conditions. Introduction Polymers and polymeric composites are widely used as structural materials for their lightweight and specific energy absorption capabilities. Syntactic foams (SF) belong to a class of polymer composites made by dispersing hollow microballoons in a matrix material. For energy absorption applications it is important that mechanical response of SF under stress wave loading conditions be understood. Material failure due to high strain-rate loading is often initiated by shear localization. Such a scenario can be realized experimentally when a plate specimen is impacted (a) (b) on its side by a square punch. This results in shear bands originating at the punch tip. One way of Figure 1: Analogy between (a) compressively loaded semi-infinite double crack and (b) punch loaded semi-infinite plate characterizing dynamic failure in such a scenario is by evaluating dynamic stress intensity factor history at the punch tips [1]. This is done by making use of the analogy between the problems, a square punch loaded semiinfinite plate and two semi-infinite edge cracks being loaded under compression [2, 3], as shown in Fig. 1. In this work, SF plates made of different microballoon volume fractions (10%, 20%, 30% and 40%) in epoxy matrix were dynamically loaded by a square punch and the stress intensity factor histories at the punch corner were evaluated. The transient in-plane deformations of the SF plates were measured optically using a high-speed digital camera and 2D Digital Image Correlation (DIC) method. The dominant y-displacement field was then used to compute the analogous stress intensity factor history near the punch corner.

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214 Experimental procedure A schematic of the experimental setup used is shown in Fig. 2. It consists of a 6 foot long, 1 inch diameter aluminum 7075 bar butt against the plate sample, a 12 inch long, 1 inch diameter striker also made of aluminum 7075 held inside the barrel of a gas gun. The edges of the sample and the end of the bar are machined flat to achieve a uniform areal contact. One face of the plate sample is painted with a random black and white speckle pattern for DIC measurements. The striker is launched towards the long-bar using the gas-gun. When the striker contacts the long-bar, a compressive stress wave propagates through the bar and gets transmitted into the SF plate. The sample deformation is photographed using a rotating mirror-type Cordin550 high-speed digital framing camera at a rate of ~250,000 frames per second. The recorded sets of deformed and undeformed images are then correlated to obtain in-plane displacements using DIC.

Figure 2: Schematic of the punch experiment setup Verification of the setup Before testing the SF samples, the experimental setup and the analysis procedure were verified on a sheet of PMMA, an extensively studied polymer. The dominant (y-direction) displacement field in the PMMA sample obtained from the experiment was compared to a finite element simulation of the punch experiment using the measured stress pulse in the longbar. The comparison is shown in Fig. 3. Evidently, the experimental and computational results agree rather well, thus validating the experimental setup. In addition, as this displacement field is analogous to the crack opening displacement field given by [4], 𝑢� =

��

��

�

�

�

��� �𝑘 𝑠𝑖𝑛 �� � + � 𝑠𝑖𝑛 �

��� �

�

�

� + 𝑠𝑖𝑛 � ��, �

�

it can be used to compute the dynamic stress intensity factor Figure 3: Experimental and finite element y-displacement history at the punch corner. In the above equation, 𝑢� is the contours in a punch loaded PMMA plate. Contour levels crack opening displacement, (𝑟, 𝜃) are the polar coordinates are in micrometers. with the origin at the punch corner, 𝑘 = (3 − 𝑣)⁄(1 + 𝑣) for plane stress where 𝜇 and 𝑣 are shear modulus and Poisson’s ratio respectively and 𝐾� is the punch tip dynamic SIF. The normalized 𝐾� history thus obtained from the experiment is compared to the analytical results reported by an earlier work [3]. As seen in Fig. 4, the experimental and analytical results are in agreement which serves as an additional verification of the setup and the analysis procedure used in this work. Dynamic punch-tip stress intensity factor histories in PMMA and syntactic foam Sheet samples made of PMMA and SF samples with different volume fraction of microballoons in epoxy matrix were dynamically loaded using a square punch and the corresponding in-plane displacements were obtained as described earlier. The punch tip displacement fields were analyzed using crack analogy for evaluating punch tip dynamic stress intensity factor

215 (𝐾� ) histories. Figure 4 shows 𝐾� history corresponding to the case of 154 mm x 124 mm x 5.5 mm PMMA sheet loaded by a short pulse of duration ~57 µs ( 𝐶� 𝑡⁄2𝑎 = 6.6) produced using a short striker bar. The horizontal axis of the plot represents normalized time scale, where 𝐶� denotes the dilatational wave speed in PMMA, 𝑡 is time after the beginning of the stress pulse in the sample and 2𝑎 is width of the punch. In the plot, the stress intensity factor 𝐾� increases monotonically up to 𝑡~56 𝜇𝑠 after which it begins to drop-off, coinciding with the end of the stress pulse input to the sample. Note that in this experiment the pulse ended well before reflected stress waves return to the punch tip from the far-edges of the sample.

Figure 4: 𝐾� history in PMMA (stress pulse duration ~ 57 µs; ( 𝐶� 𝑡⁄2𝑎 ~6.6))

Figure 5: 𝐾� history in SF30 (stress pulse duration ~ 200 µs; (𝐶� 𝑡⁄2𝑎 ~19)

In Fig. 5 results from a similar experiment carried out using a syntactic foam sheet (130 mm x 50mm x 5.7 mm) impacted on the 130 mm side of the sample are shown. In order to load the sample to failure, a sufficiently long duration stress pulse lasting about ~200 µs ( 𝐶� 𝑡⁄2𝑎 ~19) is used. After a rapid increase in 𝐾� up to (𝐶� 𝑡⁄2𝑎 ~1.5) a softening response ensues until a distinct drop at approximately ( 𝐶� 𝑡⁄2𝑎 ~5.5) is seen. The onset of material nonlinearity at the punch tip is likely responsible for the former whereas failure events (crack initiation, shear band formation) occurring at the punch tip cause the latter. Similar experiments are currently being carried out to confirm the repeatability of these measurements, understand the failure events involved, and study other homogeneous and graded SF architectures. Summary A long-bar setup to study the in-plane dynamic response in punch loaded polymeric syntactic foam plates was developed and calibrated. The method of DIC was used to obtain in-plane displacements. The feasibility of obtaining the dynamic stress intensity factor histories at the punch corner from the measured in-plane displacements was demonstrated. Future work will include, understanding the failure events during the loading history, studying the effects of volume fraction of microballoons in the SF sheets as well as compositional gradation of microballoons in the direction of impact. References 1. Chen, L. and R.C. Batra, Material instability criterion near a notch-tip under locally adiabatic deformations of thermoviscoplastic materials. Theoretical and Applied Fracture Mechanics, 30(2): p. 153-158, 1998. 2. Roessig, K.M. and J.J. Mason, Dynamic stress intensity factors in a two dimensional punch test. Engineering Fracture Mechanics, 60(4): p. 421-435, 1998. 3. Rubio-Gonzalez, C. and J.J. Mason, Experimental investigation of dynamic punch tests on isotropic and composite materials. Experimental Mechanics, 41(2): p. 129-139, 2001. 4. Westergaard, H.M., Bearing pressure and cracks. ASME J. Appl. Mech., 6: p. 49-53, 1939.

Dynamic Strain Measurement of Welded Tensile Specimens Using Digital Image Correlation

Kathryn A. Dannemann1, Rory P. Bigger1, Sidney Chocron1, Ken Nahshon2 1 Southwest Research Institute, San Antonio, TX 2 Naval Surface Warfare Center Carderock Division, West Bethesda, MD Abstract The high strain rate behavior of a welded interface was evaluated using a direct tension Kolsky (Hopkinson) bar setup. The weld and the Heat Affected Zone (HAZ) were included in the gage section of the tensile specimen to evaluate the effects of structural constraint (vs. weld metal only). The welds of interest are under-matched Metal Inert Gas (MIG) butt welds between identical aluminum alloys. Limitations on specimen geometry and maintaining the weld bead intact were imposed to provide a specimen that was most representative of the material and application. Strain determination for the welded specimen with non-uniform cross section was challenging using traditional techniques. A Digital Image Correlation (DIC) system was employed for dynamic strain measurement during high strain rate tensile testing. The results obtained with the DIC system are compared with strain gage data and post-test deformation measurements. Numerical simulations were also employed to aid with interpretation. These will be presented in further detail at the conference. Background Welded aluminum construction is utilized in high speed naval vessels for weight reduction. Understanding the behavior of these welded joints, especially at high strain rates, is critical for design of ship structures. The 5000 and 6000 series aluminum alloys used in marine applications (e.g., 5083-H116 and 6082-T6 alloys) show significant strength decline when fusion welded [1,2]. The strength decline for under-matched welds in structures must be considered as plastic deformation will often localize at a weld during structural deformation [1,3]. Although the mechanical behavior of aluminum weld metals have been evaluated, testing has generally been conducted on coupons extracted from the weld [4,5]. The effects of structural constraints on the weld are not usually considered in mechanical characterization studies of weldments. A key objective of the present investigation was to evaluate the high strain rate behavior of welded specimens (vs. weld metal only) and quantify the strain differential vs. virgin specimens. The test specimens contained the weld bead and the Heat Affected Zone (HAZ) on either side of the weld. Material The aluminum alloy of interest is 5083-H116. Under-matched welds were fabricated for this material using a manual Metal Inert Gas (MIG) welding process. A 9.5-mm thick plate was cut into two pieces, machined down to 6.35 mm along two edges, and welded together using 5183 filler wire. Specimens were machined from the welded plate; both virgin and welded specimens were evaluated. Although some high strain rate test data is available for the Al 5083 alloy [6], monolithic material from the same plate was tested to provide a baseline for comparison with the welded specimens. Similar specimen designs were employed for the baseline and welded specimens. A standard tensile specimen could not be employed for the high strain rate tension tests owing to the longer gage length (30-mm minimum) required to accommodate the weld, and HAZ material on either side of the weld, in the specimen gage section. An additional design requirement was the need to retain the weld crown in the specimen to ensure that fracture occurs in a manner typical of a production weldment. Numerical simulations were conducted to aid with specimen design, which was further dictated by the thickness of the aluminum stock. A flat specimen design was used to eliminate the need for machining near the weld. Specimen design details and testing methodology were discussed in our 2010 SEM annual conference paper.

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Experimental Procedure High strain rate tension tests were conducted at SwRI using a Hopkinson bar system with 25-mm diameter maraging steel bars that allows direct tension loading of the specimen. Both welded and virgin specimens were tested. The maximum strain rate achieved for the high strain rate tests was approximately 800 s-1, owing to the -4 -1 long specimen gage length. The low strain rate (~10 s ) tension tests were also conducted using an MTS servo-hydraulic machine. Quasistatic tests were performed on specimens with identical geometries to the high strain rate test specimens to allow a direct comparison of the test results. Determination of stress and strain in the welded specimens was not straightforward owing to the non-uniform cross section for these specimens. Failure of the tensile specimen must occur on the initial stress pulse to obtain an accurate strain measurement. Stress was estimated based on the smallest cross-sectional area; strain was estimated using a nominal gage length. Strain gages were applied to some specimens in the HAZ region on either side of the weld; data were used to “calibrate” the numerical simulations that were conducted to aid with interpreting the experimental results. Post-test measurements on the specimen also provided strain estimates. Specimen deformation was monitored during high strain rate testing using a 3D digital image correlation (DIC) system. Two high speed Phantom cameras were used to acquire the images for the DIC measurements. The DIC system provides a non-contact measurement of strain during dynamic deformation. The strains were determined from the deformation measurements using the ARAMIS software [7]. Results The high strain rate tensile test results show similar strengths for welded vs. base metal specimens. The welded specimens failed at a lower strain than the parent material for both quasistatic and dynamic tests. The magnitude of this difference is dependent on the failure location (i.e., weld vs. HAZ). The strain profiles obtained with the ARAMIS system allowed a direct comparison of the response of the welded and virgin specimens. Representative strain profiles for a welded specimen are shown in Figure 1. A strain concentration occurs near the toe of the weld early in the high strain rate test; failure of the welded specimens occurred at strains approaching 45%. The high ductility of the 5083 alloy was evident, with local strains as high as 70% being measured for base material coupons under high-rate tension. The strain values derived from the DIC measurements are similar to those obtained from post-test elongation measurements of the specimens.

HAZ

Weld

HAZ t = 0.191 ms

t = 1.594 ms

t = 3.096 ms

Figure 1. Strain profiles for a welded specimen during a high strain rate tensile test (strain rate ~ 800 s-1.)

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Conclusions The tensile tests conducted as part of this work are the first known evaluation of the response of welded test specimens (vs. specimens excised from weld metal). It was found that using a DIC system for strain measurements provided strain contours and real time strain-rate measurements that provide insight into the failure process of complex and highly inhomogeneous welded coupons. Acknowledgments The authors acknowledge the Office of Naval Research Aluminum Structural Reliability Program, under the direction of Dr. Paul Hess, for funding a portion of this work. Appreciation is also extended to Mr. Art Nicholls (SwRI) and Mr. Darryl Wagar (SwRI) for their assistance with specimen design and testing, and Mr. Albert Brandemarte (NSWCCD) for his assistance with metallographic evaluations. Mr. Tim Schmidt of Trilion Quality Systems is acknowledged for assistance with specimen preparation for the DIC measurements. References 1. M.D. Collette, “The Impact of Fusion Welds on the Ultimate Strength of Aluminum Structures”, 10th International Symposium on Practical Design of Ships and Other Floating Structures”, Houston, TX, 2007. 2. L. Zheng, D. Petry, H. Rapp, T. Wierzbicki, “Characterization and Fracture of AA6061 Butt Weld”, Thin-Walled Structures, Vol. 47, Issue 4, p. 431-441 (2009). 3. B.C. Simonsen, R. Tornqvist, “Experimental and Numerical Modeling of the Ductile Crack Propagation in Large-Scale Shell Structures”, Marine Structures, Vol. 17, p. 1-27 (2004). 4. Y.J. Chao, Y. Wang and K.W. Miller, “Effect of Friction Stir Welding on Dynamic Properties of AA2024-T3 and AA7075-T7351”, Welding Research Supplement, 196s-200s, (2001). 5. R.W. Fonda, P.S. Pao, H.N. Jones, C.R. Feng, B.J. Connolly, A.J. Davenport, “Microstructure, Mechanical Properties and Corrosion of Friction Stir Welded Al 5456”, Materials Science and Engineering A519, p. 1-8, (2009). 6. A.H. Clausen, T. Borvik, O.S. Hopperstad, A. Benallal, “Flow and Fracture Characteristics of Aluminum Alloy AA5083-H116 as Function of Strain Rate, Temperature and Triaxiality”, Materials Science and Engineering A364, p. 260-272, (2004). 7. T. Schmidt, J. Tyson, K. Galanulis, “Full Field Dynamic Displacement and Strain Measurement Using Advanced 3D Image Correlation Photogrammetry”, Experimental Techniques, Vol. 27, No. 3, p. 41-44 (2003).

Ultra high speed full-field strain measurements on spalling tests on concrete materials.

F. Pierron, P. Forquin* LMPF, Arts et Métiers ParisTech, Rue Saint-Dominique B.P.508, 51006 Châlons en Champagne Cedex, France * LEM3, Paul Verlaine University, Ile du Saulcy, 57045 Metz Cedex 01, France [email protected], [email protected]

ABSTRACT. This paper presents the use of an ultra high speed camera in conjunction with full-field measurements (grid method) to measure strain fields in a concrete specimen submitted to a spalling test (inertial tensile test). A Shimadzu camera is used to capture the images with a time resolution of 2 microseconds. Strain and acceleration fields are then obtained by spatial and temporal differentiation. It is then shown how the data can be used in the Virtual Fields Method to identify Young’s modulus using the acceleration forces as a distributed volume load cell. The identified modulus values are consistent with expected values and the onset of damage can be detected as a sudden decrease in Young’s modulus. This is a first step towards better description of damage at high strain rate in such brittle materials using full-field measurement approaches.

1. Introduction Dynamic testing of materials often relies on uni-axial loads in the form of tensile or compressive waves, with the load measured through the use of slender metallic bars (input and output) equipped with strain gauges. This is the so-called Kolsky or Split Hopkinson Pressure Bar (SHPB) set-up [1]. For brittle materials, specific tests have been developed to generate brittle failure such as the 1-point bending test, for instance [2]. One of the difficulties is to avoid introducing the tensile load directly because of stress concentrations in the gripping area that may produce unwanted premature failure. One way around this is to introduce a compressive wave through an impact on one side of the specimen and use the other side as a free surface to reflect the compressive wave which then becomes a tensile wave. This is often called a spalling or spallation test. A complete procedure has recently been devised to determine the tensile strength on concrete materials using a cylindrical test specimen impacted on one of its edges [3]. The present paper is based on this set-up. The objective of this article is to explore the feasibility to perform quantitative full-field strain measurements using an ultra high speed imaging camera. Different types of cameras can capture high spatial resolution images at temporal resolutions of 1 μs or below but the technology is still not fully adapted to quantitative use of such images for full-field strain measurements [4-7]. Here, a Shimadzu HPV1 camera has been used and performances are briefly assessed. Then, images of a grid bonded onto the specimen have been taken during a spalling test. Spatial phase shifting has been used to derive displacement from which strain and acceleration fields have been calculated. The main novelty of this paper is to show that inertia forces can be used to identify Young’s modulus with the Virtual Fields Method without any external force measurement, in the same spirit as in [5,6].

2. Experimental procedure and camera performance 2.1 Experimental procedure The specimen tested here is a dry concrete (density 2.27) cylinder with dimensions reported on Fig. 1. It has been flattened out using a grinder so that a cross-grid could be bonded using the procedure reported in [8]. The grid images were processed with spatial phase shifting to extract displacements, as detailed in [9]. The frame rate was such that between two consecutive images, the displacement remained less than one grid period, thus avoiding the necessity to use phase unwrapping (temporal unwrapping). The specimen was impacted using a striker projected by an air gun. The striker was an aluminium cylinder 70 mm long and 45 mm in diameter (1.2 kg) and its speed at impact was 5.9 m.s-1. Experimental details can be found in [3].

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Figure 1: Specimen dimensions (mm) and field of view.

Figure 2: Experimental set-up.

2.2 Camera performance A Shimadzu HPV1 camera was used to capture images of the grid bonded onto the specimen. An interframe time of 2 μs was selected and 102 images were recorded. A global view of the experimental set-up is given on Fig. 2. The technology of the camera is based on a dedicated CCD chip with on-board memory. A schematic view of the chip is given on Fig. 3. One of the problems with this camera is its low fill-factor in the horizontal direction, as can be seen on Fig. 3 where the photodiodes (in pink) are well separated in the horizontal direction, meaning that the light is sampled over only a fraction of the field of view. This results in distorted images of the grids, as shown on Fig. 4 where the pattern varies in the x-direction. The actual fill factor is 14% only in the x-direction and 76% in the y-direction. This gives rise to moiré effects when the specimen is loaded and the resulting strain resolution in the x-direction is way above the small levels of strains undergone by the specimen (of the order of 0.05%) before fracture. This is the reason why the camera was rotated to align the vertical direction of the camera with the y-axis in Fig. 1. Unfortunately, this results in a loss of pixels since the CCD has 312 pixels in x and only 260 in y. One should note however that this spatial resolution is significantly higher than for a standard CMOS high speed camera such as the Photron SA5 that only provides 128 by 48 pixels at the same frame rate. The final data set consists in 24 by 36 independent displacement measurement points.

Figure 3: Schematic of the ISIS CCD chip [10].

Figure 4: Image of a regular grid deformed by low fill factor in the horizontal direction

223

Spatial grey level standard deviation

The grid has a 1 mm pitch and each period is sampled by 6 pixels, resulting in a field of view of 37 by 24 mm, near the free surface, as shown in Fig. 1. Another problem with this camera is that although it is technically an 8-bit one, one must avoid grey levels above 100. The reason is shown in Fig. 5 where the standard deviation of uniform scene images is plotted as a function of image number for two illuminations. For the light images, a periodical evolution of the standard deviation appears every twelve images, with dramatic issues towards the last images whereas with the same scene captured at low lighting, this problem is much reduced. Therefore, the images used in this study have a mean intensity below 100 grey levels. The first 10 images of the test when the stress wave has not reached yet have been used to evaluate the noise. A typical displacement noise map is given in Fig. 6. The standard deviation of these displacement maps is about 1 μm, which is one thousandth of the pitch (or 0.0017 pixel), comparable to standard CCDs. Since the strain levels are very low, some smoothing is necessary to improve the strain resolution. Here, diffuse approximation was used [11] with a radius of 8 data points. This results in a noise strain standard deviation of 1.3 10-4. Finally, acceleration maps were also derived from raw displacement maps with a temporal second order polynomial smoothing over a centred sliding window of 7 images, leading to a standard deviation of 4.104 m.s-2.

6 5,5 5 4,5 4 3,5 3 2,5 2 1

12

23

34

45

56

67

78

89 100

Image number Figure 5: Standard deviation of grey levels, uniform scene. Blue for dark images and red for bright images.

Figure 6: Vertical displacement in mm, noise.

3. Measurement results The results in terms of displacement, strain and acceleration are presented in Figs. 7 to 9. Looking at the displacements (Fig. 7), one can see that the stress wave reaches the field of view area at about 40 μs. Then, the average displacement curve exhibits a sharp acceleration stage and then a more linear evolution with a slight deceleration after 80 μs. Looking at the spatial distributions, one can clearly see the compressive wave arriving from the bottom of the specimen (very clear at 60 μs) and then reverting at the free edge (80 μs). One can also see a macro-crack forming at about 90 μs, apparent on the last map in Fig. 7 to the bottom left of the specimen. This displacement history is confirmed by the strain one in Fig. 8. After a compressive peak reaching -3.10-4 (with local values of the order of -7.10-4 consistent with strain gauge values recorded on the same specimen), the strain becomes tensile and rapidly leads to specimen damage. At the onset of the macro-crack, the local strain reaches about 7.10-4. Finally, the acceleration maps (Fig. 9) show an acceleration stage caused by the compressive wave up to about 2.105 m.s-2 (locally up to 3.105 m.s-2) and then a sharp drop followed by a deceleration stage just before the onset of macro-cracking. The present results are of very good quality considering the rather low spatial resolution of the camera and the different issues relating to its dedicated CCD chip. It should be emphasized that the use of the grid method providing an independent measurement point each 6 pixels (area of 36 pixels2) brings here a significant advantage over digital image correlation for which an area of typically 20 by 20 pixels would have had to be used (400 pixels2) to reach a comparable resolution, hence about 10 times more independent data points with the grid method. The objective of the rest of paper is to show that the measurements can be used quantitatively to identify Young’s modulus using inertial forces derived from the acceleration maps, as in [5,6] but for different materials and test configurations.

224

Figure 7: Displacement history (in mm).

Figure 8: Strain history.

4. Identification procedure with the Virtual Fields Method The Virtual Fields Method (VFM) is a well known procedure to identify materials mechanical constitutive parameters using full field measurement [12] and has been applied to many different problems over the years. The VFM is based on the principle of virtual work which, in the case of small displacement, can be written as follows:

∫ σ : ε dV + ∫ a.u dV = ∫ f .u dS + ∫ b.u dV *

V

*

V

*

∂V

*

V

(1)

225

where V is a volume of the solid, ∂V the surface of its boundary, σ the stress tensor, a the acceleration vector, f the surface forces acting on ∂V, b the body forces acting on V and dV, dS, dm, are the elementary volume, surface and mass respectively. u* is a differentiable vector field defined over the whole solid volume and ε* is the tensor obtained from u* as ε ij* = 1/ 2(∂ui* / ∂x j + ∂u *j / ∂xi ) .

Figure 9: Acceleration history (in m.s-2). Often u* is considered as a virtual displacement so the integrals of Eq. 1 can be interpreted as the virtual work of the internal forces, the virtual work of the inertial forces, the virtual work of the boundary forces and the virtual work of the volume forces respectively. However, mathematically, the principle of virtual work is equivalent to the dynamic equilibrium equation and u* is an arbitrary function that can be defined by the user. In linear elasticity, the selection of at least as many virtual fields as unknowns to be identified leads to a linear system that can be inverted if the virtual fields are well chosen (see [12]). In static, and in absence body forces, Eq. 1 becomes:

∫ σ : ε dV = ∫ f .u dS *

*

(2)

∂V

V

It is therefore necessary to include a force information in order to identify the whole set of stiffnesses. Failing to do so will only provide relative stiffness information (like Poisson’s ratio, for instance). However, in dynamics (and in absence of body forces), the equation becomes:

∫ σ : ε dV + ∫ a.u dV = ∫ f .u dV *

V

*

*

By selecting suitable virtual fields, it is possible to zero the term

(3)

∂V

V

∫ f .u dS , which means that the external applied forces are *

∂V

∫

*

forced out of the equation. However and contrary to the static case, the term a.u dV can be used to provide a force inforV

mation (here, inertial forces). Indeed, this term only depends on the acceleration field that can easily be obtained by temporal differentiation of the displacement fields. The first implementation of this idea is due to Grédiac and Paris [13] for elastic stiffnesses and was recently extended to viscoelasticity in [14]. However, in both cases, the loading was harmonic and accel-

226

eration was proportional to displacement. More recently, the first applications of the VFM in high strain rate testing were published on composites [6] and metal [5]. In the composites paper, it was shown that Young’s modulus and Poisson’s ratio of a quasi-isotropic laminate could be identified using the present approach (ie, without any force measurement), using a SHPB set-up. In the second, a complex three point bending impact test was successfully analyzed with the principle of virtual work. However, for both studies, the test configuration implied several contact points where load was introduced, leading to restrictions on the selection of the virtual fields, and preventing quantitative identification in [5]. Here, the test is inertial and therefore a good candidate to apply the same approach. The challenge however lies in the very low strain levels. In order to identify the material’s Young’s modulus, the following virtual field will be used over the field of view sketched in Fig. 1:

⎧ ε*xx = 0 ⎪ * ⎨ε yy = f ' ( y) ⎪ ε* = 0 ⎩ xy

⎧⎪ u *x = 0 ⎨ * ⎪⎩u y = f ( y) with

(4)

f (0) = 0 so that ∫ f .u *dS = 0 and f continuous and differentiable. Because of the nature of the test and the material, ∂V

it will be supposed here that the test leads to a uniaxial (if heterogeneous) state of stress. Therefore, Eq. 1 becomes:

− E ∫ ε yyf ' ( y)dV = −ρ ∫ a y f ( y)dV V

(5)

V

where E is supposed to be constant over the specimen (at the macroscopic scale). The second hypothesis is that the strain and acceleration fields only depend on y in the specimen. Therefore, Eq. 5 can be integrated directly over x and z. If S is the cross section of the specimen, then Eq. 5 becomes: L

L

− ES∫ ε yy f ' ( y)dy = −ρS∫ a y f ( y)dy 0

(6)

0

where L is the length of the field of view. Finally, after simplifying by S, each of the integrals in Eq. 6 can be approximated by a discrete sum using the actual measurements:

L − E ∫ ε yy f ' ( y)dy ≈ − E nx V

nxny

L

L ε f ' ( y ) and − ρ∫ a y f ( y)dy ≈ −ρ ∑ nx i =1 0 i yy

i

nxny

∑ a f (y ) i =1

i y

i

(7)

where nx and ny are the number of measurement points in the x and y directions respectively. One can then multiply and divide each term by ny, which leads to:

− En y

L nxny

nxn y

∑ εiyyf ' ( yi ) = −ρn y i =1

L nxny

nxny

∑ a f (y ) i =1

i y

i

(8)

Simplifying by L and ny and introducing the spatial average function, the final equation giving E is:

E=ρ where a y f ( y ) =

1 nxny

nxny

∑ a iyf ( yi ) and ε yyf ' ( y ) = i =1

a yf (y ) ε yyf ' ( y )

1 nxn y

nxny

∑ε i =1

i yy

(9)

f ' ( y i ) are the spatial average over the field of view of

the functions under the bar. This is a very simple expression giving E directly from measurements reported in the previous section. A value is obtained for each image during the test. The last problem consists in selecting the virtual fields. Looking at Eq. 9, it is clear that a good virtual field will be such that a y f ( y ) and ε yy f ' ( y ) are as large as possible. One way to en-

227

sure this in an easy manner is to define f(y) using the actual measurements. To do so, at each time t when an image is recorded, the y displacement map is averaged over x, leading to a vector depending only on y, and is then expanded in the x direction to provide a virtual field that only depends on y, as required by Eq. 4. Finally, a constant value is added so that the virtual displacement is zero at y=0. Fig. 10 shows the actual and virtual y displacement fields at 60 μs.

Figure 10: Actual and virtual y-displacement field at t=60 μs.

5. Identification results Fig. 11 shows the two terms of Eq. 9 as a function of time. The term containing the acceleration is only significant between 50 and 67 μs and again between 75 and 87 μs. The second term (containing strains) is significant between 50 and 67 μs too and after 75 μs. Therefore, the identification procedure will only be valid between 50 and 67 μs and between 75 and 87 μs. This is reported in Fig. 12 where the identified Young’s modulus is plotted against time and invalid identifications are shaded in blue. One can see that in the first time interval, the modulus is constant at about 28 GPa, with the reference from wave speed and density giving 29 GPa, a very nice match. In the second time interval, the modulus is significantly lower and decreases with time, indicating that tensile damage has probably already occurred in the form of micro-cracks, eventually leading to a macro-crack at about 90 μs. These results are very encouraging. The next step is to refine the measurements in order to identify a damage law and derive local failure stresses.

Figure 11: Evolution of the two terms of Eq. (9). Blue and green shaded areas representing invalid time intervals for identification

Figure 12: Identified Young’s modulus.

5. Conclusions The main conclusions of this study are as follows. • The Shimadzu HPV1 high speed camera is based on a dedicated CCD chip that suffers from two main problems. First, the very low fill factor in the horizontal direction generates some measurement issues. This is critical for the grid method but would also be a problem for image correlation, particularly for patterns having high spatial fre-

228

• • • •

quency contents. Then, saturation effects appear for high grey level values which means that dark images have to be used for full-field measurements. Provided that precautions are taken to avoid the two previous issues, good quality full-field measurements can be obtained. Here, a strain resolution of 1.5 10-4 was obtained with a spatial resolution of about 8 mm. The virtual fields method can be employed to identify stiffnesses by using the inertial forces derived from the acceleration maps without any need for force measurement. This is a significant advantage since all the data needed are born by the recorded images, regardless of how the load is introduced. Young’s modulus of the dry concrete material tested in a spalling configuration was successfully identified during the compressive stage of the test and decreasing values were obtained during the tensile stage just before the macrocrack probably indicating diffuse damage in the form of micro-cracks. These results need to be confirmed by further tests and microscopic investigation of the failed specimens.

References [1] J.E. Field, S.M. Walley, W.G. Proud, H.T. Goldrein, C.R. Siviour, Review of experimental techniques for high rate deformation and shock studies, International Journal of Impact Engineering, 30, pp.725–775, 2004. [2] S. Belenky, D. Rittel, A simple methodology to measure the dynamic flexural strength of brittle materials, Experimental Mechanics, available online, 2011. DOI 10.1007/s11340-010-9453-0. [3] B. Erzar, P. Forquin, An experimental method to determine the tensile strength of concrete at high rates of Strain, Experimental Mechanics, 50, pp. 941–955, 2010. [4] V. Tiwari, M.A. Sutton, S.R. McNeilL, Assessment of high speed imaging systems for 2D and 3D deformation measurements: methodology development and validation, Experimental Mechanics, 47, pp. 561–579, 2007. [5] F. Pierron, M.A. Sutton, V. Tiwari, Ultra high speed DIC and virtual fields method analysis of a three point bending impact test on an aluminium bar, Experimental Mechanics, available online, 2011. DOI 10.1007/s11340-010-9402-y. [6] R. Moulart, F. Pierron, S.R. Hallett, M.R. Wisnom, Full-field strain measurement and identification of composites moduli at high strain rate with the virtual fields method, Experimental Mechanics, available online, 2011. DOI 10.1007/s11340-010-9433-4. [7] M.S. Kirugulige, H.V. Tippur HV, Measurement of fracture parameters for a mixed-mode crack driven by stress waves using image correlation technique and high-speed digital photography, Strain, 45, pp. 108–122, 2009.

[8] J.-L. Piro, M. Grédiac, Producing and transferring low spatial-frequency grids for measuring displacement fields with the moiré and grid methods, Experimental Techniques, 28, pp. 23–26, 2004. [9] Y. Surrel, Customised phase shift algorithms. In: P.K. Rastogi and D. Inaudi, Editors, Trends in optical non-destructive testing and inspection, Elsevier, Oxford, 2000. [10] T. G. Etoh, C. Vo Le, Y. Hashishin, N. Otsuka, K. Takehara, H. Ohtake, T. Hayashida, H. Maruyama, Evolution of ultra-high-speed CCD imagers, Plasma and Fusion Research, 2:S1021, 2007. [11] S. Avril, P. Feissel, F. Pierron, P. Villon, Comparison of two approaches for controlling the uncertainty in data differentiation: application to full-field measurements in solid mechanics, Measurement Science and Technology, 21, 015703 (11 pp), 2010. [12] M. Grédiac, F. Pierron, S. Avril, E. Toussaint, The virtual fields method for extracting constitutive parameters from full-field measurements : a review, Strain, 42, pp. 233-253, 2006. [13] M. Grédiac, P.-A. Paris, Direct identification of elastic constants of anisotropic plates by modal analysis: Theoretical and numerical aspects, Journal of Sound and Vibration, 195 (3), pp. 401-415, 1996. [14] A. Giraudeau, F. Pierron, B. Guo, An alternative to modal analysis for material stiffness and damping identification from vibrating plates, Journal of Sound and Vibration, 329, pp. 1653-1672, 2010.

Contact Mechanics of Impacting Slender Rods: Measurement and Analysis Anthony Sanders 1 , M.S., Ira Tibbitts 1 , B.S., Deepika Kakarla 1 , B.E., Stephanie Siskey 2 , B.S., Jorge Ochoa 3 , Ph.D., Kevin Ong 2 , Ph.D., and Rebecca Brannon 1 , Ph.D. 1 University of Utah, Dept. of Mechanical Engineering, 2134 MEB, Salt Lake City, UT 84112 2 Exponent, Inc., 3401 Market St., Suite 300, Philadelphia, PA, 19104 3 Exponent, Inc., 15375 SE 30th Place, Suite 250, Bellevue, WA 98007

ABSTRACT To validate models of contact mechanics in low speed structural impact, slender rods with curved tips were impacted in a drop tower, and measurements of the contact and vibration were compared to analytical and finite element (FE) models. The contact area was recorded using a thin-film transfer technique, and the contact duration was measured using electrical continuity. Strain gages recorded the vibratory strain in one rod, and a laser Doppler vibrometer measured velocity. The experiment was modeled analytically using a quasi-static Hertzian contact law and a system of delay differential equations. The FE model used axisymmetric elements, a penalty contact algorithm, and explicit time integration. A small submodel taken from the initial global model economically refined the analysis in the small contact region. Measured contact areas were within 6% of both models’ predictions, peak speeds within 2%, cyclic strains within 12 microstrain (RMS value), and contact durations within 2 µs. The accuracy of the predictions for this simple test, as well as the versatility of the diagnostic tools, validates the theoretical and computational models, corroborates instrument calibration, and establishes confidence that the same methods may be used in an experimental and computational study of the impact mechanics of artificial hip joints.

1. Introduction The problem of analyzing the impact of slender rods has previously been addressed in several classical works [1-3]. Recent approaches have included substructure analysis [4] and modal analysis combined with a Hertzian contact law [5,6]. FEA has been applied to problems of a single impacted rod [7,8] and two impacting rods [9], with results that have shown close fidelity to analytical models. Considering experimental approaches, contact duration between impacting metallic spheres and rods has been measured using electrical continuity [10,11]. Strain gages have been used to measure the strain waves in impacted rods [1,12-15]. Recently, laser vibrometry has been employed to measure the transient velocity on the surface of a rod impacted by a sphere [6,8,12,13]. Notably, the problem of a sphere striking the end of a long rod was formulated using a system of delay differential equations [12,13]. Viewing the effectiveness of this approach, one aim of the present work is to extend this recently demonstrated approach to the case of two impacting rods, and to add experimental validation of the predicted contact mechanics. To date, neither the contact area nor contact stress generated by such impacts has been adequately analyzed and experimentally validated. It is difficult to experimentally record a small, transient elastic contact area; even so, practical methods have been described [16,17], including a recent one of our own design that uses inexpensive materials: grease and photocopier powder [18]. Accordingly, the second aim of the present work is to measure the contact area generated between impacting spherically tipped rods. The results will validate the analytical use of a Hertzian contact law to describe the forcedisplacement relation at the impact site. Hertzian theory can provide comprehensive contact mechanics predictions [19,20], but its accuracy depends upon assumptions that may be ill-suited to some impact problems. FEA also provides a means of examining impact-induced contact stress; however, sufficient mesh refinement in the case of small contacts may require small elements that entail high computational cost. In the submodelling technique in FEA, the results of a coarsely meshed global model are applied as boundary conditions to a submodel of a small area of interest that requires a refined mesh. Submodelling has been applied in FEA where a contact area of interest was a small portion of a larger model [21,22]. Hence, two approaches, analytical and FEA, are suited to simulating the two-rod impact problem on both the macro and micro-scales. Likewise, experimental techniques may validate the simulations’ results on both scales. Therefore, this work aims to demonstrate both simulation approaches and to compare their predictions with experimental outcomes, at both scales. T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_34, © The Society for Experimental Mechanics, Inc. 2011

229

230

2. Methods and materials 2.1. Analytical model The analysis begins with the schematic in Fig. 1. Rod 1 with speed s1 travels axially and impacts stationary Rod 2. The rods are parallel, and the impact is centric. Dimensions di give the diameters, ri the tip radii, and Li the lengths. The rods’ coordinates, x1 and x2, are measured inward from the radiused tip, and the rods are modeled as homogeneous, linear elastic, and isotropic; accordingly, the material properties are the densities ρi, elastic modulii Ei, and Poisson’s ratios νi.

Rod 1

d1

s1 ρ1, E1, ν1 L1

Rod 2

r1 r2

s2 = 0

d2

ρ2, E2, ν2

x1

L2

x2

Fig. 1 Schematic for the axial, centric impact of two slender rods The governing equation of each rod is approximately the 1D wave equation [1]. In Rod 2, the general solution is [23]: u2 ( x2 , t ) = f 2 ( t − x2 c2 ) + g 2 ( t + x2 c2 )

(1)

Here, u2 is the displacement and c2=√E2/ρ2 is the longitudinal wave speed. This solution represents two waves: righttraveling unknown function f2( ) and left-traveling unknown function g2( ) [1]. A similar equation applies to Rod 1, although a reversal of the left/right description applies since x1 is positive leftward. The stress-free boundary condition at x2=L2 implies the strain-free condition ∂u2/∂x2=0, which yields g′2(t+L2/c2)= f ′2(t−L2/c2), where (′) denotes differentiation with respect to the entire argument. This expression holds at an offset instant to=t–L2/c2, which yields: g 2′ (t ) = f 2′ ( t − 2 L2 c2 )

(2)

At the impacted end (x2=0), the strain is related to the stress via Hooke’s law: ∂u2/∂x2=F(t)/E2A2, where F(t) is the contact force during impact and A2 is the cross-sectional area. Applied to Eq. (1), this yields:

f 2′ (t ) = g 2′ ( t ) − c2 F (t ) E2 A2

(3)

This assumes that the contact force is uniformly distributed across the impacted end, which is inaccurate in the vicinity of the contact, but is nevertheless appropriate to model the wave motion far from the point of contact based on St. Venant’s principle. In the contact region, the contact force is related to the displacement by a Hertzian contact law [13]: F(t)= −K[δ(t)]3/2, where K is the Hertzian contact stiffness (addressed below), and δ(t) is the compression due to impact; the negative sign yields a compressive (negative) force and stress. The compression is the difference in rod displacements at the impacted ends: δ(t)= −u1(0,t) −u2(0,t), which treats the impacted ends’ displacements as uniform over the cross-section of each rod. In Rod 1, the strain rate at the free end (x1=L1) is zero: ∂u1/(∂x1∂t)=[−f″1(t−x1/c1)+ g″1(t+x1/c1)]/c1=0. This holds at an offset instant to=t–L1/c1, which yields: g1′′ (t ) = f1′′ ( t − 2 L1 c1 )

(4)

Evaluating the strain rate at the impacted end (x1=0) yields: ∂u1/(∂x1∂t)=[ −f″1(t)+ g″1(t)]/c1= F′(t)/E1A1. Thus: f ′′ ( t ) = g ′′ ( t ) + 3 ( c K E A ) δ(t ) δ (t )

(5)

The rate of compression in Eqn. (5) is derived from the compression relation, and it is given by: δ (t ) = − f ′(t ) − g ′(t ) − f ′(t ) − g ′ (t )

(6)

1

1

1

2

1

1

1 1

2

2

The governing system of differential equations of the system is thus given by Eqns. (2)-(6). The initial conditions are: f1 (0) = 0,

g1 (0) = 0,

f 2 (0) = 0,

g 2 (0) = 0,

f1′ (0) = 0,

g 1′ (0) = − s1

(7)

The Hertzian stiffness is [24]: K = 43 E * R *

with

E * = ⎡⎣(1 − v12 ) E1 + (1 − v22 ) E 2 ⎤⎦

−1

and

R * = r1 r2 ( r1 + r2 )

(8)

Various kinematic quantities may be determined using suitable derivatives of the wave equation for each rod. For instance, the speed and the strain at the midpoint of Rod 2 are:

u2 ( L2 2, t ) = f 2′ ( t − L2 2c2 ) + g 2′ ( t + L2 2c2 )

ε 2 ( L2 2, t ) = − c12 f 2′ ( t − L2 2c2 ) +

1 c2

g 2′ ( t + L2 2c2 )

(9)

231

The contact radius, a, and the peak contact pressure within that area, P, relate to the contact force as follows [24]:

a = 3 3FR* 4 E *

P = 3F 2πa 2

(10)

Solutions to the governing equations were computed using numerical integration using Simulink (Mathworks, Natick, MA). Integration was performed using the MATLAB function ode45 [25]. The time delay was implemented using the Transport Delay function block. The model also computed kinematic quantities, e.g. Eq. (9).

2.2. Finite element model The 3D global model comprised the geometries of both rods. The meshes were generated using HyperMesh (Altair, Troy, MI) and consisted of hexahedral elements with a longitudinal edge length of 1.0 mm and an average cross-section edge length of 0.25 mm (Fig. 2). The material model was linear elastic to represent the steel from which the rods were made (Sec. 2.3 below). Contact constraints were implemented using a penalty algorithm. The submodel comprised the first 14 mm (measured from the impact tips) of both rods. This cutoff length was where the subsurface stresses diminished to near-zero magnitudes at the time of peak contact force in the global results. Three mesh refinements were used to examine convergence. The element aspect ratios were approximately 1:1:1, and the average edge length was successively halved in the refinement steps: from 0.25 (Coarse), to 0.125 (Mid), to 0.0625 mm (Fine) (first 2 models, Fig. 2b and 2c). The Fine submodel was additionally simplified as a half-symmetry model. The physical configuration of the rods is detailed in Table 1. The impact speed, 2.197 m/s, was a value recorded during one of the experimental trials. The finite element solver Abaqus/Explicit (Abaqus v. 6.8, Simulia, Providence, RI) was used to perform the global and submodel analyses. The 8-node linear hexahedral element type with uniform strain and hourglass control (C3D8R; reduced integration element) was implemented for both rods. a) Global model

c) Mid submodel

b) Coarse submodel

∅12.7 mm

Total: 193 k elements 203 k nodes

Total: 1.4 M elements 2.4 M nodes

Total: 1.5 M elements 1.5 M nodes

Fig. 2 Cross sections of the FE meshes in each rod: a) Global 3D model, b) Coarse submodel, c) Mid submodel. Submodels were also cylindrical, but shown as halved to display element density Table 1 Physical configuration details of the finite element model Rod 1 length 250.09 mm

Rod 1 r1 35 mm

Rod 2 length 700.99 mm

Rod 2 tip flat

Diameter 12.70 mm

Density 7.803 g/cc

Young’s modulus 204.3 GPa

Poisson’s ratio 0.30

Impact speed 2.197 m/s

2.3. Experimental techniques Both rods were made from a single lot of precision ground A2 steel drill rod. The spherical tip of Rod 1 was finished using a concave 35 mm radius cast iron lap charged with diamond particles. The flat tip of Rod 2 was lapped against a granite surface plate using fine grit silicon carbide sandpaper. (Due to the flat tip, r2 →∞ in Eq. (8), so R*=r1.) There were three specimens of Rod 1 to allow repeat trials, and one of Rod 2. The rods were hardened and tempered to Rc 60. The density was determined from the volume and the mass of one specimen; length was measured using a height gauge, diameter using a micrometer, and mass using an analytic balance. The isentropic elastic properties were measured using the impulse excitation method, ASTM

232

E 1876, using a Grindosonic MK5 instrument (Lemmens, Lueven, BLG). Further, the experiments were designed to maintain contact stresses within the material’s linear elastic region; the criterion P < 1.1Sy, where Sy is the uniaxial yield stress [26] was upheld by the experimental design. The impact experiments were performed in a drop-tower test machine (Dynatup 8250, Instron, Massachusetts, USA) (schematic Fig. 3). The machine provides a motorized latch block that suspends a sled. Upon computer command, the sled may be released from the latch block into free fall guided by twin columns. To the sled was mounted a tubular fixture that suspends Rod 1; the rod was spaced off the tube’s interior surface by 2 oiled o-rings. The rod’s weight was suspended by a thin ring of tape whose diameter was slightly greater than the tube’s ID; otherwise, Rod 1 was distally unconstrained. Rod 2 was suspended in a tubular fixture attached to the test machine base; this fixture also spaced its rod from the interior surface via oiled o-rings. Both fixtures provided approximately 6 cm of clearance behind the rods’ distal ends, spaces into which the rods could slide freely after impact. Rod 2 was partly supported on its distal end by a plastic plug lightly press-fit into the tube; the plug could fall freely into the fixture’s clearance space when Rod 2 was impacted. The position of the lower fixture was adjustable to permit manual alignment of the rods to achieve parallel, centric impact. The velocity of the sled was measured using an infrared sensor fixed to the drop tower that sensed passage of a flag mounted to the sled. The sensor was positioned to detect velocity at the impact position, and it was assumed that Rod 1’s velocity was equal to the sled’s velocity. In repeat trials, the impact speed varied slightly (±0.02 m/s) because the drop height was not precisely repeatable.

latch block velocity flag

sled Rod 1 fixture

velocity sensor 24 g wire

Rod 1

trigger

10 kΩ columns

drop height

+ −

Rod 2

laser vibrometer

strain gages

clearance

3V

Rod 2 fixture

Fig. 3 Schematic of drop tower impact test machine, with both rods, their fixtures, and trigger circuit; data acquisition and strain gage circuit not illustrated

Both rods were wired into an electrical circuit by taping to each a 24 gauge, multi-filament wire. The circuit charged Rod 1 to 3 V relative to Rod 2 using an electrical power supply. Continuity between the rods during impact created a voltage across a 10 kΩ resistor, and the voltage was used to trigger data acquisition and to measure the duration of impact. The velocity of Rod 2 was measured at its midpoint using a 3D laser Doppler vibrometer (CLV-3D, Polytec, Germany). The vibrometer provided 3 separate, orthogonal velocity components, but only the component parallel to the rod’s surface was recorded. Strain was measured at Rod 2’s midpoint using two 1000Ω foil strain gages (WC-06-125AC-W/C, MicroMeasurements, Raleigh, NC, USA) oriented to measure axial strain and wired into opposing arms of a Wheatstone bridge. This circuit design doubled the bridge sensitivity compared to a circuit with only one active gage. The bridge was powered and its output signal was amplified using a high bandwidth signal conditioner (2310B, MicroMeasurements). Use of 1000Ω gages permitted maximal excitation of the bridge (15 V); so, the amplifier gain was set relatively low (~110), which

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enhanced the amplifier’s frequency response quality (-3 dB bandwidth of 230 kHz). The bridge and amplifier were calibrated using a shunt calibration procedure [27]. The three measurement signals were recorded at 443 kHz using a 16 bit analog-todigital (A/D) converter (USB1604HS, Measurement Computing, Norton, MA, USA) controlled by a laptop computer. To enhance A/D accuracy, the input range of each A/D channel was programmed to limits just greater than the maximal signal value; thus, the ranges for velocity and strain were ±10 V, and ±0.5 V, respectively. A record of the contact between the rods was made using a “fingerprinting” technique. The tip of Rod 2 was given a thin coat of bearing grease. The tip was wiped repeatedly (16 times), each time using a clean piece of paper towel, to leave a scant grease film. The tip of Rod 1 was cleaned with warm, soapy water and thoroughly rinsed and dried. During contact, a thin spot of grease transferred from Rod 2 onto Rod 1. After the test, the entire tip of Rod 1 was sprinkled with black photocopier toner powder. The powder was blown off with an aerosol duster can. A monoparticle layer of powder (the “fingerprint”) remained adhered to the thin transfer layer of grease. This patch was then microscopically measured and photographed using an optical coordinate measuring machine (‘CMM’, Nexiv VMR 3020, Nikon, JPN). The CMM detected edge points by analyzing contrast levels in the digital image of the contact patch; 64 points were found at uniform spacing around the patch’s perimeter, and these were used to compute the radius and circularity of a best-fit circle.

3. Results The recorded velocity and strain at the midpoint of Rod 2 are graphed with the analytical and FEA predictions in Fig. 4. Also, the contact force from both models is superimposed with the contact trigger voltage. Non-scaled transducer voltage data from the experimental velocity and strain records (Fig. 4d) demonstrate the sufficiency of the A/D sampling rate. Fig. 5 gives photos of the recorded contact patch on the tip of Rod 1, along with a contour plot of the contact pressure from the global FE model. The images show the circular contact patch that has been revealed by black toner powder adhering to the thin layer of grease transferred from the tip of Rod 2. a)

c)

Analytical

FEA

Exp’t

b)

Analytical

FEA

Exp’t

d)

Strain

Speed

Fig. 4 Global model results: a) Speed and b) Strain at midpoint of Rod 2, vertical line at trigger instant; c) Contact force and trigger signal; d) Samples of non-scaled voltage data points for speed and strain

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b)

a)

c)

Fig. 5 a) and b): Images of a typical contact patch, ~∅2.2 mm, recorded by the “fingerprinting” technique: a) from handheld camera, b) from optical CMM, original magnification 37×. c) Plot of contact pressure from global FEA, same scale as b)

Table 2 compares experimental to analytical model results in 2 trials of each Rod 1 specimen. The circularity (as defined by ASME Y14.5M) of the measured contact patch was 7-8% of the radius in Specimens 1 and 3, but 11-12% in Specimen 2, perhaps indicating that Specimen 2 had more form error in its spherical tip. The predicted contact radius was at most 5% less than measured. The peak speed at the midpoint of Rod 2, consisting of values ≥1.0 mm/s, was averaged over the first five plateaus in the record; the maximum difference was 2.2%. The root-mean-square (RMS) of the difference in strain was computed over the first five periods; the maximum RMS difference was 11.6 microstrain. The FEA model used the impact speed of Rod 1, Spec. 1, Trial 3. Table 3 shows the contact force and contact area for the global model and the three meshes in the submodel. The contact radius (from r=√(area/π)) of the Fine submodel was 1.14 mm (+2.8% vs. expt.). Fig. 6 compares the analytical and FEA results of the radial stress component, σr, as a function of radial coordinate. The stresses were extracted from the element integration points closest to the surface, and the analytical results were computed at identical points using formulas in [26]. Table 2 Results from two trials of each Rod 1 specimen. ‘Analytical’ gives value and difference from experiment. ‘RMS Δ’ is the root-mean-square difference Rod 1 specimen 1 2 3

Trial # 1 3 4 5 2 3

Contact radius (mm) Impact speed Experimental Analytical (m/s) radius Circularity radius 2.208 0.098 1.129 1.093 (-3.2%) 2.197 0.091 1.109 1.090 (-1.7%) 2.178 0.140 1.143 1.087 (-4.9%) 1.848 0.117 1.059 1.029 (-2.8%) 2.166 0.090 1.120 1.085 (-3.2%) 2.128 0.083 1.115 1.079 (-3.2%) 3.2% Average Δ (absolute value)

Rod 2 midpoint Avg. peak speed (mm/s) RMS Δ strain Exp’t Analytical (microstrain) 1.114 1.105 (-0.8%) 6.0 1.105 1.095 (-0.9%) 11.6 0.921 0.930 (+0.9%) 3.4 1.114 1.089 (-2.2%) 7.4 1.078 1.070 (-0.7%) 5.0 1.1% 6.7

Table 3 Contact force and area from the global model and three refinement levels of the submodel Model Contact force (N) Contact area (mm2) Global 5,549 4.61 Submodel-Coarse 5,219 4.80 Submodel-Medium 5,368 4.44 Submodel-Fine 5,492 4.10 Analytical / Experimental 5,543 3.73 / 3.86 -0.92% 9.9% / 6.2% Δ, Fine vs. Analytical / Exp’t

235 a)

b)

Analytical

FEA

Analytical

FEA

Fig. 6 Comparison of analytical and FEA subsurface radial stress result, σr, from element integration points

4. Discussion and Conclusions Both of the analysis models aimed to give high fidelity predictions of the global structural response. The analytical model predicted the peak Rod 2 midpoint speed within 0.7-2.2% error. Likewise, the model’s strain prediction had an RMS error (over 5 periods of the vibration) of 3.4-11.6 microstrain (Table 2). Though not tabulated, the error of the model’s contact duration prediction was only 1-2 µs, which is approximately the value of the sampling period. The FE model yielded similar fidelity, with the plots of its results nearly overlying those from the analytical model in Fig. 4. Similarly accurate predictions of global structural response, from both analytical and FE models, have been reported for the case of a ball striking a long rod [8,12,13], though without direct measurement of the contact duration and contact area. The chief aim of submodelling was to provide a refined FEA focused on the contact mechanics. The peak contact radius during impact was ~1 mm, so the 0.25×0.25×1.0 mm elements in the global model were expected to yield relatively coarse resolution of the contact stress and area, since contact stress fields are quite localized [26]. The use of three submodel mesh refinements with successively halved element lengths has been previously recommended [21]. In the first submodel, the contact force differed by -330 N from the global model; the change occurred because mesh refinement reduced the stiffness of the contact surfaces while nodal displacements from the global model were applied to the submodel boundary. This effect diminished with subsequent mesh refinements, finally yielding a contact force <1% different from the analytical model and contact area 6.2% different from the experimental measurement. Ongoing work is expected to improve the FEA results in contact area and in the convergence of particular stress values; the existence of some non-convergent stress values in the halfsymmetry Fine submodel has revealed that some nodes were minutely displaced from the symmetry plane. The impact of slender rods provides a means for examining fundamental characteristics of the transient dynamics of impacting bodies. These include material property effects, speeds of wave propagation, and contact mechanics. A basic understanding of these dynamic phenomena, as they occur in the approximately 1D domain of slender rods, is an important pre-requisite to advanced impact analysis and testing involving more complicated structures. In contrast with the case of a ball–rod impact, the case of two impacting rods requires consideration of vibrations in both bodies, which may be more representative of complicated impact scenarios. In our laboratory, this study has served as a means to verify and validate analysis and laboratory techniques for studying the transient dynamics of artificial hip joints. An artificial hip may experience small (e.g. 2 mm) separations of the ball from the socket [28], followed by rapid relocation that causes high and damaging contact stresses [29]. The hip study’s objective is to identify peak contact force and stress during the rapid relocation phase, and these quantities cannot be directly measured. Therefore, they are being computed using an FE model of a corresponding dynamically actuated structure in an in-vitro relocation simulation. To validate the FE model, the structure’s response to the inputs is measured using laser vibrometry and strain gages with high-bandwidth amplification, as used for the two-rod impact study. Thus, the present work has validated measurement and analysis techniques in a rudimentary test case, so that they may be applied confidently to a challenging biomedical problem. The engineering approach to impact problems in other fields may benefit similarly by preparatory testing and analysis of the low speed impact of slender rods.

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Acknowledgements The authors are grateful to Jeff Kessler for laboratory assistance, in particular with the drop tower impact test machine.

References [1] Timoshenko, S.P., and Goodier, J.N., Theory of Elasticity, McGraw Hill, New York, 1970. [2] Thomson, W.T., Laplace Transformation, Prentice Hall, Englewood Cliffs, 1960. [3] Goldsmith, W., Impact: The Theory and Physical Behaviour of Colliding Solids, Edward Arnold, Ltd., London, 1960. [4] Guo, A., and Batzer, S., "Substructure Analysis of a Flexible System Contact-Impact Event," Journal of Vibration and Acoustics, 126(1), 126-131, 2004. [5] Marghitu, D.B., and Boghiu, D., "Spatial Impact of a Flexible Link Using a Nonlinear Contact Force," Atlanta, GA, USA, 90, 103-110, 1996 [6] Schiehlen, W., and Seifried, R., "Three Approaches for Elastodynamic Contact in Multibody Systems," Multibody System Dynamics, 12(1), 1-16, 2004. [7] Trowbridge, D.A., et al., "Low Velocity Impact Analysis with Nastran," Computers and Structures, 40(4), 977-984, 1991. [8] Seifried, R., and Hu, B., "Numerical and Experimental Investigation of Radial Impacts on a Half-Circular Plate," Multibody System Dynamics, 9(3), 265-81, 2003. [9] Wei, H., and Yida, Z., "Finite Element Analysis on Collision between Two Moving Elastic Bodies at Low Velocities," Computers and Structures, 57(3), 379-82, 1995. [10] Stoianovici, D., and Hurmuzlu, Y., "A Critical Study of the Applicability of Rigid-Body Collision Theory," Transactions of the ASME. Journal of Applied Mechanics, 63(2), 307-16, 1996. [11] Bokor, A., and Leventhall, H.G., "The Measurement of Initial Impact Velocity and Contact Time," Journal of Physics D: Applied Physics, 4(1), 160-163, 1971. [12] Hu, B., and Eberhard, P., "Simulation of Longitudinal Impact Waves Using Time Delayed Systems," Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME, 126(3), 644-649, 2004. [13] Hu, B., et al., "Comparison of Analytical and Experimental Results for Longitudinal Impacts on Elastic Rods," Journal of Vibration and Control, 9(1-2), 157-74, 2003. [14] Sundin, K.G., and Ahrstrom, B.O., "Method for Investigation of Frictional Properties at Impact Loading," Journal of Sound and Vibration, 222(4), 669-77, 1999. [15] Ueda, K., and Umeda, A., "Dynamic Response of Strain Gages up to 300 Khz," Experimental Mechanics, 38(2), 93-98, 1998. [16] Hertz, H., "On the Contact of Rigid Elastic Solids and on Hardness", in Miscellaneous Papers by H. Hertz, MacMillan, London, 1882. [17] Diaconescu, E.N., et al., "A New Experimental Technique to Measure Contact Pressure," Ponte Vedra Beach, FL, USA, 121-128, 2003 [18] Sanders, A.P., and Brannon, R.M., "Determining a Surrogate Contact Pair in a Hertzian Contact Problem," Journal of Tribology, accepted for publication [19] Fabrikant, V.I., "A New Symbolism for Solving the Hertz Contact Problem," Quarterly Journal of Mechanics and Applied Mathematics, 58, 367-81, 2005. [20] Sackfield, A., and Hills, D.A., "Some Useful Results in the Classical Hertz Contact Problem," Journal of Strain Analysis for Engineering Design, 18(2), 101-105, 1983. [21] Cormier, N.G., et al., "Aggressive Submodelling of Stress Concentrations," International Journal for Numerical Methods in Engineering, 46, 889-909, 1999. [22] Rajasekaran, R., and Nowell, D., "On the Finite Element Analysis of Contacting Bodies Using Submodelling," Journal of Strain Analysis for Engineering Design, 40, 95-106, 2005. [23] Drumheller, D.S., Introduction to Wave Propagation in Nonlinear Fluids and Solids, Cambridge University Press, Cambridge, 1998. [24] Johnson, K.L., Contact Mechanics, Cambridge University Press, Cambridge, 1985. [25] Shampine, L.F., and Reichelt, M.W., "The Matlab Ode Suite," SIAM Journal on Scientific Computing, 18(1), 1, 1997. [26] Fischer-Cripps, A.C., Introduction to Contact Mechanics, Mechanical Engineering Series, Springer-Verlag, New York, 2000. [27] "Tech Note Tn-514: Shunt Calibration of Strain Gage Instrumentation," Vishay Micro-Measurements, 2007. [28] Lombardi, A.V., et al., "An in Vivo Determination of Total Hip Arthroplasty Pistoning During Activity," J Arthroplasty, 15(6), 702-709, 2000. [29] Mak, M.M., et al., "Effect of Microseparation on Contact Mechanics in Ceramic-on-Ceramic Hip Joint Replacements," Proc Inst Mech Eng [H], 216(6), 403-408, 2002.

Solenoid Actuated, Rail Mounted, Aircraft Payload Release Mechanisms

Clark L. Reynolds Dynetics, Inc. 1002 Explorer Boulevard Huntsville, AL 35806 John A. Gilbert Professor of Mechanical Engineering Department of Mechanical and Aerospace Engineering University of Alabama in Huntsville Huntsville, AL 35899 ABSTRACT The spring loaded detent mechanisms currently used to retain missile-type payloads require a high forward load in order to overcome a spring before release can be achieved. However for smaller aircraft, such as UAVs, there is a pressing need to develop alternate mechanisms which require reduced release forces to avoid excessively loading the vehicle. This paper describes two mechanisms which were designed and tested to achieve this goal. Both of the prototypes rely on a solenoid which changes the boundary conditions and alters the required release force. The advantages and disadvantages of each design are discussed. INTRODUCTION Aircraft serve many purposes. Some are used as mass transit vehicles, some as cargo vehicles, and some as weapons platforms. For weapons platform aircraft, payloads may be carried internally within a cargo hold or externally via some type of mounting interface. For this study, it is an externally mounted payload that will be considered. Payloads of this type are commonly called “stores” and are typically mounted under a wing or on a pylon. These stores may be permanently mounted, such as extended range fuel tanks, or deployable, such as missiles or emergency drop stores. Figure 1, for example, shows a Brimstone weapon system comprised of a re-usable launcher with three missiles [1]. For those stores that are deployable, any number of methods may be used to retain the payload during flight. If the mechanism is such that forward motion is required to initiate payload release, the associated thrust loads are transferred to, and reacted by, the aircraft. One method for retaining a missile type payload is Fig. 1 Brimstone Weapon System [1] that of a spring loaded detent mechanism. But the latter requires the payload to be subjected to a high forward load in order to overcome a spring before payload release can be

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_35, © The Society for Experimental Mechanics, Inc. 2011

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238 obtained. Such a thrust load is not problematic for large aircraft. However, as UAVs gain popularity, it is desirable to carry and deploy existing stores from ever smaller airframes. As the airframes get smaller, the ability to compensate for a load due to a thrusting payload becomes an important consideration. Thus, there is a pressing need to develop alternate mechanisms which require reduced release forces to avoid excessively loading the vehicle. This paper describes two mechanisms which were designed and tested to achieve this goal. Both of the prototypes rely on a solenoid which changes the boundary conditions and alters the required release force. The advantages and disadvantages of each design are discussed. Due to the nature of this topic, actual launcher dimensions, missile thrust values, loads, and safety factors are not available. Consequently, generic missile information obtained from various publicly available online resources is used to describe the investigation [2]; and, generic design requirements obtained from the Federal Airworthiness Regulations (FARs) are used as a basic staring point [3]. BENCHMARK In order to carry wing-mounted payloads on various aircraft, it is necessary to provide an attachment system which performs several functions. Actual design requirements are many and varied, but for the purposes of this study, the primary requirements are to: 1) restrain the payload during flight, 2) restrain the payload in the event of a crash, and 3) reliably permit release of the payload. Before embarking on a mission to improve an existing design, the form and function of the benchmark must be fully understood. Further, before beginning an analysis or testing effort for an improved design, that concept must be defined. One current design for an aircraft release mechanism that meets the aforementioned goals is a simple spring-lever detent concept. With this design, “shoes” are attached to the missile. These shoes are designed to fit inside of a rail that is attached to the aircraft. The shoes keep the payload restrained in all Degrees Of Freedom (DOFs) except the forward/aft direction. Aft direction restraint is provided by “stops” inside the rail. Forward restraint is provided by a spring loaded detent mechanism such as that shown in Fig. 2.

Fig. 2 Baseline Detent Mechanism

239 This simple mechanism works by requiring forward motion of the missile to effect release. As the missile moves forward, the shoe presses against the sloped face of the detent lever. This causes the lever to attempt to rotate about its pivot point. The spring pack provides an opposing force and thus a restraint to the payload. In order for the missile to actually be released, enough thrust (load) must be applied to overcome the detent spring pack. This is a simple and reliable mechanism. However, it has several potential issues such as: 1) spring rate variations, 2) the sloped detent face can cause shoe wear, and 3) thrust loads to overcome the detent must be reacted somewhere. The first of these issues is the variability in spring rates. Because this design works by virtue of spring deflection, the governing equation for the thrust load required for missile release is F = Kx. Of the two variables, x can be controlled via manufacturing tolerances, but K is a spring rate of an off-the-shelf commercial component. Variability in that spring rate is not easily determined due to lack of manufacturing data, and it can have a large impact on the resulting release force. As an example, assume that the travel distance required for missile release is 6.35 mm (0.25 in.) and that the nominal release load is 2223 N (500 lb). Further assume that the detent lever geometry is such that the shoe travel distance required for release is matched by the compression distance of the spring pack. This would mean that the required spring rate is 350 kN/m (2000 lb/in). A variation of only 10% in the spring rate would cause the payload release force to be in a range of 2000 N (450 lb) to 2445 N (550 lb). Such a variation, considered alone, is not large. But if the payload only has a thrust capacity of about 2667 N (600 lb) and it also has a 10% variability, the possibility exists for a situation to arise in which the payload could not overcome the detent and would not be deployed. The second issue refers to wear and tear that could occur while the payload is simply being carried. Some payloads may be installed on the aircraft, flown, and then returned unused. For some payloads, it may actually be a rare event that they are ever deployed. In such situations, the sharp corner of the shoe would constantly be in contact with the detent face. Normal vibration of aircraft and payload could wear away both the shoe and the detent. In extreme cases, the shoe could potentially wear to a point that it rattles inside the rail. Another side effect of this wear is that, as the shoe is worn down, ever decreasing force is required to retain the payload because the very interference providing the restraint is being continually worn away. While this permits ever easier payload release, it also reduces the detent’s ability to retain the payload during high “g” maneuvers or during a crash event. The third of these issues is the most fascinating one. Because a thrust load is required to effect release of the missile; the thrust load must be reacted by the aircraft in some way. Consider a missile that weighs approximately 445 N (100 lb). Assume that flight maneuvers might impose inertial loading on the order of 2 or 3 g. Further assume that the payload must be retained in the event of a crash load as high as 5 or 6 g and that a safety factor as high as 2 could be required. For these conditions, this payload would require a detent capable of retaining as much as 2 x 6g x 445 N (100 lb), or 5334 N (1200 lb). While it is entirely feasible for a missile to have the thrust required for this example, a detent which requires the payload to actually thrust up to such a high level means that the aircraft carrying it must be able to withstand and compensate for that high load. This could be a problem for a small UAV, especially with a payload mounted at a wingtip. This is the primary concern in the present study. DESIGN REQUIREMENTS Before introducing the proposed designs, some basic design requirements need to be defined. The following requirements are assumed: Missile weight is 110 lb Minimum restraint force must be 1g Detent must withstand a 6g load Safety Factor of 1.5 is required Must preclude a hang fire Launch occurs in straight and level flight

(similar to Brimstone [2]) (nose down aircraft attitude) (per FAR 23.337 [3]) (per FAR 23.303 [3]) (arbitrary requirement) (simplifying assumption)

Two mechanically different concepts are proposed to address the aforementioned issues. While these designs are mechanically different, they both operate on a fundamentally similar concept. That concept is a simple change of boundary conditions.

240 CANTILEVERED BEAM CONCEPT As illustrated by the schematic diagram shown in Fig. 3, the first concept is based on a simple cantilever beam. Restraint to the payload is provided in the same manner as with the baseline design; a shoe presses against a sloped detent face. But for this concept, rather than an off-the-shelf helical spring, the restraint force is provided by deflection of the cantilever.

Fig. 3 Cantilevered Beam Detent Concept The lifting force will reach a maximum when the shoe travels far enough to completely lift the beam up and out of the way and no longer impedes payload release. The configuration at the moment of release is shown in Fig. 4.

Fig. 4 Cantilevered Beam Detent Concept at Payload Release By carefully selecting the size and material for the beam, a low release force can be obtained. However, while a low force is desired for release, the payload must also be safely restrained against flight maneuvers as well as crash loads. If nothing

241 more is done with the beam and it is sized such that the release force is less than the weight of the payload, even a simple nose down attitude could result in inadvertent release. Consequently, to increase the release force, a simple intermediate support can be added to the cantilevered beam (see Fig. 5). Depending upon the placement of the intermediate support along the beam, the increased release force can be fine-tuned.

Fig. 5 Cantilevered Beam Detent Concept at Inadvertent Release By incorporating a solenoid into the design, the intermediate support can be applied to or removed from the beam depending on the solenoid state. In Fig. 6, for example, the solenoid is unpowered and a vertical lever positions a roller against the cantilever beam to provide the intermediate support. This condition will provide the maximum possible release force for the design. As such, this configuration will be referred to as the “Hi-Lock” condition.

Fig. 6 Cantilevered Beam Detent Concept, Solenoid De-Energized The position of the intermediate support is chosen such that the Hi-Lock release force is equivalent to the current baseline design release force. Therefore, should the solenoid fail for any reason, the configuration would be functionally equivalent to the existing design.

242 Fig. 7, on the other hand, shows the condition that occurs when power is applied to the solenoid. In this case, the vertical lever is forced to rotate such that there is no contact between the roller and the cantilever beam. This condition will provide the minimum possible release force for this design. As such, this will be referred to as the “Lo-Lock” condition.

Fig. 7 Cantilevered Beam Detent Concept, Solenoid Energized GATED SLOT CONCEPT The second concept is based on the idea of a variable width slot. The payload is restrained by a latch which can translate fore and aft along two slots. As illustrated in the schematic diagrams included as Fig. 8, one slot has a wide section near the end of travel. When the latch travels forward from the position shown to the left and reaches the widened portion of the slot, illustrated to the right, it will rotate to allow payload release.

Fig. 8 Gated Slot Detent Concept With Slots – Payload Constrained (Left); Payload Released (Right)

243 If part of the widened portion of the slot is selectively hidden or exposed by a gate, the travel length required for payload release will be altered. When if the forward travel of the latch is restrained with a spring pack, the release force required will also be altered. Figure 9 illustrates this concept.

Fig. 9 Gated Detent Concept By incorporating a solenoid into this design and attaching it to a plunger pin the widened region of the slot can selectively be either filled or exposed. Thus the slot is “gated” and the position of the gate determines how much latch travel, and therefore release force, is required for payload release. In Fig. 10, for example, the solenoid is unpowered and the plunger attached to the solenoid is used to fill in a portion of the slot in which the lower detent lever roller rides. With the plunger in this extended position, the slot is effectively made longer because the detent lever cannot rotate until it travels enough to reach the pocket at the end of the slot. This condition will provide the maximum possible release force for this design. As such, this is the “Hi-Lock” condition for this concept.

Fig. 10 Gated Slot Detent Concept, Solenoid De-Energized

244 Fig. 11, on the other hand, shows the condition that occurs when power is applied to the solenoid. In this case, the plunger attached to the solenoid is retracted and the full length and width of the slot is exposed. With the plunger in this position, very little detent travel is required before the lower pin reaches the wide portion of the slot. This condition will provide the minimum possible release force for this design. As such, this will be referred to as the “Lo-Lock” condition.

Fig. 11 Gated Slot Detent Concept, Solenoid Energized PROTOTYPING AND TESTING A preliminary analysis was performed to ensure that a solenoid actuated device could be built which would meet the design requirements [4]. Detailed analyses were not necessary at this early stage. However, enough analysis to get baseline sizing data was required. Multiple prototypes were built and tested to evaluate both concepts. Figures 12 and 13 show two, as-built units.

Fig 12 Cantilever Beam Prototype

Fig. 13 Gated Slot Prototype

As illustrated in Fig. 14, tests were performed by placing the prototype designs into a common rail and using a pull tester to

245 determine release force. The tester has no instrumentation other than a dial indicator that shows the maximum load applied by the test apparatus.

Fig. 14 Pull Tester Numerous test runs were made on the cantilevered configuration. During testing, several different shoes were used as each shoe is only valid for a limited number of uses due to wear and tear. Additionally, various changes were made to the prototype using the built-in adjustment features. Results from the configuration shown in Fig. 12 are included in Table 1. It should be noted that the “Lift Height” is the vertical distance required for the shoe to slide under the detent. This is a measured value and is different for every shoe tested. Results from the configuration shown in Fig. 13 are shown in Table 2. As with the cantilever beam concept, numerous test runs were made and several different shoes were used to evaluate the gated slot configuration. Additionally, various values of high spring preload (corresponding to the “Hi Pk Hght”) and gap between spring packs (“Spacer”) were made to the prototype using the built in adjustment features.

246 DISCOVERIES/CRITIQUE/ISSUES While both prototypes worked rather well and provided distinct Lo-Lock and Hi-Lock release forces, each concept had issues discovered during the testing phase. The first hurdle to overcome with the cantilever beam concept was payload preload. The rail used for testing had its own spring pack which ensures release of an electrical connector attached to the payload. That spring pack imparts a substantial preload into the detent mechanism. Because the default condition of the detent is to be in a Hi-Lock condition, the beam’s intermediate support is designed to be in contact with the beam in that condition. Consequently, any preload on the detent imparts a preload into the intermediate support. Packaging and weight constraints require the use of small solenoids. These small solenoids have a relatively low amount of force that they can generate. As a result, the combination of low solenoid capability and a preloaded payload occasionally caused the solenoid to stick in the Hi-Lock condition. The sticky solenoid problem could be reduced by introducing a slight gap between the roller lever and the beam. But, that gap would also alter the Hi-Lock release force because the intermediate support, in effect, would not be in place until some forward travel of the payload had been accomplished. As significant as the sticky solenoid problem was, it was actually trivial compared to the tolerance stack-up problem encountered. In hindsight, this is one problem that should have been somewhat obvious. Recall that the underlying method by which the cantilever beam concept works is by means of the shoe lifting the beam in a vertical direction. Now consider that the shoe itself has a thickness tolerance as does the rail slot in which it rides. Further consider that the solenoid detent assembly sits inside the rail. The assembly housing bottom also has a thickness tolerance. Within the solenoid assembly, the beam is positioned vertically via an adjustment block. The common theme here is that there are numerous vertical direction tolerances that have to be considered and compensation must be made for them. Because the rails and shoes are in-service items, only the solenoid assembly tolerances can be controlled. And only by placing shims between the assembly and the rail or between the beam and the assembly can any adjustment be made. All of these vertical dimensions influence the beam lift height. That lift height is ultimately what controls the release force. Consequently, while the cantilever beam itself can be manufactured to high tolerances, the ultimate release force depends on numerous items beyond the control of the solenoid assembly. In essence, the flaw in this design is not the basic concept, but rather that the relevant tolerances all stack up in the same direction as the direction required to generate the release force. There were actually few issues discovered with the gated slot design. Unlike the cantilever beam concept, the gated slot’s release force does not depend on a vertical lift height. As such, the tolerances of the payload are generally decoupled from the final release forces generated. Nevertheless, some issues did show up during testing. The first of these was pin bending. Due to the schedule, prototype manufacturing and analysis actually occurred in parallel. As a result, the pins on which the detent lever rides were actually sized too small. Very high strength materials were therefore used in order to avoid scrapping hardware. Nevertheless, on some of the units, a couple of pins showed evidence of bending. Because the gated slot design uses pins sliding along a slot, bearing stresses are very high at the pin-slot interface. These high stresses lead to galling, especially given the materials that were necessarily chosen at the time. While not excessive, galling was evident in some test articles and became worse as testing progressed. To a large extent, pin bending and galling were expected due to the concurrent engineering process dictated by schedule. However, the main issue discovered was, as similarly discovered with the cantilever beam, a result of low solenoid force capability combined with a tolerance stack-up.

247 As before, the default condition of the gated slot design is in a Hi-Lock state. In this nominal state, the low spring pack governs until the low spring guides bottom out on the high spring nuts. In order to ensure that a Lo-Lock release is possible, the gap between the spring packs must be slightly greater than the travel distance required for the lower pins to reach the LoLock open gate. However, because the gate is not actually open until the solenoid plunger is retracted, it is possible, and in fact expected, that the payload can travel forward until the low spring bottoms out. If this happens while the solenoid is deenergized, the lower lever pins will actually travel forward until they attain a position where they would normally have reached an open slot, but instead will reach, and begin riding on, the gate. When this occurs, it is possible for the tolerances to be such that sufficient load is introduced into the lower lever pin that it bears on the slot gate and more force is required to retract the gate than the solenoid can provide. CONCLUSIONS As with any rapid prototyping and testing program, this one had its share of issues, problems, and challenges to overcome. While the goal was obviously to produce as near a production-ready prototype as possible, the actual objectives were much more realistic. Because this project was intended to be a test project from the very beginning, the actual release forces obtained were not of prime importance. Rather, the primary goal was only to demonstrate that a solenoid actuated detent mechanism could be produced which would provide a selectable Lo-Lock or Hi-Lock release condition. This goal was accomplished. Furthermore, while neither concept exactly achieved the desired release forces in either Hi-Lock or Lo-Lock conditions, both concepts were generally in the target region. As a result of the testing, the cantilevered beam design was judged to be inadequate due to the previously discussed tolerance stack-up issues. There are simply too many variables beyond the control of the solenoid assembly for the beam design to be able to compensate. When it was observed that the beam concept would not be as robust as hoped, a highly compressed schedule was required to develop and test an alternative design. Thus the gated slot was conceived. Most all of the issues with the gated slot concept were due to the compressed schedule. Were a second attempt to be made to further develop the gated slot concept, issues thus far discovered could readily be addressed. Most notably, the pins could be made larger to preclude bending and the ends could be flattened so that the bearing stresses on the slot are much lower. Testing of the gated slot showed that the release force obtained can be significantly changed by adjusting spring pack preload and gap. With additional time to procure high tolerance springs and to design additional adjustability into the preload and gap parameters, the target release forces should be readily attainable. REFERENCES 1. Wikipedia contributors. File: Missile MBDA Brimstone.jpg. Wikipedia, The Free Encyclopedia. Reprinted with permission, see: http://en.wikipedia.org/wiki/File:Missile_MBDA_Brimstone.jpg (accessed 02/25/2011). 2. Wikipedia contributors. Brimstone missile. Wikipedia, The Free Encyclopedia. http://en.wikipedia.org/wiki/Brimstone_missile (accessed 02/25/2011). 3. Risingup Aviation. Federal Aviation Regulations Index of all Federal Aviation Regulation Parts. http://www.risingup.com/fars/info (accessed 02/25/2011). 4. Reynolds, C.L., “Alternative Designs for a Solenoid Activated, Rail Mounted, Aircraft Payload Release Mechanism,” M.S.E. Thesis, University of Alabama in Huntsville, Huntsville, Alabama, 2009.

Finite Element Modeling of Ballistic Impact on Kevlar 49 Fabrics Deju Zhu1, Barzin Mobasher2, S.D. Rajan3 1

Postdoctoral fellow, Department of Mechanical Engineering, McGill University, Montreal, QC, Canada, E-mail: [email protected] 2 Professor, Ph.D., P.E., Department of Civil, Environmental and Sustainable Engineering, Arizona State University, Tempe, AZ, 85287, E-mail: [email protected] 3 Professor, Ph.D., Department of Civil, Environmental and Sustainable Engineering, Arizona State University, Tempe, AZ, 85287, corresponding author, E-mail: [email protected] ABSTRACT This paper presents an improved material model suitable for Kevlar 49 fabric which was implemented into the commercial explicit Finite Element (FE) software LS-DYNA through a user defined material subroutine (UMAT). The fabric constitutive behavior in the current material model was obtained from new experimental data in the principal material directions (warp and fill) under static loading. Two different modeling configurations, i.e. single FE layer and multiple FE layers were used to simulate the ballistic tests conducted at NASA Glenn research center. Both the shear properties of the fabric and the parameters used in Cowper-Symonds (CS) model which accounts for strain rate effect on material properties were optimized to achieve close match between the FE simulations and experimental data. The residual velocity of the projectile, the absorbed energy by the fabric after impact, and the temporal evolution and the spatial distribution of the fabric deformation and damage were closely examined. Sensitivity analysis was carried out to study the effect of the failure strain of the fabric and the coefficient of friction on the simulation results. Keywords: Ballistic impact; Kevlar fabric; Strain rate effect; Finite element model 1. Introduction Fabrics based on high-performance fibers are extensively employed in variety of ballistic and impact protection applications. Despite the fact that over the past two decades, there has been a great deal of work done on understanding the ballistic behavior of these fabrics using various analytical and numerical techniques, the design of fabric armor systems remains largely based on the employment of extensive experimental test programs and empiricism. While such experimental programs are critical for ensuring the utility and effectiveness of the armor systems, they are generally expensive, time-consuming and involve destructive testing. Consequently, there is a continuing effort to reduce the extent of these experimental test programs by complementing them with the corresponding computation-based engineering analyses and simulations [1]. In our previous study [2], a material model was developed to include non-linearity in the stress-strain response and strain rate effect on the material response. The model was incorporated into the LS-DYNA through a UMAT and was validated by comparing the results against experimental ballistic tests conducted at NASA Glenn Research Center. The fabric layers were represented by one single finite element (FE) layer; hence it was not able to capture the effect of friction between the fabrics layers which is actually an important factor for determining the ballistic behavior of the fabric. Later on, a multi-layer model was built to consider the friction between the fabric layers [3]. But all these models were built on the experimental data where the fabric was loaded up to the strain of about 4% [4, 5], and the post-peak behaviors were assumed to follow certain patterns without experimental validation. New experimental data show that the fabric can deform up to 20% before the complete failure, the energy absorption ability will dramatically increase due to the large strain capacity [6]. Research on the mechanics properties of aramid yarns and woven fabrics has been reported by some authors. Amaniampong and -4 -1 Burgoyne [7] studied the effect of gage length and strain rate from 3x10 to 0.003 s on the failure stress and

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_36, © The Society for Experimental Mechanics, Inc. 2011

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250

failure strain of Kevlar® 49 yarns. Yarn strength decreases slightly as the gage length increases; whereas the ® failure strain of the Kevlar 49 yarns was independent of the gage length, however decreased slightly as the strain rate increased. Zhu et al. [8-11] conducted dynamic tensile testing on Kevlar® 49 single yarn and fabrics using a servo-hydraulic high speed machine [12-18] and found that the Young’s modulus, tensile strength, maximum strain and toughness increased with increasing strain rate over a range of 20 to 170 s-1. In the current study, the fabric constitutive behavior was obtained from the new experimental data in the principal material directions. Based on the new constitutive behavior, the UMAT was modified and new material parameters were identified for the simulations. Single layer and multi-layer models were used to simulate the ballistic tests conducted at NASA Glenn research center. Finally, sensitivity analysis was carried out to study the effect of failure strain and friction coefficient on the simulation results of the multi-layer model. 2. Constitutive Modeling 2.1 Constitutive Behavior of the Fabric In this study simplified assumptions were made to fully capture the complexities of the stress-strain behavior in the principal material directions (warp and fill). The fabric has negligible stiffness perpendicular to both fabric material directions and hence those properties were assumed to be zero. No coupling effect between the material directions was assumed – the Poisson’s ratios were assumed to be zero. The constitutive behavior used in the material model in stiffness incremental form is shown in Eq. (1):   11        22        33      12        31      23  

0

0

0

0

0 0

E22 0

0 0

0 0

0 0

0

0

0

0

0

0

0

2G12

0

0

0

E11

0 0

2G31 0

0   11    0    22  0    33    0   12  0    31    2G23     23 

(1)

Material direction 11 refers to the main longitudinal direction of the fabric or warp direction, direction 22 refers to the direction along the width of the fabric or fill direction, and direction 33 refers to the direction perpendicular to both warp and fill directions. E33 is taken as zero simply because shell elements are used to model the fabric. The values for E11, E22, G12, G31, and G23 are a function of several factors including current stress and strain, stress and strain history, and strain rate. The determination of these material properties will be discussed in the following section. 2.2 Material Parameters of the Fabric New tensile tests were conducted in both warp and fill directions until the load carrying capacity of the fabric reached most zero, the results show that the fabric can deform up to 20% before the complete failure. Figures 1(a) and 1(b) show the new stress-strain curves used in the model based on quasi-static tensile test results using 200 mm x 50 mm (length x width) swath specimens and the material model developed for the fabric in warp and fill directions, respectively. Note that there are four distinct regions in the constitutive behavior: crimp region, linear pre-peak region, linear post-peak region and non-linear post-peak region. In the crimp region, the stress increase is relative low due to the straightening of the woven structure of the fabric. When the crimp is removed the straightened yarns start to behavior linearly and take more loads, reaching linear pre-peak region. When the stress level reaches the tensile strength of the fabric the yarns start to break and the stress of the fabric decreases dramatically until reaching a transition point which is about 70 MPa (linear post-peak region). After the transition point the stress decreases gradually to almost zero when the strain reaches to about 0.2 mm/mm, representing the non-linear post-peak region.

251

Linear Pre-peak Region

300 Linear Post-peak Region

200 100 0

Non-linear Post-peak Region

0.04

0.08

0.12

Strain, mm/mm

0.16

Experiment Model

400

Linear Pre-peak Region

300 Linear Post-peak Region

200 100

Crimp

0

Fill Direction (22)

(b)

500

Stress, MPa

400

Stress, MPa

600

Warp Direction (11) Experiment Model

(a)

500

0

0.2

Non-linear Post-peak Region Crimp

0

0.04

0.08

0.12

Strain, mm/mm

0.16

0.2

Fig. 1 Stress-strain curves and model in (a) warp and (b) fill directions Based on the stress-strain curves in the warp and the fill directions, it was found that the elastic stiffness in prepeak region of warp direction is identical to that of fill direction, and the crimp stiffness for warp and fill directions is 0.06 and 0.20 times of the elastic stiffness in pre-peak region, respectively. The stiffness in linear post-peak region of warp and fill directions is 2.2 and 5.6 times of the elastic stiffness in pre-peak region. The crimp strain of the warp direction is about 2.6 times larger than that of the fill direction. And the peak stress of the warp direction is 15% lower than that of the fill direction. There is a slight difference in the strain at peak stress and the stiffness of linear post-peak region. In the current material model, the elastic stiffness and strain at peak stress were assumed to be a function of the strain rate using Cowper-Symonds model as follows:

Eiiadj

    Eii  1  ii   CE 

 iimax( adj )



 iimax

1 PE

 iimax  1  C 

(2)

  

1 P

(3)

where Eii (i = 1, 2) is the static elastic stiffness, Eiiadj is the adjusted elastic stiffness considering strain-rate effect,

ii is the strain-rate, CE and PE are the Cowper-Symonds factors for elastic stiffness,  iimax is the strain at peak stress,  iimax(adj ) is the adjusted strain at peak stress, iimax is the maximum strain rate experienced by the element in each respective direction, and C and P are the Cowper-Symonds factors for the strain at peak stress. Picture frame test has been conducted to determine the shear behavior of Kevlar 49 fabric [6]. In the experiment study, the fabric was sheared at quasi-static loading rate without any pretension and it has very low shear resistance. But in real ballistic scenario, the fabric under impact will experience large tension force during shear deformation. The tension force in the fabric will dramatically influence the shear resistance of the fabric by altering the conditions of the yarn interaction (crimp, yarn compression, normal force at cross-over points), and hence the friction. Duan et al. [19] investigated the friction effects on the ballistic impact behavior of plain weave fabric and found that friction contributed to delaying fabric failure and increasing impact load which allowed the fabric to absorb more energy. If the shear properties of the fabric obtained by picture frame test were used directly in FE simulation, the fabric behaves like a rubber-like material with very large deformation. As the relation between shear properties and tension in fabric is not clear, the shear properties used in the FE simulation was adjusted until the deformation of the fabric in simulation was similar to that of the experiment, and then was optimized to obtain the smallest error in absorbed energy between the simulations and experiments. The in-plane shear stress increment was computed as follows: 12   12G12 (4)

252

2.3 Validation of the Continuum Model To validate the material model, the interaction between the projectile and the fabric was analyzed using the two different modeling configurations (single and multi-layer models) and two aspects of these interactions were closely examined: (a) the absorbed energy by the fabric after the impact, and (b) the temporal evolution and the spatial distribution of fabric deformation and damage. The loss of projectile kinetic energy (absorbed energy), ΔEpk is computed as the kinetic energy of the projectile before impact minus the kinetic energy of the projectile after impact, as follows: 1 (5) Epk  Ei  Er = m v i2  v r2  2 where m is the mass of the projectile, vi is the projectile initial velocity, and vr is the projectile residual velocity. If presented in percentage, it is given by:





Epk (%)  (Ei  Er )/Ei  100  v i2  v r2 /v i2  100

The percent difference in absorbed energy was computed as: exp sim D (%)  Epk (%)  Epk (%) where

exp E pk

is the absorbed energy in experiments, and

sim E pk

(6) (7)

is the absorbed energy in simulation. A positive

percent difference corresponds to the FE simulation under-predicting the absorbed energy and a negative percent difference corresponds to the FE simulation over-predicting the absorbed energy. 3. Simulation of Ballistic Impact 3.1 Simulation of Single-Layer Model Figure 2(a) shows the FE model of the ring and the fabric. The steel ring was modeled with 6.35 mm x 6.35 mm x 25.4 mm hexagonal elements since the ring is not of interest with respect to the FE analysis results. The fabric was modeled with a uniform mesh containing 6.35 mm x 6.35 mm shell elements. One layer of shell elements was used to represent the fabric irrespective of the actual number of fabric layers. The fabric model was meshed using two different parts. The fabric directly in contact with penetrator is given separate part id than rest of the fabric. This type of configuration facilitates tracing of energy balance for this area separately. The shorter, thicker projectile (old) was modeled with 3.81 mm uniform tetrahedral elements for the tip and 5.08 mm x 3.97 mm x 5.14 mm hexahedral elements for the body. The longer, thinner projectile (new) was modeled with 2.54 mm uniform tetrahedral elements for the tip and 3.81 mm x 2.98 mm x 3.80 mm hexahedral elements for the body. Figure 2(b) shows the FE mesh for both projectiles. Both the ring and the projectiles are made of stainless steel, and are modeled as a linear elastic material with Johnson-Cook model to consider the strain rate effect.

New projectile Old projectile

(a)

(b)

Fig. 2 FE models: (a) ring and single FE layer, (b) projectile FE mesh Figure 3 shows the comparison of absorbed energy between the experiments and single-layer model simulations. Note that one model predicts the exact absorbed energy in which the projectile was contained by the fabric layers (LG657), three over-predict the absorbed energy with error less than 5%, and eighteen under-predict the absorbed energy with error between 0.4% and 21.1%. When the entire suite of 22 test cases is considered, the FE simulation under-predict the absorbed energy by an average of 5.7% with a standard deviation of 7%. The

253

model LG656 under-predicts the absorbed energy by the largest amount of 21.1%, followed by LG655 with 19.6% difference. These two are 32-fabric layer test cases with a high projectile velocity relative to the other test cases. In addition to comparing the absorbed energy, the temporal evolution and the spatial distribution of fabric deformation and damage were also compared between the experiment and the FE simulations. Figure 4 shows the deformations of the experiment and the simulation for LG594. In the test case LG594 the longer and thinner projectile was used. The fabric broke at the impact location and the projectile was bent due to the resistance of the fabric. Overall the simulation captures the general deformed shape and damage of the fabric quite well. 120.0

Absorbed Energy, %

100.0

Experiment Simulation

80.0 60.0 40.0 20.0 0.0

Fig. 3 Comparison of absorbed energy between experiments and single-layer model simulations

Experiment

Simulation

Fig. 4 Deformation comparison between experiment and simulation for LG594

254

3.2 Simulation of Multi-Layer Model In the multi-layer model, shell elements were used to represent the fabric and solid elements were used to represent the steel ring and steel projectiles. The fabric was modeled with a uniform mesh containing 6.35 mm x 6.35 mm shell elements. One layer of shell elements was used to represent the four fabric layers. Thus for an test case with 8 fabric layer, there will be two FE layers with shell element thickness of one fabric layer multiplied by 4, or 0.28 mm x 4 = 1.12 mm. With this methodology, the friction between the fabric layers can also be considered. In the model the center of the shell elements was placed at a distance of one half the thicknesses of shell element away from the ring and one shell element away from adjacent shell layer to facilitating contact between them at the start of the analysis. The steel ring and projectiles were modeled in the same way as in the single-layer model. All the simulations of the multi-layer model were run using the same material properties as the single layer model. Figure 5 shows the FE model of the ring and the fabric for a 16 fabric-layer (4 FE-layer) model.

(a)

(b)

Fig. 5 Multi-layer FE model of the ring and the fabric: (a) overall view, (b) close-up view of the fabric Figure 6 shows the comparison of absorbed energy between the experiments and multi-layer model simulations. The test cases LG403, LG410, and LG618 are excluded in the analysis because LG403 and LG410 only have four layers of fabric (one FE-layer) and in LG618 the projectile hits the fabrics in plat, causing significant bending of the projectile during impact. Note that one simulation predicts the exact absorbed energy in which the projectile was contained by the fabric layers (case LG657), and eighteen simulations under-predict the absorbed energy with error between 1.9% and 37%. When the 19 test cases are considered, the FE simulation under-predicts the absorbed energy by an average of 14.5% with a standard deviation of 11.2%. Similar to the single layer model, the temporal evolution and the spatial distribution of fabric deformation and damage were also compared between the experiment and the multi-layer FE simulation. Figure 7 shows the deformations of the experiment and the simulation for test case LG656. The projectile penetrates the fabric layers and the simulated deformation of the fabrics follows the similar pattern to the experiment. 120

Absorbed Energy, %

100

Experiment Simulation

80 60 40 20 0

Fig. 6 Comparison of absorbed energy between experiments and multi-layer model simulations

255

Experiment Simulation Fig. 7 Deformation comparison between experiment and simulation for LG656 3.3 Comparison of Single-Layer and Multi-Layer Models The comparison between the LS-DYNA simulations using single-layer and multi-layer models and the experiments is listed in Table 1. The single-layer model under-predicts 18 out of the 22 models and the percent different in the absorbed energy increases with increasing number of fabric layers (increasing thickness of FE layer). The multi-layer model under-predicts 18 out of the 19 models, and the percent different in the absorbed energy also increases with increasing number of fabric layers, as shown in Figure 8. The possible reason is that the outer layers of FE model break prematurely, resulting in relative less resistance again the projectile when comparing with the models with less FE layers. The single-layer model performs better than the multi-layer model as the single-layer model predicts the ballistic tests with less error and standard deviation in terms of the percent difference in absorbed energy. On the average, both of the models under-predict the absorbed energy and are conservative. Difference in Absorbed Energy, %

40.0 35.0 30.0

Single-Layer Model Multi-Layer Model

25.0 20.0 15.0 10.0 5.0 0.0 -5.0

Fig. 8 Comparison of the percent difference in absorbed energy between the single-layer and multi-layer models

256

Table 1 Comparison between experiments and FE simulations for single-layer (SL) and multi-layer (ML) models Model

exp E pk (%)

sim E pk (%) (SL)

D (%)

LG403 LG410 LG404 LG409 LG424 LG594 LG609 LG610 LG611 LG612 LG618 LG620 LG689 LG692 LG429 LG432 LG405 LG411 LG427 LG655 LG656 LG657

11.3 9.8 16.1 17.6 20.1 67.0 18.4 16.9 22.4 16.1 58.4 57.8 46.6 53.7 38.4 47.4 69.6 78.2 56.0 46.1 76.5 100.0

7.1 4.9 15.1 15.3 18.8 69.9 12.5 16.5 12.1 10.6 61.9 50.3 45.0 46.8 29.3 37.3 52.6 69.3 60.7 26.5 55.4 100.0

4.2 4.9 1.0 2.3 1.3 -2.9 5.9 0.4 10.2 5.5 -3.5 7.5 1.6 6.8 9.1 10.1 17.1 8.9 -4.7 19.6 21.1 0.0

Average Minimum Maximum Std. Dev.

sim E pk (%) (ML)

D (%)

-

-

13.3 13.7 18.2 61.5 9.6 9.2 13.1 11.8 47.2 25.7 34.9 22 25.4 37.5 41.2 40.5 15.1 49.8 100

5.7

2.8 3.9 1.9 5.5 8.8 7.7 9.3 4.3 10.6 20.9 18.8 16.4 22 32.1 37 15.6 31 26.6 0 14.5

-4.7 21.1 7.0

0.0 37.0 11.3

3.4 Sensitivity Analysis A sensitivity analysis of the material model was performed to determine the multi-layer model’s sensitivity to the failure strain of the fabric and the friction coefficients of fabric-to-fabric and ring/projectile-to-fabric. The failure strain of the fabric used in the material model was 0.35 for the shell elements. Simulations of all the test cases were run using alternative values of 0.40 and 0.5. Figure 9(a) shows the comparison of the difference in absorbed energy when the different failure strain values are used. Note that there is no difference for all the case when the values of 0.40 and 0.50 are used for the failure strain of elements, and very small difference for the majority of the cases when 0.35 is used. Four difference cases have been carried out to study the effect of friction coefficients of the steel ring/projectile to fabric, and the fabric to fabric, as listed in Table 2. The difference in the absorbed energy varies when different coefficients of friction are used in the simulation, as shown in Figure 9(b). Note that the case LG656 is most sensitive to the friction coefficient compared with other cases, followed by LG427 and LG594. Table 2 Case study of the effect of friction coefficients case 1

case 2

case 3

case 4

fabric-to-fabric

0.2

0.3

0.3

0.4

ring/projectile-to-fabric

0.1

0.1

0.2

0.2

257 45.0

Difference in Absorbed Energy, %

(a)

40.0

e_fail=0.35

35.0

e_fail=0.40

30.0

e_fail=0.50

25.0 20.0 15.0 10.0 5.0

0.0

60.0

Difference in Absorbed Energy, %

(b)

50.0

Case 1

40.0

Case 3

30.0

Case 2 Case 4

20.0

10.0 0.0 -10.0 -20.0

Fig. 9 Comparison of the difference in absorbed energy with different (a) failure strains and (b) coefficients of friction 4. Conclusions The continuum model developed in previous research has been improved by modifying the stress-strain constitutive behavior and the shear properties. The material model was validated by comparing the FE simulation with the NASA ballistic test results. Both single and multi-layer models were generated in the modeling configuration. The single layer model is computationally efficient and predicts the ballistic tests with less error than the multi-layer model, while the latter is able to consider the friction between fabric layers. The models are validated by comparing the residual velocity of the projectile and the absorbed energy by the fabric after the impact, and the temporal evolution and the spatial distribution of fabric deformation and damage. The sensitivity analysis shows that the material model is more sensitive to the coefficients of friction than to the failure strain within the investigated range. Acknowledgements The authors wish to thank William Emmerling, Donald Altobelli and Chip Queitzsch of the Federal Aviation Administration's Aircraft Catastrophic Failure Prevention Research Program for their support and guidance. Funding for this effort was provided by the FAA.

258

Reference [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

Grujicic M, Bell WC, Arakere G, He T and Cheeseman BA. A meso-scale unit-cell based material model for the single-ply flexible-fabric armor. Materials and Design, 30, 3690-3704, 2009. Stahleker Z, Sankaran S, Mobasher B, Rajan SD and Pereira JM. Development of reliable modeling methodologies for engine fan blade-out containment analysis, Part II: finite element analysis. Int J Impact Eng, 36(3), 447-459, 2009. Bansal S, Mobasher B, Rajan SD and Vintilescu I. Development of fabric constitutive behavior for use in modeling engine fan blade-out events. ASCE Journal of Aerospace Engineering, 22(3), 249-259, 2009. Sharda J, Deenadayalu C, Mobasher B and Rajan SD. Modeling of multi-layer composite fabrics for gas turbine engine containment systems. ASCE Journal of Aerospace Engineering, 19(1), 38-45, 2006. Naik D, Sankaran S, Mobasher B, Rajan SD and Pereira JM. Development of reliable modeling methodologies for engine fan blade-out containment analysis. Part I: experimental studies. Int J Impact Eng., 36(1), 1-11, 2009. Zhu D, Mobasher B, Rajan SD. Characterization of Mechanical Behavior of Kevlar 49 Fabrics. Society for Experimental Mechanics - Annual Conference & Exposition on Experimental and Applied Mechanics, Uncasville, CT, June 13-15, 2011. Amaniampong G and Burgoyne, CJ. Statistical variability in the strength and failure strain of aramid and polyester yarns. Journal of Material Science, 29, 5141-5152, 1994. Zhu D, Mobasher B and Rajan SD. Experimental study of dynamic behavior of Kevlar 49 single yarn. Society for Experimental Mechanics - Annual Conference & Exposition on Experimental and Applied Mechanics, v3, p. 542-547, 2010. Zhu D, Mobasher B, and Rajan SD. Dynamic tensile testing of Kevlar 49 fabrics. ASCE Journal of Materials in Civil Engineering, 2010, in press, http://dx.doi.org/10.1061/(ASCE)MT.1943-5533.0000156. Zhu D, Mobasher B and Rajan SD. High Strain Rate Testing of Kevlar 49 Fabric. Society for Experimental Mechanics - 11th International Congress and Exhibition on Experimental and Applied Mechanics, v1, p.34-35, 2008. Zhu D, Mobasher B and Rajan SD. Image Analysis of Kevlar 49 Fabric at High Strain Rate. Society for Experimental Mechanics - 11th International Congress and Exhibition on Experimental and Applied Mechanics, v2, p.986-991, 2008. Zhu D, Rajan SD, Mobasher B, Peled A and Mignolet M. Modal analysis of a servo-hydraulic high speed testing machine and its application to dynamic tensile testing at an intermediate strain rate. Experimental Mechanics, 2010, in press, http://dx.doi.org/10.1007/s11340-010-9443-2. Zhu D, Mobasher B and Rajan SD. Characterization of dynamic tensile testing using aluminum alloy 6061-T6 at intermediate strain rates. ASCE Journal of Engineering Mechanics 2010 (accepted). Zhu D, Peled A and Mobasher B. Dynamic tensile testing of fabric-cement composites. Construction and Building Materials, 25(1), 385-395, 2011. Zhu D, Mobasher B, Silva FA and Peled A. High Speed Tensile Behavior of Fabric-Cement Composites. International Conference on Material Science and 64th RILEM Annual Week, Proceedings pro075: Material Science - 2nd ICTRC - Textile Reinforced Concrete - theme 1, p. 205-213, 2010. Silva FA, Zhu D, Mobasher B, Soranakom C and Toledo Filho RD. High speed tensile behavior of sisal fiber cement composites. Materials Science and Engineering: A, 527(3), 544-552, 2009. Silva FA, Butler M, Mechtcherine V, Zhu D and Mobasher B. Strain rate effect on the tensile behaviour of textile-reinforced concrete under static and dynamic loading. Materials Science and Engineering: A, 528(3), 1727-1734, 2011. Mechtcherine V, Silva FA, Butler M, Zhu D, Mobasher B, Gao S and Mäder E. Behaviour of strainhardening cement-based composites under high strain rates. Journal of Advanced Concrete Technology, 9(1), 51-62, 2011. Duan Y, Keefe M, Bogetti TA and Cheeseman BA. Modeling friction effects on the ballistic impact behavior of a single-ply high-strength fabric. Int J Impact Eng., 31, 996-1012, 2005.

Optimal Pulse Shapes for SHPB Tests on Soft Materials

Mike Scheidler, John Fitzpatrick and Reuben Kraft Weapons and Materials Research Directorate U.S. Army Research Laboratory Aberdeen Proving Ground, MD 21005-5066 ABSTRACT For split Hopkinson pressure bar (SHPB) tests on soft materials, the goals of homogeneous deformation and uniform uniaxial stress in the specimen present experimental challenges, particularly at higher strain rates. It has been known for some time that attainment of these conditions is facilitated by reducing the thickness of the specimen or by appropriately shaping the loading pulse. Typically, both methods must be employed. Pulse shapes are often tailored to deliver a smooth and sufficiently slow rise to a constant axial strain rate, as this promotes equality of the mean axial stress on the two faces of the specimen, a condition referred to as dynamic equilibrium. However, a constant axial strain rate does not eliminate radial acceleration, which may result in large radial and hoop stresses and large radial variations in the radial, hoop and axial stresses. An approximate analysis (assuming homogeneous deformation and incompressibility) indicates that these radial inertia effects would be eliminated if the radial strain rate were constant. Motivated by this result, we consider loading pulses that deliver a constant radial strain rate after an initial ramp-up. The corresponding axial strain rate is no longer constant on any time interval, but for sufficiently thin specimens the resulting departure from dynamic equilibrium may be small enough to be tolerable. This is explored here by comparing the analytical predictions for the conventional and “optimal” loading pulse shapes with corresponding numerical simulations of SHPB tests on a soft, nearly incompressible material. 1. Introduction The split Hopkinson pressure bar (SHPB), also known as the Kolsky bar, is widely used to characterize the strain rate sensitivity of inelastic materials in a state of compressive uniaxial stress for moderate to large strains. The review articles by Gray [1], Gama et al. [2], and Ramesh [3] and the recent book by Chen and Song [4] are recommended for background on this technique. The standard specimen shape for SHPB tests is solid disk, which (particularly for soft specimens) is relatively thin compared to the diameter in order to reduce axial inertia effects. The analysis and simulations in this paper are confined to this specimen geometry. The standard analysis of SHPB test data relies on the assumption of uniform uniaxial stress and uniform biaxial strain throughout the specimen. These conditions, as well as a nearly constant nominal strain rate, can often be achieved after an initial ring-up. However, a number of technical challenges arise when the specimen is extremely soft; cf. [1]–[4] and also Gray and Blumenthal [5], Chen et al. [6], Song and Chen [7], Moy et al. [8], Song et al. [9], and Sanborn [10]. Inertial effects in soft specimens may result in highly non-uniform conditions and a stress state that is far from uniaxial, particularly if the strain rate is sufficiently high. To minimize the effects of axial inertia, loading pulse shapes are typically tailored to deliver a smooth and sufficiently slow rise to a constant nominal (or engineering) axial strain rate. This promotes axial uniformity of the stress and strain components and, in particular, equality of the mean axial stress on the two faces of the specimen, a condition referred to as dynamic equilibrium. The constant axial strain rate condition also simplifies calibration of constitutive models for which the stress depends explicitly on the strain rate, e.g., plasticity models with rate dependent yield stress. However, a constant nominal axial strain rate does not eliminate the radial acceleration of the specimen, which may result in large radial and hoop stresses and large radial variations in the radial, hoop and axial stresses. Since only the mean axial stress is measured in an SHPB test, the full stress state cannot be inferred in such cases; in particular, the deviatoric stress (which typically exhibits the largest rate effects) cannot be determined. Consequently, the measured axial stress is not useful for constitutive model calibration unless the data can be “corrected” for radial inertia effects. An approximate analysis (assuming an incompressible specimen that deforms homogeneously) indicates that the radial inertia effects would be eliminated if the nominal radial strain rate were constant. Motivated by this result, we T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_37, © The Society for Experimental Mechanics, Inc. 2011

259

260 consider loading pulses that deliver a constant nominal radial strain rate after an initial ramp-up. The corresponding axial strain rate is no longer constant on any time interval; in fact, it is necessarily decreasing on the time interval over which the radial strain rate is constant. But for sufficiently thin specimens the resulting departure from dynamic equilibrium may be small enough to be tolerable. This is explored here by comparing the analytical predictions for the conventional and “optimal” loading pulse shapes with corresponding numerical simulations of SHPB tests. An inertial correction for the deviatoric stress is also compared with the deviatoric stress in the simulation. The analysis on which these the inertial corrections and the optimal loading pulse shapes are based applies only to nearly incompressible specimens, that is, materials for which the bulk modulus is several orders of magnitude larger than the shear modulus. This is not necessarily a drawback, since radial inertia effects are expected to be more pronounced for nearly incompressible specimens. Examples of soft, nearly incompressible materials include many rubbers, biological materials with high fluid content (in particular, brain tissue), and tissue surrogates such as gelatins. The analysis assumes that the specimen is isotropic, although no particular form for the constitutive relation is required. On the other hand, an isotropic, nonlinear elastic constitutive relation was used for the specimen in the numerical simulations, namely, a compressible version of the Mooney-Rivlin model with the bulk modulus of water and a shear modulus typical of a gelatin, so that the ratio of bulk to shear modulus exceeds 104 . 2. Homogeneous, Axisymmetric Deformations Our analysis is confined to the deformation and stress in the specimen, which is assumed to be a (relatively thin) solid disc. We use a cylindrical coordinate system aligned with the axis of symmetry of the specimen and the pressure bars. Coordinates of the specimen in the undeformed reference configuration are denoted by (R, Θ, Z); coordinates of the corresponding material point in the deformed state are denoted by (r, θ, z). For a general axisymmetric deformation, r = rˆ(R, Z, t) , θ = Θ , z = zˆ(R, Z, t) , (1) where t denotes time. The radial, hoop, axial and shear components of the Cauchy or true stress tensor σ (taken positive in compression) are denoted by σrr , σθθ , σzz and σrz . The other two stress components, σrθ and σzθ , are zero for axisymmetric deformations since the specimen is assumed to be isotropic, but in general (e.g., if the specimen bulges) σrz need not be zero. The Cauchy stress measures force per unit deformed area. The nominal (engineering, 1st Piola-Kirchhoff) stress tensor Σ measures force per unit original (undeformed) area. The axial and radial components of this tensor (which are the only ones referred to in the sequel) are denoted by ΣzZ and ΣrR and are likewise taken positive in compression. We make the approximation that the specimen deformation is also homogeneous, in which case (1) reduces to r = λ r (t)R ,

θ = Θ,

z = λz (t)Z + z0 (t) .

(2)

The initial radius of the specimen is denoted by R0 , and the deformed radius at time t is denoted by r0 (t): r0 (t) = λ r (t)R0

and

r/r0 (t) = R/R0 .

(3)

For a material point located initially at radius R, the deformed radius r at time t is given by (2)1 , so the radial velocity and acceleration of the specimen at this radial location are given by

. .r = λ. r (t) R = λ r (t) r ,

.. ..r = ..λ r (t) R = λ r (t) r ,

λ r (t)

λ r (t)

(4)

where a superposed dot denotes a time derivative. At any instant t, the radial velocity and acceleration of the specimen increase linearly with the radius. They are zero on the axis and attain their largest absolute value at the (stress-free) lateral surface, where r 0 (t) = λ r (t)R0 and r 0 (t) = λ r (t)R0 . From the relations (2), we have

.

.

λr =

..

∂r r = =: λ θ ∂R R

..

and

λz =

∂z . ∂Z

(5)

It follows that λ r , λ θ and λz are the stretches (local ratios of deformed to undeformed length) in the coordinate directions. Since we are concerned with possibly large specimen deformations, we must be careful to distinguish between various finite deformation measures of strain. The two most commonly used measures in the Hopkinson

261 bar literature are the nominal (or engineering) strain, which is the change in length per unit original length, and the logarithmic (or true) strain. The nominal and logarithmic measures of radial, hoop and axial strain are defined by er = 1 − λ r ,

eθ = 1 − λ θ ,

e z = 1 − λz ,

(6)

εr = − ln λ r ,

εθ = − ln λ θ ,

εz = − ln λz ,

(7)

and respectively. Since the radial and hoop stretches are equal, so are the corresponding strains: eθ = e r

and

εθ = εr ;

(8)

so we will focus on the radial and axial stretches and strains. The nominal and logarithmic strain measures approach each other in the small strain limit, but differ substantially for large strains. We will work primarily with the nominal strains in this paper, as this simplifies most of the results. Note that stretches are 1 in the undeformed state, greater than 1 in extension, and between 0 and 1 in compression. The strains have been defined so that they are positive if the particular coordinate direction is in compression and negative if it is in extension.1 In an SHPB test the specimen is compressed in the axial direction, so that 0 < λz < 1 and 0 < ez < 1. Since the lateral surface is unconfined, the specimen is free to expand radially and we expect that λ r > 1 and hence er < 0. From (6) and (7), we see that the axial strain rates and stretch rates are related by

. .εz = − λz = e.z

. . ez = −λz ,

λz

1 − ez

.

(9)

Analogous relations hold for the radial strain and stretch rates. Now consider the stress state. Let p denote the pressure and s the deviatoric part of the Cauchy stress tensor. Then σrr = srr + p ,

σθθ = sθθ + p ,

srr + sθθ + szz = 0 ,

σzz = szz + p ,

p = 13 (σzz + σrr + σθθ ) .

(10) (11)

Since the radial and hoop stretches coincide and the specimen is isotropic, the radial and hoop stresses are equal and the shear stress is zero: σrr = σθθ , σrz = 0 . (12) Thus a homogeneous axisymmetric deformation results in a biaxial stress and strain state; the principal axes of stress and strain are the axis of symmetry and any axes orthogonal to it. In this case, the relations (11) simplify to srr = sθθ = − 12 szz ,

p = 13 (σzz + 2σrr ) .

(13)

On substituting the relation (13)2 for p into the relation (10)3 for σzz , we obtain the equivalent relations σzz = 32 szz + σrr ,

szz = 23 (σzz − σrr ) .

(14)

For axisymmetric deformations of an isotropic specimen, the balance of radial momentum is given by2

..

σrr − σθθ ∂σrz ∂σrr + + = −ρ r , ∂r r ∂z

(15)

where ρ is the density in the deformed state. In view of (12), for homogeneous deformations this reduces to

..

∂σrr = −ρ r . ∂r

(16)

Since the lateral surface of the specimen is stress free, σrr = 0 when r = r0 (t). Then on integrating (16) from an arbitrary radius r to r0 (t) and using (4) and (3), we obtain [ [ )2 ] ( ( )2 ] r R σrr = 2 σ rr (t) 1 − = 2 σ rr (t) 1 − , (17) r0 (t) R0 1 The standard sign convention in the continuum mechanics literature takes stress and strain components as positive in tension. The opposite sign convention is typically used in the compression Hopkinson bar literature, and we have followed that convention here. 2 The minus sign on the right side of (15) is a consequence of the convention that stresses are positive in compression.

262 where

..

ρR02 λ r (t) λ r (t) . (18) 4 Working directly with (17), we find that σ rr (t) is the mean value of σrr over the deformed volume of the specimen at time t. Since the deformation is homogeneous, this is also the mean value of σrr over the undeformed reference configuration. And since σrr is axially uniform, σ rr is also the mean value of σrr over any deformed (z = constant) or undeformed (Z = constant) cross-sectional area. Since σθθ = σrr , we conclude that the radial and hoop stresses vary quadratically with the radius but are axially uniform. They attain a peak absolute value of 2σ rr (t) at the centerline (r = R = 0) and reduce to zero at the lateral surface. They have the same sign as the radial stretch acceleration λ r , but without additional assumptions neither the sign nor the magnitude of λ r can be inferred. In any case, it is clear that the presence of nonzero radial and hoop stresses is a radial inertial effect, that is, a consequence of the fact that the ρ r term in (16) is not necessarily negligible. It follows that in the quasi-static limit, that is, in the limit as r (or λ r ) approaches zero, σ rr (t) = 0. Hence, as expected, in a quasi-static test we have a uniaxial stress state (σrr = σθθ = 0), and by (13)2 and (14)2 the pressure and deviatoric stress are completely determined by the axial stress: p = 13 σzz and szz = 23 σzz . On the other hand, it is clear from (13)2 and (14)2 that these quasi-static estimates for p and szz will be in error if σrr is sufficiently large relative to σzz . σ rr (t) =

..

..

..

.. ..

3. The Incompressibility Approximation The Jacobian of the deformation, denoted by J, is the determinant of the deformation gradient F and represents the local ratio of deformed to undeformed volume. For a general deformation, J is the product of the principal stretches; for a homogeneous axisymmetric deformation this yields J = λ r λ θ λz = λ r2 λz . Since the focus of this paper is on nearly incompressible specimens, we will make the approximation that the deformation is volume-preserving. Then J = 1, which is equivalent to any of the following relations: 1 λr = √ , λz

λz =

1 , λ r2

εr = − 12 εz .

(19)

Thus the radial stretches or strains are determined by axial stretches or strains, and vice versa. In this case the true and nominal stress components are related by3 σzz = λz ΣzZ = (1 − ez )ΣzZ ,

σrr = λ r ΣrR .

(20)

Now recall the relations (9) between the axial components of strain rate and stretch rate (with analogous relations for the radial components). By (19) we obtain the following additional relations between the radial and axial rates:

. −λ.z . −er = λ r =

3/2 2λz

.

=

ez 3/2 2λz

.

ez = , 2(1 − ez )3/2

. .ez = −λ.z = 2λ r ,

.

.

εr = − 12 εz .

λr3

(21)

From the relations on the left in (21), we find that the radial strain and stretch accelerations are given in terms of the axial stretch or strain rates by ( )2 ( )2 ez 1 λz 3 λz 1 3 ez + = + . (22) − er = λ r = − 2 λz3/2 4 λz5/2 2 (1 − ez ) 3/2 4 (1 − ez ) 5/2

.. ..

We also have

.

..

..

1 λr = √ λz

.

..

[

] 1 ( )2 1 εz + εz , 2 4

..

.

(23)

although this relation is less useful than (22). For later use in the discussion of optimal pulse shapes, we note that the nominal axial strain acceleration is given in terms of the radial stretch rates by ( )2 λr λr ez = − λz = 2 3 − 6 . (24) λr λ r4

..

..

..

.

Now consider the radial stress. Since the deformation is volume-preserving, ρ = ρ0 , the density in the undeformed state. On substituting this and the relations (19)1 and (22) for λ r and λ r into (18), we obtain the following

..

3 These

relations follow from the general relation σ = J −1 ΣF T , or from the relations between deformed and undeformed area.

263 7 ´ 107

5000 •

Strain Acceleration Hs-2 L

Strain Rate Hs-1 L

••

Axial strain acceleration ez

6 ´ 107

Axial strain rate ez 4000

•

Radial stretch rate Λr

3000

2000

1000

••

Radial stretch acceleration Λr

5 ´ 107

••

Normalized radial stress Λr Λr 4 ´ 107 3 ´ 107 2 ´ 107 1 ´ 107

0

0

50

100

150

200

250

Time HΜsL

(a)

0

300

0

50

100

150

200

250

300

Time HΜsL

(b)

Figure 1: Axial strain and radial stretch rates (a) and accelerations (b) for a smooth ramp-up to a constant nominal axial strain rate ez of 2500/s after 135 µs. Also shown in (b) is the normalized mean radial stress (dashed line).

.

expressions for the mean radial (and hoop) stress in an isotropic specimen undergoing a homogeneous, volumepreserving, axisymmetric deformation: [ ( )2] ] [ 3 λz ρ0 R02 2 ez ρ0 R02 ρ0 R02 −2 λz 3 (ez )2 = . (25) λ r (t) λ r (t) = σ rr (t) = + + 4 16 λz2 λz3 16 (1 − ez )2 (1 − ez )3

.

..

..

..

.

The radial (and hoop) stress distribution is then given by (17). For a given radial or axial strain history, we see that the mean radial stress at any instant is proportional to ρ0 R02 . Thus radial inertial effects can be reduced by decreasing the radius of the specimen.4 Note that the only material property appearing in (25) is the density ρ0 ; in particular, this estimate for the radial stress is independent of the constitutive relation for the specimen. The relation (17), with σ rr (t) given by the expression on the right in (25), is equivalent to relations in Dharan and Hauser [11], Warren and Forrestal [12], and Scheidler and Kraft [13]. Now consider a conventional smooth loading pulse for an SHPB test with rise time t1 . We take time t = 0 to be the instant at which the loading pulse arrives at the specimen-incident bar interface, so that ez (0) = 0. For a smooth loading pulse the nominal axial strain rate and strain acceleration are initially zero also, ez (0) = 0 and ez (0) = 0; and ez (t) increases smoothly with t up to time t1 , after which ez is remains constant at the test strain rate ez1 = ez (t1 ) > 0.5 At early times, the strain acceleration term in (25), ez , dominates as ez increases from its initial value of zero; ez eventually reaches a peak (positive) value and then decays to zero as ez approaches the plateau strain rate ez1 . This peak in ez results in a corresponding early peak in the radial stress. However, while the radial stress initially decreases after this peak, it does not decay to zero with ez since the (ez )2 term in (25) is positive. In fact, since ez is increasing, the (1 − ez )3 term in the denominator is decreasing, and hence the 3(ez )2 /(1 − ez )3 term is strictly increasing, even in the plateau region when ez (t) = ez1 . Consequently, σ rr (t) begins to increase just prior to time t1 and continues to do so while ez remains constant, i.e., there is a strain amplification effect on the inertially generated radial stress. These features are illustrated in Figure 1, where the rise time t1 = 135 µs and the plateau value of the nominal axial strain rate is ez1 = 2500/s. For this strain rate history, ez (t1 ) = 0.17 and ez (300µs) = 0.58. Note that by (25)1 , the product λ r λ r is the mean radial stress σ rr normalized by the factor ρ0 R02 /4; it is this normalized mean radial stress that is plotted in the figure. Also note that by (25)1 and (20)2 , the radial stretch acceleration λ r is the mean value of the nominal radial stress ΣrR normalized by the same factor.

.. .

.

.

. ..

.

. ..

..

..

.

.

. .

.

.

.

. . ..

..

For sufficiently soft materials and sufficiently high strain rates, the inertial effects discussed above must be taken into account when analyzing the data from SHPB tests. In this regard, the following relations for the mean values of the stress components are useful: σ zz = 32 szz + σ rr ,

szz = 23 (σ zz − σ rr ) ,

σ zz = (1 − ez )ΣzZ .

(26)

4 However, reducing the specimen radius also reduces the signal to the transmission bar, so substantial reduction in specimen size must be accompanied by a corresponding reduction in the diameter of the pressure bars. This is one of the motivations for the use of miniaturized Hopkinson bars. 5 For simplicity, we neglect the subsequent unloading stage in both the analysis and the numerical simulations.

264 4 ´ 107

Strain Acceleration Hs-2 L

Strain Rate Hs-1 L

2500

2000

1500

1000

•

Axial strain rate ez •

Radial stretch rate Λr

500

0

0

50

100

150

200

250

300

Time HΜsL

(a)

••

Radial stretch acceleration Λr

2 ´ 10

7

••

Normalized radial stress Λr Λr

1 ´ 107 0 - 1 ´ 107 - 2 ´ 107

350

••

Axial strain acceleration ez

3 ´ 107

0

50

100

150

200

250

300

350

Time HΜsL

(b)

Figure 2: Axial strain and radial stretch rates (a) and accelerations (b) for a smooth ramp-up to a constant nominal radial stretch rate of 1747/s after 135 µs. Also shown in (b) is the normalized mean radial stress (dashed line). These follow from (14) and (20)1 , and may be regarded either as volumetric averages or as cross-sectional averages; the two are equivalent if, as will be assumed here, the specimen is in dynamic equilibrium so that the axial stresses are axially uniform. The well-known relation on the right expresses the mean axial Cauchy stress in terms of the axial strain and the mean axial nominal stress, both of which are measured (or inferred from measurements) in an SHPB test. The relation on the left in (26) implies that if the early spike in σ rr is sufficiently large relative to szz (a situation that could occur for sufficiently high strain rates and sufficiently soft specimens), then a corresponding spike in the measured axial stress is to be expected. These inertial spikes have indeed been observed in SHPB tests on soft, nearly incompressible materials; cf. [4], [8], [9], and [10]. The middle relation in (26), which is equivalent to the relation on the left, indicates that an “inertial correction” must be applied to the (quasi-static) uniaxial stress relation szz = 23 σzz when σ rr is sufficiently large. Since σ rr can be determined from the measured axial strain rate ez by (25)3 , the relation (26)2 could provide a means to estimate the axial component of the deviatoric stress,6 provided the assumptions on which our analysis is based are approximately valid and the difference between σ zz and σ rr is not so small that it is in the noise level. Then the radial and hoop components of deviatoric stress can be determined from (13)1 .

.

4. Optimal Loading Pulses Since a constant nominal axial strain rate does not eliminate inertially generated radial and hoop stresses, it is reasonable to seek axial strain rate histories that do so. On setting σ rr (t) = 0 in (25), we see that the expression on the right yields an ODE for ez : 2 ez + 3(ez )2 /(1 − ez ) = 0. However, it is simpler to proceed as follows. From (25)1 we see that σ rr = 0 iff λ r = 0 iff λ r is constant iff er is constant. Since the loading pulse arrives at t = 0 and since λ r = 1 in the undeformed state, we take λ r (0) = 1. For a smooth loading pulse we cannot impose the constant radial strain rate condition initially. Instead we want λ r (0) = 0 and λ r (0) = 0, with λ r (t) increasing smoothly and monotonically with t up to some time t1 , after which λ r remains constant:

..

..

.

.

.

. .

.

..

.

λ r (t) = λ r (t1 ) > 0 ,

.

for t ≥ t1 .

.

(27)

..

.

Then λz (t) = 1/λ r2 [cf. (19)2 ], and from λz we can determine ez , ez and ez ; alternatively, we can determine ez and ez directly from λ r and its rates by using (24) and the middle relation on (21). Since λ r (t) is positive for t > 0, so is ez (t) [cf. (21)]. However, λ r (t) = 0 for t ≥ t1 , so by (24) we see that ez (t) < 0 for t ≥ t1 ; hence ez (t) is decreasing for t ≥ t1 . Since the initial condition λ r (0) = 0 implies ez (0) = 0, it follows that ez (t) must increase from zero to a peak value at some time tp < t1 , after which ez decreases. In spite of this non-standard feature, the resulting nominal axial strain rate history ez (t) will have the property that for t ≥ t1 , σ rr (t) = 0 and hence [cf. (17)] σrr (r, z, t) = 0 throughout the specimen.

..

..

.

.

.

.

.

..

.

.

.

6 This is essentially what was done in Sanborn [10] for SHPB tests on various rubbers. His “corrected” axial stress is σ zz − σ rr and hence is equivalent to the 32 szz term in (26)1 .

265

Stress (Pa)

x 10

5

8

σzz

6

σθθ

σrr

1.5 szz

4 2 0 150

200

250

300 350 Time (µs)

400

450

Figure 3: Mean values of the axial, radial and hoop stress and of 3/2 the axial deviatoric stress, for the axial strain rate history in Figure 1. The radial and hoop stresses are indistinguishable. For comparison with the more conventional strain rate history considered in Figure 1, we take the rise time to the constant radial strain rate to be the same as the previous rise time to the constant axial strain rate, namely t1 = 135 µs; and we chose the plateau value λ r (t1 ) = 1747/s, since this results in ez (t1 ) = 2500/s. The resulting axial strain and radial stretch rates are shown in Figure 2.a. The corresponding strain and stretch acceleration histories are shown in Figure 2.b along with the normalized mean radial stress, which is nearly indistinguishable from λ r . For this strain rate history, ez (t1 ) = 0.20 and ez (350µs) = 0.55.

.

.

..

9. Numerical Simulations To test the optimal pulse shaping and the inertial correction theory, we performed numerical simulations of hypothetical SHPB tests on a soft, nearly incompressible, solid specimen using the Lagrangean, 3-D finite element code PRESTO from Sandia Laboratories. The initial radius of the specimen was R0 = 6.35 mm and the initial thickness was L0 = 1.45 mm, giving a length-to-diameter ratio of 0.11. The incident and transmission bars were included in the simulation, and the specimen-bar interfaces were treated as frictionless. We used 1000 mm long aluminum bars with a radius of 19.05 mm. The specimen and bar dimensions (except for the bar lengths) were taken from the experimental study on gelatins by Moy et al. [8]. An isotropic, nonlinear elastic model was used for the specimen (a compressible version of the Mooney-Rivlin model). The model was calibrated to give rough agreement with the large strain, quasi-static, uniaxial compression data on the 20% ballistic gelatin tested in [8].7 This resulted in a small strain shear modulus of 80 kPa. Since 20% ballistic gelatin is 80% water, we used the bulk modulus of water, 2.3 GPa; the ratio of bulk to shear modulus is 2.9 × 104 . The loading wave was generated by an imposed axial velocity history vz (t) at the far end of the incident bar. This velocity was obtained from the desired nominal axial strain rate ez using the approximate relation

.

.

vz (t) ≈ 12 L0 ez (t) ,

(28)

which neglects the motion of the specimen-transmission bar interface and assumes the particle velocity doubles on reflection from the specimen-incident bar interface. The nominal axial strain rate histories used in computing vz (t) were those from Figures 1 and 2, but since the relation (28) is only approximate, the actual strain rates and strain accelerations in the specimen would differ somewhat from those in the figures even if the specimen deformation were approximately homogeneous. Figures 3 and 4 plot the histories of σ zz , σ rr , σ θθ and

3 2 szz .

These are the (volumetric) mean values of the radial,

7 Ballistic gelatin is a viscoelastic material, and the quasi-static tests showed some increase in stress with increasing strain rate. The model was calibrated so that the axial stress-strain curve was slightly above the curve for highest (quasi-static) strain rate of 1/s.

266 5

2.5

x 10

Stress (Pa)

2 1.5

σzz σrr

1

σθθ

0.5

1.5 szz

0 −0.5 150

200

250

300 350 Time (µs)

400

450

Figure 4: Mean values of the axial, radial and hoop stress and of 3/2 the axial deviatoric stress, for the axial strain rate history in Figure 2. The radial and hoop stresses are indistinguishable. hoop and axial stress and 3/2 the mean value of the axial deviatoric stress, as computed in the simulations. Recall that σ zz = 32 szz for a uniaxial stress state; cf. (26)1 with σ rr set to zero. The times in Figures 3 and 4 are shifted relative to those in Figures 1 and 2, since in the simulations t = 0 is the instant at which the velocity is applied at the far end of the incident bar. Figure 3 is for a nominal axial strain rate history given (approximately) by that in Figure 1.a, that is, when ez is eventually constant. The radial and hoop stresses are indistinguishable, and they increase substantially after the inertial spike. The compressed specimen began squeezing out beyond the bars at around 453 µs.

.

.

Figure 4 is for a nominal axial strain rate history given (approximately) by that in Figure 2.a, that is, when λ r is eventually constant (an optimal pulse shape). The radial and hoop stresses are again indistinguishable, but now they drop to nearly zero after the inertial spike, that is, once λ r reaches its plateau value, which occurs at a nominal strain of 0.20. Thereafter, σ zz is very close to 32 szz . Both of these facts indicate that a nearly uniaxial stress state has been achieved. When comparing Figures 3 and 4, keep in mind that the scales on the vertical axis are different. The inertial spike in Figure 4 is only slightly larger than that in Figure 3. For the case considered in Figures 1 and 3,

.

x 10

5

s

Stress (Pa)

8

zz

szz|uniaxial

szz|corrected

6 4 2 0 150

200

250

300 350 Time (µs)

400

450

Figure 5: The computed mean axial stress (szz ) compared with two estimates for it. Figure 5 provides a check on the inertial correction (26)2 for the axial deviatoric stress: szz = 23 (σ zz − σ rr ). In the

267 figure legend, szz denotes the mean axial deviatoric stress szz as computed in the simulation. szz |uniaxial denotes an estimate for the deviatoric stress that is only valid for a uniaxial stress state, namely 23 σ zz . It clearly disagrees with szz and shows the error that would result in neglecting inertial effects in the analysis of the data. Finally, szz |corrected denotes the estimate for the deviatoric stress obtained from the inertial correction above, using σ zz computed in the simulation and σ rr determined from (25)3 . It is quite close to the actual mean value for most of the simulation. Near the end of the simulation the conditions departed substantially from those on which the analysis was based: the specimen bulged and was no longer in dynamic equilibrium. 10. Discussion and Conclusions The results of the analysis and the preliminary numerical simulations indicate that radial inertial effects in SHPB tests on soft, nearly incompressible materials could be eliminated (after a preliminary inertial spike) by tailoring the loading pulse so that the nominal radial strain rate in the specimen is eventually constant. The corresponding axial strain rate history is easily determined analytically and can be easily (though only approximately) imposed computationally. Whether such loading pulses can be generated experimentally with use of pulse shapers remains to be seen. Recently, Casem [14] has developed a technique for tailoring loading pulse shapes by means of graded impedance striker bars. This method, in conjunction with conventional pulse shapers, may possibly provide a means to generate the pulse shapes considered here. References [1] Gray III, G. T., Classic Split-Hopkinson Pressure Bar Testing. In H. Kuhn and D. Medlin, editors, ASM Handbook Vol. 8, Mechanical Testing and Evaluation, pages 462–476. American Society for Metals, Materials Park, Ohio, 2000. [2] Gama, B. G., Lopatnikov, S. L. and Gillespie Jr., J. W., Hopkinson bar experimental technique: A critical review. Appl. Mech. Rev., 57:223–250, 2004. [3] Ramesh, K. T., High strain rate and impact experiments. In W. N. Sharpe, editor, Springer handbook of Experimental Solid mechanics, chapter 33, pages 1–30. Springer, New York, 2009. [4] Chen, W. and Song, B., Split Hopkinson (Kolsky) Bar. Springer, New York, 2011. [5] Gray III, G. T. and Blumenthal, W. R., Split-Hopkinson Pressure Bar Testing of Soft Materials. In ASM Handbook Vol. 8, Mechanical Testing and Evaluation, pages 488–496. American Society for Metals, Materials Park, Ohio, 2000. [6] Chen, W., Lu, F., Frew, D. J. and Forrestal, M. J., Dynamic compression testing of soft materials. Exp. Mech., 69:214–223, 2002. [7] Song, B. and Chen, W., Split Hopkinson pressure bar techniques for characterizing soft materials. Latin Am. J. Solids Struct., 2:113–152, 2005. [8] Moy, P., Weerasooriya, T., Juliano, T. F., VanLandingham, M. R. and Chen, W., Dynamic Response of an Alternative Tissue Simulant, Physically Associating Gels (PAG). In Proc. of the 2006 SEM Annual Conference, 2006. [9] Song, B., Ge, Y., Chen, W. W. and Weerasooriya T., Radial Inertia Effects in Kolsky Bar Testing of Extra-soft Materials. Exp. Mech., 47:659–670, 2007. [10] Sanborn, B., An Experimental Investigation of Radial Deformation of Soft Materials in Kolsky Bar Experiments. Master’s thesis, Purdue Univ., Weast Lafayette, IN, 2010. [11] Dharan, C. K. H. and Hauser, F. E., Determination of stress-strain characteristics at very high strain rates. Exp. Mech., 10:370–376, 1970. [12] Warren, T. L. and Forrestal, M. J., Comments on the effect of radial inertia in the Kolsky bar test for an incompressible material. Exp. Mech., 50:1253–1255, 2010.

268 [13] Scheidler, M. and Kraft, R., Inertial effects in compression Hopkinson bar tests on soft materials. In Proceedings of the 1st ARL Ballistic Protection Technologies Workshop, 2010. [14] Casem, D. T., Hopkinson bar pulse-shaping with variable impedance projectiles—an inverse approach to projectile design. Technical Report ARL-TR-5246, US Army Research Laboratory, 2010.

Dynamic Tensile Characterization of Foam Materials

Bo Song, Helena Jin, Wei-Yang Lu Sandia National Laboratories, Livermore, CA 94551-0969, USA Polymeric foams have been widely utilized in packaging and transportation applications due to their light weight but superior energy absorption capabilities. Here the superior energy absorption is usually recognized when the foams are subjected to compression. The polymeric foams possess load-bearing capability under large deformation in compression; whereas, such a load-bearing capability may be lost when the foams are subjected to tensile loading [1]. Dynamic compressive response of foam materials has been extensively characterized, mostly with Kolsky compression bar techniques. However, dynamic tensile characterization of foam materials has 3 been less touched due to the difficulties in dynamic tension techniques. In this study, we employed a 0.26×10 3 kg/m PMDI foam as an example to explore the dynamic tensile characterization of foam materials with our newly developed Kolsky tension bar [2, 3]. The Kolsky tension bar developed at Sandia National Laboratories, California has been presented at 2010 SEM Annual Conference [2]. This Kolsky tension bar has been utilized to characterize alloys, e.g., 4330-V steel [3]. However, the techniques used for the alloys may not be directly transferred to polymeric foams due to drastic difference in mechanical characteristics between alloys and polymeric foams. Characterization of foam materials always challenges experimental techniques even in dynamic compression tests. Dynamic tensile characterization of foam materials is much more challenging. For example, the foam tensile specimen needs to be sufficiently big. The large diameter requires a Kolsky bar system with a larger diameter, while a longer gage section may not satisfy the uniform deformation requirement in Kolsky bar experiments. Therefore, the specimen dimension needs to be carefully determined. Unlike the button specimen sandwiched between the bars in Kolsky compression bar tests, the foam tensile specimen needs to be firmly attached to the ends of both bars. Stress concentration is another concern in foam tensile characterization. Both require additional attention in specimen design. In this study, we designed the foam specimen shown in Fig. 1 for Kolsky tension bar experiments. It is noted that the diameter for the threads portion of the foam specimen is 25.4 mm, which is larger than the bar diameter of 19.1 mm. We employed a pair of couplers to attach the foam specimen to the bar ends, the photograph of which is shown in Fig. 2.

Fig. 1. Foam tensile specimen

Fig. 2. Photograph of the testing section

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_38, © The Society for Experimental Mechanics, Inc. 2011

269

270 As shown in Fig. 1, the foam tensile specimen has a gage length of 6.35 mm. This gage length was determined based on the preliminary tests. In preliminary experiments, we applied digital image correlation (DIC) techniques on a 9.53-mm-long foam specimen to estimate the elastic wave speed and to monitor the uniformity of deformation in the foam specimen. DIC analysis yields the elastic wave speed of approximately 525 m/s in the foam specimen. This elastic wave speed is important to synchronize stress and strain histories in data reduction. Figure 3 shows the deformation history of the 9.53-mm-long foam specimen obtained from DIC analysis. The DIC results show that the 9.53-mm-long foam specimen did not achieve uniform deformation until t = 46.2 µs. After 46.2 µs, the specimen is nearly under uniform deformation until macroscopic crack initiation at t = 79.2 µs. To be conservative, we reduced the specimen gage length from 9.53 mm to 6.35 mm, which enables earlier uniform deformation (or stress equilibrium). It is noted that the classic “2-wave” “1-wave” method may not be applicable to verify stress equilibrium in the foam experiment due to the weak transmitted signal and unreliable reflected pulse. The DIC analysis presented here becomes an effective alternative method to monitor the process of deformation uniformity in the foam specimen.

T = 0 µs

T = 19.8 µs

T = 39.6 µs

T = 46.2 µs

T = 72.6 µs

T = 79.2 µs

Fig. 3. Axial Deformation (Exx) History in the 9.53-mm-long Foam Specimen Since the reflected pulse may not be reliable for accurate calculation of specimen strain [1], we employed a laser-beam system, as shown in Fig. 2, to directly measure the incident bar end displacement. The specimen strain can be calculated with the following equation, t

ε (t ) =

c′ ⋅ (∆L(t ) − C0 ∫ ε T (τ )dτ ) Ls

0

(1)

where ΔL is the laser beam output; εT is transmitted signal; C0 is the steel bar elastic wave speed; Ls is the specimen gage length, Ls = 6.35 mm; c’ is correction coefficient. As shown in Fig. 1, the foam specimen has a dumbbell shape consisting of a gage section and a transition (non-gage section) from threads to the gage section. When the specimen is subjected to tension, both gage and non-gage sections are elongated. The term, t

ΔL(t ) − C0 ε T (τ ) dτ , in Eq. (1) describes the overall displacement over both gage and non-gage sections.



0

However, the displacement of the non-gage section should not be accounted for the strain calculation of the specimen. Here, the coefficient, c’, is used to correct this effect. If we assume the foam specimen is always in linear elasticity and in stress equilibrium, the coefficient c’ was approximated as a constant, c’=0.4, for the foam

271

specimen specified in Fig. 1. This means only 40% of the overall deformation contributes to the deformation over the gage section. It is noted that, in order to measure the weak transmitted pulse, we employed a pair of semiconductor strain gages on the transmission bar. Figure 4 shows the oscilloscope records of the incident, reflected, and transmitted signals (Fig. 4(a), as well as the laser-beam system output (Fig. 4(b)).

(a)

(b) Fig. 4. Oscilloscope records

Figure 5 shows the resultant tensile stress3 3 strain curve of the 0.26×10 kg/m PMDI foam -1 specimen at the strain rate of 400 s . The foam specimen exhibits a nearly linear elastic behavior with a small failure strain of about 1.5%. It is clearly shown that the tensile stress-strain response of the foam specimen is quite different from the compressive response. The “classic” compaction response in compression changes to be brittle in tension for the PMDI foam material.

ACKNOWLEDGEMENTS Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

Fig. 5. Tensile stress-strain curve at 400 s

-1

REFERENCES nd

1. Gibson, L. J., and Ashby, M. F., 1999, Cellular Solids, Structure and Properties, 2 ed, Cambridge. 2. Song, B., Antoun, B. R., Connelly, K., Korellis, J., and Lu, W.-Y., 2010, “A Newly Developed Kolsky Tension Bar,” In: Proceedings of 2010 SEM Annual Conference and Exposition on Experimental and Applied Mechanics, Indianapolis, IN, June 7-10, 2010. 3. Song, B., Antoun, B. R., Connelly, K., Korellis, J., and Lu, W.-Y., 2011, “Improved Kolsky Tension Bar for High-rate Tensile Characterization of Materials,” Measurement Science and Technology, 22, 370304 (7pp), in press

On Measuring the High Frequency Response of Soft Viscoelastic Materials at Finite Strains

Sean Teller, Rod Clifton, and Tong Jiao School of Engineering, Brown University 182 Hope St., Providence, RI 02912 *Corresponding author: [email protected] ABSTRACT In such applications as repairing damaged vocal folds to restore normal phonation, viscoelastic properties of normal tissues and candidate replacement materials should be measured at phonation frequencies (100 – 1000 Hz) and at the large strains (up to 30%) that occur during speech and singing. Previously the authors have developed a torsional wave experiment to measure the complex moduli, in shear, for vocal folds subjected to small strains at phonation frequencies. This method has now been extended to finite deformations by sandwiching a thin disk of vocal fold tissue (lamina propria), or replacement material, between two rigid plates. The lower plate is driven by a galvanometer at phonation frequencies and small rotations. A second stiffer material is placed between the upper plate and a third plate attached to an upper galvanometer that oscillates sinusoidally at low frequency and large rotation. At periodic peak rotations of the upper galvanometer, the lower galvanometer superimposes infinitesimal oscillations at a series of higher frequencies. The magnitude and phase of rotation of the middle plate yield the viscoelastic properties of the test specimen for infinitesimal deformations at high frequency superimposed on finite deformations at low frequency. Preliminary results show th e potential of the new test. INTRODUCTION Vocal folds are a soft, layered, viscoelastic mucous membrane that are stretched horizontally across the larynx. The folds a re composed of a vibratory layer, known as the lamina propria (LP), sandwiched between the vocalis muscle and an epithelium membrane [1,2]. During phonation, the LP are driven into a shear-dominated wavelike motion at frequencies between 75 and 1000 Hz and large strains of up to 30%[3]. Due to overuse, abuse, and surgery, the LP can become damaged, and scar tissue can form hindering the use of the vocal folds and adversely affecting voice quality[4,5]. Tissue engineering holds great promise for the repair and replacement of damaged tissue, but the unique bio chemical and mechanical properties of the vocal folds create challenges in the development of candidate replacement materials and testing of natural tissues. As demonstrated in [6], typical shear rheometry is not applicable for vocal fold tissues (and similar synthetic materials) due to the deformation not remaining uniform through the thickness of the sample at phonation frequencies. The Torsional Wave Experiment (TWE)[6] was developed to overcome these difficulties. It has enabled the authors to measure the complex modulus of soft synthetic materials [6-10] and natural tissues [10] over the lower part of the range of phonation frequencies. Although the TWE has proven useful, to completely characterize natural and synthetic materials the shear modulus at strains comparable to those experienced in situ are needed. To this end, consider the experimental setup shown in Figure 1, the Finite Strain TWE (FSTWE). A thin, cylindrical disk of the viscoelastic sample material, in blue, is sandwiched between two rigid plates. Above the middle plate is another cylindrical disk of a known material, also sandwiched between two rigid plates. Both the top and bottom plates are attached to galvanometers. During a test of a sample, the top galvanometer is slowly rotated through a large angle, rotating the midd le plate and inducing a strain field in both materials. At peak rotation of the top galvanometer, the bottom galvanometer oscillates sinusoidally at phonation frequency (much higher than the top galvanometer) sending a torsional wave through the sample. The upper galvanometer continues its periodic rotation, unloading the sample. Upon return to the peak rotation, the lower galvanometer again oscillates, but at a different frequency. Measurement of the amplitude of rotation of the middle plate, and the phase difference between the middle plate and the bottom plate, yield the viscoelastic tangent modulus in shear at a finite strain.

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_39, © The Society for Experimental Mechanics, Inc. 2011

273

274

Fig. 1 Schematic for finite strain TWE. Sample material is blue, a known elastic material is brown, and rigid plates are white To simplify the solution to the problem, the known material should possess three qualities: the material should be elastic, significantly stiffer than the sample material, and should be stiff enough in torsion so that the stress distribution is uniform throughout the thickness of the sample. The elastic requirement is needed so that analysis of the proposed test is simplified and tractable. The stiffness requirement is necessary so that the majority of the rotation applied by the upper galvanometer is transferred to the middle plate. These factors are discussed further below in the “Experimental Considerations” section. MATHEMATICAL DESCRIPTION OF FINITE STRAIN TORSIONAL WAVE EXPERIMENT Following the example of [6], the additional torsional moment T1(z,t) due to the shear wave propagating through the sample at any cross section z and time t is a

T1(z, t ) = 2π ∫ r 2 τ1(z, r, t )dr 0

(1)

where a is the radius of the sample, r is the radial coordinate, and τ1(z,r,t) is the shear stress. A subscript ‘1’ indicates that the quantity described is that of the sample material, while a subscript ‘2’ denotes properties and quantities in the known elastic material. Assuming the material is viscoelastic[11], and that the deviation from the large strain can be approximated by a tangent modulus, the shear stress is given as

τ1 (z, r , t ) =

t

∫ 0G1(t − t ')γ1(z, r, t ')dt '

(2)

where G1(t) is the viscoelastic tangent shear modulus, γ1 (r, z, t ) is the shear strain history, and a superposed dot indicates a derivative with respect to time. The deformation applied is assumed to be pure torsion, and the amount of rotation of the cross section is θ1 (z , t ) . From the kinematic constraint of pure torsion, the shear strain history is then given as ∂v (z, r, t ) ∂θ (z , t ) ∂Ω1(z, t ) γ1(z, r, t ) = 1 =r 1 =r (3) ∂z ∂z ∂z where v1(z,r,t) is the circumferential velocity and Ω1(z,t) is the angular velocity. Substituting Equations (2) and (3) into (1), the torque is then t ∂Ω (z, t ) T1 (z, t ) = J 1 ∫ G1(t − t ′) 1 dt ′ (4) 0 ∂z where J 1 =

πa 4 is the polar moment of inertia of the cross section. Considering the balance of angular momentum yields 2

the equation

ρJ 1

∂Ω1 (z , t ) ∂t

=

∂T1 (z , t ) ∂z

Combining Equations (4) and (5), the equation of motion for the sample is

.

(5)

275

ρ1

∂Ω1(z, t ) ∂t

=

t

∫0 G1(t − t ')

∂2Ω1(z, t ) ∂z 2

dt ′.

(6)

The solution of (6) requires two boundary conditions: the first is the imposed rotation at the bottom plate, and the second is due to the inertial force of the rigid plate and the torque applied from the known material. These are described as

θ1 (0, t ) = θ0 exp(i ωt ) ρ0 I 0

∂Ω1(h, t ) ∂t

t

∂Ω1(z , t ′)

0

∂z

= −J 1 ∫ G (t − t ′)

(7)

dt ′ − θ1 (h, t ) z =h1

J 2G2 h2

(8)

where θ0 and ω0 are the amplitude and frequency of the applied rotation; ρ0 is the mass density and I0 the polar moment of inertia of the middle plate; J2, G2, and h2 are the polar moment of inertia, shear modulus, and thickness of the known material, respectively. Equations (6)-(8) are most easily solved in Laplace space. Taking the Laplace transform of these equations yields, respectively, ∂2θ1(z, s ) (9) ρ1s θ1 (z, s ) = G1(s ) ∂z 2 θ0 θ1(0, s ) = (10) s − i ω0 ∂θ (z, s ) JG (11) ρ0I 0s 2 θ1(h1, s ) = −J 1G1(s )s 1 − θ1(h1, s ) 2 2 ∂z z = h h2 1

 has been replaced by sθ . The solution to Equation (9) is where a superposed tilde (~) denotes the Laplace transform, and Ω 1 1 θ1 (z, s ) = A cosh λz + B sinh λz where λ =

(12)

ρ1s . The unknown coefficients A and B are found using the boundary conditions (10) and (11). Utilizing the  G1(s )

boundary condition at the lower galvanometer, one obtains

A(s ) =

θ0 s − i ω0

.

(13)

Substituting (11) into (12) yields

J 2G2 cosh λh1 h2 B = −AB = −A . JG ρ0I 0s 2 sinh λh1 + sJ 1G1 (s )λ cosh λh1 + 2 2 sinh λh1 h2 The solution (12) is then inverted, using the definition of the inverse Laplace transform ε +i ∞ 1 exp(st )θ1 (z, s )ds θ1 (z, t ) = ∫ 2πi ε −i∞ ρ0I 0s 2 cosh λh1 + sJ 1G1 (s )λ sinh λh1 +

(14)

(15)

where ε is a positive real number to the right of the imaginary axis, so that the real part of all the poles are less than ε. The contour can be closed in the left half plane at ∞, and it can be shown that there are no contributions from branch cuts. Then, from the Residue theorem, the integral in (15) is 2πi times the sum of the residue at the pole s = iω0 To simplify the expression for the solution, we use the definitions: G1* (ω0 ) =| G1* (ω0 ) | e i δ(ω0 ) = i ω0G1 (i ω0 )

λ(i ω0 ) = i ω0

ρ1 * G1 (ω0 )

(16) (17)

276

ξ = ω0h1

ρ1 * | G1 (ω0 )

(18)

|

δ z sin h1 2 z δ βˆ = zˆβ = cos h1 2

αˆ = zˆα =

(19)

where G*(ω0) is the complex shear modulus, which can be represented in terms of its magnitude |G*(ω0)| and the phase shift δ(ω0)[12]. The solution can be further simplified by noting that

λz = i ξ ( αˆ − i βˆ ) . λh1 = i ξ ( α − i β )

(20)

θ1(z, t ) = θ0eiω0t ( cosh λz − B sinh λz )

(21)

The solution to Equation (6) can now be written as with the complex constant B B =

N r + iN i Dr + iDi

=

N r Dr + N i Di Dr2

+

Di2

+i

N i Dr − N i Dr Dr2 + Di2

= Br + iBi

(22)

where

N r = −d cos(ξα) cosh(ξβ ) − c β cos(ξα) sinh(ξβ ) − cα sin(ξα) cosh(ξβ ) − k cos(ξα) sinh(ξβ ) N i = −d sin(ξα) sinh(ξβ ) − c β sin(ξα) cosh(ξβ ) + cα cos(ξα) sinh(ξβ ) − k sin(ξα)sinh(ξβ ) Dr = −d cos(ξα) sinh(ξβ ) − c β cos(ξα) cosh(ξβ ) − cα sin(ξα) sinh(ξβ ) + k cos(ξα)sinh(ξβ ) Di = −d sin(ξα) cosh(ξβ ) − c β sin(ξα)sinh(ξβ ) + cα cos(ξα) cosh(ξβ ) + k sin(ξα)cosh(ξβ ).

(23)

are real numbers. The constants c1, d, and k are given by

c = ω0J 1 ρ1 | G * (ω0 ) | d = ρ0I 0 ω02 JG k = 2 2. h2

(24)

With Equation (20) the solution can be written as θ1(z , t ) = θ0 ⎣⎡ E cos ω0t + F sin ω0t ⎦⎤ + i θ0 ⎡⎣ P cos ω0t + Q sin ω0t ⎤⎦

(25)

where (see footnote 1)

E = cos(ξαˆ) cosh(ξβˆ) − Br cos(ξαˆ) sinh(ξβˆ) + Bi sin(ξαˆ) cosh(ξβˆ) F = − sin(ξαˆ) sinh(ξβˆ) + B sin(ξαˆ) cosh(ξβˆ) + B cos(ξαˆ) sinh(ξβˆ) r

i

P = sin(ξαˆ) sinh(ξβˆ) − Br sin(ξαˆ) cosh(ξβˆ) − Bi cos(ξαˆ) sinh(ξβˆ) Q = cos(ξαˆ) cosh(ξβˆ) − Br cos(ξαˆ) sinh(ξβˆ) + Bi sin(ξαˆ) cosh(ξβˆ).

(26)

In considering the full solution, it is helpful to assume that the excitation at the bottom plate is the real part of the harmonic excitation described in (7). Then, the real part of the solution (25) becomes θ1 (z , t ) = θ0M (z , ω0 ) cos ⎡⎣ ω0t − φ(z , ω0 ) ⎤⎦

(27)

where M(z,ω0) and φ(z, ω0) are the amplification factor and phase shift at height z and excitation frequency ω0. They are defined as 1

Note that errors in [6] have been corrected here

277

M (z, ω0 ) =

E2 + F2

(28)

F tan ⎡⎣ φ(z , ω0 ) ⎤⎦ = . E

(29)

From measurements of the amplitude and phase of rotation of the middle plate relative to the bottom plate, the real and imaginary parts of the complex shear modulus can be calculated. DISCUSSION To facilitate the discussion of the above solution for the finite strain torsional wave experiment, we define several new dimensionless variables in addition to the dimensionless frequency ξ defined in (18). Similar to [6], we introduce the parameter μ, a measure of the inertia of the sample relative to that of the middle plate, as ρ0 I 0 ω02 ρI ρ1 d = = 0 0 ω0h1 = μξ. c | G1* (ω0 ) | ω0J 1 ρ1 | G1* (ω0 ) | J 1h1ρ1

(30)

Similarly we introduce κ, a measure of the elastic torsional stiffness of the known material to that of the sample, as JG J 2G2h1 k 1 1 = 2 2 = * c h2 ω J ρ | G * (ω ) | ω J 1 | G1 (ω0 ) | h2 0h1 0 1 1 1 0

| G1* (ω0 ) | ρ1

=

κ . ξ

(31)

From Equations (18), (30), and (31) the final solution (27) can be shown to be a function of only five parameters: ξ, δ, μ, κ, and zˆ . Figure 2 shows the dependence of the amplification factor at the middle plate ( zˆ = 1 ) on the dimensionless frequency ξ. Figure 2a shows how the loss angle affects the amplification factor, while 2b shows the effect of the relative torsional stiffness κ. Here, a value of κ=0 indicates that the known material is not present. Due to the increasing damping and energy loss in the system, increasing loss angles decreases the amplification factor with little change in the resonance frequency. As the relative stiffness of the sample decreases, the resonance frequency increases. Figure 3 shows the phase angle at the middle plate as a function of frequency ξ while varying κ. Similar to Figure 2b, this figure shows that increasing the relative stiffness increases the resonance frequency of the system.

Fig. 2 Amplification factor at the middle plate as a function of frequency ξ and (a) loss parameter δ and (b) parameter κ. For both figures: inertia parameter μ=3.0, sample parameters a1=3 mm, h1=0.5 mm, ρ=1000 kg/m3, |G1*|=1 kPa, middle plate parameters ρ0 =1040 kg/m3, l=2.629 mm, h0=2mm. (a) For the known material: G2 = 10 kPa, a2 = 3 mm, h2 = 0.5 mm. Torsion constant κ=10. (b) δ=0.3

278

Fig. 3 Phase angle φ as a function of frequency ξ and parameter κ. Values are the same as those used in Figure 2(b)

Rotation of the middle plate is measured using an optical lever technique depicted in Figure 4. One surface of a hexagonal plate is polished to a mirror finish and coated with a thin (~200 nm) layer of metal to provide a reflective surface. A laser beam is brought in and reflected off the mirrored surface. This reflection then passes through a spherical lens that first focuses then expands the beam. The beam then passes through a cylindrical lens, which creates a thin vertical laser line at the focal plane of the lens. As the plate oscillates, the laser line moves back and forth over a mask-covered photodiode detector that gives a voltage change proportional to the rotation of the middle plate. In this way, small rotations are measured to accuracies of the order of 3.5 milliradians. To determine the unknown material properties |G1*(ω0)| and δ(ω0), the amplification factor is recorded over a wide range of frequencies that includes the resonance frequency via the optical lever technique described above. The amplification factor is then fit with the linear viscoelastic model analytical solution described by (27)-(29). This method assumes a constant complex modulus over this range of frequencies, but small changes with respect to frequency are expected and have been observed in the current method [6-10]. To vary frequencies at which the moduli are determined, the specimen sizes can be varied to affect the resonant frequencies. EXPERIMENTAL CONSIDERATIONS Returning to the discussion in the “Introduction” of the choice of the known upper material, we first focus on the requirement that the material is sufficiently stiff to allow for the assumption of a uniform strain distribution in the upper material, so that a stress wave analysis is not needed for interpreting the deformation of this material. This condition is typically satisfied in mechanical testing by ensuring that the natural frequency determined by the round trip transit time through the thickness of the specimen is much greater than the driving frequency of the applied loading. For an elastic cylinder in torsion this requirement is [13] c G2 / ρ2 (32) f  s2 = 2h2 2h2 where f is the driving frequency, and the properties of the upper material, denoted by subscript ‘2’ are as follows: cs2 is the shear wave speed, G2 is the shear modulus, ρ2 is the mass density, and h2 is the height. For practical purposes it is also important that the strain in the sample be much greater than in the upper material, otherwise the loading configuration would be too soft to impose the deformation on the sample, i.e. the lower material. At the middle plate, we have two conditions that must be met: the total torque applied by both materials must be equal and opposite, and the angle of twist (denoted as ψ ) of the top material must be much smaller than that of the sample, expressed as ψ2  ψ1 . If the loading of the sample material to finite strain is considered to be done quasi-statically and for simplicity, the sample is modeled as neo-Hookean then one can use Rivlin’s solution [14] for large rotations of a circular cylinder to relate the rotations and impose the constraint on the relative rotations:

ψ2 ψ1

=

4 G1J 1h2 1 9 G2J 2h1

(33)

279

Fig. 4 Optical lever technique schematic. Sample and known material in blue and rigid plates are white where G1 is the neo-Hookean modulus, taken to be equal to the magnitude of the complex modulus, |G1(ω0)|. The quotient (33) is similar to the inverse of the parameter κ defined in (31). Together, Equations (32) and (33) indicate that the upper material should have a high shear modulus and a thin, wide geometry. In addition to these constraints, if κ is too large, the amplification factor (both at the resonance peak and the entire response) at the middle plate is greatly reduced. This makes finding the resonance frequency and measuring the amplification factor difficult. To avoid this difficulty, κ should be below 100. Because viscoelastic properties are sought throughout the entire phonation range, the maximum value for ω is 1000 Hz. Typical vocal fold specimens have values for ρ, |G(ω0)|, h1, and a that are approximately 1000 kg/m3, 1000 Pa, 0.5 mm, and 2.5 mm, respectively. Based on the above limitations on potential values for two materials are shown in Table 1. Table 1 Material properties and geometrical sizes of candidate upper materials. All upper sample materials had a radius of 2.5 mm G2 (kPa) ρ2 (kg/m3) h2 (mm) κ Material 1 10 2000 0.25 20 Material 2 100 2000 0.5 100 In addition to the requirements above, the candidate upper material should be easily made, and tunable based on the size and expected properties of the sample material. Three potential materials that are frequently used in cell cultures are polyacrylamide (PA) gels[15-18], polydimethylsiloxane (PDMS)[19], and hyaluronic acid (HA) based gels[20]. These all have material properties in the ranges that are described in Table 1, and meet the ease of use requirements. CONCLUSION A novel experiment to determine the material properties at finite strains of soft viscoelastic materials has been developed, with the intent to test vocal folds at physiologically relevant strains and frequencies. A brief discussion of the mathematical model shows the relevancy and potential of the test. Briefly, the test allows for the stress-strain curves of a single sample to be measured in one experimental configuration. The frequency dependence can be determined by varying the sample size of the specimen to affect the resonance frequency. Implementation and development of the experimental apparatus is ongoing; preliminary results are expected to be presented at the conference. ACKNOWLEDGEMENTS The work was funded by a sub-contract from the University of Delaware on NIH grant R01 008965.

280

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Gray S. D., 2000, “Cellular Physiology of the Vocal Folds.,” Otolaryngologic Clinics of North America, 33(4), pp. 679-98. Hirano M., 1981, “Structure of the Vocal Fold in Normal and Disease States: Anatomical and Physical Studies,” ASHA Rep. Titze I. R., 1994, Principles of Voice Production, Prentice Hall, New Jersey. Hirano S., 2005, “Current Treatment of Vocal Fold Scarring,” Current Opinion in Otolaryngology & Head and Neck Surgery, 13(3), pp. 143-147. Benninger M. S., Alessi D., Archer S., Bastian R., Ford C., Koufman J., Sataloff R. T., Spiegel J. R., and Woo P., 1996, “Vocal Fold Scarring: Current concepts and management,” Otolaryngology - Head and Neck Surgery, 115(5), pp. 474-482. Jiao T., Farran A., Jia X., and Clifton R. J., 2009, “High Frequency Measurements of Viscoelastic Properties of Hydrogels for Vocal Fold Regeneration.,” Experimental Mechanics, 49(2), pp. 235-246. Grieshaber S. E., Nie T., Yan C., Zhong S., Teller S. S., Clifton R. J., Pochan D. J., Kiick K. L., and Jia X., 2011, “Assembly Properties of an Alanine-Rich, Lysine-Containing Peptide and the Formation of Peptide/Polymer Hybrid Hydrogels,” Macromolecular Chemistry and Physics, 212(3), pp. 229-239. Farran A. J. E., Teller S. S., Jha A. K., Jiao T., Hule R. A., Clifton R. J., Pochan D. P., Duncan R. L., and Jia X., 2010, “Effects of matrix composition, microstructure, and viscoelasticity on the behaviors of vocal fold fibroblasts cultured in three-dimensional hydrogel networks.,” Tissue Engineering. Part A, 16(4), pp. 1247-61. Jha A. K., Hule R. a, Jiao T., Teller S. S., Clifton R. J., Duncan R. L., Pochan D. J., and Jia X., 2009, “Structural Analysis and Mechanical Characterization of Hyaluronic Acid-Based Doubly Cross-Linked Networks.,” Macromolecules, 42(2), pp. 537-546. Jia X., Yeo Y., Clifton R. J., Jiao T., Kohane D. S., Kobler J. B., Zeitels S. M., and Langer R., 2006, “Hyaluronic acid-based microgels and microgel networks for vocal fold regeneration.,” Biomacromolecules, 7(12), pp. 3336-44. Fung Y. C., 1993, Biomechanics: Mechanical Properties of Living Tissues, Springer. Pipkin A. C., 1986, Lectures on Viscoelasticity Theory (Applied Mathematical Sciences), Springer. Clifton R. J., Jia X. Q., Jiao T., Bull C., and Haln M. S., 2006, “Viscoelastic Response of Vocal Fold Tissues and Scaffolds at High Frequencies,” Mechanics of Biological Tissue, G.A. Holzapfel, and R.W. Ogden, eds., Springer, New York, pp. 445-455. Rivlin R. S., 1948, “Large Elastic Deformations of Isotropic Materials. III. Some Simple Problems in Cylindrical Polar Co-Ordinates,” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 240(823), pp. 509-525. Engler A., Richert L., Wong J., Picart C., and Discher D., 2004, “Surface probe measurements of the elasticity of sectioned tissue, thin gels and polyelectrolyte multilayer films: Correlations between substrate stiffness and cell adhesion,” Surface Science, 570(1-2), pp. 142-154. Beningo K. a, Lo C.-M., and Wang Y.-L., 2002, “Flexible polyacrylamide substrata for the analysis of mechanical interactions at cell-substratum adhesions.,” Methods in cell biology, Elsevier Science, pp. 325-39. Kandow C. E., Georges P. C., Janmey P. A., and Beningo K. A., 2007, “Polyacrylamide Hydrogels for Cell Mechanics,” Methods in Cell Biology, VOL 83., Elsevier, pp. 29-46. Lo C.-M., Wang H.-B., Dembo M., and Wang Y.-L., 2000, “Cell Movement Is Guided by the Rigidity of the Substrate,” Biophysical Journal, 79(1), pp. 144-152. Tanaka Y., Morishima K., Shimizu T., Kikuchi A., Yamato M., Okano T., and Kitamori T., 2006, “Demonstration of a PDMS-based bio-microactuator using cultured cardiomyocytes to drive polymer micropillars.,” Lab on a chip, 6(2), pp. 230-5. Park Y. D., Tirelli N., and Hubbell J. a, 2003, “Photopolymerized hyaluronic acid -based hydrogels and interpenetrating networks.,” Biomaterials, 24(6), pp. 893-900.

The Blast Response of Sandwich Composites With a Graded Core: Equivalent Core Layer Mass vs. Equivalent Core Layer Thickness

Nate Gardner and Arun Shukla Dynamic Photomechanics Laboratory, Dept. of Mechanical, Industrial & Systems Engineering University of Rhode Island, 92 Upper College Road, Kingston, RI 02881, USA [email protected] ABSTRACT In the present study, the dynamic behaviors of two types of sandwich composites made of E-Glass Vinyl-Ester (EVE) face sheets and Corecell TM A-series foam were studied using a shock tube apparatus. The materials, as well as the core layer arrangements, and overall specimen dimensions were identical; the only difference arises in the core layers, where one configuration has equivalent core layer thickness, and the other configuration has equivalent core layer mass. The foam core itself was layered based on monotonically increasing the acoustic wave impedance of the core layers, with the lowest wave impedance facing the shock loading. A high-speed side-view camera system along with a high-speed back-view Digital Image Correlation (DIC) system was utilized to capture the real time deformation process as well as mechanisms of failure. Post mortem analysis was carried out to evaluate the overall blast performance of these two configurations. The results indicated that with a decrease in areal density of ~ 1 kg/m2 (5%) from the sandwich composites with equivalent core layer thickness to the sandwich composites with equivalent core layer mass, an increase in deflection (20%), in-plain strain (8%) and velocity (8%) was observed. 1. INTRODUCTION Sandwich structures have important applications in the naval and aerospace industry. Their high strength/weight ratio, high stiffness/weight ratio and high energy absorption capabilities play a vital role in their applications, especially when they are subjected to high-intensity impulse loadings, such as air blasts. Their properties assist in dispersing the mechanical impulses that are transmitted to the structure, and thus protect anything located behind it [1-3]. Core materials play a crucial role in the dynamic behavior of the sandwich structures when they are subjected blast loadings. Common cores are made of metallic and non-metallic honeycombs, cellular foams, balsa wood, PVC, truss and lattice structures. Extensive research exists in the literature regarding the dynamic response of sandwich structures consisting of the various core materials and geometric arrangements. Dharmasena et al. [3], Zhu et al. [4], and Nurick et al. [5] have tested sandwich structures with a metallic honeycomb core material. Tagarielli et al. [6] has investigated the dynamic response of sandwich beams with PVC and balsa wood cores. Radford el al. [7] has conducted metal foam projectile impact experiments to simulate a blast loading on sandwich structures with metal foam cores. McShane et al. [8, 9] have investigated the underwater blast response of sandwich composites with a prismatic lattice (Y-frame, corrugated), as well as simulated an air blast, using metal foam projectiles, on sandwich composites with a pyramidal lattice cores. These studies have indicated that advanced sandwich structures can potentially have significant advantages over monolithic plates of equivalent mass in absorbing the blast energy, whether in air or underwater. In recent years, functionally graded materials, where the material properties vary gradually or layer by layer within the material itself, have gained much attention. The numerical investigation by Apetre et al. [10] on the impact damage of sandwich structures with a graded core (density) has shown that a reasonable core design can effectively reduce the shear forces and strains within the structures. Consequently, they can mitigate or completely prevent impact damage on sandwich composites. Li et al. [11] numerically examined the impact response of layered and graded metal-ceramic structures. They found that the choice of gradation has a great significance on the impact applications and a particular design can exhibit better energy dissipation properties. In their previous work, the authors experimentally investigated the blast resistance of sandwich composites with stepwise graded foam cores [12, 13]. Results indicated that monotonically increasing the wave impedance of the foam core, thus reducing the wave impedance mismatch between successive foam layers, will introduce a stepwise core compression, greatly enhancing the overall blast resistance of sandwich composites. T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_40, © The Society for Experimental Mechanics, Inc. 2011

281

282 In [12] two types of core configurations were studied and the sandwich composites were layered / graded based on the wave impedance of the given foams, i.e. monotonically and non-monotonically. In [13] four types of core configurations were investigated and the sandwich composites had a monotonically graded core based on increasing wave impedance, and the core gradations consisted of one, two, three and four layers respectively. The specimen dimensions and overall thickness were held constant, and the individual core layers had equivalent core layer thickness, i.e. with two layer core gradation, each core layer was 19 mm, while with four layer core gradation, each core layer was 9.5 mm. The present study is an extension of the author’s previous work and focuses on the blast response of sandwich composites with equivalent core layer mass. By using sandwich composites with equivalent core layer mass, the overall areal density of the specimen is reduced in comparison to its sandwich composite counterpart with equivalent core layer thickness The quasistatic and dynamic constitutive behaviors of the foam core materials were first studied using a modified SHPB device with a hollow transmitted bar. The sandwich composites were then fabricated and subjected to shock wave loading generated by a shock tube. The materials, as well as the core layer arrangements, and overall specimen dimensions were identical; the only difference arises in the core layers, where one configuration has equivalent core layer thickness, and the other configuration has equivalent core layer mass. The shock pressure profiles, real time deflection images, and post mortem images were carefully analyzed to reveal the mechanisms of dynamic failure of these sandwich composites. Digital Image Correlation (DIC) analysis was implemented to investigate the real time deflection, in-plane-strain, and particle velocity. 2. MATERIAL AND SPECIMEN 2.1 SKIN AND CORE MATERIAL The skin materials utilized in this study are E-Glass Vinyl Ester (EVE) composites comprised of E-glass fiber (0.61 kg/m2 areal density) and a vinyl-ester matrix. The plain weave woven roving E-glass fibers of the skin material consisted of 8 layers and were placed in a quasi-isotropic layout [0/45/90/-45]s. The core materials used in the present study are CorecellTM A series styrene foams manufactured by Gurit SP Technologies. The two types of CorecellTM A foam were A300 and A800 with density 58.5 and 150 kg/m3 respectively. The cell structures of the two foams are very similar and the only difference appears in the cell wall thickness and node sizes, which accounts for the different densities of the foams. The SEM images of the cell microstructures can be seen in Fig. 1.

A300

A800

Shock Wave

A300

Shock Wave

A800

100 mm A300

100 mm

A800 Fig. 1 Cell microstructures of core materials (a) Equivalent Thickness (b) Equivalent Mass Fig. 2 Specimen configurations and loading direction

2.2 SANDWICH PANELS WITH STEPWISE GRADED CORE The sandwich panels were produced by VARTM-fabricated process. The panels were 102 mm (4 in) wide, 254 mm (10 in) long with 5 mm (.2 in) front and back skins. The core consisted of two layers of foam. For the sandwich panels with equivalent core layer thickness, the first and second core layers of foam, A300 and A800 respectively, were both 19 mm (.75 in).These panels had an areal density of approximately 18.50 kg/m2. For the sandwich panels with equivalent core layer mass, the first core layer of foam, A300, was 25.4 mm (1 in), while the second core layer, A800, was 12.7 mm (.5 in). These panels had an areal density of approximately 17.60 kg/m2. Fig. 2 shows the specimen configurations for the sandwich composites with a core of (a) Equivalent Thickness and (b) Equivalent Mass.

283 3. EXPERIMENT SETUP AND PROCEDURE 3.1 MODIFIED SPLIT HOPKINSON PRESS BARS WITH HOLLOW TRANSMITTER BAR Due to the low wave impedance of CorecellTM foam materials, core material tests were performed by a modified SHPB device with a hollow transmission bar. It has a 304.8 mm (12 in)-long striker, 1600 mm (63 in)-long incident bar and 1447 mm (57 in)-long transmitter bar. All of the bars are made of a 6061 aluminum alloy. The nominal outer diameters of the solid incident bar and hollow transmission bar are 19.05 mm (0.75 in). The hollow transmission bar has a 16.51 mm (0.65 in) inner diameter. At the front and at the end of the hollow transmission bar, end caps made of the same material as the bar were press fitted into the hollow tube. By applying pulse shapers, the effect of the end caps on the stress waves can be minimized. The details of the analysis and derivation of equations can be found in ref [14]. The cylinderical specimens with a dimension Φ10.2 mm (0.4 in) X 3.8 mm (0.15 in) were used for test. 3.2 SHOCK TUBE Fig. 3 shows the shock tube apparatus with muzzle detail, which was utilized to obtain a controlled blast loading. It has an overall length of 8 m, consisting of a driver, driven and muzzle section. The high-pressure driver section and the low pressure driven section are separated by a diaphragm. By pressurizing the high-pressure section, a pressure difference across the diaphragm is created. When this difference reaches a critical value, the diaphragms rupture. This rapid release of gas creates a pressure wave that develops into a shock wave as it travels down the tube to impart dynamic loading on the specimen. The final muzzle diameter is 76.2 mm (3 in). Two pressure transducers (PCB102A) are mounted at the end of the muzzle section 160 mm apart, with the closest transducer 20 mm away from the specimen. The support fixtures ensure simply supported boundary conditions with a 0.1524 m (6 in) span. In the present study, a simply stacked diaphragm of 5 plies of 10 mil mylar sheets with a total thickness of 1.27 mm was utilized to generate an impulsive loading on the specimen with an incident peak pressure of approximately 1 MPa, a reflected peak pressure of approximately 5 MPa and a wave speed of approximately 1050 m/s. For each configuration, at least three samples were tested. A high-speed side-view camera system along with a high-speed back-view Digital Image Correlation (DIC) system was utilized to capture the real time deformation process as well as mechanisms of failure. Both camera systems were operating at 20,000 fps, with an interframe time of 50 µs and the high speed photography system can be seen in Fig. 4.

Shock tube Muzzle Detail and Specimen Fig. 3 Shock tube apparatus

Fig. 4 Digital Image Correlation (DIC) Set-up

3.3 DIGITAL IMAGE CORRELATION (DIC) Digital Image Correlation (DIC) was utilized to obtain the real time response of the sandwich composites. A speckle pattern was placed on the back face sheet of the specimens. Two high speed digital cameras, Photron SA1, were placed behind the shock tube to capture the real time deformation and displacement of the sandwich composite, along with the speckle pattern. During the blast loading event, as the specimen bends, the cameras track the individual speckles on the back face sheet. Once the event is over, a graphical user interface was utilized to correlate the images from the two cameras and generate real time deflection, strain (in plane and out of plane), and particle velocity.

284 4. EXPERIMENTAL RESULTS AND DISCUSSION 4.1 DYNAMIC CONSTITUTIVE BEHAVIOR OF CORE MATERIALS

Table1. Yield strength of core materials Core Layer

A300

A800

Quasi-Static Yield Stresses (MPa)

0.60

2.46

High Strain-Rate Yield Stresses (MPa)

0.91

4.62

Fig. 5 Quasi-static and high strain-rate behaviors TM of the two types of Corecell A Foams Fig. 5 shows the quasi-static and high strain-rate behavior of the different types of Corecell TM A foams. The quasi-static and dynamic stress-strain responses have an obvious trend for the different types of foams. Lower density foam has a lower strength and stiffness, as well as a larger strain range for the plateau stress. The high strain-rate yield stresses and plateau stresses are much higher than the quasi-static ones for the same type of foams. The dynamic strength of A800 increases approximately 100% in comparison to their quasi-static strength, while A300 increases approximately 50%. The improvement of the mechanical behavior from quasi-static to high strain-rates for all core materials used in the present study signifies their ability to absorb more energy under high strain-rate dynamic loading. Table 1 shows the quasi-static and high strain-rate yield stresses respectively. 4.2 RESPONSE OF SANDWICH COMPOSITES WITH GRADED CORES 4.2.1 REAL TIME DEFORMATION

Equivalent Mass

Equivalent Thickness

The real time side view deformation image series of the sandwich composites with equivalent core layer thickness and equivalent core layer mass under shock wave loading are shown in Fig. 6 respectively. The shock wave propagates from the right side of the image to the left side and some detailed deformation mechanisms are pointed out in the figures.

Fig. 6 Real - time side - view deformation of sandwich composites under shock wave loading

285 For the sandwich composites with equivalent core layer thickness, it can be observed that at t = 100 μs indentation failure of the core has initiated. This means that compression has initiated in the first core layer of foam (A300). By t = 400 μs heavy core compression is evident in the A300 foam core layer and core cracking can be seen in the A800 layer where the supports are located. At t = 700 μs delamination between the front face sheet and the foam core can be observed, both at the top and the bottom of the specimen. Also the core cracks have propagated from the back face sheet to the front face sheet. By t = 1600 μs, heavy core cracking and skin delamination are visible, along with heavy compression in the core (A300 only). For the sandwich composites with equivalent core layer mass, it can be observed that at t = 100 μs indentation failure of the core has initiated. This means that compression has initiated in the first core layer of foam (A300). By t = 400 μs, the A300 layer has continued to compress, and core cracking can be seen in the A800 layer where the supports are located. By t = 700 μs the core cracks have propagated from the back face sheet to the front face sheet. Also at this time, skin delamination between the front skin and foam core has initiated (top and bottom of specimen). By t = 1600 μs, heavy core cracking and skin delamination are visible, along with heavy compression in the core (A300 only). In both core configurations, equivalent core layer thickness and equivalent core layer mass, the deformation mechanisms were identical. Both configurations exhibited a double-winged deformation shape which means both configurations were under shear loading. Indentation failure was followed by compression of the first layer of foam (A300) and core cracking, and finally delamination between the front face sheet and foam core. The extent of the damage mechanisms varies between configurations, but the time at which the damage mechanisms were observed is identical. The mid-point deflections of the front face (front skin), interface 1 (between first and second core layer), and the back face (back skin) for both configurations, directly measured from the real – time side – view deformation images, are shown in Fig. 7 respectively. For the sandwich composites with equivalent core layer thickness (Fig. 7a), it can be seen that at t = 1600 μs the front skin, interface 1 and the back face deflect to approximately 46 mm, 32 mm and 32 mm respectively. Since the A300 foam core layer is located between the front skin and interface 1, the difference in deflection between the front skin and interface 1 shows the amount of compression in the A300 layer. Therefore it is evident that the A300 foam compresses approximately 14 mm, which is 75% of its original thickness (19 mm). Also note that the deflection curves for interface 1 and the back face follow the same trend and deflect to the same value at t = 1600 μs (32 mm). Therefore, no compression was observed in the A500 core layer of foam. For the sandwich composites with equivalent core layer mass, the mid-point deflections are shown in Fig. 7b. It can be seen that at t = 1600 μs, the front skin, interface 1, and the back skin deflect to approximately 60 mm, 41 mm and 41 mm respectively. Since the A300 foam core layer is located between the front skin and interface 1, the difference in deflection between the front skin and interface 1 shows the amount of compression in the A300 layer. Therefore it can be observed that the A300 foam compresses approximately 19 mm, which is 75% of its original thickness (25.4 mm). Also note that the deflection curves for interface 1 and the back face follow the same trend and deflect to the same value at t = 1600 μs (41 mm). Therefore, no compression was observed in the A500 core layer of foam.

(a) Equivalent Thickness

(b) Equivalent Mass Fig. 7 Mid-point deflection curves for both configurations

Fig. 8 Comparison of back face deflections Fig. 8 shows a comparison of the mid-point deflections for the back face of each configuration. It can be seen in the figure that at t = 1600 μs, the back face of the sandwich composites with equivalent core layer thickness deflects approximately 20% less than the back face of the sandwich composites with equivalent core layer mass.

286 4.2.2 DIGITAL IMAGE CORRELATION (DIC)

Equivalent Mass

Equivalent Thickness

The real-time response of the sandwich composites was generated using 3-D Digital Image Correlation and the results are shown in Fig 9 - Fig. 11. Fig. 9 shows the full-field deflection contours for both the sandwich composites with equivalent core layer thickness and the sandwich composites with equivalent core layer mass. It is evident from the figure that the back face deflection along the central region of both configurations is in excellent agreement with the results generated using the high-speed deformation images.

t = 0 μs

t = 100 μs

t = 400 μs

t = 700 μs

t = 1000 μs

t = 1600 μs

Fig. 9 Real time full-field deflection (W) contours for both configurations Through DIC analysis, using the inspection of a single point in the center of the back face sheet, the data for mid-point inplane strain and particle velocity during the entire blast loading event was extracted. The results of the in plane strain and particle velocity are shown in Fig. 10 and Fig. 11 respectively. Fig. 10 shows the in-plane strain of both configurations. It can be seen that the sandwich composite with equivalent core layer thickness exhibits 2.2% strain, while the sandwich composite with equivalent core layer mass exhibits 2.4% strain. The back face particle velocity can be observed in Fig. 11. The back face of the sandwich composites with equivalent core layer thickness reach a maximum mid-point particle velocity of 31 m/s, while the sandwich composites with equivalent core layer thickness reach a maximum back face particle velocity of 34 m/s. Table 2 summarizes the back face DIC results for both configurations.

Fig. 10 In-plane strain (εyy) for both configurations

Fig. 11 Particle velocity (dW/dt) for both configurations

287 Table 2. Summary of DIC results Deflection

In-plane Strain

Velocity

(kg/ m )

(mm)

(%)

(m/s)

Eq. Thickness

18.5

32

2.2

31

Eq. Mass

17.6

41

2.4

34

Difference

5%

22%

8%

8%

Configuration

Areal Density 2

4.2.3 POST MORTEM ANALYS After the shock event occurred, the damage patterns in the sandwich composites were visually examined and recorded using a high resolution digital camera and are shown in Fig.12. For both the sandwich composites with equivalent core layer thickness and equivalent core layer mass, there were two main cracks located at the support position. Delamination is visible between the front face and the foam core, as well as between the bottom layer of foam core and back face sheet. Also compression was observed in the A300 core layer. The locations and damage mechanism were identical for both configurations; the only difference was in the extent of damage observed. Equivalent Thickness

Fiber Delamination

Core cracking

Core Compression

Delamination Core cracking

Equivalent Mass

Fiber Delamination

Core Compression

Delamination

(a) Front face sheet (blast side)

(b) Foam and PU core

(c) Back face sheet

Fig. 12 Visual examination of sandwich composites after being subjected to high intensity blast load 5. Summary The following is the summary of the investigation: (1) The dynamic stress-strain response is significantly higher than the quasi-static response for every type of CorecellTM A foam studied. Both quasi-static and dynamic constitutive behaviors of Corecell TM A series foams (A300 and A800) show an increasing trend. (2) Sandwich composites with two types of monotonically graded cores based on increasing wave impedance were subjected to blast loading. In order to reduce areal density, a sandwich composite with equivalent core layer mass was fabricated and its blast performance was compared to its sandwich composite counterpart with equivalent core layer thickness. Results indicated that with a reduction in areal density of approximately 1 kg/ m 2 (5%), the sandwich composites with equivalent core layer mass exhibit a higher back face deflection (22%), higher in-plane

288 strain (8%) and higher back face velocity (8%) in comparison to the sandwich composites with equivalent core layer thickness Acknowledgement The authors kindly acknowledge the financial support provided by Dr. Yapa D. S. Rajapakse, under Office of Naval Research (ONR) Grant No. N00014-04-1-0268. The authors acknowledge the support provided by the Department of Homeland Security (DHS) under Cooperative Agreement No. 2008-ST-061-ED0002. Authors thank Gurit SP Technology and Specialty Products Incorporated (SPI) for providing the material as well as Dr. Stephen Nolet and TPI Composites for providing the facility for creating the composites used in this study. References [1] Xue, Z. and Hutchinson, J.W., Preliminary assessment of sandwich plates subject to blast loads. International Journal of Mechanical Sciences, 45, 687-705, 2003. [2] Fleck, N.A., Deshpande, V.S., The resistance of clamped sandwich beams to shock loading. Journal of Applied Mechanics, 71, 386-401, 2004. [3] Dharmasena, K.P., Wadley, H.N.G., Xue, Z. and Hutchinson, J.W., Mechanical response of metallic honeycomb sandwich panel structures to high-intensity dynamic loading. International Journal of Impact Engineering, 35 (9), 1063-1074, 2008. [4] Zhu, F., Zhao, L., Lu, G. and Wang, Z. Deformation and failure of blast loaded metallic sandwich panels – Experimental investigations. International Journal of Impact Engineering, 35 (8), 937-951, 2008 [5] Nurick, G.N., Langdon, G.S., Chi, Y. and Jacob, N. Behavior of sandwich panels subjected to intense air blast: part 1Experiments. Composite Structures, 91 (4), 433 – 441, 2009. [6] Tagarielli, V. L., Deshpande, V.S., and Fleck, N.A. The high strain rate response of PVC foams and end-grain balsa wood. Composites: Part B, 39, 83–91, 2008. [7] Radford, D.D., McShane, G.J., Deshpande, V.S. and Fleck, N.A. The response of clamped sandwich plates with metallic foam cores to simulated blast loading. International Journal of Solids and Structures, 44, 6101-6123, 2006. [8] McShane, G. J., Deshpande, V.S., and Fleck, N.A. The underwater blast resistance of metallic sandwich beams with prismatic lattice cores. Journal of Applied Mechanics, 74, 352 – 364, 2007. [9] McShane, G. J., Radford, D.D., Deshpande, V.S., and Fleck, N.A. The response of clamped sandwich plates with lattice cores subjected to shock loading. European Journal of Mechanics – A: Solids, 25, 215-229, 2006. [10] Apetre, N.A., Sankar, B.V. and Ambur, D.R., Low-velocity impact response of sandwich beams with functionally graded core. International Journal of Solids and Structures, 43(9), 2479-2496, 2006. [11] Li, Y., Ramesh, K.T. and Chin, E.S.C., Dynamic characterization of layered and graded structures under impulsive loading. International Journal of Solids and Structures, 38(34-35), 6045-6061, 2001. [12] Wang, E., Gardner, N. and Shukla, A., The blast resistance of sandwich composites with stepwise graded cores. International Journal of Solids and Structures, 46, 3492-3502, 2009. [13] Gardner, N., Wang, E., Kumar, P., and Shukla, A. Performance of graded sandwich composite beams under shock wave loading. In Progress [14] Chen, W., Zhang, B., Forrestal, M.J. A split Hopkinson bar technique for low impedance materials. Experimental Mechanics, 39 (2), 81–85, 1998.

Effects of High and Low Temperature on the Dynamic Performance of the Core Material, Face-sheets and the Sandwich Composite

Sachin Gupta1 and Arun Shukla2 1 Dynamics Photomechanics Laboratory, Department of Mechanical, Industrial & Systems Engineering, University of Rhode Island, Kingston, RI 02881. Email: [email protected] 2 Dynamics Photomechanics Laboratory, Department of Mechanical, Industrial & Systems Engineering, University of Rhode Island, Kingston, RI 02881. Email: [email protected] ABSTRACT The performance of sandwich structures is highly affected by the varying environmental temperature during service, especially when they are subjected to blast loading. Typically, sandwich panels consist of polymer based composites (facesheets) and polymer foams (core material), and the properties of its components change substantially under different temperatures. In this paper, high strain rate constitutive behavior of E-glass Vinyl ester composites and CorecellTM M100 foam at different temperatures has been studied. A special temperature control chamber was designed in order to heat or cool the specimen to an assigned temperature. Once the specimen reached the target temperature, it was subjected to high strain rate loading using a SHPB apparatus. Eight different target temperatures were chosen: -40ºC, -20ºC, 0ºC, 22ºC, 40ºC, 60ºC, 80ºC and 100ºC. The sandwich composites were maintained at the target temperature before being subjected to shock-wave loading using a shock-tube. A high-speed photography system utilizing Digital Image Correlation (DIC) was used to record the real time deformation of the specimen. The results show a significant decrease in flow stress with the increase in the temperature of core material. Significant fiber-matrix delamination was observed in face-sheets at elevated ambient temperatures with little change in the value of compression modulus. For the low temperature environment, the core material shows brittle behavior resulting in core-cracking of the sandwich specimen under blast loading. INTRODUCTION Composite materials have important applications in the marine and aerospace industry. Marine structures undergo mechanical loadings under varying ambient temperatures especially when operating in extreme environments such as the Arctic Ocean and gulf areas. The environmental temperature has an adverse effect on the blast resistance of the structure. As a core material, polymeric foams are extensively used for energy absorption for high strain rate applications against ballistic impacts and blast waves [1-2]. Composite materials, such as sandwich structures, have important applications in ship structures due to their advantages, such as high strength/weight ratio and high stiffness/weight ratio. Tekalur [3] studied the dynamic behavior of sandwich structures with reinforced polymer foam cores. Wang and Gardner [4] analyzed the performance of sandwich composites stepwise graded core under blast loading. Most of the previous research only focuses on the blast resistance of various composite structures at room temperature. Recent studies [5] observe significant and complex effects of environmental temperature on the dynamic compressive behavior of syntactic foams. The numerical study on the response of a large aspect ratio sandwich panel subjected to the elevated temperatures on one of the surfaces showed that the maximum deflection of the panel increases as the surface temperature increases [6]. These studies investigated the performance of sandwich composites under static loading only. Erickson et al. [7] performed low-velocity impact experiments on sandwich composites and both found that the temperature can have a significant effect on the energy absorbed and the peak force endured by the specimen. Aktas [8] studied the impact behavior of glass/epoxy laminated composite plates at high temperatures and concluded that energy absorbing capabilities of the specimen reduces with increasing temperature. The study on the role of temperature on impact properties of Kevlar/fiberglass composite laminates by Salehi-Khojin [9] shows that maximum deflection increases with a corresponding increase in temperature. Dutta [10] tested the energy absorption and brittleness of graphite/epoxy composites at room and low temperature under low velocity impact. To the best of the author’s knowledge, there are no results of the blast performance of composite structures under extreme environments. An in-depth analysis on the effects of temperature is needed.

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The present paper experimentally studies the dynamic behavior of Corecell M100 foam core and E-glass Vinyl Ester composite face-sheets under high temperature environments using a Split Hopkinson Bar apparatus. A controlled temperature environment was designed in order to achieve the target temperatures. The shock-tube facility was used to study the dynamic behavior of sandwich composites under blast loading. A special fixture was designed to heat the sandwich composites to different temperatures. For low temperature experiments, dry ice was utilized to cool down the specimen prior to being subjected to blast loading. A high-speed photography system with three cameras was used to capture real-time motion images. Digital Image Correlation (DIC) technique was utilized to obtain the details of the deformation of the sandwich panels during the blast event. Post mortem visual observations were carefully analyzed to identify the mechanisms of dynamic failure of the sandwich composites under low and elevated temperature environments. 2. MATERIAL SPECIMEN: 2.1 Split Hopkinson Bar Specimen: Cylindrical specimens were cut from the foam material using a die having a diameter of 11.5 mm and a thickness of 3.8 mm. Specimen dimensions were carefully chosen to obtain uniform deformation while minimizing inertia effects and maximizing the number of cells in the specimen. For face-sheets, cylindrical specimens with a thickness of 3.18 mm and a diameter of 10.16 mm were used. 2.2 Sandwich Specimen: The skin material utilized in this study was E-Glass Vinyl Ester (EVE) composite. The woven roving E-glass fibers of the skin material were placed in a quasi-isotropic layout [0/45/90/-45]s. The fibers were made of the 0.61 kg/m2 area density plain weave. The resin system used was Ashland Derakane Momentum 8084 and the front skin and the back skin consisted of identical layup and materials. The core material used in the present study was CorecellTM M100 styrene foam, which was manufactured by Gurit SP Technologies for high temperature marine applications. Table 1 list important the material properties of this foam from the manufacturer’s data [11]. Foam Type

Nominal Density (kg/m3)

Compression Modulus (MPa)

Shear Elongation (%)

Corecell M100

107.5

107

52%

Table1 Material properties of the foam core [11]

The VARTM procedure was carried out to fabricate the sandwich composite panels. The overall dimensions for the specimen were 102 mm wide, 254 mm long and 32 mm thick. The foam core itself was 25.4 mm thick, while the skin thickness was 3.3 mm. The average areal density of the samples was 16.81 kg/m2. Figure 1 shows a real image of a specimen, its dimensions and the speckle patterns on both front and side face. The cell microstructure of M100 foam is shown in fig. 2. 3. EXPERIMENT SETUP AND PROCEDURE 3.1 Quasi-static Characterization:

The quasi-static compression tests were performed using a standard compression test machine (Instron Model 5582). The tests were performed following the ASTM D 1621 – 04a standard [12] using rectangular specimens (50.8 mm×50.8 mm, 19.1 mm thick) at a crosshead speed of 1 mm/min for M100 foam. 3.2 Split Hopkinson Bar Apparatus: A split Hopkinson pressure bar (SHPB) apparatus was used to test the M100 foam at high strain rates of deformation. Due to the low-impedance of CorecellTM foam materials, dynamic experiments for the core materials were performed with a

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modified SHPB setup. A hollow incident bar was used to increase the strain rate to achieve high end-strain values and a hollow transmitted bar was used to increase the transmitted signal intensity. It had a 50.8 mm long steel striker, 1828.8 mmlong incident bar and 1447 mm-long transmitted bar. Incident and transmitted bars were made of a 6061 aluminum alloy. The nominal outer and inside diameters of the both hollow incident/transmitted bar were 19.05 mm and 16.51 mm respectively. At each end of the hollow bars, end caps made of the same material were pressure fitted and pinned using aluminum pins. By using lead pulse shapers, the effect of the end caps on the stress waves was minimized. The details of the analysis and derivation of equations for analysis of experimental data can be found in Ref. [13]. 3.3 Heating Chamber for SHPB: A small heating chamber (220mm x 130 mm, 170 mm high) made of wood with a standard resistance heating wire, NickelChromium Alloy, 60% Ni / 16% Cr, was utilized to heat the chamber. An external DC power supply (0 – 30 V) with a voltage controller was utilized to control the amount of heat supplied. A heat resistant borosilicate glass sheet was used as a transparent removable face for the chamber. The chamber was calibrated using a series of experiments by supplying different voltages to the chamber and the saturation temperature value was estimated by using a K-type thermocouple. 3.4 Cooling Setup for SHPB: A low temperature cooling chamber with polystyrene rigid foam insulation was designed and flexible copper tubing of 9.25 mm outer diameter was wound into 100 mm inner diameter spirals. Cooling chamber was filled with dry ice (CO2) and nitrogen gas was supplied using Nitrogen Deliver System from one end of copper tubing. The other end was fed into same wood chamber used for heating. Different steady state temperature inside the chamber can be achieved by controlling the output pressure of nitrogen gas supplied. Shock tube Specimen

Supports 160 mm

Side-view camera system

Transducers Shock-tube muzzle Specimen SS Support frame

Back-view DIC system

Fig. 4 High-speed photography systems

Fig. 3 Shock tube apparatus

3.5 Shock Tube: The shock tube apparatus was utilized in the present study to develop controlled blast loadings. The details of this apparatus can be found in Ref [4]. Fig. 3 shows the shock tube apparatus with a detailed image of the muzzle. The final muzzle diameter is 76.2 mm. Two pressure transducers (PCB102A) are mounted at the end of the muzzle section with a distance 160 mm. The support fixtures ensured simply supported boundary conditions with a 152.4 mm span. The shock tube has an overall length of 8 m, consisting of a driver, driven and muzzle section. The high-pressure driver section and the low pressure driven section are separated by a diaphragm. By pressurizing the high-pressure section, a pressure difference across the diaphragm is created. When this difference reaches a critical value, the diaphragms rupture. This rapid release of gas creates a shock wave, which travels down the tube to impart dynamic loading on the specimen. In the present study, a simply stacked diaphragm of 4 plies of 10 mil mylar sheets with a total thickness of 1 mm was utilized to generate an impulse loading on the specimen with an incident peak pressure of approximately 0.83 MPa and a wave speed of approximately 970 m/s.

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3.6 High-Speed Photography Systems: Two high-speed photography systems were utilized to capture the real-time 3-D deformation data of the specimen. The experimental setup, shown in fig. 4, consists of a back-view 3-D Digital Image Correlation (DIC) system with two cameras and a side-view camera system with one camera. All cameras are Photron SA1 high-speed digital camera, which have the capability to capture images at a frame-rate of 20,000 fps with an image resolution of 512×512 pixels for one second time duration. These cameras were synchronized to make sure that the images and data can be correlated and compared. The 3-D DIC technique is one of the most recent non-contact methods for analyzing full-field shape, motion and deformation. Two cameras capture two images from different angles at the same time. By correlating these two images, one can obtain the three dimensional shape of the surface. Correlating this deformed shape to a reference (zero-load) shape gives full-field in-plane and out-of-plane deformations. To ensure good image quality, a speckle pattern with good contrast is put on the specimen prior to experiments. 4. EXPERIMENTAL RESULTS AND DISCUSSION:

3.5

3.5

3.0

3.0

2.5

2.5

2.0 1.5 22 C 40 C 60 C 80 C 100 C Quasi-static (22 C)

1.0 0.5 0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

Eng Stress (MPa)

Eng Stress (MPa)

4.1 Dynamic Constitutive Behavior of the Core-material:

2.0 1.5 22 C 0C -20 C -40 C Quasi-static (22 C)

1.0 0.5 0.0 0.00

0.05

Eng Strain

0.10

0.15

Eng Strain

(a) High temperature

(b) Low temperature Fig. 5 Quasi-static and dynamic behavior of M100 foam under low and high temperature

The quasi-static and dynamic stress-strain curves for M100 foam at low and high temperatures are obtained. Figure 5 shows the difference in the dynamic behavior and quasi-static behavior for M100 foam. The flow stress value under quasi-static loading is 1.5 MPa, while for dynamic testing, the flow stress value increases approximately by 100% and it shows a flow stress value of 3.1 MPa at a strain rate of 3700/s. The effect of high temperature on the stress-strain behavior of foam under dynamic loading can be observed exclusively in fig. 5(a). The plateau stress drops from 3.1 MPa to 2.3 MPa as temperature increases from 22⁰C to 100⁰C. The cell collapse occurs during the plateau stress region in polymeric foam, which is entirely dependent on the cell material. So, in polymeric foams, a decrease in the plateau stress can be expected due to thermal softening of the base material [14].

4.2 Dynamic Constitutive Behavior of the Face-sheet: Dynamic constitutive behavior of the face-sheet under quasi-static and high strain rate loading is shown in fig. 6. The compressive modulus for quasi-static loading is 2.6 GPa. As the fibers in the specimen are running in transverse direction, the compressive properties are moderated by the resin, which results into a relative lower value of compressive modulus as compared to young modulus of E-glass fibers. As the temperature is increases from -40⁰C to 100⁰C,

700 600

Eng Stress (MPa)

Fig. 5(b) shows the dynamic constitutive behavior under low temperatures. The dynamic experiments were performed at a strain rate of 2200/s. As the temperature decreases from 22⁰C to -40⁰C, the collapse of cells occurs rapidly due to low temperature environments and this result into brittle failure of cells. Thus, M100 foam exhibits brittle behavior, which is clearly demonstrated by the decreasing slope in plateau region.

500 400

300

-40 C -20 C 0C 22 C 40 C 60 C 80 C 100 C Quasi static

Strain rate = 1400/s Strain rate = 1400/s

200 100 0 0.00

0.02

0.04

Eng Strain

0.06

0.08

Fig. 6 Quasi-static and dynamic behavior of face-sheet under low and high temperature

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a decrease in the stress-strain curve is observed. The young modulus of resin is highly dependent on the temperature and a decrease in compressive modulus under dynamic testing can be explained by the decrease in young modulus of the resin used in manufacturing of face-sheets. 4.3 Response of Sandwich Composites under Room, Low and High temperature: 4.3.1 Real Time Deformation

The real time observation of the transient behavior of sandwich composites under shock wave loading at low temperature ( 65⁰C ), room temperature (22⁰C) and high temperature (80⁰C) are shown in figs. 7, 8 and 9 respectively. The shock wave propagated from the right side of the image to the left side and some detailed deformation mechanisms were pointed out in the figures. As shown in fig. 7, at room temperature environment, the sandwich composite exhibits no visible failure during the event. At 200 µs, the stress wave has propagated inside the specimen and both front and back faces of the specimen start to deform under blast loading. At t = 400 µs, the core material starts compressing and by t = 750 µs, specimen core is compressed by 5% in the lower part of the supports. The core material stops compressing after t = 1350 µs and a maximum of 9% strain is observed in the core. The specimen deforms in a double wing shape and no global bending response is observed.

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Fig. 8 shows the blast loading response of a sandwich composite subjected to a low temperature environment. The core material temperature is -65⁰C. At t = 300 µs, primary core cracks are formed around the perimeter of loading area. The lower crack initiates from the front face, while the upper crack is initiated from the back face. The crack formation is followed by the skin delamination from front and back face in the lower and upper part of the specimen respectively. The secondary core cracks are visible at t = 750 µs, and the specimen deformation behavior changes from double wing to global bending. This allows for heavy slippage between the core and face-sheet, which results into the formation of other secondary cracks. At 1600 µs, the skin delamination on the top of the back face can be observed. During the whole blast event, there is no core compression observed, which clearly demonstrates the core-hardening at low temperature environments. The blast response of a sandwich composite with a core temperature of 80ºC is shown in fig. 9. The core compression starts at t = 300 µs and the centre of the core is compressed by 6% by t = 600 µs. The bending of grid lines close to the back face shows the presence of shear stresses in the core materials near the face-sheets. The core compression increases with time and at t = 1350 µs, the core is compressed more than 30% at the center. Following the core compression, local failure of the front face-sheet is observed at t = 1350 µs, which results in the local deformation of the core material in the central area. 4.3.2 Deflection The mid-point deflection in the front and back faces, at the center point under different temperature environments were obtained from high speed side-view images and are shown in fig. 10. Comparing the back face deflections at t = 1200 µs, the sandwich specimen at room temperature deflected 17 mm, while the deflection for the specimen tested under low and high temperature environment is 23 mm and 28 mm respectively. Therefore, it can be observed that the deflection of sandwich panel under low temperature is 35% more than as compared to room temperature experiments. Also, the high temperature specimen has 65% more back face deflection than at room temperature. The front face deflections are also plotted in fig. 10 to compare the amount of core compression observed during the blast testing. The core compression and compressive strain at the center of the specimen at t = 1200 µs are listed in table 2. Under low temperature, the core compression is less as compared to Fig. 10 Front and Back face deflection room temperature, which depicts the increase in yield strength of the core curves for room, low and high temperature material under low temperature environments. At Core Compression Compressive Strain high temperature, the thermal softening of foam (mm) (%) and the local failure of the face-sheets allows for 1.89 mm 7.5% Room Temperature more core compression, which results into a 0.82 mm 3.2% Low Temperature compressive strain of 31.5% at the center of the specimen. High Temperature 8.00 mm 31.5% Table 2 Core compression and compressive strain at time t = 1200 µs

4.3.3 Digital Image Correlation (DIC) Analysis

Fig. 11 Out-of-plane velocity on the back face for room, low and high temperature

Shown in fig. 11 and 12, the real time response of the sandwich composites was generated using Digital Image Correlation (DIC). Through DIC analysis, the out-of-plane deflection and velocity during the entire blast event was extracted. The out-of-plane velocity on the back face of the sandwich specimen is plotted in fig. 11. Under room temperature, at t = 300 µs, the specimen reaches a maximum velocity of 25 m/s and it begins to decelerate later in time. A maximum velocity of 30 m/s and 34 m/s is observed in low and high temperature experiments respectively at approximately t = 350 µs. Therefore, low and high temperature experiments exhibit higher maximum velocities, but the time taken to reach maximum velocity is delayed by 50 µs. At t = 1300 µs, the room temperature experiment reaches the maximum deflection and starts reverberating, which is shown by the negative velocities after this time. Compared to the low and high temperature experiments, the specimen is still undergoing deformation with velocities of 5 m/s and 10 m/s respectively at t = 1300 µs.

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Fig. 12 shows the full-field back face deflection for high temperature experiment. By t = 1800 µs, the deflection at the midpoint of the back face is 35 mm, which is the same as the deflection of the back face calculated from the side-view images. W (mm)

t = 0 µs

t = 400 µs

t = 700 µs

t = 1000 µs

t = 1300 µs

t = 1800 µs

Fig. 12 Full field back view for high temperature experiment

4.3.4 Post-mortem Analysis After the blast event occurred, the damage patterns in the sandwich specimens were visually examined and recorded using a high resolution digital camera and are shown in fig. 13. When the sandwich specimen was subjected to shock wave loading under room temperature, the specimen shows a maximum deflection of 2.5 mm and there is no evidence of permanent damage inside core and face-sheet. When the sandwich specimen was subjected to shock wave loading under low temperature, the damage is confined to the places where the supports are located in the shock loading and a significant amount of core-cracking was observed. The core-cracks propagated completely through the foam core and a large amount of skin delamination between the core and the back face-sheet was observed. No specific damage in face-sheets and core compression is visible, and the specimen shows a deflection of 5.5 mm at center. In the high temperature environment, excessive fiber matrix delamination and fiber breakage was seen confined to the center of the shock wave loading. The fiber matrix delamination can be attributed to low heat distortion temperatures of the resin used in preparation of the face-sheets. The heat distortion temperature for Ashland Derakane Momentum is 82ºC, which is very close to the test temperature of the high temperature experiments performed. The center point of the specimen has a deflection of 17.8 mm and a core compression behind the local failure of face-sheet is also evident from images.

Fig. 13 Visual inspection of sandwich specimens after being subjected to blast loading

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5. SUMMARY a)

Under dynamic testing, M100 foam shows thermal softening and the flow stress value decreases with an increase in temperature upto 100ºC. At low temperature, foam has a brittle nature due to hardening and the plateau region shows a decreasing stress value with increasing strain. b) The sandwich specimens were subjected to blast loading at room temperature (22ºC), low temperature (65ºC) and high temperature (80ºC). The sandwich specimen under low temperature demonstrated around 30% more back-face deflection with respect to room temperature and the main failure mechanisms observed are core-cracking and skin delamination. At high temperature, a significant amount of fiber breakage and fiber matrix delamination occurs due to the low tolerance of polymer resin for high temperatures, used in preparation of the sandwich specimens. ACKNOWLEDGEMENT The authors kindly acknowledge the financial support provided by Dr. Yapa D. S. Rajapakse, under the Office of Naval Research (ONR) Grant No. N00014-10-1-0662. Authors thank Gurit SP Technology for providing materials for the study and also thank Dr. Stephen Nolet and TPI Composites for providing the facility for fabricating the materials. REFERENCES [1] Zhang., J., Kikuchi, N., Li, V., Yee, A. and Nusholtz, G., Constitutive modeling of polymeric foam material subjected to dynamic crash loading, Int. J. Impact Eng., 21 (5), 369–386, 1998. [2] Neremberg, J., Nemes, J.A., Frost, D.L., Makris, A., Blast wave loading of polymeric foam, in: Proceedings of the 21 st International Symposium on Shock Waves 1, 91–96, 1997. [3] Tekalur, S.A., Bogdanovich, A.E., Shukla, A., Shock loading response of sandwich panels with 3-D woven E-glass composite skins and stitched foam core, Composite Science and Technology 69 (6), 736–753, 2009. [4] Wang, E., Gardner, N., Shukla, A., The blast resistance of sandwich composites with stepwise graded cores, International Journal of Solid and Structures 46, 3492-3502, 2009. [5] Song, B., Chen, W., Yanagita, T., Frew, D.J., Temperature Effects on dynamic compressive behavior of an epoxy syntactic foam, Composite Structures 67 (3), 289-298, 2005. [6] Birman, V., Kardomateas, G.A., Simitses, G.J., Li, R., Response of a sandwich panel subject to fire or elevated temperature on one of the surfaces, Composites Part A: Applied Science and Manufacturing 37 (7), 981-988, 2006. [7] Erickson, M.D., Kallmeyer, A.R., Kellogg, K.G., Effect of temperature on the low-velocity impact behavior of composite sandwich panels, Journal of Sandwich Structures and Materials 7, 245-264, 2005. [8] Aktas, M., Karakuzu, R., Icten, B.M., Impact Behavior of Glass/Epoxy Laminated Composite Plates at High Temperatures, Journal of Composite Materials 44 (19), 2289-2299, 2010. [9] Salehi-Khojin, A., Bashirzadeh, R., Mahinfalah, M., Nakhaei-Jazar, R., The role of temperature on impact properties of Kevlar/fiberglass composite laminates, Composites Part B: Engineering 37 (7-8), 593-602, 2006. [10] Dutta, P.K., Low temperature compressive strength of glass fiber reinforced polymer composites, Journal of Offshore Mechanics and Arctick Engineering 116, 167-172, 1994. [11] http://www.gurit.com [12] Standard test method for compressive properties of rigid cellular plastics, ASTM Standard D 1621, 2004. [13] Chen, W., Zhang, B., Forrestal, M.J. A split Hopkinson bar technique for low impedance materials, Experimental Mechanics 39 (2), 81–85, 1998.

[14] Gibson, L.J., Ashby, M.F., Cellular Solids-Structure and Properties, Cambridge University Press, Cambridge, 1997.

INFLUENCE OF TEXTURE AND TEMPERATURE ON THE DYNAMICTENSILE-EXTRUSION RESPONSE OF HIGH-PURITY ZIRCONIUM

Daniel T. Martineza, Carl P. Trujilloa, Ellen K. Cerretaa, Joel D. Montalvoa, Juan P. Escobedo-Diaza, Victoria Websterb, and G.T. Gray IIIa a

Material Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA b

Mechanial Engineering Dept. Case Western Reserve University, Cleveland, Ohio

Abstract To create comprehensive models of mechanical deformation in Zirconium (Zr) it is important to observe the effect of high strain on the material. The mechanical behavior and damage evolution in textured, high-purity zirconium (Zr) is influenced by strain rate, temperature, stress state, grain size, and texture. In particular, texture is known to influence the slip-twinning response of Zr, which directly affects the work hardening behavior at both quasi-static and dynamic strain rates. However, while microstructural and textural evolution of Zr in compression and to relatively low strains in tension has been studied, little is understood about the dynamic, high strain, tensile response of Zr. Here, the influence of texture on the dynamic, tensile, mechanical response of high-purity Zr is correlated with the evolution of the substructure. Experiments will be conducted using dynamic-tensile-extrusion process. A bullet-shaped sample has been impacted into a high-strength steel extrusion die and soft recovered in the Taylor Anvil Facility at Los Alamos National Laboratory. Finite element modeling that employs a continuum level constitutive description of Zr will be performed to provide insight into the dynamic extrusion process. Current experimental findings will be presented.

Introduction The hexagonal close packed (HCP) metal, zirconium (Zr) lacks the symmetry and isotropy commonly exhibited in cubic materials. This anisotropy creates a unique challenge in modeling the deformation in the material. To develop accurate predictive models, extensive characterization of the mechanical response must be performed.

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_42, © The Society for Experimental Mechanics, Inc. 2011

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298 Statistically understanding the deformation behavior over a wide range of strain rates, temperatures and stress states is key. Zirconium has been the subject of significant research. A great deal of this work has focused on microstructural evolution due to the deformation dominated by a combination of twinning and slip. Additionally many of these same studies have examined the effect of texture on deformation [1]. However much of this work has been conducted under uni-axial stress conditions. While a few studies such as the one performed by Vogel et al. [2] have macroscopically observed the effects of high pressure on Zr, systematic studies on the effects of high strain and high strain rates on zirconium have not been performed. In this study, high strain and high strain rates will be established using the dynamic extrusion process developed at Los Alamos National Laboratory. Dynamic extrusion is a method for imparting relatively high strains to specimens that have been accelerated into a steel die at velocities of up to 640m/s using a modified Taylor Gun Facility built at Los Alamos National Laboratory[3]. The study of dynamic extrusion on other materials such as Cu and Ta, has found that considerable grain elongation as well as shear localization leading to void formation accommodates deformation at these strains (Cao et al. [6]). Instability develops and the fragments are extruded from the die during testing. The first portion of the study will focus on the effects of varying velocity and therefore varying strain and strain rate on ductility of Zr. Given prior understanding of zirconium, both twinning and slip are expected to dominate deformation. Twinning has been found to be vital in deformation of HCP metals as they have fewer slip systems than cubic materials (Song and Gray [4]). Zirconium has three prominent twin systems: (1-102), (11-21), and (11-22), and extensive twinning is expected to be observed in each at high strain rates (Tome et al. [5]). The next portion of the study will focus on the effects of temperature on the dynamic ductility of Zr. Relative differences as a function of temperature and texture will be examined.

Experimental Method This study examines high-purity, alpha zirconium which has been clock rolled. The average grain size is 35 µm, as measured by the Heyn method. Using Electron Back Scatter Diffraction (EBSD) a strong 0001 basal texture was observed as shown in Figure 1, which is a characteristic of the plate having been clock rolled during processing. The plate had been annealed at 550°C for one hour and as such the initial dislocation density is very low. Bullet shaped specimens of approximately 0.299” in diameter and 0.310” in length were obtained from the as-received plate.

299

Figure 1. Pole figure from EBSD scans of asannealed Zr

  Figure 2: Schematic representation of the asreceived Zr plate and the dynamic extrusion specimens sectioned from this plate as a function of orientation. IP and TT directions are identified.  

These specimens were machined in the in-plane (IP) and through thickness (TT) directions (See Figure 2).   The dynamic extrusion tests were preformed at velocities ranging from 400 to 640m/s using the Taylor Anvil Gas Gun Facility at Los Alamos National Laboratory. The samples impacted a high strength steel extrusion die with an extrusion angle of 81 degrees. The dies were made of S-7 tool steel, machined to tolerance, and heat-treated to a Rockwell hardness of 56-58. High-speed photography was used to view the in-siu extrusion process on a macroscopic level (Figure 3). The high-speed photographs were utilized to observe the elongation of the material as it exited the die and to reassemble fully extruded and soft recovered segments for post mortem characterization. Additionally, these images can be utilized to determine the exit velocity of the specimens.

300

A)

B)

C)

Figure 3. High speed photography of Zr specimens extruded in the following test conditions: A) 534 m/s, in-plane direction, 25°C; B) 535 m/s, through thickness direction, 25°C; and C) 602 m/s, through thickness direction, 250°C.

For all tests, the extruded pieces were soft captured and collected. The portion of the bullet that remained caught in the die was removed by cutting the die near the sample to subsequently weaken the die thus allowing the specimen ejection. All the soft recovered pieces as well as to verify that all pieces were captured. Measuring the length of each soft recovered piece and summing these lengths established the total elongation. Three exampled of reassembled, fully extruded segments along with the portion left in the die are shown in Fig. 4. For many of the tests, each extruded segment along with the die left in the die was examined using scanning electron microscopy (SEM). These pieces were then mounted in epoxy, ground to the mid-section, prepared using standard metallographic preparation techniques, and chemically etched with a solution of 45 ml H2O, 45 ml HNO3, and 10 ml HF for 20-25 seconds. Optical microscopy was then performed on the samples using a microscope equipped with polarized light.

a)

b)

301

c)

Figure 4. Reassembled soft captured fragments of extruded Zr in the following test conditions: A) 534 m/s, in-plane direction, 25°C; B) 535 m/s, through thickness direction, 25°C; and C) 602 m/s, through thickness direction, 250°C.

Results and Discussion Dynamic extrusion tests have been performed on both IP and TT specimens at velocities of 400-640 m/s and at room temperature and 250°C. Total elongations for each test are given in Fig. 5.

Figure 5. Elongation (mm) as a function of velocity Although in all tests the metal was not fully extruded, velocity is seen to strongly affect the fragmentation and elongation of the specimen. Total elongation was measured after reassembling the fragments in the extrusion order according to the high-speed images. From Fig. 5 it is evident that IP samples consistently display more elongation than the TT specimens. Additionally, elongation increases with increasing impact velocity regardless of specimen texture and test temperature. Finally, the role of temperature on dynamic extrusion is less clear. However,

Figure 6. SEM image of the fracture surface of an extruded specimen

temperature slightly increased the elongation in the TT specimens. Additionally, more instability is developed with higher velocity. The number of fragments varies depending on the velocity (ex. five segments

302 for IP specimen tested at 479.8 m/s whereas nine segments were recovered for an IP tested at 654.6 m/s). Scanning electron microscopy (SEM) was used to observe the exterior appearance of the soft recovered segments as well as the tips of the first and last extruded segments to examine the fracture surfaces. This analysis revealed that rather than having failed due to shear as seen in previous studies of cubic materials, the end of the fragments displayed a ductile, fracture surface similar to that seen in tensile tests (Figure 6).   Optical microscopy was performed on many of the tested specimens. Rather than the expected grain elongation, all samples experienced significant recrystallization in most of the fully extruded segments (Figure 7). However, the initial fully extruded segment and segment remaining the die displayed a range of microstructures as shown in Fig. 7.

P1 (a)

(b)

P2

P3

303 Figure 7. Optical images of the IP, 421m/s, 25°C extrusion. The microstructure of the (a) segment left in the die and (b) the first fully extruded piece display deformed grains in region 1, elongated grains in region 2, and recrystallization in region 3. Electron back scattered diffraction was utilized to examine differences in twinning and evolving texture in IP and TT specimens as a function of test velocity. Twinning was more significant in the TT specimens than in the IP specimens and more twinning was observed with increasing test velocity. The significant differences in twinning result if difference texture evolutions between the TT and IP specimens, as is shown in Figs. 8 and 9. This difference in active deformation mechanism as a function of specimen texture is also likely the reason for the differences in the development of instability as a function of texture and velocity and this likely directly influences elongation of the specimens.

(a)

(b)

(c)

Figure 8. IP case: (a) the undeformed microstructure and texture, (b) the microstructure and local texture in the segment left in the die tested at 25°C and 460m/s, and (c) the microstructure and local texture in the segment left in the die tested at 25°C and 600m/s.

304

(a)

(b)

(c)

Figure 9. TT case: (a) the undeformed microstructure and texture, (b) the microstructure and local texture in the segment left in the die tested at 25°C and 468m/s, and (c) the microstructure and local texture in the segment left in the die tested at 25°C and 603m/s velocity. Conclusions Thus far, we can conclude from this study the following about the dynamic tensile extrusion response of high purity Zr: 1.

Impact velocity strongly influences the large-strain tensile ductility of Zr and thus elongation.

2.

Impact velocity influences the development of instability and therefore the number of extruded segments during the extrusion process; higher velocities produce more fragments.

3.

Differences in the relative activation of twinning as a function of velocity and texture correlated with the observed differences in the development of instability and the total elongation of specimens.

Acknowledgements This work has been performed under the auspices of the United States Department of Energy and was supported by the Joint DoD/DOE Munitions Technology Development Program.

305 References 1.

Kaschner, G. C., and G. T. Gray III. “The Influence of Crystallagraphic Texture and Interstitial Impurities on the Mechanical Behavior of Zirconium.” Metallurgical and Materials Transactions A 31A (200): 19972003. Print.

2.

Vogel, Sven C., Helmut Reiche, and Donald W. Brown. “High Pressure Deformation Study of Zirconium.” Powder Diffraction 22.2 (2007): 113-17. Print.

3.

Gray III, G. T., E. Cerreta, C. A. Yablinsky, L. B. Addessio, B. L. Henrie, B. H. Sencer, M. Burkett, P. J. Maudlin, S. A. Maloy, C. P. Trujillo, and M. F. Lopez. “Influence of shock Prestraining and grain size on the dynamic-tensile-extrusion response of Copper: Experiments and Simulations.” Shock Compression of Condensed Matter (2006): 725-28. Print.

4.

Song, S. G., and G. T. Gray III. “Influence of Temperature and Strain Rate on Slip and Twinning Behavior of Zr.” Metallurgical and Materials Transactions A 26A (1995): 2665-675. Print.

5.

Tome, C. N., P. J. Maudlin, R. A. Lebensohn, and G. C. Kaschner. Acta Metall. 49 (2001): 3085-096. Print.

6.

Cao, F., E. K. Cerreta, C. P. Trujillo, and G. T. Gray III. “Dynamic Tensile Extrusion Response of Tantalum.” Acta Materialia 56 (2008): 5804-817. Science Direct. Elsevier Ltd, 14 Sept. 2008. Web. .

Modeling and DIC Measurements of Dynamic Compression Tests of a Soft Tissue Simulant Steven P. Mates, Richard Rhorer and Aaron Forster National Institute of Standards and Technology 100 Bureau Drive Stop 8553, Gaithersburg, Maryland 20899 Richard K. Everett, Kirth E. Simmonds and Amit Bagchi Naval Research Laboratory 4555 Overlook Ave SW, Washington, D.C. 20375 ABSTRACT Stereoscopic digital image correlation (DIC) is used to measure the shape evolution of a soft, transparent thermoplastic elastomer subject to a high strain rate compression test performed using a Kolsky bar. Rather than using the usual Kolsky bar wave analysis methods to determine the specimen response, however, the response is instead determined by an inverse method. The test is modeled using finite elements, and the elastomer stiffness giving the best match with the shape and force history data is identified by performing iterative simulations. The advantage of this approach is that force equilibrium in the specimen is not required, and friction effects, which are difficult to eliminate experimentally, can be accounted for. The thermoplastic is modeled as a hyperelastic material, and the identified dynamic compressive (non-linear) stiffness is compared to its quasi-static compressive (non-linear) stiffness to determine rate sensitivity. INTRODUCTION Tissue simulant materials are sought to provide realistic experimental devices to simulate the human body’s response to blast or impact loading that can occur in military scenarios, law enforcement and emergency response events, vehicle accidents or sporting events [1]. This approach is meant to help develop better protective equipment or procedures to prevent serious injury or death. In most practical injury scenarios, tissues are subject to dynamic loading involving large amplitude strains (20 % [2]) to vulnerable soft tissues at strain rates -1 above 10 s [3]. Numerical models of these test devices are crucial to interpreting the measurement data in these complicated tests, and efforts are underway to provide the material data needed to calibrate such models. Simpler uniaxial mechanical tests of soft tissue specimens show that the large-strain response of these materials is generally non-linear and rubber-like, and can be represented by hyperelastic models developed for polymers [2, 4, 5]. Strain rate sensitivity is also generally observed in soft tissues, prompting the use of viscoelastic models to describe the relaxation behavior [2, 6, 7]. High strain rate measurements of actual tissues and tissue simulants are often performed using a Split Hopkinson -1 Pressure Bar, or Kolsky Bar [8, 9], which can achieve large strains at uniform strain rates in excess of 100 s . The difficulties in obtaining valid high strain rate data on soft materials using Kolsky bar techniques are well documented. Soft materials take much longer to achieve mechanical equilibrium when subject to a rapidly changing load. In most practical situations, equilibrium is not established in the sample until the test is nearly over, invalidating most if not all of the test [8]. Careful selection of specimen thickness and the use of pulse shaping to increase the rise time of the load pulse have been effective methods for achieving valid dynamic test results [10]. Measuring the forces on soft samples is also challenging because they are below the typical sensitivity of Kolsky bars designed for testing metals. Special, highly sensitive force transducers placed directly on either side of the sample are required to obtain force signals [11]. Care must be taken to separate out inertial effects in these force signals to obtain the true specimen response [12]. Finally, confinement techniques have been successfully employed to force either hydrostatic or shear loading conditions at high strain rates [13]. T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_43, © The Society for Experimental Mechanics, Inc. 2011

307

308 For extremely soft materials (less than 10 Shore A), such as the material examined in our experiments, achieving force equilibrium in a Kolsky bar test is exceedingly difficult. Further, friction between the sample and the compression platens is difficult to eliminate entirely, even with generous lubrication. Under dynamic loading, these effects will tend to produce non-uniform deformation in the sample in the form of large amplitude surface waves and prominent barreling which invalidates the usual Kolsky bar assumptions of uniform, uniaxial stress and strain throughout the specimen. In low-rate compression tests, the effect of friction on the perceived stress-strain behavior can be significant. This has led some researchers to impose no-slip boundary conditions in order to more accurately determine the material response with a known, rather than ambiguous, state of friction [4, 14]. Recent advances in optical shape measurement using stereoscopic (3D) Digital Image Correlation (DIC) [15] and in high speed digital camera technology has now made it feasible to measure the shape evolution of soft specimens during dynamic testing in a Kolsky bar. Further, by combining this new measurement capability with finite element modeling, one may be able to deduce the constitutive behavior of the material at high rates using so-called inverse methods [16]. Such methods, which include minimizing the error of finite element simulations against experimental data by systematically adjusting the relevant material parameters has been used to identify the properties of metals [17-19], composites [20], ceramics [21], polymers [22] and biomaterials [23] usually under quasi-static test conditions but more recently at high rates of strain [24]. In this paper, high-speed 3D DIC is used to measure the dynamic deformation of a soft tissue simulant material while it is compressed at high strain rate using a modified Kolsky bar technique. The sample is allowed to deform non-uniformly due to specimen ring-up effects and to friction between the sample and the bar. Finite element modeling is employed to deduce the constitutive behavior of the material by systematically varying the physical parameters governing the mechanical behavior of the modeled specimen and the friction coefficient to match the DIC history and force history data. This approach was first described in [25]. In this updated work, commercial software is used to perform model sensitivity analysis and to identify optimal parameter values. The identified optimal parameters are averaged for five individual experiments and the result is then compared to the quasistatic behavior to determine rate sensitivity of the material. Standard deviations of the identified parameters are used to estimate the overall uncertainty of the identified dynamic response of the specimen. EXPERIMENTAL METHODS The elastomer was cast into 9.5 mm diameter by 6.5 mm thick specimens for dynamic compression testing. The 3 material has a density of 870 kg/m , and it is nearly transparent. Tests were conducted using a maraging steel Kolsky bar measuring 15 mm in diameter with bar lengths of 1500 mm. Forces were measured with piezo-electric dynamic force transducers placed on either side of the specimen, as shown in Fig. 1 below. The force transducers have a resolution of ± 0.1 % of full scale, or ± 0.5 N on the incident side and ± 0.25 N on the transmission side. Polished steel platens (5 mm thick by 15 mm diameter) are placed between the transducers and the sample. No lubrication was used in the experiments. Compressive pulses were created by impacting a 250 mm long, 15 mm diameter maraging steel striker bar against the incident bar at 5.5 m/s. An additional difference between these tests and more traditional Kolsky bar testing is that the upstream platen and force transducer are allowed to collectively decouple from the incident bar to become a “flyer plate.” The flyer has the advantage of producing a much larger dynamic strain in the specimen than if the transducer and platen remained fixed to the incident bar, which was desirable given the limited compression pulse width (and thus specimen strain) our Kolsky bar could generate on a sample of this size. The flyer test is accomplished by using a low-tack adhesive at the incident bar-transducer interface that is strong enough to hold the pieces in place prior to the test but too weak to resist the tensile wave reflection off the incident platen as the wave passes through the sample. A thick, soft rubber pulse shaper was employed to produce a long, gradually rising stress pulse, which helped develop a repeatable release of the flyer from the incident bar. DIC measurements of the sample deformation are acquired at 120,000 frames per second with an image size of 1 128 x 208 pixels. Commercial DIC software [26] is used to perform the image correlation measurements. The

309

Fig.1 Axisymmetric finite element model showing the arrangement of the sample, platens, load cells and transmission bar cameras use 90 mm macro (1:1 magnification) lenses with f/5.6 apertures and are placed 30 cm from the sample with a 12.5 º pan angle. The resolution is 18 pixels/mm. A random speckle pattern was created on the samples using a light dusting of flat black spray paint. Attempts were made to achieve a nominal average speckle size of 5 to 7 pixels in diameter and a coverage factor of 50 % following manufacturer’s recommendations. The speckle patterns were backlit using a Teflon reflector placed directly behind the specimen and illuminated by halogen optical fiber lamps, as shown in Fig. 2. Using this optical set-up, rigid body translations were measurable within ± 0.01 mm. Fig. 3 indicates the typical correlation region on a sample. Correlation measurements used the default software analysis settings (21 pixel windowing, 5 pixel overlap, with default smoothing). The DIC measurement data, obtained from about 200 stereo image pairs during the test, are automatically oriented to a reference plane that is defined by fitting the initial shape of the (cylindrical) specimen prior to deformation. Because of this automatic plane fit and the fact that the sample deforms axisymmetrically, the DIC coordinate system could be aligned with the axisymmetric finite element model coordinate system using simple y- and z-translations. FINITE ELEMENT MODEL The finite element model consists of the flyer assembly (force transducer plus steel platen), the sample, the 1 transmitted force transducer and platen, and the transmission elastic bar. ABAQUS/Explicit is used to perform the simulations. A portion of the modeled mesh was shown earlier in Fig. 1. The model uses axisymmetric CAX4R elements, with 352 elements in the sample and much coarser meshes in the steel parts. The maraging steel compression bars as well as the platens are modeled as linear elastic solids with the following properties: 11 3 E = 1.9x10 Pa,  = 0.29, and ρ = 8048 kg/m . The donut-shaped force transducers are modeled with the same 3 stiffness and Poisson ratio as the maraging steel but with a density of 6594 kg/m . The boundary condition for the simulation is set by specifying the velocity history of the flyer, which is obtained directly from DIC measurements on the flyer itself. Simulations are carried out until the flyer velocity vector begins to deviate from the compression axis, or “pitch,” which occurs eventually when the flyer decelerates and comes to rest against the specimen. The elastomer is modeled as an isotropic hyper-elastic material using the Marlow strain energy potential [27]. Hyper-elasticity is characterized by large, recoverable elastic strains that are characteristic of rubbery materials. Rubbery response in polymeric materials occurs in the region nestled between the visco-elastic response and viscous flow region in strain-rate space [28]. Micromechanical behavior associated with rate dependency and dissipation (viscoelastic effects), such as chain slippage and the breakage and reforming of secondary (crosslink) bonds, is relatively unimportant in the rubbery regime.

1

Commercial products are identified in this work to adequately specify certain procedures. In no case does such identification imply recommendation or endorsement by NIST, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

310

Fig. 2 Measurement set-up

Fig. 3 Typical correlation region shown on an actual image of a speckled sample from which correlation measurements are obtained The rate sensitivity of the elastomer is examined by attempting to identify a purely hyperelastic dynamic response by inverse methods and comparing this identified response to the quasi-static behavior. If the identified dynamic response is found to be much different than the quasi-static response, rate sensitivity would thereby be indicated. The baseline constitutive response of the material used in the model is derived from uniaxial stress-strain data obtained at a low strain rate (4 x 10 -4 s-1) following ASTM D 575-91 [29]. The quasi-static response, shown in Fig. 4, is characteristic of rubbery materials, showing an upturn in the stress-strain curve due to the finite deformability of even very long chain polymeric molecules. To maintain this basic character while allowing for possible rate effects, the response of the specimen is modified by a stiffness scaling factor, m, as shown in Fig. 4.

311

Fig. 4 Quasi-static compression stress-strain response of tissue simulant with polynomial fit and two alternate models with twice (m = 2) and half (m = 0.5) the stiffness of the quasi-static response (defined as m = 1) Rayliegh damping is employed in the numerical scheme to eliminate unphysical oscillations in the simulations to better mimic the dynamic response of the sample. This method is employed by the finite element code as a generic means to account for dissipation in many different materials [27]. It works by adding a small damping stress, σd, to the stress from the basic hyperelastic response that is proportional to the strain rate, by specifying a positive damping factor β:



d

(1)

Ee

The magnitude of the damping factor must be chosen with caution as it is not intended to simulate the strain rate effects due to the bulk micromechanical response of the specimen. To avoid obscuring gross strain-rate effects from those related to numerical damping, a minimal damping coefficient must be chosen that is just large enough to eliminate non-physical oscillations while not adding significant numerical stiffness to the sample. 1

A commercial software package [30] is used to perform the model sensitivity analysis and to identify the dynamic sample stiffness by comparing the simulation results to the experimental data. The software acts as GUI-driven macro that alters finite element model input data, controls the solver execution and displays and analyses the simulation results. It also approximates the finite element model response over the variable space of interest using interpolation functions, and employs optimization tools to identify optimal parameter values using objective functions describing the agreement between the simulation results and the data. The objective functions are described next. OBJECTIVE FUNCTIONS Objective functions expressing the difference between the finite element model results and the data are built individually for the force history and shape history data. Then, a single “cost” function is assembled to represent the overall agreement between simulation and data for identifying optimum parameter values. DIC shape history data are compared to the simulation nodal displacements as indicated in Fig. 1. Residuals are computed at the DIC measurement locations along the center of the correlation region parallel to the compression (y-)axis at each measurement time point (e.g. for each image pair acquired and analyzed). The modeled surface positions are interpolated to match the DIC measurement positions along the length of the specimen. The shape history objective function, ΦS, is given by:

1 S  M

 1  t 1  N  M



2   FEA DIC  Z i ,t  Z i ,t      Z iDIC   i 1   ,t    N



0.5 

   

(2)

312 In Equation 2, Z is the out-of-plane surface position of the deforming sample. The superscript FEA refers to the finite element model result, while the superscript DIC denotes experimental data (DIC measurement). Subscripts i and t refer to space and time, respectively. The transmitted force history objective function, ΦF, is given by:

F 

P

 ([ F jFEA  F jEXP ]2 ) 0.5

(3)

j 1

F is the transmission force, while EXP refers to the experimental data and FEA is again simulation results. At FEA EXP at the locations where F is available. The each time step, ΦF is calculated by linearly interpolating the F combined objective function, Φ, is assembled as follows:



 F  S SFF SFS

(4)

In the above equation, SFF and SFS are scale factors for the force and shape residuals, respectively. The scale factors are selected to weight each individual objective function such that the order of the scaled objective is 1 near the optimum point. For this study, SFF = 1000 N and SFS = 0.001. RESULTS Force-time data from a typical experiment are shown in Fig. 5 along with a sketch showing the corresponding behavior of the flyer plate during the test. Prominent features of the data are labeled in the graph. A sharp rise in the incident force marks the arrival of the incident compressive strain wave generated by the striker. As this wave reflects from the incident platen-sample interface, a tensile (negative) force is observed. This tensile load causes the bond holding the flyer to fail, releasing it into the specimen. As the specimen compresses further, the transmitted force steadily increases until the flyer arrests, causing the transmitted force to peak. Next, the sample begins to release its stored strain energy, pushing back on the flyer. Soon the flyer is thrown back against the incident bar, which all the while has been advancing forward due to the action of the trapped compression wave. This second impact is marked by the final sudden rise in both force signals. Because the free surface of the flyer carries no stress, the forces measured on either side of the specimen will never be equal. Because of this, the force signals cannot be compared to check sample equilibrium nor can they be relied upon to determine the sample stress in the usual way.

Fig. 5 Force signals recorded on the incident and transmitted side of the specimen during a flyer experiment (left) and a sketch of the flyer experiment as visualized using the axisymmetric finite element simulation (right)

313 Fig. 6 plots the overall sample strain and strain-rate versus time for the same experiment. In the test shown, almost 70 % engineering strain is achieved before symmetry breaks down. A Kolsky bar 10 times as long as the one used here would be needed to achieve this level of strain using the same sample at this strain rate without using the flyer technique. A second observation is that the strain rate is relatively uniform over most of the test. Ordinarily this is critical in normal Kolsky bar tests. Here, however, it is less so in this study because the assumption of uniform strain rate is not used in the inverse analysis to deduce the specimen response. The finite element model parameters governing the mechanical response of the sample are the stiffness factor, m, the friction coefficient, f, Poisson’s ratio, , and the damping factor, β. In principle, all of these factors can be examined simultaneously by conducting a sensitivity analysis using a large Design-of-Experiments (DOE) matrix. However, certain parameters can be specified ahead of time to reduce the number of unknowns in the problem. For example, since very soft polymeric materials are incompressible,  = 0.5 should be prescribed. However, computational stability requires that some compressibility be added, which will affect accuracy of the solution of this highly confined compression test [27]. Computational cost and accuracy were found to be adequate with  = 0.4950. The penalty for allowing compressibility is a systematic under-prediction of the out-of-plane deformation, which introduces a systematic error in the value of ΦS. Another parameter that must be prescribed is the damping factor, β. Selecting β = 0.000025 s provides realistic-looking force signal while having little overall effect on the force levels themselves, and therefore not adding a gross amount of stiffness to the material that would confuse attempts to identify a dynamic value of m. Using Eq. 1, this damping level adds a stress equal to -1 about 1 % of the zero-strain modulus of the material for a global strain rate of 400 s .

Fig. 6 Global strain and strain rate histories for a typical flyer experiment from DIC-measured flyer velocity With  and β fixed, we proceed to investigate the influence of the stiffness factor, m, and the friction coefficient, f. To this end, a Full-Factorial DOE is performed between 0.0 < f < 2.0 and 0.75 < m < 1.25 with 10 levels for each parameter. The influences of stiffness and friction on ΦF and ΦS are indicated in Fig. 7. The friction coefficient has dramatic influences on both ΦF and ΦS at low values, but at higher values the sensitivity to friction is minimal. For ΦF, the sensitivity to friction falls beyond about 0.3, while for ΦS the influence dramatically lessens above 0.5. Due to the lack of sensitivity above f = 0.5, identifying an optimal friction coefficient would be difficult and the result would likely be unreliable. Consequently, the friction coefficient is set to the limiting case of no-slip for the remaining simulations used to identify m. DOEs were executed for each experiment with 0.75 < m < 1.5 to identify the dynamic stiffness of the specimen relative to its quasi-static response. The other parameters are fixed to values discussed previously, namely: β = 0.000025 s,  = 0.495 and no-slip friction. A representative plot of the effect of m on ΦF and ΦS is shown in Fig. 8. Both ΦF and ΦS have distinct minima, though not at the same value of m. A conclusion from the latter observation is that the model is not able to achieve perfect agreement with experimental observation, leading to

314 this tradeoff between shape and force agreement. Since neither objective has a clear precedence over the other, the objectives will maintain approximately equal weighting by using the previously-defined scale factors. The simulation results of Fig. 8 are fit with a Radial Basis Function (RBF) approximation model prior to the identification step to reduce the computation time required to identify an optimum value of m [30]. As Fig. 8 shows, the RBF approximations are an excellent representation of the simulation results. A Downhill Simplex optimization method [30] is used to identify optimum values of the stiffness using the force response function for five independent experiments. The identified optimum values, mopt, are listed in Table 1. The average stiffness identified using the inverse method was m opt = 1.05 ± 0.18 (k = 2). Thus, within the observed repeatability level, the material is not strain rate sensitive, as m = 1.0 represents the quasi-static response. The 17 % uncertainty in mopt reflects random errors due to experimental and modeling approximation factors.

Fig. 7 Effect of friction coefficient, f, and stiffness, m, on (a) ΦF and (b) ΦS

Fig. 8 Typical response surface showing the effect of m on actual ΦF and actual ΦS along with RBF approximations used for calculating the optimal m

315 Table 1. Identified values of m opt Experiment

Average Strain -1 Rate [s ]

mopt

ΦF,opt [N]

ΦS,opt

Φ

1786

340

1.18

104

0.00623

6.33

1791

397

1.04

475

0.00607

6.55

1818

405

1.09

649

0.00573

6.38

1819

429

0.94

1346

0.00686

8.20

1820

422

1.02

897

0.00713

8.05

Average

399

1.05

694

0.0064

7.10

70

0.18

930

0.0012

1.86

U

*

*

U is the expanded uncertainty (k = 2) of the average value based on twice the standard deviation of the individual values in the table.

SUMMARY An inverse method was used to determine the dynamic stiffness of a prospective biomimetic elastomer using a Kolsky bar with the intention to determine whether the strain rate sensitivity mimics real tissues. High-speed digital image correlation (DIC) was used to measure the surface deformation of the sample during the test, which suffered from significant friction effects and non-uniaxial deformation due to the extreme softness of the elastomer. Further, a special flyer technique was used to compress the specimen to larger compressive strains that would otherwise have been possible for the NIST Kolsky bar. A finite element model of the experiment was constructed and the resulting simulations were compared to the DIC shape history data and to force history data. The sensitivity of the model to specimen stiffness, friction, Poisson ratio and damping were examined, and the simulation results were compared to the experimental data to identify the dynamic stiffness using an inverse method. An appropriate numerical damping coefficient was determined that produced realistic-looking forcehistory signals while avoiding adding artificial stiffness to the simulated specimen which would confound attempts to identify the true strain-energy-dependent dynamic stiffness. A Poisson ratio of 0.495 was chosen to achieve reasonable computation times, but this led to a systematic under-prediction of the outer surface displacement of the sample compared to the DIC measurements. Friction in the model was very important for capturing the actual deformation of the specimen. Interestingly, the sensitivity of the simulation to friction above f = 0.5 was minimal, so a no-slip friction condition was used. With the friction and damping conditions established, the stiffness scale factor m was observed to produce distinct minima in the force and shape residual plots in five repeat experiments. The optimal stiffness scale factor for these high strain rate tests was m = 1.05 ± 0.18, indicating very little rate sensitivity in this material up to 60 % compressive strain. Thus this particular elastomer is not strain rate sensitive, unlike what has been reported for actual soft tissues. REFERENCES [1] Roberts, J.C., A.C. Merkle, P.J. Biermann, E.E. Ward, B.G. Carkhuff, R.P. Cain and J.V. O’Connor, “Computational and experimental models of the human torso for non-penetrating ballistic impact,” Journal of Biomechanics, 40 (2007) 125-136. [2] Prange, M.T., and S.S. Margulies, “Regional, Directional and Age-Dependent Properties of the Brain Undergoing Large Deformation,” ASME J. Biomech. Eng., 124, 244-252 (2002) [3] LaPlaca, M.C., D.K. Cullen, J.J. McLoughlin and R.S. Cargill, “High rate shear strain of three-dimensional neural cell cultures: a new in vitro traumatic brain injury model,” J Biomech., 38(5) (2005) 1093-1105. [4] Roan, E., and K. Vemaganti, “The Nonlinear Material Properties of Liver Tissue Determined from No-Slip Uniaxial Compression Experiments,” Journal of Biomechanical Engineering, ASME Transactions, 129 (2007) 450456. [5] Fung, Y.C., Biomechanics:Mechanical Properties of Living Tissues, Springer-Verlag, 1993.

316 [6] Sparks, JL, and R. B. Dupaix, “Constitutive Modeling of Rate-Dependent Stress-Strain Behavior of Human Liver in Blunt Impact Loading,” Annals of Biomedical Engineering 36(11) (2008) 1883-1892. [7] Edsberg, L.E., R.E. Mates, R.E. Baier and M. Lauren, “Mechanical characteristics of human skin subjected to static versus cyclic normal pressures,” Journal of Rehabilitation Research and Development, 36 (1999). [8] Saraf, H., K.T. Ramesh, A.M Lennon, A.C. Merkle and J.C. Roberts, “Mechanical properties of soft human tissues under dynamic loading,” Journal of Biomechanics 40 (2007), pp.1960-1967. [9] Chen, W., F. Lu, D.J. Frew and M.J. Forrestal, “Dynamic Compression Testing of Soft Materials,” ASME Transactions 69 (2002), pp.214-223. [10] Song, B., and W. Chen, “Dynamic stress equilibration in split Hopkinson pressure bar tests on soft materials,” Experimental Mechanics, 44 (2004), pp.300-312. [11] Casem, D., T. Weerasooriya and P. Moy, “Inertial Effects in Quartz Force Transducers Embedded in a Split Hopkinson Pressure Bar,” Experimental Mechanics, 45 (2005), pp.368-376. [12] Song, B., Y. Ge, W.W. Chen and T. Weerasooriya, “Radial Inertia Effects in Kolsky Bar Testing of Extra-soft Specimens,” Experimental Mechanics, 47 (2007), pp.659-670. [13] Saraf, H., K.T. Ramesh, A.M Lennon, A.C. Merkle and J.C. Roberts, “Measurement of Dynamic Bulk and Shear Response of Soft Human Tissues,” Experimental Mechanics 47 (2007), pp.439-449. [14] Miller, K, “Method of testing very soft biological tissues in compression,” Journal of Biomechanics, 38 (2005) 153-158. [15] Sutton, M.A., J.-J. Orteu and H.W. Schreier, Image Correlation for Shape, Motion and Deformation Measurements, Springer, 2009. [16] Avril, S. et al., “Overview of Identification Methods of Mechanical Parameters Based on Full-Field Measurements,” Exp. Mech. 48 (2008) 381-402. [17] Mahnken, R, “A comprehensive study of a multiplicative elastoplasticity model coupled to damage including parameter identification,” Computers and Structures, 74 (2000) 179-200. [18] Hoc, T, J. Crépin, L. Gélébart and A. Zaoui, “A procedure for identifying the plastic behavior of single crystals from the local response of polycrystals,” Acta Materialia 51 (2003) 5477-5488. [19] Cooreman, S. et al., “Identification of Mechanical Material Behavior Through Inverse Modeling and DIC,” Exp. Mech. 48 (2008) 421-433. [20] Roux, S., and F. Hild, “Digital Image Mechanical Identification (DIMI),” Exp. Mech. 48 (2008), 495-508. [21] Robert, L., F. Nazaret, T. Cutard and J.-J. Orteu, “Use of 3D Digital Image Correlation to Characterize the Mechanical Behavior of a Fiber Reinforced Refractory Castable,” Exp. Mech. 47 (2007) 761-773. [22] Giton, M., A.-S. Caro-Bretelle and P. Ienny, “Hyperelastic Behavior Identification by a Forward Problem Resolution: Application to a Tear Test of a Silicone-Rubber,” Strain 42 (2006) 291-297. (it does not use DIC). [23] Kauer, M., V. Vuskovic, J. Dual, G. Szekely and M. Bajka, “Inverse finite element characterization of soft tissues,” Medical Image Analysis 6 (2002) 275-287. [24] Kajberg, J, and B Wikman, “Viscoplastic parameter estimation by high strain-rate experiments and inverse modeling – Speckle measurements and high-speed photography,” Int. J. Solids and Structures, 22 (2007) 145164. [25] Mates, S.P., et al.,”High Strain Rate Tissue Simulant Measurements Using Digital Image Correlation,” Proceedings of the 2009 SEM Annual Meeting, June 1-4, 2009, Albuquerque, NM, USA. [26] Correlated Solutions Inc., Columbia, SC, USA. [27] ABAQUS Documentation, Dassault Systèmes, Vélizy-Villacoublay, France. [28] Ward, I.M., and D.W. Hadley, An Introduction to the Mechanical Properties of Solid Polymers, Wiley, Chichester, England, 1993. [29] ASTM D575-91, 2007, “Standard Test Methods for Rubber Properties in Compression,” ASTM International, West Conshohocken, PA, 2007, DOI: 10.1520/D0575-91R07 [30] Isight Documentation, Dassault Systèmes, Vélizy-Villacoublay, France.

Measurement of R-values at Intermediate Strain Rates using a Digital Speckle Extensometry J. Huh1, Y.J. Kim1, H. Huh1 1

School of Mechanical, Aerospace & System Engineering, Korea Advanced Institute of Science Technology, 291 Daehak-ro, Yuseong-gu, Daejeon, 305-701 Korea

ABSTRACT This paper introduces a method to measure the R-value from a speckled pattern with respect to the strain rate and loading direction. The R-value is calculated from series of images taken by a high speed camera by analyzing the deformation history during static and dynamic tensile tests. A Matlab code to track designated points by digital image correlation algorithm was constructed to measure the longitudinal and transversal strain from the speckled pattern, which was created by spraying black paint on white-paint-coated tensile specimens. Using the suggested method, R-values of TRIP590 1.2t and DP780 1.0t were measured with the variation of the strain rate (0.001/sec ~ 100/sec) and loading direction (0° ~ 90°). INTRODUCTION R-value (Plastic strain ratio) is a parameter that indicates the ability of a sheet metal to resist thinning or thickening when subjected to either tensile or compressive forces in the plane of the sheet. It is a measure of plastic anisotropy and sheet metal drawability. ASTM E517 (American Society for Testing and Materials) standard suggests two procedures to measure the Rvalue, manual procedure and automatic procedure [1]. In manual procedure, the R-value is measured from the displacement between the two gauge marks and change in the specimen width after pulling the specimen axially to a certain elongation. In automatic procedure, the R-value is measured from two extensometers, one attached in the axial and the other in the transverse direction. However, neither procedure can be applied to measure the R-value at intermediate strain rates due to the difficulties in attaching extensometers and in limiting the deformation to a certain range. For this reason, there are few researches on the measurement of R-value at intermediate strain rates. One promising solution is the adoption of digital speckle extensometry, which is widely used recently to determine the strain during deformation [2-3]. This technology allows us to obtain the information on the deformation through digital image correlation of a speckled pattern. In this paper, a new method to measure the R-value at intermediate strain rates is suggested based on a digital speckle extensometry. Based on the suggested method, R-values were measured for two advanced high strength steel sheets, TRIP590 and DP780, at the strain rate ranged from 0.001/sec to 100/sec and at the loading angles from 0o to 90o. MEASUREMENT OF R-VALUES USING A DIGITAL SPECKLE EXTENSOMETRY Uniaxial tensile tests were carried out at the strain rate ranged from 0.001/sec to 100/sec. The tensile tests were carried out with Intron 5583 for the strain rate of 0.001/sec and 0.01/sec, and with HSMTM (High Speed Material Testing Machine) [4] for the higher strain rates. Two advanced high strength steel sheets for auto-body, TRIP590 1.2t and DP780 1.0t, were selected for these experiments. The dimensions of a specimen for uniaxial tensile tests were adopted from the previous research [4] and are shown in Fig. 1. The specimens were extracted at intervals of 15 o from 0o (RD; rolling direction) to 90o (TD; transverse direction). R-value is calculated from the true strains in the transverse and longitudinal direction as in Eq. (1) [5].

r

d w  (d w / d l )  d w   d t d l  d w 1  d w / d l

(1)

where dεl, dεw, dεt denotes incremental longitudinal, transverse, and through-the-thickness true strain, respectively. Incompressible condition was assumed in Eq. (1). T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_44, © The Society for Experimental Mechanics, Inc. 2011

317

318

Fig. 1 Specimen dimensions of dynamic tensile specimen for High Speed Material Testing Machine

Fig. 2 Sample images with speckled pattern taken by high speed camera (TRIP590 1.2t-45o, 10/sec)

(a) INSTRON5583

(b) High Speed Material Testing Machine

Fig. 3 Experimental setup of tensile testing apparatus with a high speed camera.. To measure the true strain in the transverse and longitudinal direction during deformation, a random speckled pattern was generated by applying black spray paint on the white-paint-coated specimen surface. Deformation was recorded by a high speed camera, Phantom v.9.0, with the maximum resolution of 800 x 640 (at 0.001/sec) and the maximum sampling rate of 6400 fps (at 100/sec). Fig. 2 shows sample images of a specimen with the speckled pattern during the deformation taken by the high speed camera, and Fig. 3 shows an experimental setup of a tensile testing apparatus with the high speed camera.

319

dyi

dxi

dyf

dxf

(a)

(b)

Fig. 4 Magnified view of Fig. 2: (a) initial state; (b) during deformation. White and Black dots are to be used to calculate longitudinal and transverse strain, respectively. The longitudinal true strain, εl, and the transverse true strain, εw, can be measured by picking out several points and tracking the displacement of the designated points on the series of images. Four designated points will be used to calculate the longitudinal and transverse true strains by Eq. (2) as in Fig. 4. To automate these procedures, a Matlab code was constructed by modifying the existing Matlab code for Digital Image Correlation and Tracking [6].

ew 

d xf  d xi d xi

, el 

d yf  d yi d yi

,  w  ln(1  ew ),  l  ln(1  el )

(2)

Fig. 5 shows a longitudinal strain versus transverse strain curve of TRIP590 at 0.001/sec. The R-value can be calculated from the linearly fitted slope with Eq. (3) since linear relationships were observed in plots for εl vs. εw at all strain rates and loading angles from RD.

r

 d w  (d w / d l ) -r  , Slope in Fig. 5  d l  d w 1  d w / d l 1 r

(3)

To obtain a reliable R-value, the measurement range was limited to the necking instability strain of the steel sheets, which is 12% for TRIP590 and 10% for DP780. The longitudinal and transverse deformations were measured in the maximum broad region to minimize a measurement error. R-values were measured at least five times for each condition for the reliability of the measurement. The comparison of engineering stress–engineering strain curves with and without applying spray paint verifies that application of spray paint does not affect the tensile properties of the material as in Fig. 6.

Transverse strain, t

0.02 0.00 -0.02 -0.04 -0.06 -0.08 0.00

590TRIP 1.2t-TD 0.001 /s 0.02

0.04

0.06

0.08

0.10

Longitudinal strain, l

Fig. 5 Longitudinal strain versus Transverse strain distribution (TRIP590-90o, 0.001/sec)

320

Engineering Stress [MPa]

1000 590TRIP 1.2t - DD, 100 /s

800 600 400

w/ spraying w/o spraying

200 0 0.0

0.1

0.2

0.3

0.4

Engineering Strain

Fig. 6 Engineering stress–strain graph with and without spraying (TRIP590-45o, 0.001/sec) R-VALUES OF AHSS SHEETS AT VARIOUS STRAIN RATES The R-value of TRIP590 and DP780 steel sheets are tabulated on Table 1 and Table 2, respectively for the given range of strain rates and loading angles. Maximum and minimum values are shown along with the average values since the measured R-values have large variation. The variation of the R-value with respect to the loading angle and strain rate are shown in Fig 7 and Fig 8, respectively to visualize the change in the R-value at various loading angles and strain rates. DP780 1.0t

Strain rate [/s] 0.001 0.01 0.1 1 10 100

r-value

1.2

0.9

1.5

0.6 0.00 0

15

30

45

60

75

TRIP590 1.2t

Strain rate [/s] 0.001 0.01 0.1 1 10 100

1.2

r-value

1.5

0.9

0.6 0.00

90

0

o

Loading angle from RD [ ]

15

30

45

60

75

90

o

Loading angle from RD [ ]

(a)

(b)

Fig. 7 Variation of the R-values with respect to the loading angle (a) TRIP590 1.2t; (b) DP780 1.0t

1.2

r-value

1.5

o

Loading angle [ ] 0 15 30 45 60 75 90

TRIP590 1.2t

0.9

0.6 0.00

o

Loading angle [ ] 0 15 30 45 60 75 90

DP780 1.0t

1.2

r-value

1.5

0.9

0.6 0.00 1E-3 0.01

0.1 1 10 Strain rate [/s]

(a)

100

1E-3 0.01

0.1

1

Strain rate [/s]

(b)

Fig. 8 Variation of the R-values with respect to the strain rate (a) TRIP590 1.2t; (b) DP780 1.0t

10

100

321 Table 1 Average, maximum and minimum values of the measured R-values of 590TRIP 1.2t steel sheets (a) 0° (RD), 45° (DD), 90° (TD); (b) 15°, 30°, 60°, 75° (a) 0° (RD)

45° (DD)

90° (TD)

Strain rate [/sec]

avg

max

min

avg

max

min

avg

max

min

0.001 0.01 0.1 1 10 100

1.02 0.97 0.94 1.02 1.06 1.10

1.05 0.99 0.97 1.05 1.08 1.11

0.99 0.92 0.92 0.98 1.02 1.08

0.76 0.72 0.70 0.79 0.81 0.87

0.81 0.78 0.75 0.80 0.84 0.90

0.73 0.69 0.66 0.77 0.80 0.83

1.06 1.03 0.98 1.06 1.10 1.15

1.09 1.08 1.02 1.09 1.14 1.18

1.03 0.98 0.96 1.01 1.06 1.13

(b) 15°

30°

60°

75°

Strain rate [/sec]

avg

max

min

avg

max

min

avg

max

min

avg

max

min

0.001 0.01 0.1 1 10 100

0.96 0.93 0.90 0.97 1.02 1.03

0.97 0.97 0.91 0.99 1.05 1.06

0.94 0.90 0.89 0.93 0.97 1.00

0.82 0.81 0.79 0.84 0.88 0.94

0.86 0.84 0.86 0.86 0.92 0.97

0.78 0.76 0.73 0.79 0.85 0.91

0.80 0.79 0.77 0.84 0.87 0.93

0.86 0.82 0.81 0.88 0.92 0.97

0.76 0.77 0.73 0.81 0.83 0.89

0.95 0.93 0.87 0.95 0.98 1.03

0.97 0.95 0.89 1.00 1.03 1.05

0.92 0.91 0.84 0.90 0.95 1.01

Table 2 Average, maximum and minimum values of the measured R-values of 780DP 1.0t steel sheets (a) 0° (RD), 45° (DD), 90° (TD); (b) 15°, 30°, 60°, 75° (a) 0° (RD)

45° (DD)

90° (TD)

Strain rate [/sec]

avg

max

min

avg

max

min

avg

max

min

0.001 0.01 0.1 1 10 100

0.79 0.73 0.66 0.81 0.82 0.86

0.81 0.78 0.72 0.83 0.86 0.89

0.76 0.65 0.63 0.78 0.77 0.85

0.99 0.93 0.86 1.02 1.05 1.09

1.07 0.99 0.92 1.07 1.08 1.12

0.96 0.89 0.81 0.99 1.03 1.05

0.80 0.78 0.70 0.84 0.86 0.89

0.81 0.82 0.75 0.87 0.89 0.92

0.79 0.75 0.65 0.82 0.84 0.85

(b) 15°

30°

60°

75°

Strain rate [/sec]

avg

max

min

avg

max

min

avg

max

min

avg

max

min

0.001 0.01 0.1 1 10 100

0.84 0.82 0.75 0.88 0.89 0.94

0.85 0.86 0.79 0.94 0.90 1.04

0.83 0.73 0.73 0.83 0.88 0.89

0.92 0.88 0.82 0.93 0.96 1.00

0.96 0.90 0.86 0.97 0.98 1.05

0.87 0.85 0.76 0.91 0.94 0.94

0.87 0.84 0.80 0.92 0.95 1.00

0.90 0.86 0.83 0.94 1.00 1.09

0.84 0.83 0.79 0.91 0.92 0.95

0.83 0.82 0.75 0.87 0.89 0.94

0.86 0.83 0.77 0.89 0.94 0.96

0.81 0.81 0.69 0.85 0.85 0.93

322 The R-value varies with different loading directions, which indicates that these sheets have in-plane anisotropy. The trends in R-value with respect to the loading angle are different for each material. In case of TRIP590, R45 is the lowest and R90 and R0 are higher. By contrast, R45 is the highest followed by R90 and R0 in case of DP780. The different trend is due to the different texture of the two materials. R-value is highly influenced by preferred crystallographic orientations within a polycrystalline material. It can be also noted that the average R-value of both AHSS sheets is lower than that of mild steel, which could lead to poor formability. For both of the material, the R-value changes with the strain rate. It decreases as the strain rate increases up to the strain rate of 0.1/sec and starts to increase above the strain rate of 1/sec. Change in the R-value has a significant effect on the size and the shape of the yield surface. Based on the experimental result, it can be expected that not only the size but also the shape of the yield surface will change with respect to the strain rate. CONCLUSION This paper introduces a new procedure to measure the R-value at intermediate strain rates based on digital speckle extensometry. Two advanced high strength steel sheets for auto-body, TRIP590 and DP780, are considered to investigate the change of the R-value with the suggested method at the strain rate ranging from 0.001/sec to 100/sec. It is observed that the R-value is strain-rate sensitive and it decreases as the strain rate increases up to the strain rate of 0.1/sec and starts to increase above 1/sec for both materials. Based on the experimental result, it can be expected that the size and the shape of the yield surface will change as the strain rate changes. ACKNOWLEDGEMENT The present work was supported by POSCO research fund for POSCO steel research laboratory. REFERENCE [1] ASTM Standard E517 - 00, 2010, "Standard Test Method for Plastic Strain Ratio r for Sheet Metal," ASTM International, West Conshohocken, PA, 2010, DOI: 10.1520/E0517-00R10, www.astm.org [2] E. Parsons, M.C. Boyce and D.M. Parks, An Experimental Investigation of the Large-Strain Tensile Behavior of Neat and Rubber-Toughened Polycarbonate, Polymer, vol. 45, pp. 2665~2684, 2004 [3] F. Laraba-Abbes, P. Ienny and R. Piques, A New ‘Tailor-Made’ Methodology for the Mechanical Behaviour Anlysis of Rubber-Like Materials: I. Kinematics Measurements Using a Digital Speckle Extensometry, Polymer, vol. 44, pp. 807~820, 2003 [4] H. Huh et al., High Speed Tensile Test of Steel Sheets for the Stress-Strain Curve at the Intermediate Strain Rate, International Journal of Automotive Technology, vol.10, no.2, pp.195~204, 2009 [5] Y. C. Liu, On the Determination of Hill’s Plastic Strain Ratio, Metall. Trans. A, vol. 14A, pp. 2566~2567, 1983 [6] C. Eberl, "Digital Image Correlation and Tracking", http://www.mathworks.com/matlabcentral/fileexchange/12413

Study of Strain Energy in Deformed Insect Wings Hui Wan1, Haibo Dong2, and Yan Ren3 Department of Mechanical and Materials Engineering Wright State University, Dayton,OH, 45435, USA Nomenclature E

e h I

r S T U

= = = = = = = = = =

Young’s module Vector of nodal coordinate Wing thickness Identity matrix Absolute coordinates of any point Shape function Kinetic energy Strain energy due to bending and twisting Curvature Density of wing = Relative variation of kinetic energy over a hovering cycle = Relative variation of strain energy over a hovering cycle

ABSTRACT Wing deformation is almost unavoidable in insect flights. In this paper, an approach is introduced to estimate the strain energy in deformed wings during freely hovering of a dragonfly. First, high-speed photogrammetry and three-dimensional surface reconstruction technique are used to quantify the wing kinematics and wing deformation. Finite elements in the absolute nodal coordinate formulation are used to estimate the strain energy associated with wing deformation. The variations of strain and kinetic energy within a stroke cycle are presented and the implications are discussed. Keywords High speed imaging; Insect wing deformation; Absolute nodal coordinate formulation; Strain energy

1. Introduction Wing flexibility and its implication on aerodynamic performance and structural response are fundamental, but challenging problems in the natural insect flight [1, 2]. The measurement of wing deformation has been extremely difficult if it is not impossible at all. However, with a few advent usages of high-speed photogrammetry in insect flight (e.g. Walter et al. 2008 [3], Dong et al. 2010 [4]), it is now possible to quantify the wing deformation with improved accuracy. In fields of materials and structure, Tiwari et al. [5] using high-speed camera measured the plate deformation and calculated Lagrangian strain field due to blast loading. In this paper, High-Speed Photogrammetry is used to record the hovering of a dragonfly (Anax Junius). The wing kinematics and deformation are obtained by three-dimensional surface reconstruction techniques [6]. The flapping wings of the dragonfly feature large rotation and large deformation. Thus, the absolute nodal coordinate formulation (ANCF) [7] based finite elements are used to estimate the strain energy induced by wing bending and twisting. ANCF is characterized by using of slope coordinates which constrain the element orientation. ANCF was first initiated by Shabana et al. [7, 8] to study the dynamics of flexible multi-body system, then this procedure was used to investigate the complex deformation threedimensional beam[9]. Recently, Dmitrochenko et al. [10, 11] extended ANCF to the plate finite element. In the following of this paper, we will first briefly introduce the high-speed camera system, followed by the fundamental theoretical background

                                                             1

Research Scientist, [email protected] Assistant Professor, [email protected] 3 Graduate student, [email protected] 2

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_45, © The Society for Experimental Mechanics, Inc. 2011

323

324

of ANCF. Then we will discuss the obtained strain energy in the dragonfly wings. The variation of strain energy during a hovering cycle is compared with that of kinetic energy.

2. Experimental Setup and Surface Reconstruction In our high-speed imaging system, three Photron Fastcam SA3 60k cameras are aligned along the principal coordinate axes, as shown in Fig. 1. Each camera in the system can capture black and white pictures with 1000FPS (frames per second) at full resolution (1024x1024) with a shutter speed of 2 us. The cameras are synchronized using daisy-chain method. The cameras are triggered by an external Transistor-Transistor Logic (TTL) signal. By configuring the cameras in a master-slave arrangement, it is possible to trigger one camera to relay the signal to the other two cameras. By using this method of triggering, we are capable of minimizing camera delay and maximizing the response speed. Our current measurements indicate the delay between cameras at a maximum of 30 nanoseconds, which for our applications can be taken as zero delay. More information about the camera system and data acquisition can be found in Dong et al. [4]. The calibration of high-speed camera data, and the detailed procedure to generate 3D reconstruction of wings and body of a hovering dragonfly are presented in [6], based on data gathered from the aforementioned photogrammetry system. A reconstructed dragonfly body and wing surfaces are shown in Fig. 2.

 

  Fig. 1 Experimental setup

Fig. 2 Reconstructed dragonfly body and wing surfaces

3. Theoretical Background Absolute nodal coordinate formulation is applied in the current study of flapping dragonfly wings, which undertake large displacement, large rotation and large deformation during strokes. The principal idea of ANCF is to use the coordinates referred to the global reference system, i.e., it introduces large displacements of finite elements relative to the global frame without using any local frame. The nodal coordinate vector includes the nodal position vector and nodal slope vector. In current paper, the reconstructed wing surfaces of dragonfly are discretized by triangular elements as shown in Fig. 3. Figure 4 presents a triangular element in a curvilinear coordinate system OP1P2. The global coordinate ( of any point in the element can be written as: (1)

,

in which the vector of nodal coordinate

(Fig. 4) in equation (1) has the form of (2)

In detail, the nodal coordinate vector

can be obtained from

,

∂ / ∂P ,

∂ / ∂P

(3)

As shown in Fig. 4, on each node of the element, there are nine degree of freedom, including three components of nodal position, and six components of nodal slope. Therefore, each element has 27 coordinates.

325

Fig. 3 Discretized wing surface after reconstruction, only the right hand side wings are show The shape function

Fig. 4 Triangular element and nodal coordinates

in equation (1) can be expressed as

S where is the 3 by 3 identity matrix. Thus

S

S

S

S

S

S

S

,

(4)

has a dimension of 3 by 27. The shape function can be given in detail as:

S

L S S

in which

Q

S

LL

LL c Q c Q

1 LLL 3 1 2

L L 2Q 2Q c Q L L c Q , L L µ L

1

3µ

(5)

L

L

(6)

and µ ℓ ℓ /ℓ , ℓ is the length of the i-th triangle edge. L is the triangular coordinates defined in elemental coordinate system. The coefficients c are determined from the local coordinate system OP1P2, their expression and the details of triangular element shape functions can be found in Dmitrochenko [11]. Ignoring the longitudinal and shear deformations in the wing, the strain energy in dragonfly wings induced by bending and twisting can be written as:

U=

,

(7)

where , , is the curvature vector, A is wing surface area. is a 3 by 3 matrix determined by the material properties. Given the normal vector of the element, the curvature components can be calculated through

=

,

(8)

where

c 2A

c ∂ ∂L ∂L 2A

(9)

The kinetic energy of the wings are simply defined as

T=

(10)

326

4. Results and Discussion The strain energy in wings of a dragonfly under hovering is being studied. To simplify the problem, the material properties of the fore-wing and hind-wing are assumed to be isotropic, and the thickness of the wings is constant. Thus, in equation (7) leads to:

1

12 1

0

0 1 0 0 1

(11)

Strain energy

Kinetic energy

The surface averaged Young’s modulus of dragonfly wings are chosen as 4.8GPa (Jongerius and Lentink [12]). The Poisson’s ratio is 0.49, as used by Combers and Daniel [1, 2]. The density of wings is 1200Kg/m3, value from [12] and [13].

a)

Start of upstroke

b) Middle of upstroke

c)

Start of downstroke

d) Middle of downstroke

Strain energy

Kinetic energy

Fig. 5 Kinetic energy and strain energy distribution of fore-wing at four time instants. In each figure above, the wing root is at the leftmost

a)

Start of upstroke

b) Middle of upstroke

c)

Start of downstroke

d) Middle of downstroke

Fig. 6 Kinetic energy and strain energy distribution of hind-wing at four time instants As shown in Fig. 3, only the wings at right hand side are studied. The kinematics and deformation of left hand side wings are assumed to be images of their right side counterparts, since the dragonfly is under hovering with negligible maneuvers. The distributions of kinetic and strain energy of fore-wing have been shown in Fig. 5 for four instants during a stroke cycle. First, wing tip has higher kinetic energy than wing root obviously. This is in contrast to the strain energy, which obtains its

327

higher value in a region surrounding the wing root. However, the region very close to the wing root actually has smaller strain energy. Therefore, the wing root of dragonfly may be more appropriately modeled as a hinged structure, instead of clamped structure. This is more understandable from biological point of view. The dragonfly muscles would have to consume a lot more energy to firmly clamp the wing root, compared to hinge it with certain degree of flexibility and leave the bending and twisting naturally adjusted by the wing structure (e.g., vein corrugation). Furthermore, small environmental variation (e.g. aerodynamic load) caused changing of wing deformation and associated disturbance on strain energy may be naturally absorbed by the wing structure itself, instead of transferring to wing root and sensed by neurons in muscles. In other words, the high strain region surrounding the wing root serves a screen that filters out the small fluctuations in environment. For large environmental variations, the high strain region may extend to the wing root. Thus, the increased strain at wing root can be sensed and actions of muscles may be taken to change flight status if needed. Figure 6 shows the energy contours of hindwing. Similar distribution is obtained compared to those of fore-wing. Figure 7 has shown the variation of integrated kinetic energy (equation (10)) and integrated strain energy (7) over a stroke cycle. To avoid the influence on the absolute value of energy due to chosen of the material parameters, we define the relative variations as follows:

,

(12)

where t is time during a stroke. and are the time averaged kinetic and strain energy over a cycle respectively. The negative and positive value of means the energy is below and above the average respectively. The minimum kinetic energy is at the start of upstroke and start of downstroke. The maximum kinetic energy is obtained at the middle of upstroke or downstroke. The kinetic energy varies from minimum to maximum twice during a flapping cycle. The strain energy, however, varies only once. The strain energy is minimum near the start of upstroke, and increases to its maximum at the beginning of downstroke (end of upstroke). It then shows a quick release during the stroke reversal, which may be used to facilitate the wing pronation.

a. Fore-wing b. Hind-wing Fig. 7 Relative variation of integrated kinetic and strain energy over a flapping cycle. Note the flapping of the fore-wing and hind-wing has certain phase difference. Here we have removed the phase difference by showing each wing starts from the initiation of upstroke Also, we have seen that the time duration of upstroke and downstroke are different for both fore-wing and hind-wing. The downstoke takes less time compared with upstroke, which indicate the dragonfly stroke downwards faster to produce enough lift. This is also confirmed by the higher pick value of during downstroke.

5. Summary High-speed photogrammetry and surface reconstruction are used to quantify the wing deformation of a freely hovering dragonfly. Then the wing surface is discretized by triangular plate elements using absolute nodal coordinate formulation. The strain and kinetic energy of dragonfly wing are then calculated over a stroke cycle. Wing root is found to be surrounded by the high strain region, which may serve as a buffer zone to screen the wing root from the small disturbance of environment. The integrated strain energy shows minimum near the supination, and gains its maximum at the beginning of downstroke to

328

facilitate the fast pronation. Both the fore-wing and hind-wing take shorter time in downstroke process, thus higher speed and lift can be generated compared to those during upstroke.

Acknowledgements Financial support from AFOSR DURIP-09 Grant #: FA9550-09-1-0460, Wright State University (WSU) Research Challenge Program, and Center for Micro Air Vehicle Study (CMAVS) at WSU are greatly appreciated.

Reference [1] Combes, S. A. and T. L. Daniel, "Flexural stiffness in insect wings. I. Scaling and the influence of wing venation." J Exp. Biol. 206, pp. 2979-2987, 2003. [2] Combes, S. A. and T. L. Daniel, "Flexural stiffness in insect wings. II. Spatial distribution and dynamic wing bending." J Exp. Biol. 206, pp. 2989-2997, 2003. [3] Walker S.M., Thomas, A.L.R, and Taylor G.K. "Photogrammetric reconstruction of high-resolution surface topographies and deformable wing kinematics of tethered locusts and free-flying hoverflies", Journal of the Royal SocietyInterface, Vol. 6, pp. 351-366, 2009. [4] Dong H.B., Koehler, C., Liang, Z.X., Wan, H., and Gaston, Z., "An integrated analysis of a dragonfly in free flight", 40th AIAA Fluid Dynamics Conference and Exhibit, AIAA2010-4390, 2010. [5] Tiwari, V., Sutton M.A., McNeill S.R., Xu, S.W., Deng, X.M., Fourney, W.L., Bretall D., "Applicatin of 3D image correlation for full-filed transient plate deformation measurement during blast loading", Internatial Journal of Impact Engineering, Vol. 36, pp. 862-874, 2009. [6]  Koehler, C., Liang, Z.X., Gaston, Z., Dong H.B., and Wan, H., "Wing Reconstruction, Deformability and Surface Topography of Free-Flying Dragonflies" to be submitted. [7] Shabana, A. A., Hussien, H and Escalona, J. L. Application of the absolute nodal coordinate formulation to large rotation and large deformation problems. ASME Journal of Mechanical Design. 120(3), 188–195,1998 [8] Shabana, A.A. Computer implementation of the absolute nodal coordinate formulation for flexible multi-body dynamics. Nonlinear Dynamics. 16, 293–306, 1998. [9] Shabana, A.A. and Yakoub, R.Y. Three-dimensional absolute nodal coordinate formulation for beam elements: Theory. ASME Journal of Mechanical Design. 123, 606–613, 2001.  [10] Dmitrochenko, O., and Pogorelov, D.Y., "Generalization of plate finite elements for Aboulute Nodela Coordiante Formulation ", Multibody System Dyanics, Vol. 10, pp. 17-43, 2003. [11] Dmitrochenko, O., and Mikkola A., "Two simple triangular plate elements based on the absolute nodal coordinate formulation", Jounal of Computational and Nonlinear Dynamics, Vol. 3, pp. 041012, 2008. [12] Jongerius, S.R. and Lentink D., "Structural Analysis of a Dragonfly Wing", Experimental Mechanics, Vol. 50, pp 13231334, 2010. [13] Vincent, J.F.V. and Wegst, U.G.K., "Design and mechanical properties of insect cuticle", Arthropod Stru Dev. Vol. 33, pp 187–199, 2004

Experimental Study of Cable Vibration Damping Arup Maji a and Yuanzhong Qiu b a

Professor, Civil Engineering Department, University of New Mexico, CENT, Albuquerque, NM 87131, U.S.A b

Graduate Research Assistant, Civil Engineering Department, University of New Mexico, CENT, Albuquerque, NM 87131, U.S.A

ABSTRACT: Measuring and understanding the damping characteristics of cables is particularly significant for structures deployed in space using cables where vibration damping is critical for structural stability. This paper describes an experimental set-up, and provides results from several tests on steel and carbon fiber cables under varying tensile forces. The natural frequencies and the modal damping ratios under different tensile forces are experimental determined; the Rayleigh damping constants are calculated. The difference of the natural frequency between test results and analytical formula are discussed. It is shown that the damping of the cable decreases as the tension force increases. For the case of the carbon fiber cables, the damping decreases as the number of turns of the carbon fiber tows use to make the cable increases. 1.

Introduction

Cables are commonly used tension members in modern flexible structures, and in many applications it is important to know their dynamic properties. Measuring and understanding the damping properties of cables is particularly significant for structures deployed in space using cables where in the absence of air material-related vibration damping is critical for structural stability. Several researchers have investigated the dynamical behavior of cables. Based on experimental and simulated data, Barbieri et al. [1] establish a procedure to identify damping of transmission line cables. Yamaguchi and Adhikari [2] analytically investigated the modal damping characteristics of a single structural cable. They used the energy based representation of modal damping as the ratio of the modal strain energy to the total potential energy. Huang and Vinogradov [3] proposed a model to consider the inter-wire slip and its influence on the dynamic behaviors of the tension cables. Though lots of researches have been conducted on cable damping, the internal damping mechanism of cables has not been fully understood. This is particularly true for the carbon-fiber cables used in the space structures. Working towards fully understand the cable vibration damping of the deployable structures. A test apparatus was built to allow the length, size and tension of the cable to be varied. Vibration tests of stainless steel and carbon-fiber cables were conducted to investigate the characteristics of cable damping. 2.

Experimental set-up

The experimental set-up shown in Fig.1 consists of a wooden frame with two metal plates that can clamp the cables to maintain the applied tension force. Figure 2 shows the load cell and the data acquisition equipment. Fig.3 shows a close-up of the fixture for applying tension forces. Each cable is first fixed to the clamp at the right end of the wooden frame (Fig.1). This is done by screwing four bolts to tighten the two steel plates that constitute the clamp shown at the right end of Fig.3 (b). The left end of the cable is passed through the clamp on the left (Fig.3b) which is initially not tightened. The left of the cable forms a loop that can be connected to a load cell, shown as Fig.3 (a). The load cell with a digital meter provides the readings of the cable tension force during the test. The other end of the load cell is connected to an all thread stainless steel rod, which

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_46, © The Society for Experimental Mechanics, Inc. 2011

329

330 is used to adjust the cable tension force by tightening the bolts at the left end, shown in Fig.3(c). Once the appropriate load is applied, the clamp on the left (Figure 3b) is tightened to hold the tension during the tests. Two accelerometers, type PCB Model 352A73, are mounted to the cable at approximately three quarters of the span and one fifth of the span length, respectively, shown as Fig.3 (b). These acceleration locations were chosen to such that the first 3 vibration modes could be detected at either transducer by avoiding the nodal points. Having the two accelerometers located on opposite sides on the cable span aims to minimize the torsion effects caused by small movements of the accelerometers during the cable oscillations.

Fig.1 Experimental set-up

(a) NI USB 9234 data carrier

(b) Load cell Fig.2 Sensor devices

(a)

(b)

(c)

Fig.3 The connections of the experimental setup parts The steel cable been tested is made of several stainless steel wires. The carbon fiber cables are made from IM7 carbon fiber (HERCULES INC, type IM7-W-12K). Seven fiber tows are twisted to form a carbon fiber cable. To fabricate the carbon

331 fiber cables, the untwisted strands are first fixed at one end, the other end of the strands are then twisted for 10, 20 and 30 turns to form three different kinds of carbon fiber cables for testing. This aims to investigate how the number of turns influences the damping of the carbon-fiber cable. The geometric and the mechanical properties of the stainless steel and the carbon fiber cables are summarized in Table. 1. 3.

Testing and data processing

The cable is excited at the center. To investigate the influence of tension on damping, the tests are conducted under different cable tension forces of 25lb, 50lb, 75lb, 100lb and 130lb. Data acquisition hardware with signal conditioning function, type NI USB 9234, is connected to a laptop. It digitizes the incoming signals to the analog output signals. The stored data of the signals is used to obtain the spectrum of the signals via LabVIEW 8.6 with the Fast Fourier Transformation (FFT). The modal damping ratio is calculated using the half-power bandwidth method in the frequency domain and the Logarithmic decrement method in the time domain, respectively. For the in-plane vibrations, five sets of data are recorded for each accelerometer. Table.1 cable properties Mass per unit length Length (in) (lbs/in) 12 0.00122 12 0.000184 12 0.000187 12 0.000195

Type of cable Steel cable Carbon fiber cable-10 turns Carbon fiber cable-20 turns Carbon fiber cable-30 turns 4.

Young’s Modulus (psi) 29×106 40×106 40×106 40×106

Experimental results

Fig.4 shows the time history signals of the carbon-fiber cable with 10 turns under tension of 75lbs, recorded by the two accelerometers at three quarters and at one fifth of the cable length. Fig.5 presents the corresponding frequency domain data. The frequencies obtained by the spectral analysis are showed in Table.2 for different tension forces of the cable. The relation between cable vibration frequency and cable tension force is plotted in Fig.6. As shown by Fig.6, the frequency of the cable vibration increses as the tensile force increases. 1500

1000

1000

Acceleration

500 0

-500

0.6

0.62

0.64

0.66

0.68

500 0 -500

0.6

0.65

-1000

-1000 -1500

0.7

Acceleration

1500

-1500

Time/(s)

Time/(s) (a) From accelerometer A

(b) from accelerometer B

Fig.4 The time-history signals for 10 turns carbon-fiber cable under tension of 75lbs

0.7

4

Acceleration(FFT Peak)

Acceleration(FFT Peak)

332

3 2 1 0

0

500 1000 Frequency/Hz

1500

3 2.5 2 1.5 1 0.5 0

0

(a) From accelerometer A

500 1000 Frequency/Hz

1500

(b) from accelerometer B

Fig.5 The spectral of signals for 10 turns carbon fiber cable under tension of 75lbs

Tension Force (lbs)

Table.2 Frequencies of cable vibration under different tensile force Frequency (Hz) Carbon-fiber cable_10 Carbon-fiber cable_20 Carbon-fiber cable_30 turns turns Frequencies (Hz) turns

25 50 75 100 130

309.87 392.63 458.67 517.74 554.68

280.96Frequencies (Hz) 369.02 430.00 Steel Cable 500.49 534.65

Stainless steel cable

266.82 360.33 418.68 485.62 502.50

Carbon fiber cable_10 turns

Carbon fiber cable_20 turns

Carbon fiber cable_30 turns

Stainless steel cable

185.104 218.252 247.820 275.036 303.326

Frequency/Hz

600 500 400 300 200 100 0

0

20

40

60 80 Tensile force/lbs

100

120

140

Fig.6 The relation curve of cable vibration frequency and tensile force The experimental results of the cable vibration frequency under varying tension force are compared to the theoretical results. The theory based on inextensibility of the cable, suggests that the first symmetric natural frequency is given by[4, 5].

1  2.86

 H l

m

(1)

333 Where: l is the length of the cable, H is the horizontal tension of the cable, and m is the mass per unit length of the cable. Fig.7 shown the plots of the trend curves of the experimental results and the theoretical values obtained by Eq. (1). It is observed from Fig.7 that the experimental results have the same trend as the theoretical results, i.e., the frequency increases as the tensile force increases. However, the measured frequencies for carbon-fiber cables are smaller than the theoretical values. The possible explanation is that for Eq. (1), it is assumed that the cable is inextensible, so the length used for the calculation is unchanged. However, the actual cable is capable of some extension, which reduces stiffness and results in the measured frequencies being smaller than the theoretical ones. Table.3 shows the damping of the cable under different tensile forces. The damping is obtained by the half-power bandwidth method in the frequency domain and the Logarithmic decrement method in the time domain, respectively. The damping values corresponding to different tension forces in Table.3 are obtained by averaging the ten damping values obtained from the data recorded at the two measurement locations during the five separately tests. Fig.8 shows the relationship between cable damping and the tensile force using the frequency domain and time domain analysis, respectively. The results demonstrate that higher tension results in lower damping, and that the results obtained by the two methods are similar. In addition, the damping value is in the range of 1-3 percents for the tested cables. For the carbon-fiber cables twisted by different turns, as the number of twists in the cable increases, the damping of the cable decreases. This could be because when the number of twist increases, the cable becomes tighter, reducing the movement and friction internal to the cable.

700.00 600.00 500.00 400.00 300.00 200.00 100.00 0.00

Theoretical

Experimental

Frequency/Hz

Frequency/Hz

Experimental

0

50

100

150

700.00 600.00 500.00 400.00 300.00 200.00 100.00 0.00

0

50 100 Tensile force/lbs

Tensile force/lbs (a) Carbon-fiber cable twisted 10 turns Experimental

Experimental Frequency/Hz

Frequency/Hz

600.00 400.00 200.00 0.00

0

50 100 Tensile force/lbs

150

(c) Carbon-fiber cable twisted 30 turns

150

(b) Carbon fiber cable twisted 20 turns

Theoretical

800.00

Theoretical

350.00 300.00 250.00 200.00 150.00 100.00 50.00 0.00

0

Theoretical

50 100 Tensile force/lbs

(d) Stainless steel cable

Fig.7 The relation curve of cable vibration frequency with tensile force

150

334 Table.3 Damping ratios (%) of different cables and damping variation (%) between frequency domain and time domain Carbon-fiber cable Carbon-fiber cable Carbon-fiber cable Stainless steel cable damping_10 turns (%) damping_20 turns (%) damping_30 turns (%) damping (%) Tension Force (lbs) Frequency Time Error Frequency Time Error Frequency Time Error Frequency Time Error domain domain (%) domain domain (%) domain domain (%) domain domain (%) 2.42

2.39

-1.240

2.27

2.27

0.000

1.68

1.74

3.571

2.265

2.29

1.104

50

2.11

2.05

-2.844

1.8

1.79

-0.556

1.24

1.32

6.452

2.007

1.96

-2.342

75

1.83

1.85

1.093

1.68

1.75

4.167

1.18

1.24

5.085

1.923

1.83

-4.836

100

1.45

1.48

2.069

1.39

1.38

-0.719

0.97

1.03

6.186

1.403

1.38

-1.639

130

1.4

1.45

3.571

1.36

1.34

-1.471

0.93

0.94

1.075

1.155

1.19

3.030

Carbon fiber cable_10 turns

Carbon fiber cable_10 turns

Carbon fiber cable_20 turns

Carbon fiber cable_20 turns

Carbon fiber cable_30 turns

Carbon fiber cable_30 turns

Stainless steel cable 3 2 1 0

0

50 100 Tensile force/lbs

150

Damping-time domain/%

Damping-frequency domain/%

25

Stainless steel cable 3 2 1 0

0

50 100 Tensile force/lbs

150

Fig.8 Variation of cable dampings with applied tension Meanwhile, in order to evaluate the dependence of damping on the vibration mode, the Rayleigh damping theory is introduced. For the Rayleigh damping theory [6], it is assumed that the damping matrix is proportional to the combination of mass and stiffness matrices as given by the expression of Eq. (2):

C   M    K  (2) Where: C  =damping matrix of the vibration system  M  =mass matrix of the vibration system

 K  =stiffness matrix of the vibration system

 and  are the constants After orthogonal transformation and reduction to n-uncoupled equation, we can obtain the following: 2     2  1 1 1  2 (3) 2        2  2 2

335 From Eq. (3), the following is obtained:  2  (    )   1 2 1 2 2 1  22 12  (4)    2( 22 11 )  2 2 2 1 

Where 1 and

2 can be determined from a spectrum analysis of the vibration of the cable system.  1 and  2 are determined

by the half-power bandwidth method. Before applying those values to Eq. (4), the values are smoothed by quadratic curve fitting. The values of  and  for different tension forces obtained by Eq. (4) are plotted in Fig.9. From the plots, it can be observed that  initially increases as cable tension increases, and then decreases. Fig.9 shows that  for both carbon-fiber cables and the stainless steel cable decreases as tension increases. Carbon fiber cable_10 turns

Carbon fiber cable_10 turns

Carbon fiber cable_20 turns

Carbon fiber cable_20 turns

Carbon fiber cable 30 turns

Carbon fiber cable 30 turns

Stailess steel cable

Stailess steel cable 0.000200

5.000 0.000 -5.000

Beta

Alpha

10.000

0

50

100

150

0.000150 0.000100 0.000050 0.000000

0

Tensile force/lbs

50 100 Tensile force/lbs

150

Fig.9 Relation curves of alpha and beta with tension force Conclusions An experimental setup was designed to measure the damping of the steel cable and carbon fiber cables twisted by different turns. The natural vibration frequency was obtained by spectral analysis, and it was compared to the theoretical solution based on the assumption that the cable is inextensible. In addition, Half-power bandwidth method and the Logarithmic decrement method were employed to obtain the damping ratios in frequency domain and time domain, respectively. The Raleigh damping constants were computed using the derived equations. From the experimental results, the following conclusions can be drawn: The frequency of the cable vibration increases as the tensile force decreases, and for the case of carbon-fiber cables, the frequency decreases as the number of twists in the cable increases; The damping of the cable decreases as tensile force increases, and as the number twists of the carbon-fiber cable increases; Also, the damping obtained from frequency domain method agrees well with the results from the time domain method. Regarding the damping coefficients based on the Rayleigh damping theory, the stiffness damping coefficient  decreases as the tensile force increases while the mass damping coefficient References:

 initially increases followed by a decrease.

336 [1] Nilson Barbieri, Oswaldo Honorato de Souza Junior, Renato Barbieri. ―Dynamical analysis of transmission line cables Part 2—damping estimation.‖ Mechanical Systems and Signal Processing, 18 (2004), pp.671–681. [2] H. Yamaguchi, and R. Adhikari, ―Energy based evaluation of modal damping in structural cables with and without damping treatment. Journal of Sound and Vibration, 181(1), pp.71–83, 1995. [3] X. Huang and O. Vinogradov. ―Inter-wire slip and its influence on the dynamic properties of tension cables.‖ Proceeding of the second (1992) International Offshore and Polar Engineering Conference, San Francisco, USA, 14-19, June, Volume (II), pp.392-396, 1992 [4] K.A. Kashani, ―Vibration of Hanning Cables.‖ Computers & Structures, Vol.31, No.5. pp.699-715, 1989. [5] H.Max Irvine, Cable Structures, 1981, MIT, ISBN 0-262-09023-6. [6] Anil K. Chopra, Dynamics of structures-Theory and Applications to Earthquake Engineering, 3 rd Edition, ISBN-978-81203-3446-5.

Dynamic Thermo-Mechanical Response of Austenite Containing Steels

V-T. Kuokkala1, S. Curtze1,2, M. Isakov1 and M. Hokka1 1

Tampere University of Technology, Department of Materials Science, P.O.B. 589, 33101 Tampere, Finland 2 Oxford Instruments Nano Analysis, Nihtisillankuja 5, 02631 Espoo, Finland

ABSTRACT Austenite can be made thermally stable at room temperature by alloying the steel for example with nickel or manganese, which brings the martensite start temperature of the alloy below RT. Also low alloy steels can contain relatively high amounts of (retained) austenite brought about by appropriate heat treatments, which increase the carbon content of some of the austenite grains to such high levels that thermal martensite transformation does not take place in them. Well known steels containing stable or metastable austenite at room temperature are austenitic and duplex stainless steels, Hadfield manganese steels, low alloy TRIP steels, and high manganese TRIP and TWIP steels. Depending on the deformation temperature and strain rate, deformation induced martensite transformation and/or twinning, or the lack of them, can lead to quite extraordinary behavior and strength and elongation combinations. In this paper, the strain hardening behavior and strain rate sensitivity of several fully or partially austenitic steels are discussed in view of their microstructural development during deformation. The discussion is based on the experiments conducted on these materials in wide ranges of strain rate and temperature with conventional materials testing machines and various Hopkinson Split Bar techniques. INTRODUCTION In their simplest form, steels are solid solutions of iron and carbon only. Because of this, probably the most important phase diagram in materials science is the iron-carbon, or rather the iron-iron carbide (Fe3C), phase diagram. It presents the different equilibrium phases and their relative amounts at different iron-carbon compositions and temperatures. By alloying also other elements in varying amounts, stability of different phases and microstructures can be changed. In plain or low alloy carbon steels, austenite is stable only at relatively high temperatures, but by proper alloying it can be made stable down to very low temperatures, as happens for example with many austenitic stainless steels. The low temperature microstructures and existing phases depend not only on alloying but also on the thermal and mechanical deformation history of the steel, i.e., the cooling rate(s), hold times at different temperatures, and deformation taking place before, during, and after cooling. For example, austenite may be thermally stable (or metastable) at low temperatures (such as room temperature), but it may undergo a phase change when it is sufficiently mechanically deformed, as happens for example with certain austenitic stainless steels, where (non-magnetic) austenite transforms mechanically into (magnetic) martensite. Good practical examples of this are found in every household: many items of cutlery and for example the kitchen sink are made of so-called 18/8 austenitic stainless steels. Their straight parts are non-magnetic, but curved (deformed) parts are clearly attracted by a permanent magnet. The effects of temperature and deformation on the austenite-to-martensite transformation are schematically depicted in Fig. 1. The rate of deformation can have several effects on the behavior and development of the microstructure of austenite containing steels. This can be due to the direct effects of strain rate on the actual deformation and strain hardening mechanisms, or to the indirect effects of deformation induced heating on the mechanical response and evolution of the steel’s microstructure during deformation (general thermal softening or, for example, changes in the material’s phase transformation and/or twinning behavior).

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_47, © The Society for Experimental Mechanics, Inc. 2011

337

338

Strain hardening behavior of crystalline materials depends primarily on the evolution of the dislocation structure, but also on the possible occurrence of twinning and/or phase transformations during deformation. The stacking fault energy γSFE of the austenite phase has a strong influence on the martensite transformation and twinning propensity, and it is generally assumed that twinning occurs when stacking faults energy is in the range 18 ≤ γSFE ≤ 45 mJ/m2 and martensite transformation at γSFE values lower than 18 mJ/m2 . When stacking fault energy exceeds ca. 45 mJ/m2, plasticity and strain hardening are controlled solely by the glide of dislocations [1]. Stacking fault energy can be adjusted by proper alloying: for example manganese and aluminum increase γSFE while silicon normally decreases it. As the stacking fault energy is also a function of temperature, changes in the ambient temperature or in the internal temperature of the deforming material due to adiabatic heating can result in changes in the deformation and strain hardening behavior of the material.

Chemical Free Energy

ΔGMγ →S α '

U’ Stress assisted

MS

Strain induced

M σS

T0 Temperature

(a)

Md

U’

Plastic deformation of   austenite, no martensite 

Austenite

 

U’

Spontaneous thermal   martensite transformation 

ΔG α '→γ

Stress 

ΔGTγ1→α '

σ Yγ Yield strength of   parent phase 

MS

MσS

Md

Temperature 

(b)

Fig. 1 Free energy of martensite and austenite (a), and dependence of the critical stress needed to initiate mechanical martensite transformation on temperature (b) [2, 3] MATERIALS AND EXPERIMENTS In this work, the mechanical behavior and microstructure evolution of several fully or partially austenitic steels were studied at different strain rates and temperatures. The fully austenitic steels were three experimental grades of twinning induced plasticity (TWIP) steels and two commercial grades of austenitic stainless steels. The steel containing initially ca. 12 % retained austenite was a commercial low alloy transformation induced plasticity (TRIP) steel. The chemical compositions of the studied steels are presented in Table 1. The materials were tested both in tension and compression at wide ranges of strain rate and temperature using conventional servo-hydraulic materials testing machines and Hopkinson Split Bar techniques with high and low temperature capabilities. The test procedures have been described in details elsewhere [4-6]. Figures 2-4 show some results relevant to the following discussion.

339

Table 1 Compositions and stacking fault energies of the studied steels Material

Mn

Al

Si

C

Cr + Mo

Ni

Nb

Cu

P

N

Fe

SFE

TWIP 1

28

1.6

0.28

0.08

<0.01

-

<0.001

-

-

-

Bal.

27

TWIP 2

25

1.6

0.24

0.08

<0.01

-

0.05

-

-

-

Bal.

20

TWIP 3

27

4.1

0.52

0.08

<0.01

-

0.05

-

-

-

Bal.

42

TRIP700 (max)

2.0

2.0

0.6

0.25

0.6

-

0.2

-

-

-

Bal.

(?)

EN 1.4301-2B

1.7

-

0.49

0.055

18.3 + 0.17

8.1

-

0.39

-

0.05

Bal.

18

EN 1.4318-2B

1.26

-

0.43

0.022

17.4 + 0.21

6.5

-

0.22

-

0.13

Bal.

15

Hadfield steel

16.5

0.04

0.51

1.15

1.83 + 0.13

0.23

-

0.09

0.06

-

Bal.

50 (?)

800

1000 800

600 500 400 300 1000 s

200

1 s -1

100 0

-1

Eng. stress (MPa)

Eng. stress (MPa)

700

600 400 1000 s -1 1 s -1

200

0.001 s -1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Eng. strain a)

0.001 s -1 0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Eng. strain b)

Fig. 2 Engineering tensile stress-strain curves at various strain rates at room temperature: a) EN 1.4301-2B b) EN 1.4318-2B

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(a)

(b)

Fig. 3 Ductility of TWIP steels as a function of temperature at strain rate 1250 s-1 (a) [1], transformation of retained austenite to martensite as a function of plastic strain in the TRIP steel at different strain rates and temperatures (b) [7]

1600

True Stress [MPa]

1400 1200 1000 sr 0.001

800

sr 1.0

600

sr 4000

400 200 0 0

0.1

0.2

True Strain [mm/mm]

0.3

 

Fig. 4 True stress-strain curves in compression for a high manganese Hadfield steel at different strain rates DISCUSSION The mechanical behavior of austenite depends on several internal and external factors, such as stacking fault energy, martensite transformation temperatures Ms and Md, size of the (retained) austenite grains, actual deformation temperature, rate of deformation, and amount of deformation. Furthermore, stacking fault energy is a function of the instantaneous temperature of the material, which again depends on both the rate and amount of deformation because the thermal conditions gradually change from isothermal to adiabatic with increasing strain rate. Figure 2 shows clearly the difference in the mechanical behavior of the two grades of austenitic stainless steel, which differ from each other essentially only by their nickel and chromium contents. The basic reason for the different behavior is the difference in the martensite transformation temperatures, which depend on the alloying. For EN 1.4301-2B both Ms and Md are below room temperature and essentially no deformation-induced

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martensite transformation takes place until necking, where very high localized strains are generated. In contrast, EN 1.4318-2B starts to show intense martensite transformation after ca. 10 % plastic strain, provided that the strain rate is low enough. However, when strain rate is increased from 10-3 s-1 to 1 s-1, martensite transformation also in EN 1.4318-2B is strongly suppressed and both grades strain harden in a similar manner. The reason for the strain rate dependent hardening behavior of EN 1.4318-2B is that at higher strain rates adiabatic heating of the specimen gradually raises the specimen temperature above the Md temperature, and the tendency towards martensite transformation decreases and finally vanishes. For EN 1.4301-2B the effect of adiabatic heating is seen as a decrease of uniform elongation with increasing strain rate, since the plastic flow stabilization due to the martensite transformation disappears. It should be noted that in Figure 2 the strain values in the high strain rate THSB-tests (strain rate 1000 s-1) are slightly overestimated since the specimen strain was calculated based on the bar motion alone. Generally speaking, the above mentioned two different effects of strain rate can also be found in a single alloy by adjusting the starting temperature of the test. For example, if temperature was decreased below room temperature, the behavior of the more stable EN 1.4301 alloy would resemble the RT behavior of the EN 1.4318 alloy. Similarly, at higher test temperatures the austenite in the EN 1.4318 alloy would become more stable and the behavior of the alloy would be similar to the EN 1.4301 grade. It should also be noted that the above described effects of strain rate are partly a consequence of the chosen test procedure. The continuous temperature increase due to adiabatic heating occurs because the tests are carried out at constant high strain rates. However, if a test with strain rate variations, such as a sudden increase in strain rate from low to high strain rates, was carried out, the resulting temperature history of the material would be different. This would also affect the martensite transformation and hence the mechanical behavior of the material. As shown in Table 1, the chemical compositions of the three TWIP steels are slightly different, which results in slightly different SFE values for each of the steels. SFE’s of TWIP 2 and TWIP 3 are close to the lower and upper limits of the SFE range, where extensive twinning is expected to happen, while TWIP 1 is more or less in the middle of this range. This explains why the ductility of TWIP 1 in Fig. 3a is clearly higher than that of the other two. Usually the ductility of steels increases with increasing temperature, but for the TWIP steels ductility decreases dramatically when temperature is approaching 100 °C. This can be explained by the positive temperature dependence of the stacking fault energy as shown in Fig. 5, i.e., at higher temperatures the TWIP effect is gradually lost.

Fig. 5 Dependence of the stacking fault energies of TWIP steels on temperature [1] Figure 3b shows how retained austenite in the TRIP steel is transformed to martensite during deformation at different strain rates and temperatures. As the figure shows, the phase transformation rate is clearly higher at lower temperatures, which is readily explained by the decreasing stacking fault energy and increasing chemical driving force for the phase transformation with decreasing temperature. At room temperature, the phase

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transformation rate is clearly higher at low strain rates than at high strain rates, which could be explained by the adiabatic heating effects. This difference, however, is observed already at low plastic strains where adiabatic heating should not yet be significant, and therefore the effect of strain rate on the phase transformation rate is probably more complex. The compression behavior of the Hadfield manganese steel at different strain rates is shown in Fig. 4. From the plot it is obvious that the strain hardening behavior is not essentially affected by the strain rate, even though the apparent strain rate sensitivity of the material is considerably high. Simple magnetic measurements indicate that after the tests the samples are slightly ferromagnetic, which implies that deformation-induced martensite transformation occurs to some extent in this material. It seems, however, that twinning is a far more important hardening mechanism in Hadfield steels than martensite transformation. Strain rate does not seem to have a notable effect on the extent of martensite transformation (as indicated by the above mentioned rather crude magnetic measurements), but after 30 % plastic deformation the hardness of the material is 400 HV at low strain rates but 500 HV at high strain rates. The surfaces of deformed samples show quite a lot of deformation twins, which also suggests that the main hardening mechanisms, in addition to ordinary strain hardening caused by the increasing dislocation density, is twinning. An additional strain hardening mechanism that has been connected with Hadfield steels is the dynamic strain aging. CONCLUSIONS The importance of steels stems from the extraordinary variety of properties that can be obtained by alloying and thermo-mechanical heat treatments of the basic mixture of iron and carbon. The special role of austenite, in turn, is due to its ability to undergo phase changes driven by both chemical and mechanical energy. In a similar way, mechanical twinning of the austenite phase can be utilized to enhance the strength and ductility of austenite containing steels, even simultaneously, as happens with TRIP and TWIP steels. The deformation and transformation behavior of austenite, including mechanical twinning, depend on both temperature and strain rate. Especially deformation-induced heating of the material at higher strain rates can lead to quite extraordinary behavior, as seen for example in the case of certain (unstable) grades of austenitic stainless steels. It should also be kept in mind that because of the relatively poor heat conductivity of austenitic stainless steels, these effects can take place already at quite low strain rates. REFERENCES [1] Curtze, S. and Kuokkala, V-T., Dependence of tensile deformation behavior of TWIP steels on stacking fault energy, temperature and strain rate, Acta Materialia 58, 5129-5141, 2010 [2] Wayman, C.M. and Bhadeshia, H.K.D.H., Phase Transformations, Nondiffusive. In: R.W. Cahn and P. Haasen, (Eds.).Physical Metallurgy. Fourth edition. Elsevier Science Publishers, Amsterdam, The Netherlands, pp. 15071554, 1996 [3] Olsen G.B. and Cohen, M., A Mechanism for the Strain-Induced Martensitic Nucleation of Martensitic Transformations, Journal of Less-Common Metals, 28 (1), pp. 107 – 118, 1972 [4] Apostol, M., Vuoristo, T. and Kuokkala, V-T., High temperature high strain rate testing with a compressive SHPB, Journal de Physique IV France, vol 110, 459-464, 2003 [5] Hokka, M., Curtze, S. and Kuokkala, V-T., Tensile HSB testing of sheet steels at different temperatures, SEM 2007 Conference Proceedings, June 4-6, 2007 [6] Curtze, S., Hokka, M., Kuokkala, V-T. and Vuoristo, T., Experimental analysis of the influence of specimen geometry on the tensile Hopkinson Split Bar test results of sheet steels, MS&T 2006 Conference proceedings, October 15-19, 2006 [7] Hokka, M., Effects of strain rate and temperature on the mechanical behavior of advanced high strength steels, Dr. Tech. Thesis, Tampere University of Technology, 2008

Investigation into the Spall Strength of Cast Iron

G. Plume, C.-E. Rousseau, University of Rhode Island Mechanical, Industrial, and Systems Engineering 92 Upper College Rd., 203 Wales Hall, Kingston, RI, 02881 [email protected] ABSTRACT The spall strength of cast iron has been investigated by means of planar plate impact experiments conducted in a vacuum. A single stage gas gun was utilized to drive projectiles to velocities between 100 and 300 m/sec, resulting in low to moderate shock loading of the cast iron specimens. Measurement of the stress histories were made with the use of commercial manganin stress gauges that were imbedded between the back face of the cast iron specimen and a low impedance backing of polycarbonate. Spall strength values were calculated utilizing the measured peak stress and minimum stress pullback signals captured in the stress history. Spall strengths were found to vary between 0.98 and 1.45 GPa for the cast iron tested. Postmortem analysis of recovered specimen has provided insight into the evolution of spall failure in cast iron and shed light on the varying nature of the spall strength values calculated. It was determined that the lower bound of strength values were associated with small scale microfailure, while the upper bound values corresponded to complete spall fracture. 1.0 Introduction Spallation is a dynamic material failure that occurs due to tensile stresses generated by the interaction of two release waves. The failure mechanism of spallation has been widely investigated since it was first identified by Hopkinson in 1914. In recent years experimental techniques such as high velocity plate impacts, explosive drives, and laser ablation have frequently been employed to induce spallation. Records of the wave profiles created by these dynamic events are usually taken in the form of free surface particle velocities or stress histories. From these temporal records, the spallation event is typically quantified in terms of the material’s spall strength. As the name suggests, spall strength defines a material’s ability to resist spallation, which in turn is a measure of a material’s high-rate tensile strength. In order to better visualize the failure mechanism of spallation, a time distance diagram for the case of a symmetrical impact of two plates can be found in Fig. 1. When two materials collide at a rate sufficient to exceed the Hugoniot Elastic Limit (HEL), elastic waves along with plastic shock waves of compressive nature propagate from the impact surface respectively at the longitudinal and bulk wave speeds. The elastic waves reflect off the free surface of the impactor as well as the low impedance surface of the specimen-window interface as elastic tensile release waves. When the slower moving shock wave fronts reach these surfaces, they reflect as tensile “rarefaction fans”. If the impactor is half the thickness of the test specimen, these rarefaction waves interact at the mid plane of the specimen to form a tensile build up, as depicted by the shaded region of Fig. 1. If this tensile build up is sufficiently large, a spall plane is initiated. The remaining portion of the rarefaction fan that had reflected off of the specimen-window interface now reflects off the newly created spall plane, returning to the specimen-window interface as a compressive reloading signal. A material’s spall strength can be investigated by relating the maximum and minimum stress signals transmitted to the window as seen in Fig. 1. The difference between these two signals is often termed the pullback signal. The actual arrival time of the pullback signal is difficult to predict because the spall process requires some incubation of the interacting tensile stress waves before a spall plane is generated. The time lag of the pullback signal is therefore directly related to the incubation time necessary to generate a spall plane. A typical transmitted signal is shown in Fig. 2, in which the record of a spall experiment from a manganin stress gauge embedded between the specimen and window is presented. In this gauge record the maximum and minimum stress signals are clearly labeled and the velocity pullback is exemplified.

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_48, © The Society for Experimental Mechanics, Inc. 2011

343

344 8

Impactor

Polycarbonate Window

Specimen

7

Cutoff Time for 1D Stress Criterion

Spall Plane

MIN Stress Signal

5

MAX Stress Signal

4

3

2

Specimen-Window Interface (Manganin Stress Gauge Location)

1

Elastic Waves Plastic Waves 0 -10

-5

0

5

10

15

20

25

30

6

7

Distance (mm)

Fig. 1. Distance-time diagram for a symmetrical plate impact spall experiment 1 0.9

σmax

0.8 0.7

Pullback Signal 0.6 Stress (GPa)

Time (micro-sec)

6

σmin

0.5 0.4 0.3 0.2 0.1

HEL 0 0

1

2

3

4

5

Time (micro-sec)

Fig. 2. Transmitted stess record for a typical spall experiment

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In this study the spall strength of cast iron is investigated in an effort to familiarize and gain experience in the design and execution of spall experiments. Experiments are conducted utilizing high velocity “symmetrical” plate impact experiments. In the case of these experiments, the term ‘symmetrical’ signifies that the impactor and test specimen are made of the same material and have an identical contact surface. Stress histories from these experiments are captured by means of manganin stress gauges embedded between the back surface of the test specimen and a low impedance polycarbonate backing. It can be noted that the impact conditions depicted in Fig. 1 can be directly correlated to the experiments conducted in this paper. The slopes of the wave representative lines in Fig. 1 are directly generated from their respective wave speeds where the time axis is in μsec, and the distance axis is in mm. This correlation can be noted when comparing Fig. 1 and 2. 2.0 Theory There exist numerous equations in the literature for the calculation of spall strength. Generally, these equations relate the relative “velocity pullback” or “stress pullback” signals observed in a spalling event to values of spall strength. In works by Gather, he points out shortcomings of the traditional methods of spall strength calculation [1], and provides the foundation for a spall strength calculation based on the Hugoniot, which is claimed to yield a much better approximation for the release isentrope. Gather’s work has been summarized by Chen [2], and an expression for the calculation of spall strength is given as:

1 2

⎛

σ sp = σ max ⎜⎜1 + ⎝

Zs Zw

⎞ 1 ⎛ Z ⎟⎟ + σ min ⎜⎜1 − s ⎠ 2 ⎝ Zw

⎞ ⎟⎟ ⎠

(1)

where σmax is the magnitude of the initial compressive wave, σmin is the magnitude of the minimum pullback signal, and Zs and Zw are the impedances of the specimen and window material, respectively. For better visualization of the “pullback signal” used in this calculation, visit Fig. 2, in which a typical gauge record is presented with the maximum and minimum stress signals clearly labeled. The specimen and window impedances can be calculated by the product of their respective initial densities and appropriate wave speeds. Special care must be exercised when calculating these impedances in regards to choosing the appropriate wave speed. If the material response preceding spall is elastic, a longitudinal elastic velocity should be used. On the other hand, if the response is hydrodynamic, a bulk wave speed must be used. The bulk wave speed of a material can be calculated from the longitudinal and shear wave speeds by use of the equation:

4 C b = C l2 − C s2 3

(2)

It has been widely recognized that when elastic waves are important to the analysis of a hydrodynamic event an equivalent wave speed can be utilized. This equivalent wave speed, proposed by Stepanov and Romanchenko [3], is the harmonic mean of the longitudinal and bulk wave speeds, given by:

Ce =

2C l C b (Cl + Cb )

(3)

It must be pointed out that due to the design of these experiments stress records of the spallation events are only captured in terms of the stress transmitted to the low impedance window. In order to infer the incident stress that passes through the cast iron sample from the transmitted stress captured by the stress gauge, the following well-known relation can be used:

σI =

Zs + Zw σT 2Z w

(4)

Much like before, special care must be exercised when choosing the appropriate wave speed in the calculation of the impedances of the specimen and window. Looking back to Eq. (1), it must be noted that the relationship seen in Eq. (4) is

346

already taken into account in the calculation of spall strength. Therefore the transmitted stress values determined from the stress gauge histories can be used directly when calculating spall strength using Eq. (1). In the study of a material’s spall strength it is important to note that values calculated using Eq. (1) can vary quite substantially for the same material undergoing different impact severities. In order to shed light on this varying nature, it must be pointed out that the level of material damage plays a strong role in the value of spall strength calculated. It has been demonstrated in [4] that in cases involving a very high concentration of damage nucleation sites, calculation of the critical failure stress produces a significant overestimation of the spall strength. On the other hand, an underestimation is generated from cases where microcracks have just begun to develop in the spall plane with spall fracture remaining incomplete. Taking into account the kinetics of damage initiation and evolution, a material’s spall strength can be classified into two critical levels of maximum tensile stress amplitude: one level representative of micro-damage generation, and another corresponding to complete spall failure [4]. 3.0 Experimental Design In order to aid in the design of the experiment, values for the longitudinal and sheer wave speeds as well as densities of both the cast iron and polycarbonate were required. Due to the fact that cast iron can have a wide spread of properties resulting from its specific casting, these properties were measured directly for the tested material. Longitudinal and shear wave speeds of the materials used were measured utilizing 10 MHz and 2.5 MHz contact transducers, respectively, in a pulse-echo mode. Transducers and power supply were manufactured by Panametrics. From the longitudinal and shear wave speeds, Eqs. (2) and (3) were utilized to respectively calculate the bulk and equivalent wave speeds. Densities were calculated from measurements of the volume and mass of the test samples. A summary of the properties determined for the specific cast iron and polycarbonate used can be found in Table 1. Table 1- Properties of materials used Material

ρo (kg/m3)

co (m/sec)

cs (m/sec)

cb (m/sec)

ce (m/sec)

Cast Gray Iron Polycarbonate

7690 1178

4400 2260

2500 910

3320 2000

3790 2120

Plate impact experiments were conducted in a vacuum chamber evacuated to 20 torr with the use of a 50 mm bore single stage helium driven gas gun. The impactors and tested specimens were 45 mm in diameter and respectively 5 mm and 10 mm thick. 20mm thick polycarbonate backing windows were used. Looking back to Fig. 1, it is noted that a 20mm thick polycarbonate backing window allows for full capture of the pullback signal before any transmitted waves to the window have time to reflect off of the window’s free surface. PVC sabots were used to carry the impactors down the 2 m long gun barrel, while specimens were supported, awaiting impact, by means of small PVC rings that were glued to a metallic sample holder. In order to help attain better impact planarity, sabots were designed so as to impact a specimen before they fully exited the barrel of the gun. Flyer velocities were captured by means of two laser detectors positioned .5 in. and 1.5 in. from the specimen’s front face. Velocity was inferred from the time between interruptions and the distance between the two detectors. 50Ω manganin stress gauges were embedded between the back surfaces of the test samples and 20 mm thick polycarbonate backing windows with the use of Buehler Epo-Thin Epoxy. A 300g mass was placed on top of the specimen package during bonding to ensure consistent embedding of the gauge during the 18 hour cure time of the epoxy. The gauges and power supply were manufactured by Dynasen. Details pertaining to their calibration and stress calculations can be found in [5]. For ease of post data analysis, special consideration was used to ensure that the gauges remained in their linear elastic regime. It has been shown that manganin gauge hysteresis can be avoided if the peak stress level is kept below 1.5 GPa [6]. Through use of equation (4) with the elastic wave speed it was determined that the maximum impact stress must be bounded by an upper limit of 10.3 GPa. Assumption of a bulk wave speed produces a slightly lower bound of 8.9 GPa. Due to the large amount of attenuation associated with shock waves it is thought that an upper bound between 8.9 and 10.3 GPa is reasonable. A digital oscilloscope manufactured by Tektronix was used to capture the output signal of the stress gauge power supply and the laser detectors. The oscilloscope and gauge power supply were both triggered when the impactor passed the first laser detector.

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A simple cutoff time criterion for the experiments was estimated under the assumption that the specimen is in a state of onedimensional stress until radial release waves originating from the sides interact at the center of the specimen. This cutoff time was calculated to be about 6.8 sec after impact. Caution must be exercised when using stress history information after this time due to the complications of the three-dimensional stress wave analysis. Looking back to Fig. 1, this time is clearly depicted, indicating that it is well beyond the expected arrival time of the pullback signal. Impacted specimens were captured for post mortem analysis utilizing a soft recovery technique by means of clay bricks coupled with a shock absorbing catch box. Recovered specimens were cut through the thickness in order to expose the spall plane for analysis. Cut specimen were sanded and polished to a 3 micron finish for clearer viewing under an optical microscope. Spall planes were categorized into 3 groups: microfailure start, microfailure and partial macrofailure, and complete spall fracture 4.0 Results and Discussion Histories of the transmitted stress to the polycarbonate windows can be found for four seperate tests in Fig. 3. A summary of the four tests depictied in Fig. 3 can be found in Table 2. In this table the impact velocity, estimated impact stress (σ =1/2ρcVfl), mimimum and maximum stress signals, the pullback signal severity, and spall stength values calculated using Eq. (1) can be found. Looking at the results, it can be seen that impact velocities ranged from 120-265 m/s, producting values of spall strength that ranged from 0.98-1.45 GPa. The spall strength of cast iron can be expected somewhere in the range of 1-2 GPa as determined in [7]. It seems reasonable that the slower impact speeds produced lower spall strength values when one remembers that the varying nature of spall strength values is dependent on the type of failure exhibited at the spall plane. At faster impact speeds it would be expected that the material would approach complete failure, and hense, as stated earlier, a calculation of strength would produce an overestimation of the actual material’s critical tensile strength. 0.7

0.6

Compressive Stress (GPa)

0.5

0.4

0.3

0.2

Test 1 (265 m/sec) Test 2 (230 m/sec) Test 3 (190 m/sec) test 4 (120 m/sec)

0.1

0 0

1

2

3

4

5

6

Time (micro-sec)

Fig. 3. Transmitted stess records

7

8

9

10

348

Test # 1 2 3 4

Table 2- Summary of results for plate impact experiments Impact σmin Δσ σIMPACT σmax Velocity (GPa) (GPa) (GPa) (GPa) (m/sec) 265 230 190 120

4.48 3.89 3.21 2.03

0.61 0.57 0.45 0.29

0.44 0.40 0.31 0.15

0.17 0.17 0.14 0.14

σsp (GPa) 1.45 1.41 1.14 0.98

The evolution of spall failure in the cast iron tested can be seen in Figs. 4, 5, and 6. Fig. 4 is an image of the spall plane from experiment 4, which was tested at an impact velocity of 120 m/s. In this figure, the onset of spallation with isolated microfailure is examplified. Figure 5 is an image of the spall plane from experiment 3 in which microfailure and partial macrofailure can be observed. It can be seen that there is a much greater opening mode to the cracks found in the cast iron tested at 190 m/sec than that tested at 120 m/s. However true, microfailure remains isolated and cracks have not conected to form a complete spall fracture for the specimen tested at 190 m/s. Fig. 6 is an image from experiment 1 which shows complete spall fracture of the specimen impacted at 265 m/s. The spall plane from experiment 2, exhibited a complete spall fracture much like that in Fig. 4. Overestimations of the spall strength are generated when complete failure occurs as were the case for experiments 1 and 2. On the other hand, an underestimate begins to present itself as material failure is limited to isolated microcracking as found in Fig. 4 for experiment 4.

Fig. 4. Spall plane from test #4 demonstrating the onset of microfailure

349

Fig. 5. Spall plane from test #3 demonstrating the evolution of microfailure and partial macrofailure

Fig. 6. Spall plane from test #1 demonstrating complete spall fracture 5.0 Concluding Remarks A series of plate impact experiments were executed in order to investigate the spall strength of cast iron. Impact velocities ranged between 100 and 300 m/s yielding estimations of spall strength that respectivly ranged from 0.98 to 1.45 GPa.

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Analysis of the spall planes provided insight into the resons for these variations in strength values. It has been shown that the lower strength values correspond to the onset of spallation, while the upper strength values correspond to complete spall fracture. Future metallographic analysis of the spall fracture zones is planed for the recovered impacted specimen in order to further clasify the fracture mechanisms involved in the evolution of spallation in cast iron. Future research including microhardness measurements of the fractured specimen may provide further insight into the changes in material microstructure and properties due to shock loading events. 6.0 Accknoledgments Work presented in this paper was made possible by funding from the Department of Energy. 7.0 References [1] [2] [3] [4] [5] [6] [7]

Gathers, R.G., “Determination of spall strength from surface motion studies,” J. Appl. Phys., Vol. 67, No. 9, pp. 4090-4092, (1990). Chen, D., Y. Yu, Z. Yin, H. Wang, G. Liu, “On the validity of the traditional measurement of spall strength,” Int. J. Impact Eng., Vol. 31, pp. 811-824, (2005). Romanchenko, V.I., G.V. Stapanov, “Dependance of critical stresses on the loading time parameters during spall in copper, aluminum, and steel ,” J. Appl. Mech. Tec. Phys., Vol 21, No.4, pp.555-561, (1980). Anderson, W.W., F.J. Cherne, M.A Zocher, “Material Properties under Intensive Dynamic Loading,” Springer, pp 249-268 (2006). “Piezoresistive Pulse Power Supply Model CK2-50/0.050-300 Instruction Manual,” Dynasen, Inc. (1996). Rosenberg, Z., D. Yaziv, Y. Partom, “Calibration of foil-like manganin gauges in planar shock wave experiements,” Jo. Appl. Phys., Vol. 51, No. 7, pp 3702-3705, (1980). Zaretsky, E.B., “Shock response of iron between 143 and 1275 K,” Jo. Appl. Phys., Vol. 106, No. 2, pp.106-115, (2009).

Development of Brick and Mortar Material Parameters for Numerical Simulations Mr. Christopher S. Meyer, U.S. Army Research Laboratory, ATTN: RDRL-WML-H, Aberdeen Proving Ground, MD 21005-5066, 410-278-3803, [email protected] ABSTRACT Numerical simulation of brick and mortar masonry in the literature has been performed using homogenized material properties; however, discrete material properties for brick and mortar for a constitutive model built into hydrocodes have been unavailable. The Holmquist-Johnson-Cook (HJC) constitutive model for concrete captures pressure and strain rate dependent strength behavior and void crushing damage behavior of brittle solids reasonably well, is readily available in many hydrocodes, and is commonly used to simulate high-velocity penetration of concrete. Due to these considerations and the similarity of concrete to brick and mortar materials, the HJC constitutive model was selected for material parameterization of brick and mortar characterization data. This paper will describe the development of the brick and mortar material parameters— including exploration of the materials’ strength, damage, and pressure behavior—from available static and dynamic test data, and the paper will provide the HJC constitutive model material parameters for brick and mortar. The paper will also present sample numerical simulations, using the material parameters developed, and comparing the resulting material behavior with mechanical test data. INTRODUCTION The U.S. Army’s interest in urban operations has led to efforts at the U.S. Army Research Laboratory to develop an initial set of material model parameters to enable physics-based penetration simulations of high-fidelity brick and mortar masonry wall models. These material model parameters take advantage of the Holmquist-Johnson-Cook (HJC) model for concrete [1]. A search of publically released literature turned up very little involving modeling and simulation of weapon effects against brick and mortar targets. While dynamic material properties for concrete are well characterized, the dynamic material properties of brick and mortar are in the early stages of investigation. Research involving material properties for masonry for use in numerical simulations has concentrated on homogenized material properties and material models, which smear the mortar and brick together into a single material rather than discrete, heterogeneous brick and mortar materials [2], [3]. Research involving numerical simulation of masonry walls has concentrated on the structural response of the walls to blast loading [4], [5]. According to Shieh-Beygi and Pietruszczak of McMaster University, “…analysis of large masonry structures should best be conducted at a macro-level… described as a continuum whose average properties are identified at the level of constituents taking into account their geometric arrangement [6].” The homogenized material property methods described in these sources are effective for macroscopic behavior of masonry walls in response to shear loads as from earthquakes or in response to blast loading as from weapon effects. But the behavior of brick and mortar masonry in response to high-velocity penetration requires heterogeneous brick and mortar models and material parameters to better explore the shock interaction across multiple layers of materials with dissimilar sound speeds, densities, and pressure dependent strength behavior. Shieh-Beygi and Pietruszczak provide a brief review of homogenized constitutive models for brick and mortar masonry and then develop their own mesoscale constitutive model [6]. Development of a constitutive material model for brick and mortar is beyond the scope of this work, which instead developed material parameters for the existing HJC constitutive model for concrete. Considering the void-collapse strain behavior of geomaterials, it is assumed that the HJC concrete model, which is widely available in Lagrangian and Eulerian simulation codes, has been shown to produce reasonable residual velocity results in high-velocity concrete penetration simulations when compared with experiment [7], and captures the void-crushing damage, strain rate, and pressure-dependent strength properties of brick and mortar sufficiently well to simulate physics-based penetration of these materials. This work developed complete sets of HJC constitutive model material parameters for grade SW brick and type N mortar from mechanical characterization data provided by the U.S. Army Corps of Engineers Engineer Research and Development Center (ERDC) [8], [9]. DEVELOPMENT OF MATERIAL PARAMETERS

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_49, © The Society for Experimental Mechanics, Inc. 2011

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352 As described in Holmquist et al., the constitutive equation (1) describes the pressure- and strain rate-dependent strength of the subject material; equation (1) states that the material’s yield strength, when the material is confined, will increase with increasing pressure and with increasing strain rate. The constants A, B, C, and N are material parameters determined by fitting the model to test data, D is scalar damage, P* is normalized pressure and σ* is the normalized compressive yield strength of the material. Strength and pressure are normalized by dividing by the unconfined compressive strength, f’c, of the material, which was determined from unconfined, axial compression mechanical test data provided by ERDC. Strain rate is made di, typically 1.0 s-1. Damage is a value from 0 mensionless by dividing the actual strain rate, , by a reference strain rate, to 1 that describes the accumulation of damage as a percentage of full cohesive strength that the material possesses such that at D=0 the material is undamaged and exhibits its full strength, but at D=1 the material is fully damaged and retains the least confined strength. The first term of the constitutive equation (1) describes the pressure-dependent strength behavior of the subject materials, the second term describes the strain rate effects, and a third term, not shown here, describes the temperature-dependent strength of the material; temperature effects are not explored in the present work.

(1)

The pressure-dependent strength behavior of the materials was fit to uniaxial stress-strain test data provided by ERDC. The uniaxial data was found by loading confined cylindrical samples of material in the axial direction. The strength fits are shown in Figures 1 and 2 for grade SW brick and type S mortar respectively. To determine material strength constants, first the cohesive strength constant, A, was determined as the ratio of the failure strength to the undamaged strength at a particular strain rate, in this case 10-5 s-1, which was then normalized to the reference strain rate, 1.0 s-1, by scaling up using the dynamic increase factors presented for brick and for mortar by Hao and Tarasov [10]. The strain rate coefficient, C, was determined next from strain rate dependent strength data by Hao and Tarasov. Material strength—normalized to the average unconfined compressive strength of the tested materials—versus strain rate was plotted for brick and for mortar; the slope of a linear fit through the data was used for the strain rate coefficient, C, of the respective materials. Once A and C were determined, the pressure hardening constant, B, and the pressure hardening exponent, N, were fit to the uniaxial data using equation (1). Damage is defined as the accumulation of equivalent plastic strain, Δεp, and plastic volumetric strain, Δµp, over each time step; damage is expressed as equation (2) [11]. The model calculates damage by summing the equivalent plastic strain and plastic volumetric strain and dividing the sum by the plastic strain to fracture under a constant pressure. The plastic strain to fracture under constant pressure may be found from whichever is greater, EFMIN or plastic strain to fracture found from

Fig. 1 HJC material model strength fit for grade SW brick

Fig. 2 HJC material model strength fit for type S mortar

equation (3). Thus, EFMIN provides for a minimum plastic strain value that will cause the material to fracture; since cyclic unconfined compressive failure data is not available for these materials, the default value of 0.01 is used in this work for all

353 materials. Plastic strain to fracture is determined by the model from two damage constants, D1 and D2, and from the normalized pressure, P*, and normalized tensile strength, T*. The tensile strength constant for each material is found from direct pull test data. D1 was found from unconfined compression test data, and D2 was assumed to be 1.0. Damage relationships are shown in Figure 3. Further study of the fracture behavior of these materials is necessary to fully understand the effect damage has on strength, particularly the assumption of linearly increasing failure strain with increasing pressure, but fracture data is presently unavailable for brick and for mortar. (2)

(3) Damage accumulates in geomaterials under load as the material fractures (the accumulation of equivalent plastic strain) and as the voids in the porous material compress and collapse (the accumulation of plastic volumetric strain). The majority of damage is accumulated through fracturing of the material with crushing of the material accounting for a small amount of the total damage. As damage accumulates, the scalar damage parameter from equations (1) and (2), D, will approach a value of 1 (or 100%); as damage approaches 100%, the normalized cohesive strength term, A, from equation (1) approaches zero: a complete loss of cohesive strength of the material.

Fig. 3 HJC material model damage fit for HJC concrete, grade SW brick and type S mortar Holmquist et.al describes the materials’ hydrostatic pressure response to volumetric strain in three distinct regions. The material behaves differently as pressure increases. At low pressure, less than crush pressure, Pcrush, the material undergoes reversible, elastic deformation. This elastic behavior is predicted by the model with a simple linear relationship controlled by the user’s input of parameters, Pcrush and µcrush. Elastic bulk modulus, Ke, is calculated by the model as the crush pressure divided by the crush volumetric strain, µcrush. This region of elastic material behavior is known as Region I; in Region I pressure is predicted by the model according to a simple linear relationship where the pressure is equal to the applied volumetric strain multiplied by the elastic bulk modulus. Figures 4 and 5 illustrate the pressure-volume behavior of grade SW brick and type S mortar respectively; Region I is shown with insets in these figures. For brittle geomaterials, the elastic region of behavior is very small. Pcrush and µcrush were determined by fitting to hydrostatic compression and uniaxial strain mechanical test data as shown in Figures 4 and 5.

354

Fig. 4 HJC material model pressure-volume fit for grade SW brick

Fig. 5 HJC material model pressure-volume fit for type S mortar

Region II begins and the linear elastic behavior governing Region I ends with parameters, Pcrush and µcrush, which is the pressure and volumetric strain at which the material begins to undergo plastic deformation. Permanent, plastic deformation in brittle geomaterials, which are riddled with tiny pores and air voids, may be described as micro-cracking and air void crushing as pressure and volumetric strain increase beyond Pcrush and µcrush. Plastic deformation (cracking) and void collapse occur in Region II, shown in Figures 4 and 5. Region II is defined by Holmquist et al. as a transition region; the model interpolates the material behavior in this region between Regions I and III. The user of the model cannot directly control the predicted material behavior in this region other than by controlling the model parameters that govern behavior for Regions I and III. The onset of Region III, shown in Figures 4 and 5, is controlled by the parameters, Plock and µlock, which are the pressure and volumetric strain at which all air voids have been crushed out of the material. Plock and µlock were determined by fitting to hydrostatic compression and uniaxial strain mechanical test data. Region III is described in Holmquist et al. as the behavior for fully dense material, where all air voids have been crushed out of the material. In this region the material is locked and cannot compress any further in either plastic deformation or void collapse. However, since the material is under hydrostatic compression and the material has nowhere to go, the pressure begins to increase dramatically for very small changes in volumetric strain. The material behavior in this region is governed by equation (4), which is a fit between high-pressure hydrostatic compression data and shock hugoniot data [12]. In equation (4), pressure is a function of volumetric strain, µ, and of three constants used to fit a cubic equation to the data as seen in Figures 4 and 5. The use of modified volumetric strain described by Holmquist et al. shifts the high-pressure region back to the origin so that there is no apparent softening due to void collapse under very high pressure stimuli. (4) Shock hugoniot data used for the type S mortar high-pressure fit was from gas shale with an average initial density of 2.54761 g/cm3 which matched the type S mortar grain density of 2.510 g/cm3. Shock hugoniot data used for the grade SW brick high-pressure fit was from fused quartz with an average initial density of 2.204 g/cm3 which matched the grade SW brick grain density of 2.250 g/cm3. At the extremely high pressures of the shock hugoniot data, inertial effects, rather than strength effects, dominate the material behavior; thus, at these pressures, the density is more important than the composition or strength of the material. Since shock hugoniot data was not available for brick or for mortar, using materials that somewhat resemble the geomaterials is assumed to be acceptable if the density is similar, and only affects the constants for equation (4) and the high-pressure behavior of the material.

355

Fig. 6 Radially-confined uniaxial compression true stress and engineering strain test data compared with simulation results for grade SW brick

Fig. 7 Radially-confined uniaxial compression true stress and engineering strain test data compared with simulation results for type S mortar

NUMERICAL SIMULATIONS WITH MATERIAL PARAMETERS Numerical simulations using the material parameters for brick and mortar were compared with true stress and engineering strain data provided by ERDC. The shock physics code CTH [11] was used to stress a 1-cm3 cube with ten uniformly distributed cells per side. Use of a flow code such as CTH is not ideal for simple deformation induced stress and strain, but time constraints necessitated the use of CTH as a near term solution as it has the HJC model readily available and the model had been in use for geomaterial simulations. Future work is expected to use EPIC to test material parameters. However, CTH provided an acceptable first look at the efficacy of the material parameters. The material parameters for grade SW brick and type S mortar were applied to the cube and strain was applied to the cube using the prescribed deformation (PRDEF) utility in CTH. Three different mechanical tests were simulated by applying strain to the cube to produce compression against set boundary conditions. Strain was applied in different configurations of the three directions to produce the following loading scenarios: (i) uniaxial compression—strain was applied axially while the radial directions were confined, (ii) hydrostatic compression—strain was applied equally in all three directions, and (iii) triaxial compression—strain was applied in hydrostatic compression to a set level, in this case, 200 MPa, and then held constant while the axial strain was increased to failure. Failure strains were determined from mechanical testing. Figures 6 and 7 show the results of low-pressure uniaxial simulations compared with ERDC test data for grade SW brick and type S mortar respectively. Figures 8 and 9 show the results of hydrostatic simulations compared with ERDC test data for grade SW brick and type N mortar respectively. Figures 10 and 11 show the results of triaxial compression simulations compared with ERDC test data for grade SW brick and type S mortar respectively. Additionally, uniaxial simulations were strained to extremely high pressures indicative of penetration events; however high-pressure test data was only available for grade SW brick. Figure 12 shows the result of the high-pressure uniaxial simulation for grade SW brick.

356

Fig. 8 Hydrostatic compression true stress and engineering strain test data compared with simulation results for grade SW brick

Fig. 10 Triaxial compression true stress and engineering strain test data compared with simulation results for grade SW brick

Fig. 9 Hydrostatic compression true stress and engineering strain test data compared with simulation results for type S mortar

Fig. 11 Triaxial compression true stress and engineering strain test data compared with simulation results for type S mortar

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Fig. 12 High-pressure uniaxial compression true stress and engineering strain test data compared with simulation results for grade SW brick

SUMMARY AND CONCLUSIONS Table 1 contains the HJC constitutive model material parameters for grade SW brick and type S mortar. Sample numerical simulations were performed to compare the model parameters to mechanical test data. Simulations included uniaxial compression, hydrostatic compression, and triaxial compression loading scenarios. Although the HJC model has been reported to overpredict the penetration rate for high velocity impacts into concrete as well as to miss the wide variation in entry and exit hole size, it was also reported to predict residual velocity fairly well [7]. The HJC model remains one of a limited number of tools available for use in hydrocode simulations of geomaterials [11]. Since the use of an Eulerian code such as CTH is not ideal for simulations of simple strain induced deformations, additional simulations in EPIC will be used in the future for further examination of these parameters. However the CTH code does provide the PRDEF subroutine for use in examining the behavior of material models in CTH simulations, and the results of the sample numerical simulations conducted here predict stress-strain relationships reasonably well when compared with mechanical characterization test data. Sample numerical simulations were limited in scope due to time constraints and did not explore damage; further study of the damage behavior of these materials is desirable to understand the effect of fracture on material strength. Penetration experiments would provide the high-pressure and fracture data needed for comparing simulations of penetrations using these material parameters to penetration events; these parameters will be further explored if such data becomes available. Development of constitutive material models for brick and for mortar for use in high-velocity penetration simulations is ideal, but these material parameters provide a near term solution for modeling brick and mortar masonry in penetration simulations.

358 Table 1 HJC constitutive model material parameters for grade SW brick and type S mortar (*Reported in Williams, et al.) Property Initial Density* Grain Density* Sound Speed* Cohesive Strength Coefficient Pressure Hardening Coefficient Pressure Hardening Exponent Strain Rate Coefficient Compressive Strength Tensile Strength Maximum Strength Shear Modulus* Bulk Modulus* Damage Constant 1 Damage Constant 2 Minimum Fracture Strain Crush Pressure Crush Volumetric Strain Pressure Constant 1 Pressure Constant 2 Pressure Constant 3 Lock Pressure Lock Volumetric Strain

ρo ρgrain Cs A B N C f’c T SMAX G K D1 D2 EFMIN Pcrush µcrush K1 K2 K3 Plock µlock

Unit kg/m3 kg/m3 cm/s

GPa GPa GPa GPa

GPa GPa GPa GPa GPa

Grade SW Brick 1986 2250 2.56E+05 0.63646 1.568 0.8264 0.0054 0.075 0.006 17.33 5.18 5.3 0.01413 1.0 0.01 0.03519 0.00664 63 -79 56 0.773 0.132931

Type S Mortar 1604 2510 2.52E+05 0.66 1.335 0.845 0.0018 0.0123 0.0018 80.24 1.15 1.7 0.006629 1.0 0.01 0.0138 0.0075 0.3 -2 19 0.1096 0.15

ACKNOWLEDGEMENTS Special thanks to Ms. Erin M. Williams, Dr. Stephen A. Akers, and Mr. Paul A. Reed, all of ERDC, for performing mechanical characterization of the subject materials, and for providing the data to ARL, without which this work would not have been possible. REFERENCES [1] Holmquist, T.J., Johnson, G.R., and Cook, W.H, “A Computational Constitutive Model For Concrete Subjected To Large Strains, High Strain Rates, and High Pressures.” 14th Intl Symp on Ballistics, Quebec, Canada, pp. 591-600, 1993. [2] Wei, X., Hao, H., “Numerical derivation of homogenized dynamic masonry material properties with strain rate effects”, International Journal of Impact Engineering 36, pp. 522-536, 2009. [3] Zucchini, A., Lourenço, P. B., “Mechanics of masonry in compression: Results from a homogenization approach”, Computers and Structures 85, pp. 193-204, 2007. [4] Wei, X., Stewart, M. G., “Model Validation and Parametric Study on the Blast Response of Unreinforced Brick Masonry Walls”, International Journal of Impact Engineering 37, pp. 1150-1159, 2010. [5] Wang, M., Hao, H., Ding, Y., Li, Z., “Prediction of fragment size and ejection distance of masonry wall under blast load using homogenized masonry material properties”, International Journal of Impact Engineering 36, pp. 808-820, 2009. [6] Shieh-Beygi, B., Pietruszczak, S., “Numerical analysis of structural masonry: mesoscale approach”, Computers and Structures 86, pp. 1958-1973, 2008.

359 [7] Dawson, A., Bless, S., Levinson, S. Pedersen, B., Satapathy, S., “Hypervelocity penetration of concrete”, International Journal of Impact Engineering 35, pp. 1484-1489, 2008. [8] Williams, E. M., Akers, S. A., Reed, P. A., “Laboratory Characterization of Solid Grade SW Brick”, U. S. Army Corps of Engineers Engineer Research and Development Center, Geotechnical and Structures Laboratory, ERDC/GSL TR-07-24, 2007. [9] Williams, E. M., Akers, S. A., Reed, P. A., “Laboratory Characterization of Type S Mortar”, U. S. Army Corps of Engineers Engineer Research and Development Center, Geotechnical and Structures Laboratory, ERDC/GSL TR-08-10, 2008. [10] Hao, H., Tarasov, B. G., “Experimental study of dynamic material properties of clay brick and mortar at different strain rates”, Australian Journal of Structural Engineering, Vol. 8, No. 2, pp. 117-131, 2008. [11] Crawford, D. A., et al., “CTH User’s Manual and Input Instructions”, Version 8.1, Sandia National Laboratories, Albuquerque, NM, pp 115-118, 2007. [12] Marsh, S. P., ed., “LASL Shock Hugoniot Data”, University of California Press, Berkeley, CA, 1980.

Electrical Behavior of Carbon Nanotube Reinforced Epoxy under Compression

N. Heeder1, A. Shukla2, V. Chalivendra3, S. Yang4 and K. Park5 1 Dynamic Photomechanics Laboratory, Department of Mechanical, Industrial & Systems Engineering, University of Rhode Island, Kingston, RI 02881.Email: [email protected] 2 Dynamic Photomechanics Laboratory, Department of Mechanical, Industrial & Systems Engineering, University of Rhode Island, Kingston, RI 02881.Email: [email protected] 3 Department of Mechanical Engineering, University of Massachusetts Dartmouth, North Dartmouth, MA 02747.Email: [email protected] 4 Department of Chemistry, University of Rhode Island, Kingston, RI 02881. Email: [email protected] 5 Micro/Nanoscale Engineering Laboratory, Department of Mechanical, Industrial & Systems Engineering, University of Rhode Island, Kingston, RI 02881. Email: [email protected] Abstract An experimental investigation was conducted to study the effect of quasi-static and dynamic compressive loading on the electrical response of multi-wall carbon nanotube (MWCNT) reinforced epoxy nanocomposites. An In-situ polymerization process using both a shear mixer and an ultrasonic processor were employed to fabricate the nanocomposite material. The fabrication process parameters and the optimum weight fraction of MWCNTs for generating a well-dispersed percolation network were first determined. Absolute resistance values were measured with a high-resolution four-point probe method for both quasi-static and dynamic loading. In addition to measuring the percentage change in electrical resistance, real-time damage was captured using high-speed photography. The real-time damage was correlated to both load and percentage change in resistance profiles. The experimental findings indicate that the bulk electrical resistance of the nanocomposites under both quasi-static and dynamic loading conditions initially decreased between 40% - 60% during compression and then increased as damage initiated and propagated. Introduction Extraordinary mechanical properties combined with excellent transport properties make carbon nanotubes (CNTs) a promising addition to the future of smart composite materials. When CNTs are effectively dispersed within a matrix material, an electrical network within the material can be formed and serve as an internal sensor. Understanding the electrical response of CNT-reinforced nanocomposites under dynamic loading will be particularly useful in designing sensors for applications such as structural health monitoring in aircrafts, vehicle crumple zones, and smart body-armor responsive systems etc. An experimental study has been performed to understand the electrical response of CNT-reinforced nanocomposites under dynamic compressive loading. A modified four-point probe measurement system was used to measure the bulk resistance change of the nanocomposite material through a series of well-designed experiments. Quasi-static and dynamic experiments were performed to investigate the electrical response of CNT reinforced epoxy specimens. A screw-driven testing machine and a drop weight tower were utilized to load the nanocomposites. The correlation of the history between the electrical resistance change, the mechanical loading, and the high speed deformation photography was utilized to characterize the electrical response of CNT reinforced epoxy under compressive loading. Significant research has been performed to fundamentally understand the enhancement of mechanical properties due to CNT reinforcement of polymers. Given the practical potential applications of CNTs in electromechanical devices, specifically as piezoresistive sensors, the effect of mechanical deformation on the electrical properties of individual CNTs has been studied theoretically [4-7] and experimentally [8, 9]. However, a limited amount of research has been conducted in studying the electrical response of CNT/polymer composites under mechanical loading. Alexopoulos et al. [10], Nofar et al. [11], and Gao

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_50, © The Society for Experimental Mechanics, Inc. 2011

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362 et al. [12] have studied damage detection and health monitoring of composites reinforced with CNT-embedded glass fibers. Alexopoulos et al. [10] performed various incremental tensile loading-unloading steps as well as three-point bending tests on specimens with CNT fibers in the tensile region. Results indicated that CNT fibers provide unquestionable advantages for sensing and damage monitoring of non-conductive composites, when compared to the competitors, e.g. the embedded carbon fibers and a modified (doped) conductive network. Sensing ability for the investigated specimens with the CNT fiber in the compressive region was also reported [10]. Gao et al. [12] have studied the sensing of damage development in composites using CNT networks utilizing two sensing techniques: electrical resistance and acoustic emission. Resistance change and acoustic emission counts showed a bi-linear relation in detecting damage in quasi-static and cyclic tests which can be used to give additional insight toward damage evolution. Thostenson et al. [13] performed tensile tests on CNT/epoxy samples and demonstrated a highly linear relationship between the specimen deformation and the electrical resistance. This result suggests that CNT networks formed in an epoxy polymer matrix could be utilized as highly sensitive sensors for detecting the evolution of damage in advanced polymer-based composites [13]. The aforementioned studies revealed that when properly dispersed within a given matrix, an internal sensory network can be formed and utilized to detect important information such as strain and damage within the material. Despite recent progresses on the mechanical and electrical characterization of CNT-based composites, no results have been published yet regarding the electrical response of CNT reinforced polymer composites under compressive loading conditions. The present study focuses on the electrical response of CNT reinforced epoxy under quasi-static and dynamic compressive loading. A modified four-point probe measurement system was used to measure the bulk resistance change of the nanocomposite material through a series of well-designed experiments. Quasi-static and dynamic experiments were performed to investigate the electrical response of CNT reinforced epoxy specimens. A screw-driven testing machine and a drop weight tower were utilized to load the nanocomposites. The correlation of the history between the electrical resistance change, the mechanical loading, and the high speed deformation photography was utilized to characterize the electrical response of CNT reinforced epoxy under compressive loading. 2. Experimental Procedure 2.1 Material Fabrication and Specimen Due to the simplicity of casting and low curing temperature, a two-part epoxy, consisting of bisphenol-A resin and an epoxy hardener with a mixing ratio of 50g/18g, was chosen as a polymeric matrix. CNTs used for this study were multi-wall carbon nanotubes (purity > 95%). The nanotubes have outside diameters of 30 ± 15 nm, lengths of 5 to 20 microns and a specific surface area of 200 to 400 m2/g. Various weight fractions of CNTs were used in the production of samples ranging from 0.1 to 0.5 wt%. The general procedure of material fabrication is shown in fig.1. High surface energy of carbon nanotubes causes the agglomeration of nanotubes when dispersed, adversely affecting the electrical transport properties of the material in a defective manner [14]. In order to address the agglomeration issue and effectively disperse the CNTs, the present work implemented high-intensity ultrasonication and high-speed shear mixing. Premeasured amounts of Part A Resin and carbon nanotubes were mechanically stirred for 5 minutes in a copper beaker. The mixture was then placed into a shear mixer outfitted with a 3-blade propeller Mix Part A + CNTs Shear Mix Ultrasonication Vacuum stirrer and shear-mixed at 600 RPM for 30 minutes. Following shear mixing, the ultrasonication process Fig. 1 Schematic of nanocomposite fabrication procedure was applied for one hour on pulse mode, 4.5 sec on 9 sec off, with 100 kHz. The mixture was then placed into a vacuum chamber to remove any trapped air bubbles generated during the mechanical mixing process [15]. A pre-measured amount of Part B epoxy hardener, in a separate container, was also placed inside the vacuum chamber. Both solutions were placed under a vacuum for 1 hour. Once all air was removed from both solutions, they were combined and mechanically stirred for 2 minutes. The mixture was once again placed back into the vacuum chamber for 5 minutes. Finally, the CNT/epoxy solution was slowly poured into pre-manufactured wax molds and allowed to cure for 3 days under ambient conditions.

363 It is critically important to control the temperature of the mixture during the sonication process for the quality of the fabricated samples. The sonication process generates substantial heat that may damage CNTs and deteriorate the electrical properties of the final composite [16]. Moreover, too much heat could cause Sonicator Probe the epoxy to reach the flash point. To Thermocouple control the temperature of the mixture Copper Beaker during sonication, a cooling apparatus was designed and built as shown in LN2 fig.2. While the mixture in the copper Copper Coils Reservoir breaker is sonicated, liquid nitrogen Anti-freeze flows through a copper-tube coil Solution submerged in an anti-freeze solution under the beaker to maintain the Fig. 2 Cooling apparatus used to control temperature during ultrasonication mixture temperature. The flow rate of liquid nitrogen is precisely controlled while temperatures of the beaker and anti-freeze solution are real-time monitored by thermocouples. The required cooling rate for proper temperature control depends primarily on the weight fraction of CNTs in the solution. For the present fabrication, the mixture temperature was Silver Paint 12.7 mm maintained in between 18°C - 30°C, depending on the sonication duration.

25.4 mm

Silver Paint

Fig. 3 Specimen used in quasi-static and dynamic experiments

Fig.3 illustrates specimens prepared for both quasi-static and dynamic compression-loading experiments. Specimens have dimensions 12.7 mm x 12.7 mm x 25.4 mm, where the loading is exerted in the longitudinal direction of 25.4 mm length. Two V-notch channels with a depth of 0.5 mm were machined in the middle section of the specimen. The channels were used to implement a modified four-point probe method, the details of which are described in the following section. These two channels separated the length of the specimen into three sections with equal lengths of 8.5 mm.

2.2 Percolation Behavior Study Prior to experimentally studying the electrical response of the CNT/epoxy nanocomposites under loading, a detailed investigation was conducted to determine the percolation threshold of the fabricated composites under no loading conditions. This study is essential in determining the optimum duration of sonication as well as the concentration of CNTs to ensure a proper percolation network within the material. The concentration of CNTs used for determining the optimum duration of sonication was 0.5 wt.%. During the sonication process, two samples were poured into the pre-manufactured molds each hour. For each duration of sonication time, two samples were measured and the results proved to be repeatable. For the second part of the study, once an optimum duration time was determined, the concentration of CNTs used was varied from 0% to .5% and resistance measurements were taken and analyzed. 2.3 Electrical Characterization of CNT-polymer nanocomposites A novel approach utilizing the four-point probe method was implemented to measure the change in electrical resistance of the specimen. The top face, bottom face, and the two inner channels of the specimen served as the four probes. The top and bottom faces of each specimen were coated with silver paint. The two inner channels were first coated with silver paint and lead wires were then attached using a silver epoxy. A constant current was supplied through the top and bottom surfaces of the specimen to allow a uniform current through the total volume of the specimen. The two inner channels served as the two peripheral electrodes that measure the voltage drop across the middle section of the sample. The electrical resistance of the middle section can be easily determined from the input current and voltage drop across the inner probes. As the specimen underwent deformation, the instantaneous resistance of the middle section changed. Percent change in resistance was calculated during each experiment. Since the initial resistance of each specimen is slightly different due to the complex dispersion pattern of nanotubes inside, only the relative resistance change was considered while the initial resistance served as the baseline for each experiment. An example of a resistance variation measured from a group of specimens, from the same batch, ranged from 1922 Ω to 3330 Ω. Compared to the classical four-point probe measurement technique, which utilized point connections and thus likely measured the sheet resistance of the material located only in the vicinity of the electrodes [17], this method allows for an average voltage reading around the periphery of the specimen. An average voltage

364 reading is advantageous during mechanical compression to incorporate a larger volume of material. Therefore, this method better provides the means to detect changes in resistance caused by strain and damage mechanisms throughout a larger volume of material. Since the current uniformly flows through the cross sectional area, the measured resistance is an estimation of the bulk resistance of the inner section. By using this average voltage measurement technique, more consistent and accurate results were obtained during a wide range of mechanical loading schemes and consequent sample deformations.

Electrometer

Current Source

Multimeter LabView Electrometer

Fig. 4 Experimental set-up for measuring resistance change under quasi-static conditions For quasi-static and dynamic experiments, different experimental setups were employed for the resistance measurement. The experimental setup used in quasi-static experiments is shown in fig.4. A constant current source was used to generate a constant DC current flow. The CNT/epoxy specimen was sandwiched by two aluminum plates to guarantee the uniform current flow through the sample during the compression loading. The silver paint, applied to the top and bottom of each specimen, minimized contact resistance between the specimen and the plates. Proper contact between the plates and specimen was ensured by applying a small initial load to each specimen prior to each experiment. Each loading head was wrapped with a layer of electrical tape to insulate the electrical measurements from the loading apparatus. Two electrometers were used to measure the voltage at each of the two individual inner probe rings. The difference between the two voltage readings, which corresponds to the voltage drop across the two inner probes, was measured using a digital multimeter and recorded using a LabView system.

High Response Current Source

Differential Amplifier

Digital Oscilloscope

Fig. 5 Experimental set-ups for measuring resistance change under dynamic conditions The experimental setup used in dynamic experiments is shown in fig.5. A constant current source with high frequency response was used to generate a constant DC current flow under high rate deformation. The CNT/epoxy specimen was again placed in between two aluminum plates. Proper contact between the specimen and the two aluminum plates was ensured by wrapping electrical tape tightly around the aluminum plates. The electrical tape also served to insulate the electrical measurements from the drop weight tower. The voltage drop between the two inner probes was measured by a differential amplifier and recorded by a digital oscilloscope. 2.4 Quasi-Static and Dynamic Mechanical Loading The quasi-static loading was implemented by a screw-driven testing machine. Specimens were loaded at a constant rate of 0.25 mm/min. The machine compliance was calculated and subtracted from the total deformation to obtain the exact deformation of the specimen under quasi-static loading. Dynamic loading was implemented by a drop tower apparatus. The drop hammer, with a mass of 11 kg, was equipped with a planar impact face. The resulting impact velocity was approximately 3.6 m/s, with a total input energy of 70 J. A typical deformation rate of ~ 60s-1 was achieved. The load and displacement history of the loading head was recorded by the drop tower. A high-speed camera captured high-speed deformation images at a frame rate of 20,000 fps. These high-speed images were utilized to calculate the strain history of the specimen under dynamic loading as well as capture major damage mechanisms.

365 3. Results and Discussion 3.1 Percolation Behavior Study The effect of the duration of the sonication process on the base resistance of the material is shown in fig.6a. While the base resistance does not show apparent change up to three hours of sonication, it begins to drastically increase as the sonication process is further carried out. This is attributed to the nanotubes being damaged due to excessive sonication. The high local temperatures and pressures during the sonication process could damage and weaken the nanotubes, thus creating a less efficient percolation network. From this observation, sonication was carried out for 1 hour during the specimen fabrication process in the present study. 350

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0.1

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Percentage Weight Fraction

Sonciation Duration (h)

(a) (b) Fig. 6 Experimental results of static resistance measurements as a function of (a) sonication duration and (b) percentage weight fraction Fig.6b shows the effect of the concentration of CNTs on the base resistance of the material. The electrical resistance of specimens with no CNTs is prohibitively high to measure: given the compliance of the current source, the resistance of the pure epoxy sample was greater than 52 GΩ. The percolation threshold of these nanocomposites is between 0.1 and 0.2 wt.% for the given nanotubes and polymers. Previous studies have shown similar electrical percolation behavior in nanotube/epoxy composites [13, 18]. It should be noted that beyond the percolation threshold, there is no further significant improvement in the electrical conductivity. The increase of nanotube concentration greater than 0.2 wt.% does not provide better electrical conduction. Therefore, the concentration of CNTs was set to 0.2 wt.% for all experiments in the present study. 3.2 Quasi-Static Results

Eng. Stress (MPa)

Resistance (kΩ)

Fig.7 shows a typical result of the actual electrical resistance and 70 90 stress changes of a specimen as a function of engineering strain. 80 60 During the quasi-static compression, the stress of the specimen 70 monotonically increases to 75 MPa at 4% engineering strain and 50 then gradually decreases. On the other hand, the electrical resistance 60 change is inversely proportional to the change in strain. As the 40 50 compressive strain increases to 12%, the resistance decreases 47% 40 30 from ~ 60 kΩ to ~ 32 kΩ. The change in electrical resistance of a 30 material is usually a combinative result of a geometrical change and 20 a resistivity change. However, since the resistance of the matrix 20 Resistance material is very high, the CNTs exclusively conduct the electrical 10 10 Stress current within the material. When considering the negligible change 0 0 of the CNT geometry during compression, the resistance change is 0 2 4 6 8 10 12 14 caused by the rearrangement of the electrical network between Eng. Strain (%) CNTs. The effective inter-nanotube gap becomes closer with the compression, forming a more efficient conductive network. As the Fig. 7 Typical electrical response of CNT/epoxy specimen reaches its elastic limit, the material begins to spread, nanocomposite under quasi-static loading causing a bulging phenomenon located in the middle section. The combination of the material compression and spreading causes little resistance change following elastic deformation.

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A series of experiments were carried out and the change in resistance demonstrated by all specimens was repeatable. During each loading, a small threshold strain was observed in which the resistance did not change until it was surpassed. The resistance change was taken from the respective strain threshold for consistency. The resistance changes of four different specimens are shown in fig.8. The difference between the curves can be attributed to the variability and complexity of the CNT networks present within each specimen. As the material compresses, the carbon nanotubes form a more efficient electrical network between nanotubes due to the compression of the matrix material. As seen in fig.8, specimens under quasi-static loading showed a 40% - 50% decrease in resistance during uniform compression. 3.3 Dynamic Results

Decrease in Resistance (%)

60 50 40 30 20 10 0 0

2

4

6

8

10

12

14

Eng. Strain %

Fig. 8 Percent decrease in resistance of 0.2 wt.% CNT/epoxy nanocomposites of rectangular geometry under quasi-static loading

A typical electrical response and the real-time deformation images of a rectangular 0.2 wt.% CNT/epoxy nanocomposite are shown in fig.9. In general, the dynamic response of the CNT/epoxy nanocomposite is more brittle than the quasi-static one due to the higher strain rate. The electrical response is very similar to that observed during quasi-static loading prior to any visible damage occurring. However, two electrical responses are observed once the damage begins. As shown in fig. 9 and 10, respectively, one group of specimens shows abrupt increase in resistance with the forming of cracks and voids within the sample while the other group shows a more gradual change. Fig.9 shows a uniform compression of the specimen during the first 0.8 ms of impact. Meanwhile, the resistance of the specimen experiences a 63% decrease. At 0.8 ms, a crack or void initiates within the sample, as conveyed by the expansion of the right side of the sample in the inset image of fig.9. This void initiation is captured by the resistance data with a sharp increase in resistance, suggesting that the carbon nanotube network should be disrupted. Damage continues growing within the specimen and causes the resistance to continue to increase.

Change in Resistance (%)

150 100 50 0

0.0

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1.4

1.6

1.8

Time (ms)

-100

0 ms

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.4 ms

.6 ms

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Fig. 9 Percent resistance change of CNT/epoxy nanocomposite under drop weight loading with real-time deformation images. Fig.10 shows a different response of a specimen demonstrating a gradual change in resistance under dynamic drop weight loading. Similar to fig.9, the sample undergoes uniform compression up to approximately 0.75 ms. At 0.8 ms, the right and left sides of the specimen demonstrate an expansion, initiating damage in the form of a void or crack. From 0.8 ms to 0.95 ms, damage further propagates throughout the sample. However, the resistance does not abruptly jump but gradually increases. We believe that this difference may come from the non-uniform dispersion of nanotubes inside the matrix and complex initiation and propagation of damages.

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Change in Resistance (%)

40 30 20 10 0 -10 0.0

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-20 -30 -40 -50

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.6 ms

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Fig. 10 Percent resistance change of CNT/epoxy nanocomposite under drop weight loading with real-time deformation images.

4. Conclusions

Decrease in Resistance (%)

Despite the different dynamic responses of the nanocomposite upon the damage initiation, the overall resistance change under dynamic compressive loading is in the same trend as the quasi-static loading. As the compressive strain increases, the resistance monotonically decreases until damage initiates inside the material. Fig.11 shows the electrical responses of three specimens that have the same geometry and fabrication conditions. 70 Overall, the resistance of the material decreases 50 - 65 % during dynamic compressive loading. Once the material reached a critical 60 strain value, damage would begin to grow within the specimen and 50 cause the resistance increase, either abruptly or gradually depending on the damage propagation. 40 30

The present paper describes the electrical response of multi-wall 20 carbon nanotube reinforced nanocomposites under quasi-static and 10 dynamic compressive loading conditions. An effective method to obtain low resistant CNT/epoxy nanocomposites was devised and 0 used to fabricate the specimens. A modified four-point probe 0.0 0.2 0.4 0.6 0.8 1.0 method, using line and face contacts rather than point contacts, was Time (ms) implemented to measure more consistent and accurate results Fig. 11 Percent decrease in resistance of CNT/epoxy during mechanical loading. A screw-driven testing machine and a nanocomposites of under dynamic loading drop weight tower were implemented for quasi-static and dynamic loading. The history between the electrical resistance change, the mechanical loading, and the high-speed deformation photography are correlated to characterize the electrical response of CNT reinforced epoxy under compressive loading. From the experimental results, the electrical resistance of the specimen shows a significant decrease both under compressive quasi-static and dynamic loading conditions. A 40 – 60% decrease in resistance was observed. This is due to a more efficient carbon nanotube network forming under compressive loading caused by the compression of the epoxy matrix. The obtained results in the present study provide insights on the electrical behaviors of CNT reinforced nanocomposites under compression loadings, and will thus be beneficial in the development of novel sensors. Acknowledgements This work was supported by the National Science Foundation (NSF) under grant number CMMI 0856133 References

368 [1] Park C, Onuaies Z, Watson KA, Pawlowski K, Lowther SE, Connell JW, Siochi EJ, Harrison JS, St. Clair TL (2002) Polymer-Single Wall Carbon Nanotube Composites for Potential Spacecraft Applications. Material Research Society 706 [2] Dalton AB, Collins S, Razal J, Munoz E, Ebron VH, Kim BG, Coleman JN, Ferraris JP, Baughman RH (2004) Continuous carbon nanotube composite fibers: properties, potential applications, and problems. Journal of Materials Chemistry 14: 1-3 [3] Kang I, Heung YY, Kim JH, Lee JW, Gollapudi R, Subramaniam S, Narasimhadevara, S, Hurd D, Kirikera GR, Shanov V, Schulz MJ, Shi D, Boerio J, Mall S, Ruggles-Wren M (2006) Introduction to carbon nanotube and nanofiber smart materials. Composites Pt B 37(6): 382-394 [4] Crespi V, Cohen M, Rubin A (1997) In Situ band gap engineering of carbon nanotubes. Physical Review Letters 79: 2093-2096 [5] Kane CL, Mele EJ (1997) Size, shape, and low energy electronic structure of carbon Nanotubes. Physical Review Letters 78: 1932-1935 [6] Nardelli M, Bernholc J (1998) Mechanical deformations and coherent transport incarbon nanotubes. Physical Review B 60: R16338-16341 [7] Rochefort A, Salahub D, Avouris P (1998) The effect of structural distortions on the electronic structure of carbon nanotubes. Chemical Physics Letters 297: 45-50 [8] Bezryadin A, Verschueren A, Tans S, Dekker C (1998) Multiprobe transport experiments on individual single-wall carbon nanotubes. Physical Review Letters 80: 4036-4039 [9] Paulson S. et al (1999) In situ resistance measurements of strained carbon nanotubes. Applied Physics Letters 75: 29362938 [10] Alexopoulos ND, Bartholome C, Poulin P, Marioli-Riga Z (2009) Structural health monitoring of glass fiber reinforced composites using embedded carbon nanotube (CNT) fibers. Composites Science and Technology 70(2): 260-271 [11] Nofar M, Hoa SV, Pugh MD (2009) Failure detection and monitoring in polymer matrix. Composites subjected to static and dynamic loads using carbon nanotube networks. Composites Science and Technology 69(10): 1599-1606 [12] Gao L, Thostenson ET, Zhang Z, Chou TW (2009) Coupled carbon nanotube network and acoustic emission monitoring for sensing of damage development in composites. Carbon 47(5): 1381-1388 [13] Thostenson ET, Chou TW (2006) Carbon Nanotube Networks: Sensing of Distributed Strain and Damage for Life Prediction and Self Healing. Advanced Materials 18: 2837-2841 [14] Kabir ME, Saha MC, Jeelani S (2007) Effect of ultrasound sonication in carbon nanofibers/polyurethane foam composite. Materials Science and Engineering A 459(1-2): 111-116 [15] Evora VMF, Shukla A (2003) Fabrication, characterization, and dynamic behavior of Polyester/TiO 2 nanocomposites. Materials Science and Engineering A 361(1-2): 358-366 [16] Ma P, Siddiqui NA, Marom G, Kim J (2010) Dispersion and functionalization of carbon Nanotubes for polymer-based nanocomposites: A review. Composites Part A: Applied Science and Manufacturing 41(10): 1345-1367 [17] Smits FM (1958) Measurements of Sheet Resistivity with the Four-Point Probe: Bell System Technical Journal 37(3): 711-718 [18] Bauhofer W, Kovacs JZ (2009) A review and analysis of electrical percolation in carbon nanotube polymer composites. Composites Science and Technology 69(10): 1486-1498

Effect of Curvature on Shock Loading Response of Aluminum Panels

a

Puneet Kumara, James LeBlancb and Arun Shuklaa*. Dynamic Photomechanics Laboratory, Department of Mechanical, Industrial and Systems Engineering, University of Rhode Island, Kingston, RI 02881, USA. b Naval Undersea Warfare Center (Divison Newport), Newport, RI. *Corresponding author email: [email protected]

Abstract Accidental explosions or bomb blasts cause extreme loadings on civilian and military structures. These structures have different curvatures, which affect their blast mitigation properties. Thus a controlled study has been performed to understand the effect of curvature on blast mitigation. The shock tube apparatus was utilized to obtain a controlled blast loading. Aluminum 2024 T3 panels having three different curvatures (infinity, 304.8 mm, and 111.76 mm) were used in the experiments. All the panels had un-deformed dimensions of 203.2 mm x 203.2 mm x 2 mm. A fully clamped boundary condition was applied. Digital Image Correlation (DIC) techniques were applied to obtain full-field in-plane and out-of-plane deformation data and in-plane strain on the back face of the panel. The results show that the panel with a 304.8 mm radius of curvature had a better blast resistance as compared to the other two panels. 1. Introduction Accidental explosions or bomb blasts cause extreme loading on structures. These structures have both flat faces and curved faces. Thus it’s important to understand the effect of curvature on the blast mitigation. Aluminum panels having three different radius of curvature are subjected to blast loading using a shock tube to study their dynamic response. Post-mortem analysis has been conducted on the blast loaded panels to evaluate the effectiveness of the material to mitigate blast loading. The aim of this study is to analyze the damaged area, midpoint transient deflection, and other characteristics of the dynamic response of panels subjected to a controlled blast loading. Jacinto et al. [1] and Stoffle et al. [2] have focused on the study of the dynamic response of thin metallic plates subjected to varying levels of shock loading. Particularly, Jacinto et al. [1] attached accelerometers to the non-impact face of the plates in order to measure the dynamic response, whereas Stoffle et al. [2] used a capacitance scheme to measure the centre deflection during loading. Experimental studies performed by Nurick et al. [3], [4] and Wierzbicki et al. [5] examined the large deformations and failure modes of thin plates subjected to a blast loading. The plates were loaded with a pressure pulse of short duration generated by explosive charges. 2. Experimental Procedure 2.1 Material Details The three different panels used during these experiments include a flat panel, panels having 304.8 and 111.76 mm radius of curvatures (specimens are shown in fig. 1). Each experiment is repeated three times. The specimens are 203.2 mm long x 203.2 mm wide x 2 mm thick. Panels were made out of 2024 Aluminum T3.

Figure 1: Specimens 2.2 Shock loading apparatus The shock tube apparatus used in this study to obtain the controlled dynamic loading is shown in fig. 2. A complete description of the shock tube and its calibration can be found in [6]. In principle, the shock tube consists of a long rigid cylinder, divided into a high-pressure driver section and a low pressure driven section, which are separated by a diaphragm. T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_51, © The Society for Experimental Mechanics, Inc. 2011

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370 By pressurizing the high-pressure section a pressure difference across the diaphragm is created. When this pressure differential reaches a critical value, the diaphragm ruptures. The following rapid release of gas creates a shock wave, which travels down the tube to impart dynamic loading on the specimen. The specimen is held in a fixture that ensures the proper specified boundary conditions. When the shock wave impacts the test panel at the end of the tube, the gas is superheated and the wave is reflected at a higher pressure than that of the initial shock front. An extensive derivation of the theoretical equations for shock tubes has been previously established in literature and is briefly discussed in the following section [7]. In basic shock wave theoretical formulations the following assumptions are generally used to describe the gas flow: 1. The gas flow is one-dimensional. 2. The gas is ideal and has constant specific heats. 3. Heat transfer and viscosity effects are neglected. 4. Diaphragm rupture is instantaneous and does not disturb the subsequent gas flow. Using conservation of energy, mass, and momentum as described by Wright [7], the following relationships for pressure, temperature and density across a shock front can be derived:

P2 2 M 12  (  1)  P1  1 T2 {2 M 12  (  1)}{(  1) M 12  2}  T1 (  1) 2 M 12

2 M 12 (  1)  1 (  1) M 12  2 P1 , T1 , 1 are pressure, temperature and density ahead of the shock front and P2 , T2 , 2 are the pressure, temperature and density behind the shock front,  is the adiabatic gas constant and M 1 is the mach number of the shock

where

wave relative to the driven gas.

Figure 2: The URI shock tube facility. The shock tube utilized in the present study has an overall length of 8 m, consisting of a driver, driven, converging and muzzle sections. The diameter of the driver and driven section is 0.15 m. The final muzzle diameter is 0.07 m. Two pressure transducers (fig. 3), mounted at the end of the muzzle section measure the incident shock pressure and the reflected shock pressure during the experiment. All of the aluminum panels are subjected to the same level of incident pressure in these experiments. A typical pressure profile obtained at the transducer location closer to the specimen is shown in fig. 4.

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Figure 3: Schematics of the muzzle of the shock tube and fixture

Figure 4: A typical pressure profile 2.3 Loading conditions The specimens utilized in this experimental study are held under clamped-clamped boundary conditions on two edges and simply supported on the other two edges prior to blast loading. The size of the specimens is 203.2 mm by 203.2 mm by 2 mm. The dynamic loading is applied over a central circular area of 76.2 mm in diameter. 2.4 Digital Image Correlation (DIC) Technique The digital image correlation technique is one of the most recent non-contact methodologies for analyzing full-field shape and deformation [8]. It involves the capture and storage of high speed images in digital form and subsequent post-processing of these images to get the full-field shape and deformation measurements. The DIC system involves the mapping of predefined points on the specimen to measure the full-field shape and deformation. Capturing the three dimensional response of the plates requires that 2 cameras be used in a stereo configuration and they must be calibrated and have synchronized image recording throughout the event. The calibration of the cameras is performed by placing a grid containing a known pattern of dots in the test space where the glass sample is located during the test. This grid is then translated and rotated in and out of plane while manually recording a series of images. As this grid pattern is predetermined, the coordinates of the center of each dot is extracted from each image. The coordinate locations of each dot extracted uniquely for each camera allows for a correspondence of the coordinate system of each camera. The DIC is then performed on the image pairs that are recorded during the shock event. Prior to testing the back face of the sample is painted white and then coated with a randomized speckle pattern (Figure 5). The post processing is performed with the VIC-3D software package which matches common pixel subsets of the random speckle pattern between the deformed and un-deformed images. The matching of pixel subsets is used to calculate the three dimensional location of distinct points on the face of the panel throughout time. A speckle pattern is placed on the back face of the (as seen in fig. 5). Two high speed digital cameras, Photron SA1s, are positioned behind the shock tube apparatus to capture the real time deformation and displacement of the panel, along with the speckle pattern. The high speed cameras are set to capture images at 20,000 frames per second (inter frame time of 50 s). During the blast loading event, as the panel responds, the cameras track the individual speckles on the back face sheet. Once

372 the event is over, the high speed images are analyzed using DIC software to correlate the images from the two cameras and generate real time in-plane strain and out-of-plane deflection histories. A schematic of the set-up is shown in fig. 5.

Figure 5: Schematics of DIC system There are two key assumptions which are used in converting images to experimental measurements of objects shape, deflection and strain. Firstly, it is assumed that there is a direct correspondence between the motion of the points in the image and that in the object. This will ensure that the displacement of points on the image have a correlation with the displacement of points on the object. Secondly, it is assumed that each sub-region has adequate contrast so that accurate matching can be preformed to define local image motion. 3. Experimental results 3.1 DIC Analysis The DIC technique (as discussed in section 2.4) is used to obtain the out-of-plane deflections and the in-plane strains on the back surface for all the five panels. The speckle pattern is applied onto the back face of the panels (fig. 5) which are subjected to shock loading. The high speed images captured using two Photron SA1 cameras are analyzed to get the back face deflections from the DIC as shown in fig. 6. Experiments have already been done to compare the back face deflection from the real time transient image and DIC to verify the accuracy of the DIC results. The error between the maximum deflection from DIC and real-time transient images is 4% [9]. The DIC results are within the acceptable error limits and so the DIC results can be used to better understand the failure and damage mechanism in the panel. The full-field DIC analysis for the five different glass panels is shown in fig. 6. From the time-deflection history (fig. 6) it is seen that the deflection rate is much slower in the panel having a radius of curvature of 304.8 mm. Also, the in-plane strain (fig. 7) is developed at a much slower rate in the panel as compared to the other two panels, even though the three panels were subjected to the same level of incident pressure. As seen in fig. 8, the kick-off velocity in 304.8 mm (radius of curvature) panel is smaller as compared to that in the other two panels. This kick-off velocity is an important parameter in the damage initiation. So, this shows that the panel having 304.8 mm radius of curvature has a better blast mitigation property.

373

Figure 6: Time-deflection history of the back face for the three panels

Figure 7: Time-in-plane strain history of the back face for the three panels

Figure 8: Time-Velocity history of the back face for the three panels 3.2 Macroscopic post-mortem analysis The result of post-mortem evaluation of the shock loaded glass panels is shown in Fig. 9. The damaged area in the flat plate was much larger as compared to the other two panels, whereas it was much smaller 111.76 mm radius of curvature panel. The damage in the panel having 111.76 radius of curvature was more because of indenting effect.

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(a) (b) (c) Figure 9: Post-mortem evaluation of (a) Flat Panel (b) 304.8 mm radius of curvature and (c) 111.76 mm radius of curvature 4. Conclusions Three different panels are subjected to a controlled shock loading using a shock tube. The high speed photography and DIC analysis is applied to obtain the out-of-plane deflection and in-plane strain on the back face of all the five panels. 1. The macroscopic post-mortem analysis and DIC deflection, velocity and in-plane strain analysis shows that the panel having 304.8 mm radius of curvature has a better blast mitigation property as compared to the other two panels. 2. There is critical radius of curvature beyond which the blast mitigation property of the panel starts decreasing. 5. Acknowledgement The authors acknowledge the financial support provided by the Department of Homeland Security (DHS) under Cooperative Agreement No. 2008-ST-061-ED0002. References 1. Jacinto, A., Ambrosini, R. and Danesi, R., Experimental and computational analysis of plates under air blast loading, Int J Impact Eng, pp. 927-47, 2001. 2. Stoffel, M., Schmidt, R. and Weichert, D., Shock wave loaded plates, Int J solids Struct, pp. 7659-80, 2001. 3. Nurick, G., et al., Deformation and tearing of blast loaded stiffened square plates, Int J Impact Eng, pp. 273-91, 1995. 4. Nurick, G. and Shave, G., The deformation and tearing of thin square plates subjected to impulsive loads - an experimental study, Int J Impact Eng, pp. 99-116, 1996. 5. Wierzbicki, T. and Nurick, G., Large deformation of thin plates under localized impulsive loading, Int J Impact Eng, pp. 899-918, 1996. 6. LeBlanc, J., et al., Shock loading of three-dimensional woven composite materials, Compos Struct, pp. 344–355, 2007. 7. Wright, J., Shock Tubes, John Wiley and Sons Inc., New York, 1961. 8. Tiwari, V., et al., Application of 3D image correlation for full-field transient plate deformation measurements during blast loading, I J Impact Eng, pp. 862-874, 2009. 9. Gardner, N. and Shukla, A., The blast response of sandwich composites with a functionally graded core and polyurea interlayer, 2010 SEM Annual Conference & Exposition on Experimental and Applied Mechanics, Indianapolis.

Deformation measurements and simulations of blast loaded plates

K. Spranghers1,a, D. Lecompteb, H. Sola and J. Vantommeb a

b

Department of Mechanics of Materials and Constructions, Vrije Universiteit Brussel (VUB), Pleinlaan 2, B-1050 Brussels, Belgium

Civil and Materials Engineering Department, Royal Military Academy (RMA), Av. De la Renaissance 30, B-1000 Brussels, Belgium

ABSTRACT Three aluminum plates are subjected to a free-air explosive loading with 40g of C4. Combining two highspeed cameras in a stereoscopic setup and the digital image correlation technique, the full-field deformation is identified. Furthermore, the identified deformation fields are compared by data computed with an explicit finite element method. The Johnson-Cook material model is used to simulate the plastic behavior of the aluminum plate. A good agreement has been found between both experimental and numerical data. Keywords 3D digital image correlation, high-speed cameras, blast loading, laboratory-scale experiments, aluminum plates.

1. INTRODUCTION Much research effort has been focused on the dynamic response of the structural components subjected to blast loading [1–4]. Some of this research has been concentrated on the response of circular and rectangular plates to impulsive loading [5–9]. In these studies, large plastic deformations are obtained and the loading impulse is experimentally determined and compared to analytical and numerical calculations. The full-field response of the specimens is mostly solely investigated numerically. In the present study the resultant displacement field of aluminum plates subjected to an explosive loading is measured using two high-speed cameras combined with the digital image correlation technique. Furthermore, the experimental data is compared to the results of finite element simulations using a commonly accepted constitutive model. 2. 3D HIGH-SPEED DIGITAL IMAGE CORRELATION Full-field measurement techniques and the digital image correlation technique (DICT) in particular are very suitable for observation of loading conditions that create complex heterogeneous deformation fields. 2D digital image correlation (2D-DIC) is an optical numerical full-field measuring technique, which offers the

1

Corresponding author. Tel.: +32 2 629 29 36. Email address: [email protected]

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_52, © The Society for Experimental Mechanics, Inc. 2011

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376 possibility to determine in-plane displacement fields at the surface of objects under any kind of loading. 2D-DIC is based on a comparison between images taken with a digital camera at different load steps. The DICT has been developed in the 80’s and has since then extensively been evaluated and improved, however, the fundamental principles of the method remain unchanged and are well described in e.g. [10]. By combining stereovision principles with 2D-DIC concepts and two synchronized high-speed digital cameras it is possible to measure three-dimensional (3D) deformations fields at high strain rate [11]. The DICT allows studying qualitatively as well as quantitatively the deformation behavior of materials under certain loading conditions. Each picture taken with a digital camera corresponds to a different load step. Two images of the specimen at different states of deformation are compared by using a pixel and its signature in the undeformed image, and searching for the pixel in the deformed image by maximizing a given similarity function. In most cases, this function is based on a least-squares formulation. The signature of a pixel can be anything that discriminates it among any other pixel signature and can be the pixel grey-value, the greyvalue derivatives or the pixel color. In this case, the pixel grey-value is used. A single grey-value is not a unique signature of a pixel; hence, neighboring pixels are used in practice. Such a collection of pixels is called a subset. The displacement result, expressed in the centre point of the subset, is an average of the displacements of the pixels inside the subset. The step size defines the number of pixels over which the subset is shifted in x- and y-direction to calculate the next result. Both parameters are expressed in pixels. The uniqueness of each signature is only guaranteed if the surface has a non-repetitive, isotropic, highcontrast pattern. Random textures of speckle patterns fulfill this constraint and can be obtained by an arbitrary speckle pattern that is applied onto the object surface, or that is offered by the texture of the specimen’s material. Possible matches at several locations are checked and a similarity score (correlation function) is used to grade them. A classic correlation function using the sum of the squared differences of the pixel values is used. The image correlation routine allows locating every subset of the initial image in the deformed image. Subsequently, the software determines the displacement values of the centers of the subsets, which yields an entire displacement field (relative to the considered reference image). 3D digital image correlation (3D-DIC) uses the same correlation concepts as described above. Two synchronized cameras coupled with stereovision principles are required to obtain three-dimensional displacement fields. The manipulation of the stereovision system is described in detail in section 3. 3. EXPERIMENTAL SET UP The experimental setup is shown in Figure 1 and located in the test bunker of the Laboratory of Analysis of Explosion Effects (LAEE) of the Royal Military Academy (RMA). The setup consists of a steel frame (mounting plate, 1m x 1m x 1.5cm) with a square captivity of 30cm by 30cm in the centre. The specimen, a thin aluminum plate (type EN AW-1050A), of 40cm by 40cm with a thickness of 3mm, is connected to the steel frame in the captivity. The connection is established by using a bolts connection with a steel plate in order to achieve a perfect fixed connection. The explosive material, 40g of C4, is suspended behind the test plate with a stand-off-distance of 25cm from the centre point in order to realize a spherical airburst wave. Two similar Photon© Fastcam Ultima APX-i2 high-speed digital cameras are mounted in a stereo configuration to record synchronized images during a free-air blast event. The full-field transient deformations of a thin aluminum plate are obtained using 3D digital image correlation (3D-DIC). The procedure used to perform the experiment is as follows. First, the aluminum test plate is covered with white paint and, a black speckle pattern is printed on the specimen to obtain high contrast images. Next, the plate is bolted together with the steel clamping plate onto the steel mounting plate in his final position. Next, the laser distance meter is clamped on a steel bar at 60cm in front of the test plate at the same height of the plate’s centre point, the charge is hang up at 25cm stand-off-distance behind the test plate and the blast pencil is positioned in a way the pressure sensor measures the same incident pressure as the plate’s centre point. Next, the high-speed cameras are placed in front of the plate at a certain object distance from the detonation point to avoid camera motions induced by the blast loading, rotated to view the specimen and maximize the common field of view of the specimen and focused. Since large out-ofplane displacements are expected during the deformation process, the aperture of both lenses is reduced and the sheet illumination is increased to compensate for the reduced aperture and maintain adequate contrast throughout the experiment. These settings ensure that the formed sheet remains in focus. Next, a calibration panel is used to calibrate the high-speed stereovision system. Next, a light intensity trigger is placed near

377 the detonation point and is adjusted in a way it will be triggered by the bright light flash of the explosion and send this starting signal to the high-speed cameras. As soon as the explosion initiates, synchronized stereo images at a frame rate of 6000fps are acquired during sheet deformation. During the experiments the plate behavior is observed with a field of view of 500mm x 500mm and an image size of 512 x 512 pixels.

Figure 1: Experimental set up. To ensure the accuracy of the results, the stereovision system is calibrated before each of the experiments as described by Tiwari et al. in [11]. The calibration panel consists of a grid of circular dots arranged in a pre-determined pattern. During the calibration this panel is moved (translated and rotated) in and out of plane and several images are acquired by both of the cameras. Once the images are acquired, coordinates of the centre of the dots in the calibration images are extracted using suitable edge detection method for all the calibration images for both the cameras. As the placement and inter-spacing of the dots in a calibration grid are pre-determined, using the knowledge of coordinate positions extracted from the calibration grid images a correspondence is drawn up between the coordinate system of both the cameras. This information is later used for feature based image registration (a process of transforming images of the object from different perspectives or at different time into one coordinate system) and measuring surface profile deformations. Once the camera calibration process is completed, actual digital image correlation is performed using undeformed and deformed image pairs to match common image subsets within the speckle patterns. Once an image subset is matched between undeformed and deformed image pairs, the stereovision system parameters are used to estimate the 3D position of points on the undeformed sheet (initial profile) and also on each deformed sheet position. Both stereo calibration and 3D image correlation are performed using Vic3D© software developed by Correlated Solutions, Incorporated [12]. 4. EXPERIMENTS AND RESULTS Two types of experiments were performed. For the first set of tests the aluminum plate is simply suspended in order to move it manually. This rigid body displacement is measured both with the cameras and a laser distance meter. By comparison of the plate’s centre point displacement the accuracy is evaluated. In the second set of tests the aluminum test plate is subjected to blast loading as described in section 3. Full-field 3D surface displacements were measured during blast loading of the aluminum sheet, with the main

378 objective of evaluating displacement measurements of thin plates subjected to blast loading using highspeed digital cameras and 3D digital image correlation. 4.1 Rigid body displacements Two sets of tests are performed. In both cases we use the two high-speed digital cameras and the laser distance meter (LDM), respectively at 50fps and 1kHz. Based on the images (512x512 pixels) captured by the cameras the displacement of every point is calculated with the 3D-DIC. From this full-field displacement field, the out-of-plane centre point displacement of the plate is extracted and compared with the displacement measured by the laser distance meter. The displacement of the plate’s centre point measured by the two systems agrees very well. The average displacement error of the 3D-DIC and the LDM equals respectively to 0.03mm and 0.33mm. 4.2 Blast loading Three blast load tests are performed as explained in section 3. The object distance and angle between the two cameras are respectively 2.5m and 39.6°. In the blast load test the cameras capture synchronized images at 6000fps. From the full-field displacement field, the out-of-plane centre point displacement of the plate is extracted. To obtain three-dimensional full-field displacement fields an area of interest (aoi) of 300x300 pixels, a 33x33 pixel subset and a subset spacing of 3 pixels are selected. Subset matching is performed using an affine subset shape function. Error assessment is based on the output variable σ, which is calculated by the correlation software and represents the confidence interval for the match of a certain point given in pixels. The displacement error, , is equal to the maximum σ value of all points for every image multiplied by the image resolution given in millimeters per pixels. The resultant displacement of the aluminum test plate’s centre point during the blast impact for tests 1, 2 and 3 are shown in Figure 3. The full-field resultant displacements for test 1 from zero to maximum deformation are shown in Figure 2. The error of the out-of-plane displacement equals to = 0,022mm, = 0.019mm and = 0,041mm for blast experiment 1, 2 and 3, respectively.

Figure 2: Full-field out-of-plane displacements (test 1).

379 5. NUMERICAL SET UP Numerical methods used to simulate blast effects problem are typically based upon a finite volume, finite difference or finite element method with an explicit time integration scheme. In this paper explicit finite element program LS-Dyna is used to predict the blast loading response of a clamped aluminum plate. In LS-Dyna, blast simulations are based on an empirical method that is developed by Kingery and Bulmash [13] where airblast parameters from spherical airbursts and from hemispherical surface bursts are predicted by equations. These equations are widely accepted as engineering predictions for determining free-field pressures and loads on structures. The Kingery-Bulmash equations have been automated in the computer program ConWep [14]. Curve-fitting techniques are used to represent the data with high-order polynomial equations, assuming an exponential decay of the pressure with time. A functional form such as the Friedlander equation can model the typical pressure history in the vicinity of a free-air explosion:

⎛ t ⎞ ⎛ at ⎞ P(t) = Pmax ⎜1 − ⎟ exp⎜ − ⎟ ⎝ t + ⎠ ⎝ t + ⎠

(1)

where t+ is the positive phase duration and the parameter a is called the waveform number and depends on Pmax (incident pressure or over pressure). Blast pressures in free-air, or incident pressures, are seldom of interest as it is the interaction of these pressures with structures, and the subsequent response of the € structure, that is the focus of attention. Mush like an acoustic wave, when a blast wave encounters a structure the sudden decrease in velocity of the shock wave, and particle velocities behind the shock, gives rise to an increase in pressure, i.e. the reflected pressure. Due to the large compressibility of air, the reflected pressure is typically mush more than doubled, Pref = CR Pinc with 2 ≤ CR ≤ 8. The reflected pressure wave also has a form similar to the incident pressure wave and can also be modelled by the Friedlander equation but with a different decay rate (waveform number). In the idealized case of ConWep there is no decay coefficient and the pressure wave is considered as a special triangular impulse because the structure is considered to be rigid and its surface infinite. ConWep is implemented in a LS Dyna algorithm for blast loads by Randers-Pehrson and Bannister and takes into account the decay coefficient that also updates the pressure history based on changes in the geometry [15,16]. The objective of this algorithm is to produce an appropriate pressure history given an equivalent TNT explosive weight. The quantities to be determined by the algorithm are: Pinc the peak incident pressure; Pref the peak reflected pressure; ta the time of arrival of the shock wave; t+ the positive phase duration; a, b the exponential decay factors for incident and reflected waves, respectively. The input values that need to be chosen are: the amount of explosive charge; R the range from charge location to the centroid of the loaded surface; and cosθ the cosine of the incident angle, angle between surface normal and range unit vector. The parameters that need to be defined by the user are the TNT-equivalent mass of the explosive and the position of the centre of the explosion in space, which defines the stand-off-distance. The Johnson-Cook material model is employed to estimate the effects of strain hardening, strain rate hardening and thermal softening. The flow stress is expressed as: m ⎛ ε ʹ′p ⎞⎛ ⎛ T − Troom ⎞ ⎞ σy = A + Bε ⎜1+ C ln ⎟⎜⎜1 − ⎜ ⎟ ⎟ ε ʹ′0 ⎠⎝ ⎝ Tmelt − Troom ⎠ ⎟⎠ ⎝

(

n p

)

(2)

where A, B, C, n, m are Johnson-Cook material parameters and εp the effective plastic strain, ε’p the effective plastic strain rate at a reference strain rate of ε’0 =1s-1 and T is the material’s temperature, Troom is the room temperature, and Tmelt is the material’s melting temperature. The EN AW-1050A aluminium plate € specimen has an initial yield point of σy,0 = 83MPa. The Johnson-Cook model parameters for EN AW1050A are A = 83MPa, B = 426MPa, C = 0.015, n = 0.350 and m = 1.0. It should be noted that these model parameters were determined using uniaxial dynamic loading conditions, with insufficient data available to improve parameter estimates for material behavior under dynamic biaxial loading conditions. The comparison of the plate’s centre point resultant displacements between the experiments and finite elements simulation is shown in Figure 3. Here we can see that the calculated displacement curves follow the experimental displacement curves. Both experimental and numerical curves have a maximum plastic

380 deformation in the first peak, followed by elastic deformations and damping. In the numerical calculations no damping is used. Furthermore, the differences between the 3 experiments are not negligible (maximum 15%), regarding the fact that they were performed under the same conditions. More efforts have to be done in order to receive repeatable results.

Figure 3: Comparison of the resultant displacements of the plate’s centre point.

6. CONCLUSIONS High-speed stereovision with 3D Digital Image Correlation has been successfully used to measure the rigid body displacement of a plate specimen. The calculated out-of-plane displacement is evaluated by the comparison with a laser distance meter and the estimated accuracy equals to 0.03mm, which is sufficient and in the order of magnitude of the plate specimens’ geometrical imperfections. More important, this high-speed stereovision system has been successfully used to monitor dynamic specimen response during free-air blast loading with a frame rate of 6000fps and an average accuracy on the out-of-plane displacement of 0.03mm. Results show that free-air blast loading induces highly localized, rapid material response. Using a well-known relationship, the material response is simulated with the finite element method, which give comparable results with the experimental data from the 3D-DIC. As a conclusion, the results confirm that dynamic stereovision systems are capable of accurately measuring surface displacement and deformation data at high rates, providing investigators with the ability to study a wide range of dynamic events, including impact, penetration and blast loading. REFERENCES [1]

P.S.Bulson, Explosive loading of engineering structures: a history of research and review of recent developments (E&FN SPON, UK 1997) p. 236 [2] E. Yandzio, M. Gough, Protection of buildings against explosions (Steel Construction Institute, UK 1999) p. 110 [3] M. Y. H. Bangash, T. Bangash, Explosion-resistant buildings: design, analysis, and case studies (Springer, USA 2006) p. 784 [4] N. Jones, Structural Impact (Cambridge University Press, UK 1989) p. 575 [5] R. G. Teeling-Smith, G. N. Nurick, International Journal of Impact Engineering 11, (1991) p. 77-91 [6] G. N. Nurick, G. C. Shave, International Journal of Impact Engineering 18-1, (1996) p. 99-116 [7] M. D. Olson, G. N. Nurick, J.R. Fagnan, International Journal of Impact Engineering 13-2, (1993) p. 279-291 [8] A. Neuberger, S. Peles, D. Rittel, International Journal of Impact Engineering 34, (2007) p. 859-873 [9] A. Neuberger, S. Peles, D. Rittel, International Journal of Impact Engineering 34, (2007) p. 874-882 [10] M. A. Sutton, J. -J. Orteu, H. W. Schreier, Image Correlation of Shape, Motion and Deformation Measurements: Basic Concepts, Theory and Applications (Springer, USA 2009) p. 321

381 [11] V. Tiwari, M. A. Sutton, S. R. McNeill, S. Xu, X. Deng, W. L. Fourney, D. Bretall, International Journal of Impact Engineering 36, (2009) p. 862-874 [12] Correlated Solutions Incorporated, VIC-3D (USA, www.correlatedsolutions.com) [13] C. N. Kingery, G. Bulmash, Airblast Parameters from TNT Spherical Air Burst and Hemispherical Surface Burst (Report ARBL-TR-02555, U.S. Army Ballistic Research Laboratory, USA 1984) [14] U.S. Department of the Army, Fundamentals of Protective Design for Conventional Weapons (Department of the Army Technical Manual, USA 1986) [15] G. Randers-Pehrson, K. A. Bannister, Airblast Loading Model for DYNA2D and DYNA3D (Report ARBL-TR-1310, U.S. Army Ballistic Research Laboratory, USA 1997) [16] Livermore Software technology Corporation, LS-DYNA User’s Manual: Nonlinear Dynamic Analysis of Structures (Version 950) (USA 1999) p. 1024

The Blast Response of Sandwich Composites with Bi-Axial In-Plane Compressive Loading

Erheng Wang1 and Arun Shukla2 1. Dept. of Aerospace Engineering, The University of Illinois, Urbana-Champaign, 104 S Wright St, Urbana, IL, 61801, [email protected] 2. Dynamic Photomechanics Lab, Dept. of Mechanical, Industrial and Systems Engineering The University of Rhode Island, 92 Upper College Road, Kingston, RI 02881, [email protected]

ABSTRACT The in-plane compressive loading in the ship hull structures during their whole service life will likely change the dynamic behavior of these structures under transverse blast loading. In the present study, the dynamic behavior of E-glass Vinyl Ester composite face sheet / foam core sandwich composites with bi-axial in-plane compressive loading were investigated under a transverse blast loading. A special test fixture which enables bi-axial in-plane compressive loading on the specimens was designed prior to transverse shock wave loading generated by a shock tube apparatus. Blast tests are carried out for three levels compressive loading. A high-speed back-view 3D Digital Image Correlation (DIC) system is utilized to acquire the real time deformation of the sandwich composites. The real time deformation and post mortem analysis were carefully conducted to study the failure mechanisms. The results show that the in-plane compressive loading induces the buckling and failure in the front face sheet. This mechanism highly reduces the blast resistance of the sandwich composites. INTRODUCTION Sandwich composites may undergo in-plane compressive loading in their application in the naval, aerospace, transportation and defense industries. For example, a ship’s hull will undergo longitudinal (in-plane) compressive loading during its service life [1], a missile’s shell will experience an in-plane inertia force when it is launched, and some residual stresses may be present in composites from the manufacturing and assembling processes. These in-plane compressive loadings will affect the dynamic response of sandwich composites when they are subjected to transverse blast loading and consequently affect their blast resistance. However, recent investigations on the behaviors of sandwich composites under blast loading mainly focus on the transverse behavior without longitudinal compressive loading [2-8]. Nurick [2], Zhu [3], and Dharmasena et al. [4] have tested sandwich structures with a metallic honeycomb core. Radford et al. [5] conducted blast experiments on sandwich composites with a metal foam core. They all found that the ability of sandwich panels to resist dynamic loading is far superior to that of monolithic metal plates with the same areal density. Tekalur et al [6] have studied the blast performance of sandwich structures with reinforced polymer foam cores. They concluded that the imparted damage was substantially reduced when Z-direction pin reinforcements (through thickness direction) were introduced into the core material. Wang et al. [7] constructed a sandwich structure with stepwise graded core materials and subjected it to transverse blast loading. They found that monotonically increasing the wave impedance of the core layers enhanced the blast resistance of the sandwich structures. Many of the above results have been summarized in Ref [8]. Some investigations on the dynamic response of pre-stressed composite structures under low-velocity transverse impact loadings can be found in literature [9-13]. Robb et al. [9] carried out the first experimental investigation on the low-velocity impact response of E-Glass reinforced/polyester laminated plates under different in-plane uniaxial and biaxial pre-stress. They found that the effects of the pre-stress were significant only at the highest level pre-

  T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_53, © The Society for Experimental Mechanics, Inc. 2011

383

384 stressed case (equivalent to 6000 µε) and the shear loading produced the greatest impact damage area on the specimen. Whittingham et al.[10] performed similar experiments on carbon-fibre/epoxy laminated plates under tensile and shear pre-stress. They also found that low pre-stressed levels (< 1500 µε) had no significant effects on the peak force of impact, absorbed energies and penetration behavior. Heimbs et al.[11] tested carbonfibre/epoxy laminated plates under an in-plane compressive pre-load. An increase in deflection and energy absorption was observed under a pre-load of 80% of the buckling load. Sun et al. [12] and Choi [13] analytically investigated the effects of pre-stress on the dynamic response of composite laminates. They found that the initial in-plane tensile load increased the peak contact force while reducing the total contact duration and deflection. The compressive load reacted oppositely. However, contact loading, such as impact loading, will induce localized damage, which is different from air or underwater blast loading. Thus, these results cannot be extended to the blast response of composite structures. There are very few theoretical and numerical studies [14-15] related to the blast response of structures with inplane compressive loading. In the author’s previous work, the dynamic behaviors and failure mechanisms of sandwich composites with uni-axial in-plane compressive loading have been studied under intensive transverse shock wave loading [19]. However, the bi-axial in-plane compressive loading are more common than uni-axial loading in reality. The present study focuses on the dynamic behavior and failure mechanisms of sandwich composites with bi-axial in-plane compressive loading while experimentally subjected to a transverse blast loading (as shown in Figure 1). A specially designed fixture was utilized to implement the uniformly passive biaxial in-plane compressive loading on the sandwich composites. Three compressive loading levels were chosen. A high-speed 3-D Digital Image Correlation (DIC) technique was used to capture the real-time full-field deformation data during blast loading. Post mortem visual observation of the test samples was also carried out to indentify the failure mechanisms.

Figure 1 A sketch of a blast loading upon a structure with in-plane compressive loading

Figure 2 Real specimen and its dimensions

Table1. Material properties of the components in sandwich composites [16, 19] Materials E-glass Vinyl Ester composite

CoreCell

TM

P600

Properties Nominal Density: 1800 kg/m3 Compressive Modulus: 13.6 GPa Compression Strength: 220 MPa 3 Nominal Density: 122 kg/m Compressive Modulus: 125 MPa Compression Strength: 1.81 MPa Shear Modulus: 56 MPa

2. MATERIAL AND SPECIMEN The sandwich specimen used in this study has two composite face sheets and a foam core. The VARTM procedure was carried out to fabricate sandwich composite panels. Figure 2 shows a sample image and its overall dimensions. The foam core itself was 25.4 mm thick, while the skin thickness was 3.8 mm. The average areal density of the samples was 17.15 kg/m2.

 

385 The skin materials are E-Glass Vinyl Ester (EVE) composites. The woven roving E-glass fibers of the skin material were placed in a quasi-isotropic layout [0/45/90/-45]s. The fibers were made of the 18 oz/yd2 area density plain weave. The resin system used was Ashland Derakane Momentum 8084 and the front skin and the back skin TM consisted of identical layup and materials. The core material was Corecell P600 styrene foams, which is manufactured by Gurit SP Technologies. Table 6-1 gives the material properties of the face-sheet that were determined using ASTM standard tests and the manufacturer’s property data for P600 foam [16]. 3. EXPERIMENT SETUP AND PROCEDURE 3.1 SHOCK TUBE The shock tube apparatus was utilized in present study to obtain the controlled blast loading. The detail of this apparatus can be found in Ref.[4]. Fig. 3 shows the shock tube apparatus with muzzle detail. The final muzzle diameter is 76.2 mm. Two pressure transducers (PCB102A) are mounted at the end of the muzzle section with a distance 160 mm. The support fixtures ensure simply supported boundary conditions with a 152.4 mm span.

Figure 3 shock tube apparatus 3.2 BI-AXIAL IN-PLANE COMPRESSIVE LOADING FIXTURE

Figure 4 Bi-axial in-plane compressive loading fixture The fixture used to apply the bi-axial in-plane compression loading on the sandwich composite panels is shown in Figure 4. It consisted of a fixed plate, holder plates, connectors, spacers and a loading head. The fixed plate was fixed on the outer frame. The other components were supported on the fixed plate using the connectors. The spacers were used to vary the angle between the axis of the connector and the horizontal line. When the loading

 

386 head pushes the specimen down, the bottom holder plate will move down. The connectors are used so that the side holder plate will shrink towards the center horizontally and consequently apply the compressive loading on the side of the specimen. The ratio between the vertical and horizontal loading vary based on the axis angle of the connector to the horizontal line. In the present study, an angle of 45 degree was chosen to ensure that the specimen was under uniform bi-axial loading. 3.3 HIGH-SPEED DIGITAL IMAGE CORRELATION (DIC) SYSTEM The high-speed photography setup is shown in Figure 5. A high-speed 3-D Digital Image Correlation (DIC) system consisted of two Phtron SA1 cameras was placed on the back side of the specimen to capture the deformation of the back face. All cameras have the ability to capture images at a framing rate of 20,000 fps with an image resolution of 512×512 pixels for a 1 second time duration. These cameras were synchronized to ensure that the images and data can be correlated and compared. The 3-D DIC technique is one of the most recent non-contact methods for analyzing full-field shape, motion and deformation. Two cameras capture two images from different angles at the same time. By correlating these two images, one can obtain the three dimensional shape of the surface. Correlating this deformed shape to a reference (zero-load) shape gives full-field in-plane and out-of-plane deformations. To ensure good image quality, a speckle pattern with good contrast was put on the specimen prior to experiments.

Figure 5 High-speed DIC system

Figure 6 Typical pressure profile from transducer 2

3.4 EXPERIMENTAL PROCEDURE The bi-axial in-plane compression loading was applied on the specimen and held at a constant level until the specimen was subjected to the transverse shock wave loading. Three in-plane compressive loading levels were chosen: 0 kN, 15 kN and 25 kN. For each compression loading, at least two samples were tested. The high-speed photography systems and the pressure sensors were synchronized and were triggered to record the pressure data and deformation images upon the arrival of shock wave. 4. EXPERIMENTAL RESULTS AND DISCUSSION 4.1 TYPICAL SHOCK WAVE PRESSURE PROFILE A typical pressure profile obtained from transducer 2 during blast loading from the shock tube is shown in Figure 6. The incoming shock wave has an incident peak pressure of approximately 1 MPa, and a wave velocity of approximately 1000 m/s (~Mach 3). This shock wave reflects from the specimen with an approximate peak pressure of 4.7 MPa and an approximate reflected velocity of 350 m/s. For a more detailed analysis of the pressure data, refer to Ref.[18].

 

387 4.2 DEFLECTION OF THE BACK SIDE The real time deflection contours of the back face of the sandwich composites are shown in Figure 7. The shock wave propagation direction is perpendicular to the paper. The support knife edges are located at the top and bottom of the images and parallel to the horizontal direction. The color stripes, which are parallel to the support line, show that the deflections of the points through the width of the panels are almost the same. This indicates that the specimens acted like a beam under the current support conditions.

Figure 7 Real time out-of-plane deflection contour of the back face of sandwich composites with different bi-axial compressive loadings 

Figure 8 Center point deflections of the back face of sandwich composites with different compressive loadings  The displacement contours show that the deflection increased with the increase of the bi-axial in-plane compressive loading. Figure 8 shows the center point deflection plots under different bi-axial compressive loading.

 

388 The center point deflection of the specimen with 25 kN bi-axial compressive loading is 70% higher than that of the specimen without the in-plane loading. The center point deflection of the specimen with 15 kN compressive loading is also 40% larger than that of the specimen without in-plane loading. It indicates that the bi-axial in-plane compressive loading highly reduces the blast resistance of the present sandwich composites. 4.3 IN-PLANE STRAIN ON THE BACK FACE

Figure 9 In-plane strain on the back face at the center point 

Figure 10 In-plane strain eyy contour of the back face of sandwich composites with various compressive loadings The in-plane strains of the center point on the back face of the sandwich specimens under different bi-axial compressive loading are shown in Figure 9. Here, the vertical direction is the y-axis. It can be seen that prior to 400 μs, the in-plane strains on the back face under different in-plane compressive loading were almost the same.

 

389 This means the deformation behaviors are identical. A similar phenomenon was also observed for uni-axial inplane compressive loading in Ref. [19]. After 400 μs, the in-plane strains of the back face with 15 kN and 25 kN bi-axial in-plane loading are similar, while they are all higher than that without in-plane loading. This indicates that the failure mechanisms of the sandwich composite with in-plane loading are different than without in-plane loading. This is also evident in the later post mortem analysis. Figure 10 shows the in-plane strain eyy contour on the back face for the sandwich composite with different compressive loadings. The contour of the in-plane strain on the back face of the sandwich composite with 25 kN bi-axial in-plane compressive loading has more concentration area and higher value than that with 15 kN loading. This clearly shows that higher in-plane loading reduces the strength of the sandwich composites. 4.2 POST MORTEM ANALYSIS The damage patterns of the sandwich composites after the shock loading were visually examined and recorded using a high resolution digital camera and are shown in Figure 11. Since the back face sheets do not show any change after the experiments, they are not shown here. From the front face-sheet images, the local damage increases with the increase of the bi-axial in-plane compressive loading. Note the yellow color is the original color of the specimen and the white color signifies fiber delamination and face-sheet buckling. For the specimen without in-plane loading, the front face only shows slight fiber delamination which is due to the large deflection bending [19]. Both the specimens with 15 kN and 25 kN of in-plane loading have one heavy failure and a fiber-delamination stripe on the front face. The front face of the specimen with 25 kN in-plane loading almost breaks into two pieces at the failure strip location and shows more fiber delamination area than that of the specimen with 15 kN in-plane loading.

Figure 11 Post mortem images of sandwich composite with various compressive loading  From the side-view images, the specimens without in-plane loading and with 15 kN of in-plane loading maintain structural integrity. The specimen with 25 kN of in-plane pre-loading catastrophically failed and broke into several pieces. The failure in the core of sandwich composites can be related to the in-plane strain on the back face, shown in Figure 10. For the specimen without in-plane loading, there are two main strain concentration regions at the upper and lower side of the center on the back face in the strain contour. They are signify the two cracks in the core. The core crack located on the right side in the side-view image is connected with the delamination between the core and the back face sheet. It is correlated to the lower strain concentration region on the back face, which shows higher strain than any other place in the strain contour. For the specimen with 15 kN of inplane loading, there is only one strain concentration region in the strain contour. It is correlated to the heavy core crack and delamination between the core and the back face. For the specimen with 25 kN of in-plane loading, there are a lot strain concentration regions present in the strain contour, which are correlated to the catastrophic failure in the core. One important failure mechanism can be observed in the back face sheet. Figure 12a shows an enlarged sideview image of the specimen with 15 kN of in-plane loading. Figure 12b and 12c show the enlarged side-view images of the specimen with 25 kN in-plane loading. It can clearly be seen that heavy delamination occurred between the glass fiber layers in the back face sheets for both the specimens loaded with 15 kN and 25 kN, respectively. This phenomenon is not present in the specimen without in-plane loading. This may be due to the presence of both the tensile wave and in-plane compressive loading within the specimen. Since the shock wave loading will induce a compressive wave propagating through the thickness of the specimen, the compressive wave will reflect from the back face of the specimen to be a tensile wave. Generally, this tensile wave itself cannot

 

390 trigger the delamination in the back face sheet due to its diminished energy after the mitigation of the foam core. However, the in-plane compressive loading will bring the instability between the fiber layers in the back face sheet. Figure 13 depicts the combining effect of the transverse tensile loading and the in-plane compressive loading. This mechanism highly reduced the delamination trigger level and consequently reduced the blast resistance of the sandwich composites.

Figure 13 Delamination under combining inplane compressive loading ands ile loading

Figure 12 Delamination in the back face of the sandwich composite with bi-axial in-plane compressive loading  5. CONCLUSIONS Sandwich composites, with E-glass Vinyl Ester composite face sheet and CoreCellTM P600 foam core, were placed under a bi-axial in-plane compressive loading prior to being subjected to a transverse shock wave loading. Three levels of compressive loading, 0 kN, 15 kN and 25kN, were chosen to study the effect of the bi-axial inplane loading on the dynamic behavior of the sandwich composites. A high-speed photography system and a 3-D Digital Image Correlation (DIC) technique were utilized to obtain full-field deformation data. The results showed that the bi-axial in-plane compressive loading induced not only the local buckling in the front face sheets but also the delamination between the fiber layers in the back face sheets. This mechanism changed the deformation mode of the sandwich composites. It is clear that higher levels of compressive loading caused more damage in the front face sheet, larger out-of-plane deflection, and higher in-plane strain on the back face sheet. Consequently, the overall blast resistance of the sandwich composites was significantly reduced. ACKNOWLEDGEMENT The authors kindly acknowledge the financial support provided by Dr. Yapa D. S. Rajapakse, under Office of Naval Research (ONR) Grant No. N00014-04-1-0268. The authors acknowledge the support provided by the Department of Homeland Security (DHS) under Cooperative Agreement No. 2008-ST-061-ED0002. Authors also

 

391 thank Dr. Stephen Nolet and TPI Composites for providing the facility for fabricating the materials used in this study. REFERENCES   [1]

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Dynamic response of porcine articular cartilage and meniscus under shock loading

Yun-Ching Juang1, Liren Tsai1, H. R. Lin2 1 National Kaohsiung University of Applied Sciences, Graduate Institute of Mechanical and Precision Engineering, Chien Kung Campus 415 Chien Kung Road, Kaohsiung 807, Taiwan, R.O.C. 2 Southern Taiwan University, Department of Chemical and Material Engineering, No.1, Nantai St., Yongkang Dist., Tainan City 710, Taiwan, R.O.C. Tel: (07)3814526 ext. 5323, Email: [email protected]

Abstract Articular cartilage and meniscus were extreme low friction, and they provide sufficient shock distribution properties under complex loading conditions. In this research, articular cartilage and meniscus samples from porcine knees were prepared and examined using Split Hopkinson Pressure Bar technique. The porcine samples were cylindrically shaped with the length-to-diameter ratio between 0.2~0.3, and PMMA bars were implemented to improve the intensity of transmitted waves through the soft material. The articular cartilage specimens were stored in a -30℃ fridge and each test was performed within 24 post-mortem hours. The experimental results showed that the stiffness of articular cartilages decreased with increasing strain rates; however, their Young’s modules did not exhibit significant difference in elastic deformation region. After each shock loading, the articular cartilages exhibited lump, fiber-like damage. The meniscus were also stored in -30℃ fridge and each test was performed within 72 post-mortem hours. The result of meniscus experiments indicated that the meniscuses were also strain rate sensitive, and their average compressive strength were 10MPa and 25MPa at 1000 s-1 and 1600 s-1 strain rates, respectively. Keyword: SHPB, dynamic response, porcine auricular cartilage Introduction Knee tissues degenerate with age when people over thirty-years old regardless of gender [1]. Some symptoms, such as anterior cruciate ligament deficient, meniscus tear and micro-crack of articular cartilage often resulted in the development of osteoarthritis [2, 3]. There were 12.1% of the population ages between 20~74 in US that had been clinically diagnosis as Osteoarthritis patients (According to National Health and Nutrition Examination Survey I) [4]. Via radiographic evidence of Osteoarthritis diagnosis, 27.8% of the Osteoarthritis occurred in knee for people over 45 years old. And the knee often broken down with age, at the age of 65 almost half of the population developed Osteoarthritis [4, 5]. The disease of Osteoarthritis has becoming a significant problem in daily life. The fundamental and quantitative analysis of articular cartilages has been developed and investigated by many scholar and researchers. Martin Stolz et al. perform the IT-AFM to obtain the articular cartilage image and to demonstrate the variation of stiffness under nano-scalar and micro-scalar (Martin Stolz et al., 2004) [6]. As the result, the digestion of proteoglycans moiety by cathepsin D had some degree effect on dynamic elastic modulus, |E*|, and stiffness was ~100-flod lower at nanometer scale than at the micrometer scale, the modulus value were 0.021 MPa and

2.6 MPa, respectively [6]. R.Y. Hori and L.F. Mockros reported that when articular cartilages were under dynamic loading a short-term shear modulus is 9~170 MPa and uniaxial confined compressive strength of human articular cartilage was 5.06 MPa [7]. P. Gudimetla, R. Crawford, and A. Oloyede found that, during normal activities, the articular cartilage underwent strain rates between 10-2~101s-1 [8]. Meanwhile, Meniscus also played an impotent role in knee structure. It increases contact area and absorb impacts, and also enhance the bearing capability of articular cartilages [2, 9]. Because of the inherent repair capability of meniscus tissues, they have received wide interests in the past three decades [2]. The aforementioned researches were mostly based on static loading conditions. However, abnormal transmission of load, micro-fracture on articular cartilage, and tear of meniscus often result in osteoarthritis [2, 3]. Aspden, R.M., J.E. Jeffrey, and L.V. Burgin define the time to peak load on the order of milliseconds for injurious impact loading plus one of the following: (1) stress rate >1000 MPa/s, (2) strain rate >500 s−1, or (3) loading rates over 100 kN/s [1, 10]. To better understand the affordability of the structures of meniscus and articular cartilage as an articulation of a knee, dynamic loading experiments were performed. In this study, dynamic response of porcine articular cartilage and

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_54, © The Society for Experimental Mechanics, Inc. 2011

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meniscus tissue were investigated with modified SHPB apparatus. The length-to-diameter ratios of specimens were between 0.1~0.3, and the specimens were store in fridge until tested. In addition, the effect of specimen preparation, acquisition, and procured processes were all considered in this research. Method and Material SHPB Dynamic response was obtained using a modified split Hopkinson pressure bar, consist of the striker, incident, transmission bar and pulse shaper the lengths were 600mm, 2000mm, 1500mm with same diameter 20mm, respectively. The semiconductor transducers strain gage used to measure the wave signal, and via amplifying, it was recorded by tektronix DPO 4104 digital phosphor oscilloscope. The SHPB is based on the one dimension wave theory, when elastic wave propagation through incident bar, partly of the elastic wave propagated to the specimen and remained to reflect to the incident bar. Nevertheless, when testing the soft material within the conventional SHPB, the dynamic equilibrium is not satisfied automatically [11-14]. For the purpose to closely approximate the dynamic stress equilibrium, conventional SHPB was using the pulse shaper to attain the constant strain rate before damaging [11-16]. In the other hand, the PMMA-bar was applied to improve the signal-noise-ratio of transmission signal. It is attributing to the wave progression in the viscoelastic bar. In consequence, the viscoelastic bar were adopted, because of the impedance differents between the metal bar and softly specimen. Knee structure The composition of knee are complex, consist with femur, tibia, fibula, cruciate ligament, articular cartilage, and meniscus, showed in figure 1. When the weight loading on the knee structure, the articular cartilage and meniscus often result some injury, and abundant evidence indicates that osteochondral injury leads to osteoarthritis [4]. Articular cartilage Articular cartilage is a hyaline cartilage structures, which provide an ultralow-friction and bearing-ability. The composed solid, porous-permeable matrix and interstitial fluid are made up by approximately 30% of solid matrix and 70% of interstitial fluid [4, 6, 18]. The Solid matrix consists primarily of type II collagen and proteoglycans. Other components include link protein; versican, biglycan and perlican etc [4, 6]. When pressure transmitted through articular cartilage, the fluid flow would result a large frictional drag on the solid matrix [18]. Two mechanisms of the articular cartilage deformation were provided during the loading condition: (1) flow-independent, friction drug, and (2) flow-dependent, interstitial fluid was flowed through the porous-permeable structures [6, 17, 18, 19]. Frictional drag of interstitial fluid permeating would balance the compressive stress, and exudates via the pore of solid matrix. As deformation

increased, the pressure would act on the solid matrix. Therefore, biphasic theory played the dominant role in the characterization of creep and stress-relaxation for articular cartilage [18]. Meniscus Meniscus is a fibrous cartilage structures, crescent-shaped, located between femoral condyles and the tibial plateau, and is in contact with an ultra-low friction of articular cartilage surface. Its primary composition includes water, collagen (90% of type I collagen, remainder consisting of types II, III and IV), glycosaminoglycans and DNA were 72%, 22%, 0.8% and 0.12%, respectively [1, 2, 20]. About one-third of peripheral meniscal border is vascular, which obtains nourishment directly by blood supply. Residual is an avascular zone, which obtains nourishment through the synovial fluid diffusion [2]. The collagen fibers were arranged along with the peripheral border [1]. When meniscus were subjected to the tensile, compressive, and shear stress during human activities, 50% loads were transmitted when knee were in extension, and 85% to 90% when knee were in flexion [1]. No matter flexion or extension, meniscus would match up with the femoral condyles distribute the weight uniformly [2, 20]. Material preparation The porcine knee were subscribed from the slaughterhouse, pre-froze it in 4℃ environment for 2 hours, and froze in a -30℃ fridge until testing. Each articular cartilage was tested within 24 post-mortem hours, and meniscus tissue was tested within 72 post-mortem hours. Using a scalpel, the meniscus was removed from the knee obtaining the complete meniscus tissue. Figure 2 (a) shows the articular cartilage specimen before examination, and (c) is a meniscus tissue. The diameter was 5.8mm to 6.3mm, the thickness was 0.7mm to 2.1mm, and it was controlled the length-to-diameter ratio of specimen in range of 0.2~0.3. In order to decrease the heat generating during cutting the porcine meniscus, the circular saw was put in a fridge before machining and a surgical scalpel was utilized to cut the specimens into slices.

Fig.1 The schematic diagram of knee structure

395

(a)

(b)

(a)

(c) (d) Fig.2 The specimen of (a) and (b) were articular cartilage before and after shock loading; (c) and (d), meniscus before and after shock loading Result and Discussion In this study, each specimen was stored in -30 ℃ and tested within 24 post-mortem hours to minimize the effect of tissue degeneration,. To improve the distribution of shock waves through the specimen, the length-to-diameter ratio were controlled in between 0.2~0.3. The dynamic compressive stress–strain curves, which presented articular cartilage and meniscus, were shown in figure 5(a) and figure 5(b), respectively. Young’s modulus and yield stress for articular cartilage were presented in table 2. According to the result, the stiffness of articular cartilage showed unique tendency and was weaker when the strain rate increased. The yield stresses of articular cartilage were 16.06MPa at strain rate 1160 s-1, and 13.76MPa at strain rate 1930 s-1. But they were highly uniform below ~0.05 strain at rising stage. Young’s modulus was also decreasing with increasing strain rate, which indicated that the articular cartilage had a weaken capability for bearing the shock loading in higher strain rate. The three primarily bases of the articular cartilage strength were from solid matrix structure, tissue fluid pressure, and friction drag when fluid flow through solid matrices [18]. The viscoelastic nature of articular cartilage is dependent with friction drag of fluid flow through the porous-permeable structure [18]. In this research, the interstitial fluid does not has enough time to flow through the porous-permeable structure which resulted in a weak dynamic viscoelastic property. Through the scanning electrons microscopy images showed in figure 3, the structure of normal articular cartilage surface is random undulate-like structure with cellur debris distributed on the surface. After shock loading, the surface were extruded and arranged with unanimity direction. Beside, from the macroscopic result in figure 2 (b), the specimen of articular cartilage would exhibit a lump, crack, and fiber-like damage under high strain rate loading.

(b)

(c) Fig.3 (a)Normal articular cartilage surfaced, (b) and (c) were articular cartilage surfaced features under shock loading, 1,000x Whereas, examination of the stress–strain curves from the results of meniscus revealed that the tissue stiffness were strain rate sensitive. Young’s modulus and yield stress for meniscus tissue were presented in table 1, which showed that the young’s modulus and yield stress both increased with increasing strain rate. The yield stress of articular cartilage is 5.55MPa at strain rate 690 s-1, and 23.83MPa at strain rate 1560 s-1. For meniscus, the Young’s modulus is 35.69MPa at strain rate 690 s-1, and 148.14MPa at strain rate 1560 s-1. According to the result, the meniscus tissue showed a stronger bearing capability than articular cartilage, and had

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a different dynamic response under compressive shock loading condition. The stress-strain curve of articular cartilage rise slowly at initial state, than rise intensively. That may be caused by the interstitial fluid evaporate form the articular cartilage specimen, and collapse at initial state, followed by intensification of the capillary pressure of the interstitial fluid. The young’s modulus and yield stress are decreasing with strain rate. In contrast, the stress-strain curve of meniscus is rising linearly at initial state than gradually stabilized. The young’s modulus and yield stress are increasing with strain rate. As shown in figure 2 (d), the specimen of meniscus was extended in lateral direction. However, the scanning electrons microscope of meniscus, figure 4, showed that its network structure also exhibited a lateral tension behavior under the axial compressive.

Table 2 Dynamic response of normal articular cartilage Strain Rate

Young's Modulus(MPa)

Yield stress(MPa)

520

10.6

6.3

1160

8.57

16.06

1360

8.29

13.50

1440

7.11

14.76

1510

6.77

12.62

1730

6.37

15.07

1770

5.65

12.26

1930

5.41

13.76

(a) (a)

(b) Fig.4(a) Normal meniscus surfaced features, 3,000x, and (b) Meniscus surfaced features under shock loading, 1,000x Table 1 Dynamic response of normal meniscus Strain Rate

Young's Modulus(MPa)

Yield stress(MPa)

690

35.63

5.55

960

36.55

10.67

1100

44.85

11.59

1380

73.21

16.16

1560

148.14

23.83

(b) Fig.5 (a) The stress-strain curve of articular cartilage: (b) The stress-strain curve of axial meniscus tissue Conclusion The porcine articular cartilage and meniscus specimen were preserved in a -30 ℃ fridge until testing, and performed the dynamic experiment after 24 and 72 post-mortem hours, respectively. Strength and young’s modulus of articular cartilage were 16.06MPa and 8.57MPa

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at strain rate 1160s-1, 13.76MPa and 5.41 at strain rate 1930 s-1. In addition, initial part of stress-strain curve of articular cartilage showed uniform feature under ~0.05 strain. Some symptoms, such as a lump, fiber-like damage were found when testing articular cartilage specimen. Contrariwise, the strength and young’s modulus of axial meniscus tissue were 5.55MPa and 35.63MPa at strain rate 690 s-1, 23.83MPa and 148.14MPa at strain rate 1560 s-1, which increased with strain rate. The scanning electrons microscope image were obtained, the articular cartilage exhibited a uniform permutation, and the meniscus revealed a lateral tension under the axial compressive loading. According to the result, they demonstrated that the meniscus tissues have a strong bearing capability than articular cartilage, and showed the different dynamic characteristics between articular cartilage and meniscus. Acknowledgments This research was supported by the National Science Council under grant no. 99-2221-E-151-014, and National Kaohsiung University of Applied Sciences in Taiwan. Reference [1] S. Kawamur, K. Lotito, S.A. Rodeo, 2003. “Biomechanics and healing response of the meniscus.” Operative Techniques in Sports Medicine, Volume 11, Number 2: pp.68–76. [2] T. Brindle, J. Nyland, D.L. Johnson, 2001. “The Meniscus: Review of Basic Principles With Application to Surgery and Rehabilitation.” Journal of Athletic Training, Volume 36, Number 2: pp.160–169. [3] Y. Fukuda, S. Takai, N. Yoshino, K. Murase, S. Tsutsumi, K. Ikeuchi, Y. Hirasawa, 2000. “Impact load transmission of the knee joint-influence of leg alignment and the role of meniscus and articular cartilage.” Clinical Biomechanics, yolume 15: pp.516-521 [4] K.A. Athanasiou, E.M. Darling, J.C. Hu, 2010. “Articular cartilage Tissue Engineering” A Publication in the Morgan & Claypool Publishers series: Synthesis Lectures on Tissue Engineering. [5] R.C. Lawrence, D.T. Felson, C.G. Helmick, L.M. Arnold, H.Choi, R.A. Deyo, S. Gabriel, R. Hirsch, M.C. Hochberg, G.G. Hunder, J.M. Jordan, J.N. Katz, H.M. Kremers, F. Wolfe, 2008. “Estimates of the prevalence of arthritis and other rheumatic conditions in the united states. Part II.” Arthritis Rheum, Volume 58, Number 1: pp. 26–35. [6] M. Stolz, R. Raiteri, A. U. Daniels, Mark R. VanLandingham, W. Baschong, U. Aebi, 2004. “Dynamic elastic modulus of porcine articular cartilage determined at two different Levels of Tissue Organization by indentation-type atomic force microscopy.” Biophysical Journal, Volume 86, Number 5 pp.3269–3283

[7] R.Y. Hori, L.F. Mockros, 1976. “Indentation test of Human Articular Cartilage.” Journal of Biomechanics, volume 9: pp.259–268 [8] R.P. Gudimetla, A.O. Crawford, 2007. "The influence of lipid-extraction method on the stiffness of articular cartilage." Clinical Biomechanics, volume 22: pp.924–931 [9] R.M. Aspden, J.E. Jeffrey, L.V. Burgin, 2002. “Impact loading of articular cartilage.” Osteoarthritis Cartilage, 10(7): pp. 588-9; author reply 590. [10] R. C. Lawrence, D.T. Felson, C.G. Helmick, L.M. Arnold, H. Choi, R.A. Deyo, S. Gabriel, R. Hirsch, M.C. Hochberg, G.G. Hunder, J.M. Jordan, J.N. Katz, H.M. Kremers, F. Wolfe, 2008. "Estimates of the Prevalence of Arthritis and Other Rheumatic Conditions in the United States.-Part II" Arthritis & Rheumatism volume 58, Number 1 :pp. 26–35 [11] B. Song, W. Chen, 2005. “Split hpokinson pressure bar technique for characterizing soft materials.” Latin American of Solid and Structures Volume 2: pp.113–152 [12] B. Song, W. Chen, 2004. “Dynamic stress equilibriation in split hopkinson pressure bar tests on soft materials ” Experimental Mechanics, volume 44, Number 3: pp.300–312 [13] B.A. Gama, S.L. Lopatnikov, J.W.G. Jr, 2004. “Hopkinson bar experimental technique: A critical review.” American Society of Mechanical Engineers, volume 57, Number 4: pp.223–250 [14] W. Chen, B. Song, D.J. Frew, M,J, Forrestal, 2003. “Dynamic Small Strain Measurements of a Metal Specimen with a Split Hopkinson Pressure Bar.” Experimental Mechanics, volume 43, Number 1: pp.20–23 [15] W. Chen, F. Lu, 2000. “A Technique for Dynamic Proportional Multiaxial Compression on Soft Materials.” Experimental Mechanics, volume 40, Number 2: pp. 226–230 [16] W. Chen, B. Zhang and M. J. Forrestal, 1999. “A Split Hopkinson Bar Technique for Low-impedance Materials.” Experimental Mechanics, volume 39, Number 2: pp.81–85. [17] A. Getgood, T.P.S. Bhullar, N. Rushton, 2009. “Current concepts in articular cartilage repair” Orthopaedics and Trauma Volume 23, Issue 3: pp. 189-200. [18] N.P. Cohen, R.J. Foster, V.C. Mow, 1998. “Composition and Dynamics of Articular Cartilage: Structure, Function, and Maintaining Healthy State.” Journal of Orthopaedic &Sports Physical Therapy, Volume 23, Number 4: pp. 203–215. [19] L.P. Li, W. Herzog, 2004. “Strain-rate dependence of cartilage stiffness in unconfined compression: the role of fibril reinforcement versus tissue volume change in fluid pressurization.” Journal of Biomechanics 37:

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pp.375–382 [20] D.P. Richards, F.A. Barber, M.A. Herbert, 2005. "Compressive Loads in Longitudinal Lateral Meniscus

Tears: A Biomechanical Study in Porcine Knees" Arthroscopy: The Journal of Arthroscopic & Related Surger, Volume 21, Issue 12 :Pp.1452-1456.

DYNAMIC RESPONSE OF BEAMS UNDER TRANSVERSE IMPACT LOADINGS Sub-title: Correlation between peak- tensile stress, Poisson’s ratio and ‘initial stress-pulse’

Dulal Goldar Distinguished Professor, Sharda University, Greater Noida, Former Principal, Delhi College of Engineering, B- 401, Jagaran Apt.,Plot-17, Sector-22, Dwarka, New Delhi- 110075, INDIA E-mail: [email protected] Abstract: Experimental results of peak- tensile stress for simply supported beams of mild- steel, aluminum, PMMA, araldite and urethane rubber subjected to low velocity impact for the same beam- striker weight ratio have been compared with interesting correspondence with Poisson’s ratios for these materials. Experimental results indicate a similitude between materials of Poisson’s ration range 0.25 to 0.40 for peak- tensile stress/ strain through ‘Characteristic Impedance’ and ‘Initial collisionvelocity’. Introduction: For a full- field stress analysis of transversely impacted beams a low- modulus rubber- like photoelastic model material is usually employed. Conversion of prototype stresses from such photoelastic model stresses so obtained always remain a vexed problem. Although in the past attempts have been made to establish suitable scaling laws for such conversions, a comprehensive scaling laws has not been established so far. Presently, an attempt is made to establish a correlationship between the peak- tensile stress (σ), Poisson’s ratio (ν) and the intensity of ‘initial stress pulse’ (ρC0v0). The peak- tensile stress (σ) generated in simply supported beams made of different materials when most of other impact parameters, namely, (i) (ii) (iii) (iv)

Beam dimensions, Location of transverse impact, Beam- striker weight ratio and Collision Velocity (vo) has been kept essentially the same.

Only homogeneous materials have been considered and the correlationship has been attempted for the materials whose Poisson’s ratio varies 0.25 to 0.40. The central theme of the present approach: A full- field analysis on a rubber- like material (PSM-4) is first conducted by the techniques of dynamic photoelasticity to get an insight into stress distribution for the problem at hand and thereby identify the points of special interest from the designers point of view; A point- wise study employing electrical resistance strain gauges is then conducted at the point of special interest on model beams made of materials like ploymethylmethacrylate (PMMA). Then it should be possible to predict the design stresses/ strains at these special points for the prototype materials for which the Poisson’s ratio varies 0.25 to 0.40.

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Although the range of Poisson’s ratio will exclude materials having very low Poisson’s ratio like Portland- cement concrete, it will include most of the commonly used metals, alloys and also thermo-plastic and thermo-hardening resins. Comparing the peak- tensile stress (σ) with the intensity of ‘initial stress pulse’ (ρC0v0) the following interesting results [(σ)/(ρC0v0) ] were obtained for the beams made up of mild steel, aluminum, PMMA, araldite and urethane rubber were 30%, 33%, 34%, 42% and 63%. It is interesting to note that the ratio of σ/ ρC0v0 indicated almost one- to –one correspondence with Poisson’s ratio (ν) for the above five materials, where, ρ is ‘mass- density’; C0 is ‘rod- wave velocity’ and v0 is ‘collision velocity’. Design of Experiments: In the current series of experiments, simply supported beams made up of the following five materials were considered, viz. (i) mild steel, (ii) aluminum, (iii) PMMA,(iv) araldite and (v) urethane rubber. All the beams had the same dimensions, namely 250 mm x 25 mm x 12.5 mm. The support condition for the beam was also identical, and a span of 120 mm was employed and it had an equal over- hang of 65 mm on either side of supports (Fig.1, 2). A cylindrical striker with hemispherical tip was allowed to fall freely and striker the mid- span point for each of five beams. The height of fall of the striker was kept at 176.4 mm in each case, while the diameter of the hemispherical tip ranged between 14 mm to 16 mm. The striker tips for impact on (i) mild steel, (ii) aluminum, (iii) PMMA, (iv) araldite beams were mild steel but for urethane rubber beam tip was made of araldite (Ciba: CY-230). It was intended to keep the ratio of the beam weight (for support to support beam length) and striker weight essentially the same, however, in practice the same ranged between 2.45 to 2.675. Accordingly, the weight of the strikers ranged between 14 gm to 115 gm. Peak- tensile strains at the lower- fiber central- span point of mild steel, aluminum, PMMA, araldite beams were measured with help of electrical resistance strain gauges (Fig.2) while for urethane rubber beam the peak- tensile stress at the lower fiber of the beam was determined from dynamic photoelastic studies (Fig.1). Since the collision velocity of the striker was low (> 2 m/s), the strain rates generated in the beams were low, and therefore the quasi- static Young’s moduli of mild steel, aluminum, PMMA, araldite were employed for converting peak- tensile strains to peak- tensile stresses. In the case of Poisson’s ratios of these materials the quasi- static values and dynamic values should not, however, be different. Accordingly, Young’s moduli and Poisson’s ratios for these beam materials were determined by conducting quasi- static uniaxial tension tests on these beam specimen themselves. In case of urethane beam, however, Poisson’s ratio specified by the manufacturer was accepted. In dynamic photoelastic study the dynamic material stress- fringe value by 10% (3). In Table- 1.the various impact parameters for each of five beams are listed:

Beam Materials

Mild steel Aluminum PMMA Araldite Urethane rubber

TABLE-1. SOME OF IMPACT PARAMETERS Striker Diameter Mass of the Beam weight/ (mm), Material of Striker (gm) Striker weight the hemispherical tip 16, mild steel 115 2.492 14, mild steel 41.5 2.468 15, mild steel 19 2.450 15, mild steel 19 2.500 14, mild steel 14 2.675

401

Experimental Techniques & Results: (a) Experimental Set Up: A system for generating impact (Fig.1a) was fabricated which essentially consisted of an electromagnet, an aluminum guide pipe and striker. The diameter of the guide pipe was slightly larger so that the striker fell freely. Experiments conducted on mild steel, aluminum, PMMA and araldite beams were essentially due to free fall of strikers and contact velocity in each case was calculated to be 1.86 m/s. However, for urethane rubber beam, experiment was conducted on different set up and actual free fall of striker could not be attended. The contact velocity calculated from experiment was found to be 1.6 m/s. These are shown in Table- 2(b). It appears that the striker somehow interfered with guide pipe and consequently its velocity of free fall was to some extent reduced in this case. A variable DC power supply was connected to the electromagnet which was supported separately over the guide pipe. This supporting system was clamped on fixture such that the centre- line of the guide pipe coincided with the centre- line of the span. After adjusting horizontality and verticality of different components of the supporting each beam specimen was placed over the wedge supports according to the marking for central- impact loading. A clear gap of 10 mm was provided between the guide pipe and beam. Each striker was kept at a height of 176.4 mm by energizing the electromagnet. A low power supply current was maintained to minimize the time delay for de-energizing the electromagnet.

Fig.1a. Legend: 1.Electromagnet (60V DC, 40mA, 1a. Guide Pipe., 2. Vertical mild steel rod for supporting electromagnet,3. Aluminum Plate,4.Fastax, 16mm framing camera,5.Monochromator,6.Optical bench,7.Light source(Sun-gun-II,800W),8.Fresnel lens & diffuser plate,9.Plane Polaroid,10.Quarter-wave plate,11.Signal from Timemarker,12.Goose-control unit

402 Dynamic Photoelasticity to Study Transverse Impact on Beams

Fig.1b. Isochrom atic Fringe Pattern Central Impact Span:120mm, Over-hang Ratio: 0.263 & I.F.T=84.11µs

Fig.1c.

120 mm Beam Span, Frame No.30, Time = 2523µs. Enlarged Photograph

(b) Measurement of Dynamic Strain: For the measurement of dynamic strain type KWR-5 (5 mm gauge length) strain gauges both active and dummy, were employed (Fig.2b) along with a storage oscilloscope. A low sweep rate was used to ensure the recording of the entire response for each beam. Several trials were made in each situation before accepting a reading of peak- tensile strain. Usually the variation was 2 to 3%. In one

403

situation, it was about 7%. The peak- tensile strain so recorded were 111µ for mild steel, 128µ for aluminum, 353µ for PMMA and 447µ for araldite beams. These values are also shown in Table- 2(b). (c) Fringe Photography: A portable diffused light polariscope (Fig.1a) was designed. An interference filter (of band pass of less than 100 Ao) was used before a high speed before a high speed camera to render the light monochromatic. A light- field arrangement of optical elements (Fig.1a. and 1b.) was employed which helped recording clearly supports and movements of falling striker on film negative. Wollensak Fastax (16 mm) framing camera operating at 12,000 fps (frames per second) was used for fringe photography. From recorded film negative the contact- velocity of striker (14 gm mass) was determined [1]. The contact velocity so obtained was 1.6 m/s and is shown in Table2(b). Enlarged prints of both continuous and discrete frames were prepared (Fig.1b, 1c) from film negative to study the history of fringe formation and identifying precisely fringe orders respectively. Based on enlarged prints of fringe photographs stress distributions along upper and lower boundaries of the beam were plotted.

+ -

-2 -1

Time from onset of Collision Top Boundary Bottom Boundary + Tensile - Compression

0 +

1

Fringe Order

757µs

2 3 4

(d) boundary Stress Distribution for 120 mm Beam- span under Central Impact Loading Fig.2. Free-

404

Fig.2a. Experimental arrangement for Measurement of Contact Force, Strain and Contact velocity

A typical frame identifying maximum fringe order at lower boundary after 757µs from the onset of collision was selected and the corresponding free- boundary stress distribution is presented in Fig.2b. In this figure the numerical value of fringe order corresponding to maximum impact stress at lower fiber, i.e. 675 gm/cm2 has been indicated. The nature of free- boundary stresses has also been indicated. Since stress distribution is symmetrical about the centre- line of the beam, the plot for the left- half of the beam only is shown. The peak stress is indicated in Table- 2(b). (e) Results: Transverse impact on beam of five different materials studied keeping the following impact parameters constant viz. (i) span- depth ratio (4.8), (ii) beam- striker weight ratio (2.5 nominal) and (iii) contact- velocity (1.86 m/s). From strain gauge study peak- tensile strains observed were converted to stresses. The following Young’s modulus (E) for different beams was obtained: 222 kN/mm2 for mild steel, 83.072 kN/mm2 for aluminum14.049 kN/mm2 for PMMA and 17.791 kN/mm2 for araldite. Dynamic photoelastic study indicated the peak- tensile stress, 675 gm/cm2 for urethane beam. From the ‘mass- density’ and ‘Young’s modulus of different materials the ‘rod wave’ velocity 5002.19 m/s, highest, for mild steel beam and is 63.185 m/s , lowest, for the urethane rubber beam. The same for other materials are indicated in Table- 2(a). Realizing the importance of ‘characteristic impedance’ of a material which is defined by the product of ‘mass- density’ (ρ) and ‘rod- wave’ velocity (Co), the same for five different

405

materials were determined. The ‘characteristic impedance’ (ρCo) [4] is 3998 gm.s/cm3, highest, for mild steel and is 6.7 gm.s/cm3, lowest, for the urethane rubber. These values are indicated in Table- 2 (b). The intensity of ‘initial stress pulse’ (ρC0v0) as defined by the product of characteristic impedance’ and the ‘collision velocity’ (v0), were calculated for beams studied, and the following values were obtained: 743,674 gm/cm2 for mild steel, 249,928 gm/cm2 for aluminum, 41,757 gm/cm2 for PMMA, 42,185 gm/cm2 for araldite and 1,065 gm/cm2 for urethane rubber. Comparing the peak- tensile stress (σ) with the intensity of ‘initial stress pulse’ (ρC0v0) the following results wee obtained for the five beams studied: peak- tensile stresses for the beams made up of mild steel, aluminum, PMMA, araldite and urethane rubber were 30%, 33%, 34%, 42% and 63%, respectively, of the corresponding ‘initial stress pulse’. These were also mentioned in Table- 2(b). It is interesting to note that the ratios of σ/(ρC0v0) indicated almost one-to-one correspondence with the Poisson’s ratio (Table- 2b) for the mild steel, aluminum, PMMA beams whereas for araldite and urethane rubber beams there was difference. The different values of Poisson’s ratio (ν) and the ratios of σ/(ρC0v0) for five beam materials were plotted and is shown in Fig.3.

Peak-tensile Stress / Initial Stress Pulse Vs. Poisson’s Ratio

Peak Tensile Stress / Initial Stress Pulse Intensity σ/(ρC0v0)

0.7

UR-

0.6 0.5 ARMildsteel

0.4 MS-

0.3

PMMA AL-

Aluminium PMMA ARALDITE Urethane Rubber

0.2

Trendline

0.1 0 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Poissons’s Ratio (ν)

Fig.3. Peak- Tensile Stress/Initial Stress Pulse Vs. Poisson’s Ratio

406

TABLE-2. (a) DYNAMIC RESPONSE OF BEAMSOF DIFFERENT MATERIALS Beam Materials

Span/Depth

Density,γ (gm/cm3)

Mild steel Aluminum PMMA Araldite Urethane rubber

4.800 4.800 4.800 4.800 4.948

7.834 2.726 1.242 1.267 1.031

Massdensity,ρ, (gm.s2/cm4) x10-3 7.993 2.782 1.266 1.292 1.052

Young’s Modulus, E (gm/cm2) 222 0.649 0.398 0.398 0.042

Rod- wave velocity, C0 =√E/ρ, (m/s) 5002.190 4829.963 1773.065 17755.134 63.185

(b) POISSON’S RATIO, CHARACTERISTIC IMPEDANCE, CONTACT VELOCITY, INITIAL STRESS- PULSE INTENSITY AND σ/(ρC0v0) Beam Materials

Mild steel Aluminum PMMA Araldite Urethane rubber

Poisson’s ratio

‘characteristic impedance’

‘collision velocity’

(ν)

(ρCo) (gm.s/cm3) 3998.0 1343.7 224.5 226.8 6.7

(v0),cm/s

0.285 0.33 0.38 0.34 0.46

186 186 186 186 160

‘initial stress pulse’ (ρC0v0), gm/cm2 743,674 249,928 41,757 42,185 1,085

Peak-tensile Strain,ε (µ) 111 128 353 447 -----

Stress,σ (gm/cm2) 222,000 83,072 14,049 17,791 675

σ/(ρC0v0) Ratio 0.30 0.33 0.34 0.42 0.63

Discussions: The term σ/(ρC0v0) (Table-2b) calls for further scrutiny. The quantity σ/(ρC0) represents particle velocity of the lower- fiber point in the direction of tension (that is, tangential to the boundary). On the other hand, v0 represents the particle velocity of the struck point in the vertical direction (i.e. the direction of the fall of the striker). Therefore, σ/(ρC0v0) also represents the ratio of the peak- particle velocity at the measured point (parallel to the beam axis) and the initial particle velocity at the struck point (perpendicular to the beam axis) (Fig.4). It is to be noted, however, that the two particle velocities mentioned above do not refer to the same point of time, and also refer to two different points of the beam. Again, the two quantities refer to velocities, though in two mutually perpendicular directions, rather than strains. Poisson’s ratio of a material on the other hand is a ratio of lateral strain and longitudinal strain measured at the same point for uniaxial state of stress, and this ratio need not change for quasi- static or dynamic loading. The similarity between the Poisson’s ratios of the three materials mild steel, aluminum and PMMA (Fig.3) with the corresponding σ/(ρC0v0) values, however, suggest that for the impact parameter under consideration both σ/(ρC0v0) and ν follow a similar mechanism in respect of deformation in the molecular scale i.e. on the behavior of materials from the point of

407

view of material science. Out of the five materials considered two, namely mild steel and aluminum have well defined atomic lattices and their deformations are essentially controlled by Condon- Morse curve. Their elasticity is essentially controlled by potential energy around the potential energy trough. The third material chosen, viz. PMMA, is long- chain polymer with practically no cross- links and low stresses and quasi- static loading it is viscoelastic in nature. The fourth material chosen, namely araldite, is net- work polymer with heavy three- dimensional cross- linking. It is also a two- phase material characterized by very strong (Infusible) primarybond and weak (Fusible) secondary- bond. For low stress and quasi- static loading at room temperature, its behavior should be essentially isotropic and possibly linearly elastic. The fifth material under consideration, namely urethane rubber, is net- work polymer with coiled- bonds and occasional cross- linking. It is a non- linearly elastic material whose elasticity is controlled by changes in entropy.

2. A comparison of peak- tensile stress and the representative ultimate tensile stress indicated the following values: 5% for mild steel, about 11% for aluminum, about 3% for PMMA and about 4% for araldite. The peak tensile stresses in these four materials were thus quite low and for mild steel and aluminum a linearly elastic response is expected. For PMMA also the stress level is low. However, the loading is dynamic and therefore, the viscoelastic effect would not be predominant; or in other words for impact parameter under consideration the response of PMMA beam can also be expected to be nearly linearly elastic. Therefore, a near one- to- one correspondence between σ/(ρC0v0) and ν for these three materials mild steel, aluminum and PMMA is in order. For araldite beam also the stress level is low, however Fig.3 indicates significant departure from one- to- one correspondence with Poisson’s ratio. This departure is attributed to two- phase nature of araldite. For urethane rubber the departure from one- to- one correspondence with Poisson’s ratio (Fig.3) is rather large.

408

3. From the results shown in Table- 2(b) the value of the parameter σ/(ρC0v0) for mild steel, aluminum, PMMA and araldite beams have been calculated and the following values have been obtained: 1.05 for mild steel, 1.00 for aluminum, 0.89 for PMMA and 1.24 for araldite. The present study, therefore, indicates that from a structural designers’ point of view the peak- tensile strain/ stress in PMMA can be converted in respect of other material mentioned above by keeping the parameter σ/(ρC0v0) ( 0.25< ν <0.40) as constant. As mentioned earlier the value of this constant need not be near about 1.0 in all situations. However, it may be expected that the value of parameter σ/(ρC0v0) (0.25< <0.40) determined for a PMMA model for similar other studies can be employed for predicting the design stresses in prototype materials. Acknowledgement: The author is thankful to the authorities of TBRL (Terminal Ballistics Research Laboratory), Chandigarh for providing all necessary facilities for high speed photography. References: 1. Goldar Dulal, “Photoelastic Studies of Transversely Impacted Simply Supported Beams”, Ph.D. Thesis, August 1981, Panjab University, Chandigarh. 2. Dally, James W.,”An Introduction to Dynamic Photoelasticity” EXPERIMENTAL MECHANICS, 20(12), Dec. 1980, pp.409- 416. 3. Dally, J.W., Riley,W.F. and Durelli,A.J.,”A Photo elastic Approach to Transient Stress Problem Employing Low Modulus Materials”, J. Applied Mechanics, Trans. ASME, Dec.,1959, pp.613- 620. 4. Kolsky, H., “Stress Waves in Solids”, Dover Publication Inc., 1963, p.34, 44 and 184.

Constitutive Model Parameter Study for Armor Steel and Tungsten Alloys

Stephen J. Schraml U.S. Army Research Laboratory Aberdeen Proving Ground, MD 21005-5066 Abstract A computational parametric study was performed to assess the influence of the selection of JohnsonCook constitutive model parameters on the numerical simulation of tungsten rods penetrating rolled homogeneous armor (RHA) steel targets. The parameter space involved two sets of Johnson-Cook model parameters for RHA, two sets of Johnson-Cook parameters for the tungsten rod material, four tungsten rod length-to-diameter (L/D) ratios, and two continuum mechanics codes. Striking velocities considered in the study ranged from approximately 1000 – 2000 m/s. The study revealed that no single combination of Johnson-Cook model parameters provides superior overall prediction of penetration depth over the other possible combinations across the full range of rod L/D considered. The study provides valuable guidance on the selection of material model parameters future studies of kinetic energy penetration.

1

Introduction

Numerical simulation is used extensively in the development of explosive warheads, kinetic energy (KE) penetrators, and armors to defeat such threats [1]. Simulations, when combined with judicious use of experimentation, can provide valuable insight into complex weapon/target interactions that is not possible through experimentation alone. The physics of shock wave propagation through various types of media, high-rate behavior of materials, and material failure are all essential characteristics of such interactions that must be captured in the computational methods. This paper documents an extensive computational study to assess the influence of existing Johnson-Cook [2] constitutive model parameters on the simulation of tungsten rods penetrating rolled homogeneous armor (RHA) steel targets. One of the initial goals of the effort was to identify a “best possible” combination of tungsten and RHA model parameters to use for simulations across a wide range of conditions of interest to the development of KE penetrators and armor systems that are effective against KE penetrators. The study considered tungsten rods with length-to-diameter (L/D) ratios of 5, 10, and 15, and striking velocities of approximately 1000 – 2000 m/s. The study also included an L/D=30 rod at a striking velocity of 1500 m/s. The metric used to evaluate the computational results was total penetration depth normalized by penetrator length (P/L). The computational results were compared to experimental data in the case of rods of L/D=5, 10, and 15. For the L/D=30 rods, the simulation results were compared to an empirical fit to an experimental database. The study results revealed that no single combination of Johnson-Cook constitutive model parameters produces the best penetration depth results across the full range of parameters considered. However, the study T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_56, © The Society for Experimental Mechanics, Inc. 2011

409

410 does provide guidance on the use of constitutive model parameters when operating within the parameter space considered. This work also provides a foundation for additional study to include the evaluation of additional constitutive models, fracture models, etc.

2

Johnson-Cook Constitutive Model

The Johnson-Cook constitutive model describes the flow stress (σ) of a metal as a function of plastic strain (εp ), plastic strain rate (ε˙p ), and temperature (T ), as defined in equations 1 and 2. In this model, the terms A, B, C, n and m are model parameters (constants), Tmelt is the melt temperature of the material, Troom is the ambient temperature, and ε˙0 is a reference plastic strain rate. The model parameters are typically derived from material characterization experiments through a fitting process intended to reproduce the trends observed in the characterization experiments while minimizing error.

σ = A+

Bεnp




T0 =

2.1

  ε˙p (1 − T 0m ) 1 + Cln ε˙0

(1)

T − Troom Tmelt − Troom

(2)

RHA Model Parameters

RHA steel is a commonly used material in armor systems and ballistic testing. It is also a de facto standard in the assessment of ballistic performance. Penetration performance of KE penetrators, explosively formed penetrators, and shaped charge jets is typically described in terms of their depth of penetration into a thick stack of RHA target plates. Furthermore, the protection capability of an armor system is typically described in terms of “RHA equivalence” which typically represents the areal density of the armor system relative to the equivalent areal density of RHA that would be required to have the same level of protection. For these reasons, the ability to accurately model the performance of RHA in ballistic events is critical to many penetrator and armor development programs. RHA is available in a variety of plate thicknesses from 6.4 mm (1/4 inch) to 152 mm (6 inches). As a result of the rolling process, thinner plates are observed to have a higher surface hardness than thicker plates. Meyer & Kleponis [3] provide an empirical relationship between RHA plate thickness and quasistatic yield strength based on experiments by Benck [4]. A common practice in modeling RHA in numerical simulations is to set the A parameter of the Johnson-Cook model according to this relationship. This practice was employed throughout the study described here. Two different sets of Johnson-Cook constitutive model parameters for RHA were evaluated in this study [3, 5]. The Meyer & Kleponis parameters were derived from material characterization tests performed by Gray et al. [6] on 50.8-mm-thick RHA plates. The Weerasooriya & Moy parameters were developed from their own characterization tests on RHA plates ranging in thickness from 19.0 mm to 76.2 mm [7]. Like Benck, Weerasooriya & Moy observed that the quasi-static yield strength varied with plate thickness, further validating the approach of varying the A parameter in the Johnson-Cook model in accordance with the

411 RHA plate thickness. Figure 1 provides a plot of flow stress vs. plastic strain at room temperature for the two sets of parameters for a quasi-static yield strength of 746 MPa which corresponds to a plate thickness of 63.5 mm. In this figure, the thin lines represent quasi-static (ε˙p =0.002/s) deformation and the thick lines represent high-rate (ε˙p =3000/s) deformation. T = 298 K Strain Rates: 0.002/s, 3000/s (σ0 = 746 MPa)

1800 1600 1400

Flow Stress (MPa)

1200 1000 800 600 400 Meyer & Kleponis Weerasooriya & Moy

200 0

0

0.2

0.4

εp

0.6

0.8

1

Figure 1. Flow stress at ambient temperature for 63.5-mm-thick RHA.

The curves in figure 1 illustrate the differences between the two sets of RHA parameters. The Meyer & Kleponis parameters and the Weerasooriya & Moy parameters produce similar work hardening behavior. Even though the trends are similar, the flow stress produced by the Weerasooriya & Moy parameters is less than that of the Meyer & Kleponis parameters. The gap between the thin (quasi-static) and thick (high-rate) curves for each parameter set indicate the rate hardening effect. The Weerasooriya & Moy parameter set has a larger C coefficient than the Meyer & Kleponis set, resulting in a slightly larger gap between the thin and thick curves.

2.2

Tungsten Model Parameters

Two tungsten penetrator materials were considered in the study. The first consisted of 90% tungsten (W), 7% nickel (Ni), and 3% iron (Fe) by volume and is denoted by 90W-7Ni-3Fe. The second consisted of 93% tungsten, 5% nickel, and 2% iron (93W-5Ni-2Fe). The Johnson-Cook parameters for these materials were derived by Johnson & Cook [2] and Weerasooriya [8], respectively. A plot of flow stress as a function of plastic strain at room temperature for the two tungsten material parameter sets is provided in figure 2. As in figure 1, the thin lines represent quasi-static deformation and the thick lines represent high-rate deformation. The curves in figure 2 show that 90W-7Ni-3Fe parameter set produces negligible work hardening as compared to 93W-5Ni-2Fe. This is a result of significantly greater

412 B and n parameters for the 93W-5Ni-2Fe parameters as compared to 90W-7Ni-3Fe. The rate hardening effect produced by the 93W-5Ni-2Fe parameters is also greater as a result of the larger C parameter and is evidenced by the greater gap between the quasi-static and high rate curves as compared to the gap between the 90W-7Ni-3Fe curves. T = 298 K Strain Rates: 0.002/s, 1500/s 3000

Flow Stress (MPa)

2000

1000

90W-7Ni-3Fe 93W-5Ni-2Fe 0

0

0.2

0.4

εp

0.6

0.8

1

Figure 2. Flow stress at ambient temperature for 90W-7Ni-3Fe and 93W-5Ni-2Fe.

3

Penetration Experiments

The goal of the study was to examine the trends in simulated penetration performance for each combination of tungsten and RHA Johnson-Cook parameter set across a wide variety of striking velocities and rod L/D. To this end, experimental data were obtained for rod L/Ds of 5, 10, and 15. The L/D=5 results were obtained from experiments by Enderlein [9] in which tungsten rods were fired into a stack of four RHA target plates at 0 degrees obliquity. The rods were right circular cylinders and were 100 mm long with a diameter of 20 mm. The target plates were each 63.5 mm thick. The L/D=10 and 15 experimental data were obtained from experiments by Magness [10]. In these experiments, 93W-5Ni-2Fe tungsten rods with hemispherical noses were fired into 152-mm cubes of RHA at 0 degrees obliquity. In all the L/D=10 and 15 experiments, the rods were machined to have a constant mass of 65 g. Experimental data were not used for the study of rods with L/D=30. Instead, the computational results were evaluated through comparison to an empirical expression for P/L. Lanz & Odermatt [11] developed this empirical relationship for penetration performance for rod L/D ranging from 11 to 32 and striking velocities ranging from 1100 to 1900 m/s. The current study considered only one striking velocity for L/D=30 (1500 m/s). Using the relationship derived by Lanz & Odermatt, the expected P/L is 0.836 for the L/D=30 rod with a striking velocity of 1500 m/s.

413

4

Numerical Simulations

Two continuum mechanics codes were used in the computational study. CTH [12] is an Eulerian finite volume code for modeling solid dynamics problems involving shock wave propagation, multiple materials, and large deformations in one, two, and three dimensions. CTH employs a two-step solution scheme – a Lagrangian step followed by a remap step. The conservation equations are replaced by explicit finite volume equations that are solved in the Lagrangian step. The remap step uses operator-splitting techniques to replace multidimensional equations with a set of one-dimensional equations. High-resolution material interface trackers are available to minimize material dispersion. The arbitrary Lagrangian-Eulerian code ALE3D [13] is the other continuum mechanics code used in the study. ALE3D uses a hybrid finite element and finite volume formulation in an unstructured grid. ALE3D is a multi-physics code with features to support simulations involving wave propagation, material deformation and fracture, heat conduction, chemical kinetics, and magneto-hydrodynamics. Simulations involving modest amounts of deformation can be run Lagrangian, in which the unstructured mesh follows the material motion. For cases of larger deformation, advection can be employed to allow the mesh to relax in order to prevent tangling. It is possible to run an ALE3D simulation as an Eulerian simulation (i.e. in the same manner that CTH operates) by employing an advection step after each Lagrangian step and relaxing the mesh to its original position. All ALE3D simulations in this study were performed in this way. A common set of characteristics was employed in all simulations in the study. The simulations were performed in two dimensions with a rectangular computational domain of uniform mesh resolution throughout. The mesh resolution was defined such that there were 10 computational cells/elements across the radius of the penetrator. A shock-particle velocity equation of state (EOS) was used for all materials in the simulations and a common set of EOS parameters was defined for all simulations. Similarly, the JohnsonCook fracture model [14] was employed for for all materials in the simulations and a common set of fracture model parameters was employed in all cases. The study involved the use of Johnson-Cook constitutive model parameters for two different tungsten rod materials. The different volume percentages of tungsten in each resulted in different densities. The density used in the study was 17.346 g/cm3 for 90W-7Ni-3Fe and 17.7 g/cm3 for 93W-5Ni-2Fe. The simulations involving the L/D=5 rod used the dimensions and cylindrical geometry that were described previously. As a result, the 90W-7Ni-3Fe and 93W-5Ni-2Fe rods in the L/D=5 simulations had slightly different masses. For the L/D=10 and 15 rods, the hemispherical nose geometry was defined to reproduce the same 65-g rod mass that was used in the experiments. In this case, the 90W-7Ni-3Fe and 93W-5Ni-2Fe rods had identical masses but slightly different dimensions. Because no experimental data were used for evaluation of the L/D=30 simulations, a notional rod geometry was employed. In this case, the rod mass was defined as 130 g with a hemispherical nose, resulting in rods that were approximately equal in diameter and double in length to the L/D=15 rods. The resulting L/D=30 rods were nominally 6.8 mm in diameter and 204 mm long.

414

5

Results and Discussion

Simulations were performed to replicate the L/D=5, 10, and 15 experiments. For each experiment, a family of simulations was performed such that every combination of continuum code (CTH and ALE3D), RHA parameter set (Meyer & Kleponis and Weerasooriya & Moy), and tungsten rod material parameter set (90W-7Ni-3Fe and 93W-5Ni-2Fe) were employed, resulting in eight simulations for each experimental measurement. Comparison of the simulation results to the experimental data are provided in the following subsections.

5.1

Results for L/D=5

Comparison of the L/D=5 simulations to the experimental results are provided in figure 3. This figure contains four plots in two rows and two columns. Each is a plot of P/L as a function of striking velocity. The plots are arranged with the 90W-7Ni-3Fe results in the left column and the 93W-5Ni-2Fe results in the right column. The ALE3D results are in the top row and the CTH results are in the bottom row. 90W-7Ni-3Fe

93W-5Ni-2Fe

1.5 1.4

Meyer & Kleponis Weerasooriya & Moy Experiment

P/L (ALE3D)

1.3 1.2 1.1 1 0.9 1.5 1.4

P/L (CTH)

1.3 1.2 1.1 1 0.9 1300

1400

1500 Vs (m/s)

1600

1700

1300

1400

1500 Vs (m/s)

1600

1700

Figure 3. Study results for L/D=5.

Common characteristics and trends are observed across all four plots in figure 3. The P/L results from the Weerasooriya & Moy parameter set are consistently greater than that of the Meyer & Kleponis set. This is a result of the overall greater strength flow strength produced by the Meyer & Kleponis parameter set. No single combination of continuum code, tungsten parameter set, and RHA parameter set matches the trend in change of P/L with increasing velocity.

415 An interesting trend observed in the L/D=5 study results is the lower simulated P/L from the 93W-5Ni2Fe parameter set as compared to the results from the 90W-7Ni-3Fe set. One would expect 93W-5Ni-2Fe to yield a greater P/L because of its higher density (and therefore greater rod mass in the case of the constantgeometry L/D=5 rod). It appears that the difference in P/L between the two tungstens is a result of the difference in the Johnson-Cook model parameters. In figure 2, the 93W-5Ni-2Fe parameter set exhibited more strain hardening than 90W-7Ni-3Fe, but had a lower flow stress in the lower range of plastic strains (εp < 0.25). Because of the limited availability of experimental data for L/D=5, the computational results do not yield a single set of parameters that provide a best match to the experimental results.

5.2

Results for L/D=10

The study results for L/D=10 are provided in figure 4. In this figure, the four plots are presented in the same 2X2 format as the L/D=5 results. The L/D=10 results show the Meyer & Kleponis parameters provide the best overall comparison to the experimental data. As in the L/D=5 results, the 93W-5Ni-2Fe tungsten parameter set produced a lower P/L than 90W-7Ni-3Fe and the P/L from the Weerasooriya & Moy RHA parameter set was consistently greater than that of Meyer & Kleponis. 90W-7Ni-3Fe

93W-5Ni-2Fe

1.4

P/L (ALE3D)

1.2

Meyer & Kleponis Weerasooriya & Moy Experiment

1

0.8

0.6

0.4 1.4

P/L (CTH)

1.2

1

0.8

0.6

0.4 1000

1200

1400 Vs (m/s)

1600

1800

1000

1200

Figure 4. Study results for L/D=10.

1400 Vs (m/s)

1600

1800

416

5.3

Results for L/D=15

The results for the L/D=15 simulations are presented in figure 5. The qualitative observations of these results are consistent with those of the L/D=5 and 10 results. Overall, all combinations of continuum code, tungsten parameter set, and RHA parameter set result in simulation results that capture the overall trends of the experimental data. Between the two RHA parameter sets, Meyer & Kleponis appears to provide the best overall comparison to the experiments. 90W-7Ni-3Fe

93W-5Ni-2Fe

1.5

P/L (ALE3D)

1.2

Meyer & Kleponis Weerasooriya & Moy Experiment

0.9

0.6

0.3 1.5

P/L (CTH)

1.2

0.9

0.6

0.3 1000

1200

1400 1600 Vs (m/s)

1800

2000

1000

1200

1400 1600 Vs (m/s)

1800

2000

Figure 5. Study results for L/D=15.

5.4

Summary of Results for L/D=5, 10, & 15

The presentation of the study results in figures 3, 4, and 5 provides a valuable qualitative comparison of the simulation results to the experimental data. These comparisons clearly illustrate the influence of the model parameters on the simulation results. However, these plots alone do not provide the quantitative information required to rank the simulation results for the purpose of selecting the best possible combination of parameter sets for simulating tungsten penetration of RHA. To obtain this quantitative measure, the sum of squares of the errors (SSE) of the simulation results compared to the experimental results was performed. In this analysis, an SSE measure was obtained for every combination of continuum code, tungsten parameter set, and RHA parameter set. The results of this analysis are summarized in table 1. These results show that for three of the four combinations of continuum code and tungsten parameter set, Meyer & Kleponis produces the lowest SSE of the two RHA parameter sets. As a result of this quantitative analysis and the trends observed in figures 3, 4, and 5, the Meyer & Kleponis

417 parameter set is found to provide the best overall results for rods with L/D in the range of 5 to 15. Table 1. Sum of Squares of Errors for L/D=5, 10, & 15. ALE3D RHA Parameter Set

5.5

CTH

90W-7Ni-3Fe

93W-5Ni-2Fe

90W-7Ni-3Fe

93W-5Ni-2Fe

Meyer & Kleponis

0.0723

0.0445

0.0260

0.0589

Weerasooriya & Moy

0.4682

0.3496

0.2172

0.0941

Results for L/D=30

Simulations were performed for the L/D=30 rods for a striking velocity of 1500 m/s. The simulation results were evaluated by comparison to an empirical model result of P/L=0.836. For each simulation, the percentage error of P/L was calculated relative to the empirical model. The results in table 2 show that the Weerasooriya & Moy RHA parameter set produces more accurate results than Meyer & Kleponis. This contradicts the results of the L/D=5, 10, and 15 rods described earlier in which Meyer & Kleponis generally produced more accurate results than the other two. Table 2. P/L Errors for L/D=30 Simulations. P/L Error (%) RHA Parameter Set

6

CTH

ALE3D

90W-7Ni-3Fe

93W-5Ni-2Fe

90W-7Ni-3Fe

93W-5Ni-2Fe

Meyer & Kleponis

-13.7

-14.9

-6.1

-11.4

Weerasooriya & Moy

2.0

1.5

7.3

2.1

Summary

This paper documents a computational study to identify the influence of the selection of Johnson-Cook constitutive model parameters on the penetration performance of tungsten rods into RHA steel targets in numerical simulations. The study encompassed two sets of model parameters for RHA, two sets for the tungsten rod material, two different continuum mechanics codes, four different rod L/Ds, and striking velocities ranging from 1000 – 2000 m/s. The original intent of the study was to identify an optimum set of tungsten and RHA model parameters for use in numerical simulations over a wide range of initial conditions. The study revealed that no single combination of model parameters provided the best overall predictive capability over the entire range of rod L/D considered. The RHA parameter set by Meyer & Kleponis produced the best comparison to experimental penetration data for rods of L/D from 5 to 15. However, for rods of L/D=30, this parameter set yielded the least accurate computational results. Even though this study did not yield an overall optimum set of RHA and tungsten Johnson-Cook parameters, it has provided valuable guidance in selection of model parameters for a variety of rod L/Ds.

418

References 1.

K.D. Kimsey, S.J. Schraml, and E.S. Hertel. Scalable Computations in Penetration Mechanics. Advances in Engineering Software, 29(3–6):209–215, 1998.

2.

G.R. Johnson and W.H. Cook. A Constitutive Model and Data for Metals Subjected to Large Strains, High Strain Rates and High Temperatures. In Ballistics, The Hague, The Netherlands, 1983. Seventh International Symposium on Ballistics.

3.

H.W. Meyer and D.S. Kleponis. An Analysis of Parameters for the Johnson-Cook Strength Model for 2in-Thick Rolled Homogeneous Armor. Technical Report ARL-TR-2528, U.S. Army Research Laboratory, Aberdeen Proving Ground, MD, June 2001.

4.

R.F. Benck. Quasi-Static Tensile Stress Strain Curves – II, Rolled Homogeneous Armor. Technical Report BRL-MR-2703, U.S. Army Ballistic Research Laboratory, Aberdeen Proving Ground, MD, November 1976.

5.

T. Weerasooriya and P. Moy. RHA Constitutive Model – Function of Hardness. Unpublished, April 2010.

6.

G.T. Gray, S.R. Chen, W. Wright, and M.F. Lopez. Constitutive Equations for Annealed Metals Under Compression at High Strain Rates and High Temperatures. Technical Report LA-12669-MS, Los Alamos National Laboratory, NM, 1994.

7.

T. Weerasooriya and P. Moy. Effect of Strain Rate on the Deformation Behavior of Rolled Homogeneous Armor (RHA) Steel at Different Hardnesses. Proceedings of the 2004 SEM International Congress and Exposition on Experimental Mechanics. Costa Mesa, CA, June 2004.

8.

T. Weerasooriya. Deformation Behavior of 93W-5Ni-2Fe at Different Rates of Compression Loading and Temperatures. Technical Report ARL-TR-1719, U.S. Army Research Laboratory, Aberdeen Proving Ground, MD, July 1998.

9.

M.W. Enderlein. Armor Backpack Effectiveness Against Rod Penetrators at Ordnance Velocities. Technical Report BRL-MR-3939, U.S. Army Ballistic Research Laboratory, Aberdeen Proving Ground, MD, 1991.

10. L.S. Magness. A Phenomenological Investigation of the Behavior of High-Density Materials Under the High Pressure, High Strain Rate Loading Environment of Ballistic Impact. PhD thesis, Johns Hopkins University, 1992. 11. W. Lanz and W. Odermatt. Penetration Limits of Conventional Large Caliber Anti Tank Guns/Kinetic Energy Projectiles. In Ballistics, Stockholm, Sweden, 1992. 13th International Symposium on Ballistics. 12. J.M. McGlaun and S.L. Thompson. CTH: A Three-Dimensional Shock Wave Physics Code. International Journal of Impact Engineering, 10:351–360, 1990. 13. A.L. Nichols. Users Manual for ALE3D: An Arbitrary Lagrange/Eulerian 2D and 3D Code System. Technical Report LLNL-SM-433954, Lawrence Livermore National Laboratory, Livermore, CA, June 2010. 14. G.R. Johnson. Status of the EPIC Codes, Material Characterization and New Computing Concepts at Honeywell. In Computational Aspects of Penetration Mechanics, New York, 1982. Springer-Verlag.

A Scaled Model Describing the Rate-Dependent Compressive Failure of Brittle Materials

1

Jamie Kimberley1, Guangli Hu1, and K.T. Ramesh1 Department of Mechanical Engineering, Johns Hopkins University, 223 Latrobe Hall, 3400 N. Charles St., Baltimore, 21218

ABSTRACT— A universal relationship is developed that describes the rate-dependent compressive strength of brittle solids based on the micromechanics of the growth of brittle cracks from populations of initial flaws. Real-time observations of crack growth provide insight to the model which captures the dynamics of interacting and rapidly growing cracks. Fundamental time and length scales involved in the problem are used to develop expressions for a characteristic stress and a characteristic strain rate in terms of material and microstructural properties. Scaling simulation results by the characteristic stress and strain rate collapses the data to a single curve in failure stress–strain rate space. This curve represents the universal response, which captures both the relatively constant failure stress at low rates as well as the dramatic increase in strength observed in experiments as the applied strain rate increases above the transition rate. The resulting model for the universal response compares well with experimental data for ceramics and geologic materials, indicating that the model has adequately captured the physics of compressive failure for a wide range of materials. INTRODUCTION— The vast majority of compressive failures of brittle solids involve both significant amounts of crack nucleation (from pre-existing flaws or defects) and significant amounts of crack propagation and crack interactions. A new ansatz for high-rate compressive failure was recently developed by Paliwal and Ramesh [1] based on real-time ultra-highspeed visualization experiments [2] coupled with theoretical investigations of massive brittle failure. In brief, that work showed that the dynamic compressive failure process is controlled by the interactions of three terms: the initial defect distribution, crack growth dynamics and crack-crack interactions, and the coupling of these three terms with the superimposed rate of loading. That model describes the increase in the strength that is often observed in brittle materials subjected to uniaxial compression at high strain rates [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. This work develops a model that captures the behavior of brittle solids in an appropriately scaled form [14].

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RELEVANT SCALES—We begin by identifying the critical length scales and timescales associated with the physical problem of a brittle solid containing a pre-existing distribution of rectilinear flaws and subjected to dynamic loading. The pre-existing flaw distribution is described in terms of a flaw density η (the number of flaws per unit area), and a pdf g( s) of the flaw size s . This results in the introduction of two length scales into the system: the mean flaw size s and the average E flaw spacing η−1 2 . Since information in an elastic material propagates at a finite velocity c 0 = (here E is the Young’s ρ € € € modulus and ρ is the mass density), the length scales above also result in two timescales. The €first is associated with the s communication time between crack tips from a single flaw t tc = . The second is the communication time between flaws, € € c0 € η−1/ 2 t 0 =€ . Crack initiation is controlled by the fracture toughness, K IC , and the fracture process occurs in a zone of length c0 € K2 l p = IC2 , defining another length scale. While this process zone size could be combined with c 0 , describing the time it takes E € dynamics of subsequent crack growth provides a more relevant for information to traverse the process zone, we feel that the lp time scale into the problem, t p = , i.e., the time it takes for a crack propagating at speed c g to cross the process zone. The € cg time that it takes for a wing crack to grow to span the average distance between flaws t link =

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η−1/ 2 , is another critical time cg

€ for Experimental Mechanics Series 99, T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society DOI 10.1007/978-1-4614-0216-9_57, © The Society for Experimental Mechanics, Inc. 2011

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420 scale. The interaction between these timescales and the timescale associated with the applied load determines the strength response of the material, σ f as a function of the loading strain rate, ε˙ . SCALED BEHAVIOR OF BRITTLE SOLIDS IN COMPRESSION—A scaled response function can be developed for all of the available results by defining a characteristic stress and a characteristic strain rate in terms of the microstructural variables € and the length € and time scales in the problem. The physics of the problem and the non-dimensional combination [15] of the key variables allows us to determine a characteristic stress σ 0 as

K ICη1/ 4 . (1) s η−1/ 2 Here the numerator is essentially the far-field stress € associated with a crack of length equal to the flaw spacing, while the denominator is the ratio of the average flaw size to the flaw separation. This characteristic stress can be thought of as the stress required to activate a wing crack sufficiently long to bridge the flaws for any given flaw density and flaw size € distribution. The timescales in the problem provide a characteristic strain rate ε˙0 defined by c K ICη1/ 4 , (2) ε˙o = 2.4 s E where the first term represents the reciprocal of the communication time along the initial flaw, while the second term € represents the far-field strain associated with a crack of length equal to the flaw spacing. Normalizing the failure stresses predicted by the Paliwal-Ramesh model [1] by the characteristic stress, and normalizing the applied strain rate by the characteristic rate collapses the data around€a single curve as shown by Fig. 1. The equation of this curve has the form, 2/ 3 (3) σ f σ o = 1 + (ε˙ ε˙o ) . The 2/3 exponent provides an excellent fit, and represents the scaling that is associated with the competition between the surface mechanism (fracture) and a volumetric quantity (the stored strain energy).

σ o = 2.4

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Fig. 1 Predicted failure strength normalized by the characteristic stress as a function of normalized strain rate (circles), The scaled model (equation 3) is represented by the solid curve This scaled model also compares very favorably with experimental observations of the rate-dependent compressive strength of brittle solids ranging from ceramics to geologic materials as shown by Fig. 2. The solid line in Fig. 2 represents the curve described by equation (3), and the experimental data are represented by the symbols. It is apparent that the scaled function represented by equation (3) is capable of describing a broad range of brittle solids under compression. The ability of the model to describe the behavior suggests that the important micromechanics has been captured. The scaled results show that the compressive strength remains nearly constant below a transition strain rate, but a rapid increase in strength develops as the strain rate is increased above the transition (characteristic) strain rate. CONCLUSIONS—The above analysis provides a micromechanics based constitutive model for describing failure strength as a function of strain rate that compares well with experimental data for a wide range of brittle materials. Because the model represents a broad range of material classes (engineering ceramics to meteorites) this new universal model has applications in many engineering and science related fields. The resulting material model is expressed in a simple functional form making it suitable for incorporation into more general analysis codes (e.g., various finite element packages). Furthermore, the ability of the model to capture the physical behavior of brittle materials under uniaxial stress leads us to believe that the

421 micromechanics based analysis used here may be applicable to more general states of stress (e.g. confined compression) that are often encountered in impact problems.

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Fig. 2 Normalized experimental data for a wide range of brittle materials including rocks, engineering ceramics, and meteorite material. The normalized data compare well with the predicted model REFERENCES [1] Paliwal, B. and Ramesh, K. T., An interacting micro-crack damage model for failure of brittle materials under compression, Journal of the Mechanics and Physics of Solids, 56(3), 896-923, (2008). [2] Paliwal, B., Ramesh, K. T., McCauley, J. W., and Chen, M., Dynamic Compressive Failure of AlON Under Controlled Planar Confinement, Journal of the American Ceramic Society, 91(11), 3619-3629, (2008). [3] Frew, D. J., Forrestal, M. J., and Chen, W., A Split Hopkinson Pressure Bar Technique to Determine Compressive StressStrain Data for Rock Materials, Experimental Mechanics, 41(1), 40--46, (2001). [4] Green, S. J. and Perkins, R. D., Uniaxial Compression Tests at Strain Rates from 0.0001/Sec. To 1000/Sec. On Three Geologic Materials., Technical Report: General Motors Technical Center Warren Mich Materials and Structures Lab MSL68-6, 1-44, (1969). [5] Kumar, A., The Effect of Stress Rate and Temperature on the Strength of Basalt and Granite, Geophysics, 33(3), 501-510, (1968). [6] Paliwal, B. and Ramesh, K. T., Effect of crack growth dynamics on the rate-sensitive behavior of hot-pressed boron carbide, Scripta Materialia, 57(6), 481--484, (2007). [7] Wang, H. and Ramesh, K. T., Dynamic strength and fragmentation of hot-pressed silicon carbide under uniaxial compression, Acta Materialia, 52(2), 355-367, (2004). [8] Lankford, J., The role of tensile microfracture in the strain rate dependence of compressive strenght of fine-grained limestone--analogy with strong ceramics, International Journal of Rock Mechanics and Mining Sciences \& Geomechanics Abstracts, 18(2), 173--175, (1981). [9] Paliwal, B., Ramesh, K. T., and McCauley, J. W., Direct Observation of the Dynamic Compressive Failure of a Transparent Polycrystalline Ceramic (AlON), Journal of the American Ceramic Society, 89(7), 2128-2133, (2006). [10] Hu, G., Ramesh, K. T., Cao, B., and McCauley, J. W., The Compressive Failure of Aluminum Nitride Considered as a Model Advanced Ceramic, Journal of the Mechanics and Physics of Solids, In Press DOI: 10.1016/j.jmps.2011.02.003 (2011). [11] Ravichandran, G. and Subhash, G., A micromechanical model for high strain rate behavior of ceramics, International Journal of Solids and Structures, 32(17-18), 2627--2646, (1995). [12] Staehler, J. M., Predebon, W. W., Pletka, B. J., and Subhash, G., Micromechanisms of deformation in high-purity hotpressed alumina, Materials Science and Engineering A, 291(1-2), 37--45, (2000). [13] Kimberley, J. and Ramesh, K. T., The Dynamic Strength of an Ordinary Chondrite, Meteoritics \& Planetary Science, Under Review(2011). [14] Kimberley, J. and Ramesh, K. T., A universal relationship for the dynamic strength of brittle solids under compression, Submitted to Acta Materialia, (2011). [15] Buckingham, E., On Physically Similar Systems; Illustrations of the Use of Dimensional Equations, Physical Review, 4(4), 345-376, (1914).

Experimental Verification of Negative Phase Velocity in Layered Media

Alireza V. Amirkhizi* and Sia Nemat-Nasser Center of Excellence for Advanced Materials, Department of Mechanical and Aerospace Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0416, USA * [email protected]

ABSTRACT Stress-wave propagation is a generally dispersive phenomenon in elastic layered and periodic media. In the frequency ranges where propagation is possible, pass bands, the wave speed is a function of frequency. In the rest of the frequency spectrum, stop bands, wave propagation is impossible. The band structure of a periodic medium can be calculated using a variety of analytical and computational methods. In this paper we present such calculations for a simple layered sample. The wave propagation through the sample was measured using an ultrasonic experimental setup. The band structure was shown to agree with numerical and analytical predictions. The group velocity was calculated based on time of travel of the pulse through samples with different thicknesses. A measure of energy velocity was introduced and measured based on the mechanical power flux through samples. The phase velocity was measured based on the phase difference observed among the transmitted waves through different thicknesses. It is shown that the phase and group velocity become anti-parallel within specific pass bands as predicted based on theory.

Introduction The propagation of elastic waves through layered media has been studied extensively before. Rytov [1] has solved the 2-phase problem in closed form. Multi-phase problems can also be solved analytically. The overall band structure shows branches with positive and negative slopes. It can be shown that the negative slope branches represent negative refraction of elastic waves. One of the manifestations of negative refraction is anti-parallel group and phase velocities, schematically shown in Figure 1. In this work, we have designed a sample with a negative branch in the ultrasonic frequency regime. The numerical calculation of the branch structure is performed both using a very accurate variational formulation and Rytov’s analytical method. The benefit of the variational formulation is best understood in higher dimensional periodic structures, for which a closed form solution may not be found [2]. A sample is fabricated based on this design and tested ultrasonically to measure its band structure, group, phase, and energy velocities. The results are presented and compared with analytical predictions.

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Fig. 1 In negative refraction, while each harmonic component it travelling to left, the overall wave packet (group) as well as the wave energy is propagating to right (or vice versa).

Sample Design and Wave Velocity A symmetric layered sample with alternating PMMA and steel buttons was fabricated. The unit cell simply consists of a hard layer embedded inside a soft medium. The band structure is calculated using Maple Software. Based on the band structure, one can calculate the phase velocity as vp=ω/k on each branch. The group velocity, on the other hand, is calculated by calculating the derivative of the curve: vg=∂ω/∂k. It is known theoretically that in most cases the energy velocity is the same as group velocity.

Experimental Setup and Analysis The experiment was performed using a high frequency function generator. An AM-modulated signal was created digitally. The signal is sent to an amplifier capable of producing high voltage for driving piezoelectric transducers. The high-voltage signal and original pulse from the function generator are independently sent to the oscilloscope. The received signal after travelling through the sample is also measured at a third channel of the oscilloscope. The signals are recorded and digitally filtered at each frequency step. The group velocity and energy velocity are derived for each frequency step by measuring the time of travel of the envelope and the energy flux as a function of time. The phase velocity is derived through Fourier analysis of the transmitted and received signals and calculating the phase difference. It is shown that for positive group and energy velocity values, the phase velocity becomes negative, indicating negative refraction.

ACKNOWLEDGEMENT This work has been conducted at the Center of Excellence for Advanced Materials (CEAM) at the University of California, San Diego. This work has been supported in part through DARPA AFOSR Grant FA9550-09-1-0709 to the University of California, San Diego.

REFERENCES [1] Rytov, S. M., Acoustical properties of a thinly laminated medium, Sov. Phys. Acoust. 2, 6880, 1956. [2] Nemat-Nasser, S., Fu, F. C. L., Minagawa, S., Harmonic waves in one-, two-, and three-dimensional composites: bounds for eigenfrequencies, Int. J. of Solids and Struct., 11, 617, 1975.

Gas Gun Impact Analysis on Adhesives in Sandwich Composite Panels Matthew Mordasky1*, Weinong Chen1,2 1

School of Materials Engineering, Purdue University. School of Aeronautics and Astronautics, Purdue University. * Corresponding author: Matthew Mordasky, 701 W. Stadium Ave. West Lafayette, IN 47907-2045 Email: [email protected] 2

ABSTRACT As the application of composite sandwich panels is ever increasing, there is an increasing interest in the impact damage that occurs on these sandwich panels.

Since these composite sandwich panels are constructed in

various manners, analysis on different panel construction practices is of interest.

Specifically, when a

sandwich composite is cured, a film adhesive or just the prepreg resin system can be used to bond the core and face sheets during the curing process.

Additionally, the cure cycle of the panel can dictate face sheet

compaction and thus fiber volume fraction, resulting in susceptibility of face sheet delaminate or disbond. Manufacturing parameters are important to the panel integrity. Through the use of a Kolsky tension bar and a gas gun with a velocity range from 20 m/s to 100 m/s, the effect of a face sheet adhesive layer and the cure cycle autoclave pressure of the panel is analyzed. The results obtained through the use of the gas gun are more representative of real world impacts while the Kolsky bar results indicate the possible damage modes of a high strain rate tensile failure in CFRP sandwich composites. INTRODUCTION The objective of designing high quality sandwich composites used in modern aircraft is to optimize the strength of the panel overall. However, the strength of the panel can be jeopardized before the panel is even used through suboptimal manufacturing methods.

Core properties and artificial disbond have been explored for

Kolsky compression testing [1]. While laminates have been tested in tension at high strain rates with a Kolsky bar [2].

However, the effect of an adhesive film and the cure cycle autoclave pressure has not been a focus

on the high strain rate properties of sandwich composites.

During the processing of sandwich structures the

curing parameters are of utmost importance. Often the prepreg manufacturer will designate the proper curing temperature cycle, however the autoclave pressure is a variable that is user dependent.

For laminates, the

autoclave pressure can be increased to around 550 kPa for better ply compaction resulting in an increased fiber volume fraction and decreased void percentage with no drawback to the overall laminate quality. With sandwich structures, the resulting product is pressure sensitive. If pressure is too high, cell dimpling and core buckling are likely to affect the panel quality and mechanical properties.

If pressure is too low poor face sheet

compaction will not occur, leading to poor resistance to delamination as well as a weak core-face sheet interface resulting in disbond. disbond.

The addition of adhesive between the core and face sheet is needed to prevent

The effect of autoclave pressure and the addition of an adhesive layer on a honeycomb core will be

the focus of this study. EXPERIMENTS AND RESULTS An IM7-8552 unidirectional prepreg was used in a 4-ply layup for both the top and bottom face sheets. Where T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_59, © The Society for Experimental Mechanics, Inc. 2011

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prescribed, an FM300 film adhesive was used to bond the face sheets to a 12.7 mm NOMEX® core with a 3

density of .064 g/cm and a cell size of 3.2 mm.

The composite panels were cured with an autoclave pressure

of 0, 34.5, 69.0 and 137.9 kPa. The panel vacuum and temperature parameters were held constant.

Figure

1 shows an example of the target and actual curing cycle of a panel cured with an autoclave pressure of 137.9 kPa. The panels produced are used for experiments on a Kolsky bar and a low velocity gas gun.

Fig. 1 A curing cycle (177⁰C) for a IM7-8552/NOMEX® sandwich panel at 137.9kPa. Cylindrical sandwich samples with a diameter of 20 mm were produced with a diamond core drill and epoxy was used to adhere the samples to the Kolsky bar. Three damage modes were identified as delamination, disbond and core tearing. Figure 2 illustrates the damage modes observed through the tension Kolsky bar.

Fig. 2 Comparison of different sandwich composite failure modes exhibited by tension Kolsky bar REFERENCE: [1] H. Mahfuz, W. Al Mamun and S. Jeelani, “High Strain Rate Response of Sandwich Composites: Effect of Core Density and Core-Skin Debond”, Journal of Advanced Materials, 34, 22-26, 2002 [2] J. Harding, L.M. Welsh, “A tensile testing technique for fibre-reinforced composites at impact rates of strain”, Journal of Materials Science, 18, 1810-1826, 1983

Damage Analysis of Projectile Impacted Laminar Composites B. S. Nashed, J. M. Rice, Y. K. Kim, V. B. Chalivendra College of Engineering University of Massachusetts Dartmouth, 285 Old Westport Road, North Dartmouth, MA 02747-2300

Abstract The bending toughness, strength retention, resistance to damage and bending stiffness of glass fiber mat, laminar composites under high strain rate impact loading conditions was studied. One of the main disadvantages of laminar composite materials is their poor interlaminar shear strength. Recent work has demonstrated a method of Z-direction reinforcement of these composites using electrostatic flocking techniques improve delamination resistance and fracture toughness without degrading the composite’s tensile strength or other in-plane properties when loaded quasi-statically. The Z-direction reinforcement is accomplished by electrostatically flocking short fibers perpendicular to and between the composite ply layers. In this study, composite samples were prepared using the flocking method in two fabrication modes by the; so-called Z-Axis “wet” and ZAxis “dry” procedures. In this work, Z-direction reinforced composite panels (including a non reinforced control) that were previously projectile impact damaged, were tested using established mechanical testing procedures. Damage areas were quantified and compared using image processing techniques. Three point bending tests were also conducted on these projectile impact damaged panels to determine and compare their bending toughness, strength retention and modulus. The results show that Z-Axis reinforcement by the flocking technique improves the overall mechanical strength and stiffness properties of glass fiber mat laminar composites. For example, Z-Axis reinforced projectile damaged and not damaged glass fiber mat composite laminates are found to have flexural strengths 9% to 15% higher and a flexural modulus (stiffness) 22% to 26% higher than comparable (not Z-Axis flock reinforced) glass fiber mat samples.

1. Introduction Reinforced composites have many applications in engineering design from sports equipment, aircraft, marine and military structures and the like. Specifically, dynamic loading conditions in composites can occur in many industry/military applications, so new methods to reinforce composites and thereby improve their mechanical properties when loaded dynamically are valuable. In addition, after a composite material structure is damaged, strength retention is also an important property. Therefore, it is important to understand the continued functional performance of fiber reinforced composites after they have been impact damaged under high strain rate ballistic loading conditions. In composites, damage due to impact usually shows up as delamination between the ply layers. In this paper, flocking technology is used to place Z-Axis reinforcing fibers between the composite’s play layers. This is as an approach used to improve the impact resistance, strength and stiffness retention for laminar composites that have undergone damage by high strain rate projectile impact loading conditions. Short fibers are flocked in the z-direction perpendicular to and between the composite layers of composites and serve as reinforcement to resist delamination. A study was conducted to determine the properties of glass fiber mat composites after they have been ballistically impacted by a projectile. Previously, Colon [1] fabricated some glass fiber mat panels that were subsequently ballistically impacted by a .45 caliber copper jacketed lead projectile at four different velocities. Using the impacted composite sample panels, damaged during the Colon study, some precise image processing was carried out to measure the type of damage and damage area in each panel. For additional studies, the 4 x 4 in panels were cut into smaller 4 x 0.6 in samples and a 3-point bending test was performed to study the flexural strength, stiffness and strain energy to failure of the Z-Axis reinforced flocked composites, according to the ASTM standard. The overall objective of this present study is to analyze and characterize the projectile impact resistance of Z-Axis reinforced composites and their non-Z-Axis reinforced control samples. Specifically,

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428 computer based image analysis technology was used to determine the surface area of the composite’s inner damage region caused by the projectile impact. The mechanical properties of the glass fiber mat composites were measured through 3-Point Bending mechanical tests based on the ASTM standard D790.

2. Experimental Materials The glass fiber mat material is 458 g/m2 (1.5 ounce/ft2) chopped strand mat, sold by Fibre Glast Development Corp., Brookville, OH. Each layer has a random fiber arrangement and is 1.14 mm (0.045 in) thick. The 2 part epoxy is also from Fibre Glast, 2000 Epoxy Resin and 2120 Epoxy Curative. 4 x 0.6 in. rectangular samples were cut and prepared out of every 4 x 4 in. panel. In each projectile velocity group, 7 to 3 replicates were tested for the non, dry and wet flocked panels as specified in the standard. All samples consist of eight layers of glass fiber mat. Dry flocked glass fiber mat samples have a 3.7 mm average thickness, while the wet flocked glass fiber mat samples have a 4.3 mm average thickness, which is a 16.2 % increase in thickness. The damaged specimens used for the 3-point bending experiment are all located within the damage boundaries as shown in Figure 1. The specimens were cut starting approximately 0.8 in from the edge of the sample, and then the width was decided based on the ASTM D790 standards. This procedure was followed precisely to ensure the consistency and accuracy of the experiment results. The non-damaged specimens were cut from the edges of the 73 m/s panels, since they had the lowest damaged areas. The reason the panels at 73 m/s were chosen, was to avoid as much mircofracture damage that may have taken place by the projectile impact as possible. 3. Damage Area Analysis Projectile damaged glass fiber mat 4x4 in panels were inspected and different damage area boundaries were observed. It was clear that the delamination in the layers occurs in two different layers on each side of a typical panel. The two areas were classified as surface damage area and inner damage area, as shown in Figure 1. Furthermore, when the samples were placed on a light table, it was noticed that some of the areas from one side, projectile entrance, did not match the boundaries from the other side, projectile exit side. In Figure 1, it is clear that the tested sample have two areas overlapping each other.

Figure 1: Surface and Inner Damage Areas in a Non-Flocked Glass Fiber Mat 4x4 in. Composite panel

429 A sequence of image processing steps, as shown in Figure 2, was followed precisely to prepare the sample and accurately calculate the area. In this study, only the inner damage areas were calculated, the surface damage areas were previously measured by Colon [1]. The image processing was done first using software called GIMP. Then MATLAB was used to calculate the area. GIMP is an open source GNU Image Manipulation Program [2]. A Resynthesizer plug-in was added to GIMP to remove the sample identification text [3]. Once the text was removed, the next step was to prepare the edges of the inner, larger, area. Using a built-in function in GIMP called “Edge-detect”, option “Sobel” the image edges were now more visible and ready for MATLAB processing. In MATLAB, a program called “CREASEG”, by Dietenbeck, T, et., al. [4] was used to calculate the areas. CREASEG is software for the evaluation of image segmentation algorithms based on level-set. The program uses an image processing function in MATLAB known as “ROI”, Region of Interest”. The idea behind ROI is to pick an area and then manipulate it separately out of the whole image, in this case the inner damage area. CREASEG comes with several algorithms that can detect the edges of the ROI. After trial and error, a method called “Chan & Vese” was used [5]. The output area is first given in pixels and then converted to square centimeter.

Figure 2: Image Processing in Measuring Surface Area

3.a Glass Fiber Mat Damage Analysis After inspection of the panels, it was observed that the damage areas on the projectile penetration entrance side of the samples are different from the damage areas on the exit side. Therefore, four different areas were calculated using the described image processing procedures. Figure 3, presents the total damage area for flock reinforced glass fiber mat samples, including 2 areas on the entrance and 2 areas on the exit sides with standard deviation bars.

430

Figure 3: Total Damage Area for Glass Fiber Mat Samples at Various Projectile Impact Velocities with standard deviation bars As illustrated in Figure 3, glass fiber mat samples were tested at four different projectile velocities. The speed of 152.4 m/s is classified as above ballistic limit (ABL), in this case the projectile passed completely through all the samples 100% of the time. The other three noted projectile velocities are classified as below ballistic limit (BBL), since the projectile did not pass through any of the samples. For each projectile velocity, three glass fiber mat configurations were tested (a) Z-Axis wet flocked (b) Z-Axis dry flocked samples. Comparing two of the groups, 152.4 m/s and 137.2 m/sec, the 152.4 group has less damage area than the 137.2 group, and the reason is that the 152.4 group is ABL. In an ABL sample, the bullet passes all the way through the sample, and therefore less of its kinetic energy is converted to strain energy in the composite sample material. With the 137.2 m/s group where the projectile comes to rest inside the sample it is concluded that all of the projectile’s kinetic energy converts to strain energy that is absorbed within the fracturing composite sample material. This created more delamination damage in the composite sample. As the projectile’s impact speed decreases to 113 m/s and 73 m/s, the damage area decreases dramatically due to the overall decreased kinetic energy of the impacting projectile itself. Moreover, it is clear that in general the projectile causes more impact damage in non-flocked samples, especially at high speeds as shown by the summary of percent decrease in damage area compared to the non-flocked samples in Table 1. Based on the damage area analysis, the Z-Axis reinforcement by flocking technique improves the damage performance of glass fiber mat composites under high strain rate impact loading. Table 1: Percentage Decrease in Damage Compared to Non-Flocked “Control” Samples

Projectile Velocity

Wet Flocked

Dry Flocked

73 m/sec

28.4 %

25.3%

113 m/sec

0.6%

8.5%

137.2 m/sec

16.1%

14.3%

152.4 m/sec

15.2%

10.8%

431 4. Three Point Bending Test Description The flexural properties of this material were measured following the ASTM D790-03 standard pertaining to testing of plastics [6]. Two types of 4 x 0.6 in. rectangular samples were cut and prepared out of every 4 x 4 in. panel called nondamaged and damaged samples. The damaged sample is loaded from the projectile penetration entrance side. The entrance side was used because we are testing the samples for strength retention and if the material is struck again it will probably be from the same side. In each projectile velocity group, 7 to 3 replicates were tested for the non, dry and wet flocked panels as specified in the standard. 4.a Flexural Stress to FailureTesting:

f = 3PL2

2 bd

 = Stress in the fibers at midpoint, MPa [Psi]

P = Load at a given point on the load-deflection curve, N [lbf] L = Support Span, mm [in.] b = Width of beam tested, mm [in.], and d = Depth of beam tested, mm [in.] 4.b Flexural Strain to Failure Testing: f = 6 Dd L2 f = Strain in the outer surface, mm/mm [in./in.], D = Maximum deflection of the center of the beam, mm [in.], and d = Depth, mm [in.] L = Support Span, mm [in.] Using these equations, the flexural stress and the flexural strain to failure was calculated. 5. Three-Point Bending Test of the Non-damaged Samples 5.a Flexural Strength Analysis Based on ASTM D790 standards, specimens were prepared, and tested. Flexural stress and strain were calculated based on the equations in sections 4a and 4b. The maximum flexural stress values for non-flocked, wet flocked and dry flocked samples are shown in Figure 4 and are an indication of flexural strength. Wet flocked and dry flocked samples show a 9.27% and 15.24% increase in flexural strength, respectively, when compared to the non-flocked (control) glass fiber mat samples. Based on this analysis, the Z-reinforcement “wet” and “dry” flocking techniques improves the flexural strength of non-high strain rate impact damaged glass fiber mat specimens.

432

Figure 4: Flexural Strength for Non-Damaged Glass Fiber Mat Composites with standard deviation bars

5.b Strain Energy Analysis Strain energy is stored within an elastic solid when the solid is deformed under load and is an indication of the bending toughness. In the absence of energy losses, the strain energy is equal to the work done on the solid by external loads, and can be obtained by measuring the area under the force-extension curve, as shown for a typical force-deflection test curve in Figure 5. In this case the area was calculated numerically and includes the complete area under the curve from the start (at zero extension) to the end of the test when failure of the sample occurred.

Figure 5: Area Under the Curve Measures the Strain Energy During a 3-Point Bending Test.

433 Figure 6 compares strain energy results for various glass fiber mat composite samples. As seen, during the three-point flexural bending the non-flocked glass fiber mat samples did not store as much energy as the Z-Axis flocked samples. Specifically, the wet flocked and dry flocked samples show a 16.9% and 19.5% increase in strain energy, respectively, compared to the non-flocked glass fiber mat samples. Therefore, Z-Axis reinforcement by flocking dramatically increases the bending toughness of glass fiber mat composites.

Figure 6: Bending Toughness for Non-Damaged Glass Fiber Mat Samples with standard deviation bars 5.c Flexural Modulus Analysis Flexural modulus is the ratio of stress to strain due to flexural deformation, and is an indication of the bending stiffness. It is determined from the slope of a stress-strain curve produced by a flexural test based on ASTM D790. In Figure 7, flexural modulus is plotted for non, wet and dry flocked Z-Axis reinforced glass fiber mat samples. The wet flocked and dry flocked samples show a 22% and 26.5% increase in flexural modulus, respectively, compared to the non-flocked glass fiber mat (control) samples. Therefore, flocking significantly increases a glass fiber mat composite’s stiffness.

Figure 7: Bending Stiffness for Non-Damaged Glass Fiber Mat Samples with standard deviation bars

434 Comparing all three composite sample configurations, it is observed that that the Z-Axis reinforced “dry” flocked glass fiber mat composite showed the most enhanced flexural strength, bending toughness and bending stiffness mechanical property improvement. 6.1 Three-Point Bending Test of Impact Damaged Glass Fiber Mat Samples As previously mentioned, glass fiber mat samples were impact tested at four different projectile speeds. Three point bending tests were carried out on the damaged samples using the ASTM D790 standard. The damaged samples were center point loaded on the projectile penetration entrance side. 6.1.a Flexural Strength Analysis for Impact Damaged Laminates Three point bending test samples were prepared from the damage area of 4x4 in panels. Figure 8 shows the maximum flexural stress attained with these specimens. The wet and dry flocked samples in the 73 m/sec and 152.4 m/sec groups show an increase in the flexural stress value when compared to the non-flocked samples. For the 113 m/sec projectile velocity group only the dry flocked sample shows a slight increase in the flexural stress value when compared to the non-flocked samples. For the 137.2 m/sec. group only the wet flocked sample showed an increase in the flexural stress value when compared to the non-flock reinforced samples. A summary of the percentage increase in flexural strength compared to the non-flock reinforced samples is shown in Table 2.

Figure 8: Flexural Strength for Damaged Glass Fiber Mat Samples with standard deviation bars Table 2: Percentage Increase in Flexural Strength Compared to the Non-Flocked Reinforced Samples Projectile Velocity

Wet Flocked

Dry Flocked

73 m/sec

10.6%

15.1%

113 m/sec

9.7%

*

1.2%

137.2 m/sec

7.3%

5.8%*

152.4 m/sec

8.6%

29.1%

* Percentage Decrease

435 A comparison of the maximum flexural stress of the non-flocked, non-damaged samples in Figure 4 (291.96 MPa) to the stress of the dry flocked damaged samples in velocity groups 73and 113 in Figure 8 (greater than 291.96 MPa), shows that the flexural strength of these non-damaged samples is retained after damage for the dry flocked samples in these two velocity groups. 6.1.b Strain Energy and Flexural Modulus Strain energy and flexural modulus were measured for the impact damaged glass fiber mat samples. In Figure 9 and Table 3 the wet and dry flocked samples show an increase of the strain energy compared to the non-flock reinforced sample, up to 26.7 % for the dry flocked samples. Moreover, the 73 m/sec. and 113 m/sec. groups show a relatively higher strain energy comparing to the other two groups and relatively small damage areas as shown in Figure 3. That is expected, since comparing Figure 3 to Figure 9 for all 4 groups, there is an opposite pattern. It is clear from earlier observations, as the damage areas decreases the strain energy increases. The flexural modulus in Figure 10 has the same pattern as the strain energy in Figure 9, and both have an opposite pattern to the damage areas in Figure 3. The samples with lower damage area, has higher stiffness and strain energy. A summary of the percentage increase in strain energy and flexural modulus compared to the non-flocked samples is presented in Tables 3 and 4, respectively. Based on the strain energy and flexural modulus results, Z-Axis reinforcement by the flocking technique has improved a glass fiber mat composite’s bending toughness (to failure) and flexural modulus increases by as much as 26.7% and 40 % with a dry flock reinforced sample, respectively.

Figure 9: Bending Toughness for Damaged Glass Fiber Mat Samples with standard deviation bars

Table 3: Percentage Increase in Bending Toughness to Failure Compared to the Non-Flocked Samples Projectile Velocity

Wet Flocked

Dry Flocked

73 m/sec

5.1%

8.7%

113 m/sec

3.4%

16.3%

137.2 m/sec

12.1%

26.7

152.4 m/sec

9.7

24.7%

436

Figure 10: Bending Stiffness for Projectile Damaged Glass Fiber Mat Samples with standard deviation bars Table 4: Percentage Increase in Bending Stiffness Compared to the Non-Flock Reinforced Samples Projectile Velocity

Wet Flocked

Dry Flocked

73 m/sec

6.8%

14.9%

113 m/sec

11.7%

34.6%

137.2 m/sec

5.8%

18.6%

152.4 m/sec

17.5%

40.6%

A comparison of the strain energy of the non-flocked, non-damaged samples in Figure 6 (2 J) to the strain energy of the dry flocked damaged samples in velocity groups 73and 113 in Figure 9 (greater than 2 J), shows that the bending toughness of these non-damaged samples is retained after damage for the dry flocked samples in these two velocity groups.

A comparison of the flexural modulus of the non-flocked, non-damaged samples in Figure 7 (11 GPa) to the flexural modulus of the dry flocked damaged samples in velocity groups 73and 113 in Figure 10 (greater than 11 GPa), shows that the bending stiffness of these non-damaged samples is retained after damage for the dry flocked samples in these two velocity groups. 7. Results and Conclusions Wet and dry flocked, Z-reinforced glass fiber mat panels were ballistically impacted by a projectile under a high strain rate. The loading was accomplished by shooting a .45 caliber copper jacketed lead projectile at four different velocities at 4” X 4” composite test samples. One of the ballistic velocities was above the ballistic limit and 3 velocities were below the ballistic limit. After inspecting the damaged panels, four distinct damage areas on the front and back of the panels were identified and measured. The total damage area for each panel was measured and then panels were then cut into specimens for mechanical testing. The mechanical properties of a series of Z-Axis reinforced glass fiber mat composites were compared using a 3-point bending test. The results of the analysis showed that:

437 

Z-reinforcement of glass fiber mat composites using the flocking technique improves the damage performance for both damaged and non-damaged glass fiber mat composites by reducing the observed damage area caused by projectile impact. Composite Z-Axis fiber reinforced samples prepared using the dry flocking process performed the best.



Z-reinforcement flocking increases the flexural strength for both damaged and non-damaged glass fiber mat composites with dry flocking having the highest maximum flexural strength in both cases.



Z-reinforcement flocking increases bending toughness of glass fiber mat composites with dry flocking having the highest bending toughness at failure in both cases.



Z-reinforcement flocking lowers damage areas which as a result increase composite stiffness and bending toughness in both the dry and wet flocked composites.



Z-reinforcement dry flocked, damaged samples exhibited excellent flexural strength, bending toughness and bending stiffness retention when compared to non-flocked, non-damaged samples at 73 m/s and 113 m/s projectile velocities.

Acknowledgements The authors wish to thank Hector Colon for his help in sample preparation and Armand Lewis for sharing his expertise in flocking and testing.

References [1] “Projectile Impact Behavior of Flock Fiber Z-Reinforced Composites” by Hector Colon, Masters Thesis, University of Massachusetts Dartmouth, to appear. [2] GIMP Documentation Team, “GNU Image Manipulation Program User Manual”. http://docs.gimp.org, 2010. [3] “Resynthesizer” by Dr. Paul Harrison, PhD, “http://www.logarithmic.net/pfh/resynthesizer” [4] Dietenbeck, T.; Alessandrini, M.; Friboulet, D.; Bernard, O., “CREASEG: Free Software for the Evaluation of Image Segmentation Algorithms Based on Level-Set”. IEEE International Conference on Image Processing. Hong Kong, China, mechanical testing2010. [5] Chan, T and Vese, L, “Active Contours Without Edges”. IEEE Transactions on Image Processing, Volume 10, pp. 266277, February 2001. [6] ASTM Standard D790, "Standard Test Methods for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials," ASTM International, West Conshohocken, PA, 2003, DOI: 10.1520/D0790-03, www.astm.org.

   

Rate Sensitivity in Pure Ni Under Dynamic Compression Krishna N. Jonnalagadda, Mechanical Engineering, Indian Institute of Technology Bombay, India

ABSTRACT Recent spurt in the use of Ni and Ni based alloys (e.g., Ni-W) with microstructure control for various applications has resulted in a significant amount work done on their mechanical behavior and rate sensitivity. However, there is no available data on the dynamic mechanical behavior of commercially pure Ni. In this paper, we present dynamic compression results from Kolsky bar experiments conducted on bulk coarse grained Nickel (CG Ni) in the strain rate range of 2600 – 8400 /s. The results show significant strain rate sensitivity and strain hardening, which are comparable to the behavior observed in other FCC materials such as copper and aluminum. Also, the cold worked Ni used in these high strain rate experiments shows higher flow stress values compared to previous studies conduced on annealed pure CG Ni.

Introduction There have been a lot of studies on the behavior of nanostructured Ni under quasi-static and cyclic loading. Previous work showed that by controlling the microstructure (e.g., grain size), the strength of Ni and Ni based alloys can be increased [1 -3]. There are also studies conducted on the pure CG Ni to understand its mechanical response under various loading conditions. Muller [4] investigated the dynamic deformation of coarse-grained Ni under annealed condition and it was found that the flow stress in Ni is a strong function of strain rate and temperature. This behavior was also found in other FCC materials, e.g., Al [5] and Cu [6] by various researchers. Magnusen et al. [7] studied the deformation of sintered Ni and Ti under varying strain rates to investigate rate sensitivity in the context of material porosity. In this work, we are interested in understanding the dynamic mechanical behavior of commercially pure CG Ni, its rate sensitivity and the microstructure changes that occur due to high strain rate loading. Both the undeformed and deformed microstructures are investigated via XRD and EBSD, and the observations are reported in this paper.

Experiments The material used in these experiments was commercially pure Ni in the form of rod stock. This starting material was cold worked and extruded. The dynamic compression specimens were prepared by turning the rod and then cutting the cylindrical discs using a wire EDM. The compression direction of the specimens was aligned with the axis of the rod. The specimens were then polished on both the flat ends using silicon carbide and diamond lapping papers. The final dimensions of the specimen were 2.72 mm in length and 5.02 mm in diameter. This gives a length/diameter aspect ratio of ~0.54, acceptable for Kolsky bar experiments. The polished specimens were subjected to dynamic compression in a conventional Kolsky bar setup with the bars made of maraging steel of diameter 12.5 mm. A schematic of the compression Kolsky bar setup is shown in Figure 1. The setup consists of an incident/input bar, transmitted/output bar and a momentum trap bar aligned coaxially [8]. A projectile is launched using a compressed air gas gun onto the incident bar generating a stress wave, whose characteristics depend on the projectile bar used. By varying the projectile length (50 mm to 200 mm) and their velocities the CG Ni samples were loaded in the strain rate range of 2600 – 8400 /s. A tiny amount of MoS2 grease was used to pulse shape the signal and remove the projectile-input bar contact issues. In all the experiments conducted, the total accumulated strain was controlled to be around 30% in order to perform postmortem microstructure studies to understand the deformation mechanisms active under dynamic compression at various strain rates. This was achieved by controlling the pulse width for different strain rates of loading in these experiments through calculations of projectile velocity from the gas gun pressure. To understand the deformation mechanisms and microstructural changes occurring due to dynamic loading of Ni specimens, scanning electron microscopy studies were conducted in conjunction with EBSD (electron backscattered diffraction) imaging. The bulk texture changes in the material were studied via XRD (X-ray diffraction) inverse pole figures. The undeformed and deformed specimens were lapped and electrochemically polished to obtain good quality surfaces for EBSD and XRD imaging. During the experiments, the polished specimen was aligned at 70 degrees to the electron beam and the

  Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, T. DOI 10.1007/978-1-4614-0216-9_61, © The Society for Experimental Mechanics, Inc. 2011

439

440 surface was scanned at a magnification of 400X with a working distance of 20 mm and a step size of 200 nm. At present, imaging was performed only on the surface normal to the loading (extrusion) direction. Several regions of the specimen surface were scanned to extract the grain size, grain misorientation, and texture data. The idea here is to observe extent of deformation and possibly dynamic recrystallization, if any, in the cold worked CG Ni subjected high strain rate loading. Projectile

Input Bar

Vertical Posts

Output Bar

Specimen

Momentum Trap

Figure 1: Schematic of a compression Kolsky bar setup with projectile and momentum trap

Results and Discussion The results from high strain rate experiments on coarse-grained Ni specimens are shown in the form of stress vs. strain curves in Figure 2. These are the first systematic study results from the dynamic compression of CG Ni at high strain rates above 102 /s. The spanned strain rate range was 2600 – 8400 /s and is typical of any conventional Kolsky bar system. This strain rate was calculated by averaging over the reflected pulse in the incident bar. It is observed that the flow stress increases with strain rate as expected and the material strain hardens as the strain increases. The flow stress increased from 634 MPa to 700 MPa between the two extremes of the strain rate range. The flow stress at the lower end of the strain rates is much higher than the values reported by Dalla Torre et al. [1] who reported values between 300 – 400 MPa for strain rates up 102 /s for CG Ni sheets. From Muller’s work [4] the room temperature flow stress at 5% strain was found to be around 320 MPa. This difference in flow stress can be attributed to prior cold work and grain size of the material used in this study as compared to annealed material used earlier by other reserachers. The total accumulated strain from all the experiments varied from 26 – 33%. This allows for a comparison of microstructure over the entire strain rate range at more or less constant strain. Within this range of accumulated strain no evidence of shear banding was observed as in the case of Al and Al based alloys [9]. The rate sensitivity of the material was calculated using m = ( log σ / log ε
)ε and a log-log plot of the flow stress vs. strain rate is shown in figure 2(b). The flow stress was obtained at a constant strain of 5% from which m = 0.066 was calculated. This rate sensitivity is higher than the rate sensitivity reported for low strain rates by Lian et al. [10] for CG Ni and also those reported earlier by Magnusen et al. [7], and Muller [4]. This was again expected because most coarse-grained FCC metals [56] exhibit this increase in rate sensitivity with strain rate, especially near ~103 /s and this rate transition in flow stress was often attributed to the increase dislocation accumulation [11]. However, the high strain rate experiments conducted here span only one decade of strain rates and for more accurate calculations of the rate sensitivity factor data from both the lower and higher strain rate experiments is required. The EBSD results for both the undeformed and deformed specimens are shown in Figure 3. From the image analysis the average grain size in the undeformed specimens was found to be ~32 micrometers. This grain size compares well with the material used by Magnusen et al. [7], however their materials was processed through powder sintering and was in annealed condition. From the Figure 2(a) it can be seen that the CG Ni tested in our experiments is randomly textured with significant amount of grain misorientation caused by the cold work. Some signatures of the existence of twins can also be seen. At this time it was not possible to ascertain if they are coherent or incoherent twins and how they formed. It is necessary that this material be annealed and experiments repeated to understand the role of these twins in the deformation at high strain rates. In the undeformed material the texture is predominantly (001) and (111) and the distribution was found to be even over several locations of the specimen surface and in some areas the texture was (001) and (101) dominated. This shows that the texture over a large of the sample was random and this was expected, as the surface is perpendicular to the extrusion direction.

441

(a)

(b)

Figure 2: (a) True stress vs. true strain curves from dynamic compression experiments on CG Ni, (b) rate sensitivity plot on the log-log scale between strain rate and flow stress at 5% strain.

(a) (b)   Figure 3: (a) Undeformed inverse pole map showing the grain structure and crystallographic orientation in the plane normal to the loading direction, (b) a similar inverse pole figure for a deformed samples loaded at 6700 /s and 25% accumulated strain.

 

442 The deformed sample shown in Figure 3(b) was from the sample tested at a strain rate of 6700 /s and 25% accumulated strain. Again the texture was found to be random and in particular inverse pole figure shown here, the grains were oriented along all the three directions. From the detailed analysis of the EBSD images it was found out that the grain orientation spread (GOS) and also the grain average misorientation angle (GAM) increased in the deformed specimen compared to undeformed specimen. However, the misorientation angle spread narrowed in the deformed sample as compared to the undeformed sample and a majority of the angles were less than 10 degrees indicating low angle grain boundaries. The twin density in the deformed sample increased and this may be due to the high strain rate deformation. Currently, work is under progress to investigate the transverse or extrusion direction texture to understand the anisotropy effects of cold work process that causes the grains to elongate in the extrusion direction. Bulk texture measurements conducted on both undeformed and deformed specimens showed that the texture changes after deformation.

Conclusion The high strain rate behavior of commercially pure coarse-grained Ni has been investigated. It was found that compared to annealed Ni (both coarse grained and ultra fine grained) investigated earlier by other researchers the strain rate sensitivity is higher for this material. From the cold worked condition, the CG Ni exhibited strain hardening behavior beyond the accumulated strains of 30 % for all strain rates of loading, which indicates that the material can be further work hardened to increase its flow strength. Compared to previous studies on similar grain sized material this material exhibited twice the flow stress at 5% strain, which is significant from applications point of view. Further studies on annealed and nanostructured material are required to completely understand the structure-property correlations under dynamic deformation.

  Acknowledgements The author thanks Prof. K.T.Ramesh and Matt Schaffer of The Johns Hopkins University for providing the resources for the experiments conducted in this work. Also, we thank Prof. A. H. Chokshi for providing the CG Nickel used in the experiments.

References   1

F. Dalla Torre, H. Van Swygenhoven and M. Victoria, “Nanocrystalline Electrodeposited Ni: Microstructure and Tensile Properties,” Acta Materialia, 50(15), 2002.

2

R. Schwaiger, B. Moser, M. Dao, N. Chollacoop and S. Suresh, “Some Critical Experiments on the Strain-Rate Sensitivity of Nanocrystalline Nickel,” Acta Materialia, 51(17), 2003.

3

M. J. N. V. Prasad, S. Suwas, and A. H. Chokshi, “Microstructural Evolution and Mechanical Characteristics in Nanocrystalline Nickel with Bimodal Grain-Size Distribution,” Materials Science and Engineering A, 503, 2009.

4

T. Muller, “High Strain Rate Behavior of Iron and Nickel,” J. of Mechanical Engineering Science, 14, 1972.

5

K. Sakino, “Strain rate dependence of dynamic flow stress considering viscous drag for 6061 aluminium alloy at high strain rates,” Journal De Physique Iv, 134, 2006.

6

P.S. Follansbee and U. F. Kocks, “A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable,” Acta Metallurgica, 36, 1988.  

7

P.E. Magnusen, P.S. Follansbee and D. A. Koss, “The influence of strain rate and porosity on the deformation and fracture of Titanium and Nickel,” Metallurgical Transactions, 16A, 1985.

8

K.T. Ramesh, Handbook of Experimental Mechanics, Springer, 2009.  

9

E.L. Huskins, B. Cao, K.T. Ramesh, “Strengthening mechanisms in an Al-Mg alloy,” Materials Science Engineering A, 527, 2010.  

10

J. Lian, C. Gu, Q. Jiang, and Z. Jiang, “Strain rate sensitivity of face centered cubic nanocrystalline materials based on dislocation deformation,” Journal of Applied Physics, 99, 2006.

11 J. W. Edington, “The influence of strain rate on the mechanical properties and dislocation substructure in deformed copper single crystals,” Philosophical Magazine, 19(162), 1969.

 

 

SEM 2011 Annual Conference & Exposition on Experimental and Applied Mechanics, June 13– 16 2011, Uncasville, Connecticut, U.S.A.

Temperature Effect on Drop-Weight Impact of Woven Composites Yougashwar Budhooa,*, Benjamin Liawb, Feridun Delaleb, a

Department of Engineering and Technology, Vaughn College of Aeronautics and Technology, 86-01, 23 Ave, East Elmhurst, Queens, NY 11369, USA b

Department of Mechanical Engineering, The City College of New York, 140th and Convent Ave., New York, NY 10031 ABSTRACT

This paper investigates the effect of temperature on hybrid woven composite panels (100mm×100mm×25mm) which underwent drop-weight impact at five different test temperatures: −60°C, −20°C, room temperature, 75°C and 125°C. The studies were conducted by using experimental and 3-D dynamic finite element approaches. The specimens tested were made of plain-weave hybrid S2 glass and IM7 graphite fibers imbedded in toughened epoxy (cured at 177°C). The composite panels were impacted using an instrumented drop-weight impact tester. The time-histories of impact-induced dynamic strains and impact forces were recorded. The damaged specimens were inspected visually and using the ultrasonic C-scan method.

INTRODUCTION Materials used in engineering structures may experience a severe change in environmental conditions during its service life. To best deal with these conditions, engineers often turn to composite materials. The main idea of composite material is to combine different materials to produce a new material with performance unattainable by the individual constituents. It gives flexibility to the designer to tailor the new material with properties to obtain peak performance for a particular application. One important environmental condition that can adversely affect a material performance is temperature. Budhoo and others study the effect of high and low temperatures response of graphite woven composites [1]. A few other studies have been done on composites where emphasis had also been placed on the effect of temperature on the behavioral response of the material [2-5].This study will focus on the effect of temperature on glass woven composites when impacted at low temperature. It will also investigate the effect of hybridization on composites when impacted at high and low temperatures.

*

Corresponding author. Tel.: +1 646 496 6102; Fax: +1 212 650 8013. E-mail address: [email protected] (Y. Budhoo).

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_62, © The Society for Experimental Mechanics, Inc. 2011

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444

EXPERIMENTAL PROCEDURE Materials: The individual constituent materials combined to form the composite material used in this research are, IM-7 graphite (IM7GP 6000) and S2-glass (S2-4533 6000) woven fabrics placed in SC-79 toughened epoxy resin matrix. The S2-Glass woven fabric and SC-79 epoxy matrix form the non-hybrid laminate called GL. The hybrid composite will be formed by a GL laminate sandwiched between two laminates of graphite composite (GR), this resultant hybrid laminate is called GR/GL/GR. The GL and GR/GL/GR panels both have a dimension of 100mm×100mm×25mm. GL consists of 37 layers of glass fabric. The hybrid GR/GL/GR specimens contain 8 layers of graphite fabric skin on each side, while the glass core has 16 layers of glass fabric

Experimental setup:

The low-velocity (drop-weight) impact study was conducted at five different temperatures: −60°C, −20°C, room temperature (R.T.), 75°C and 125°C. All the impact tests were performed using a drop-weight impact tester. Figure 1 shows the schematics of the experimental set-up for the low-velocity impact tests. The impact energy was fixed at 30J for all tests, which corresponds to an impact velocity of 3.12 m/s. In this study the shape of the impactor nose was hemispherical with a diameter of 16 mm. With an attached environmental chamber, an open coil heater provided the high temperatures, while the low temperatures were achieved through the use of liquid nitrogen. Specimens were clamped circumferentially with a 76 mm diameter fixture, where the clamp was considered to be a fixed-fixed support. Using a data acquisition system, the time histories of impact loads were measured and recorded using a load cell located just above the impact nose. The impact velocity was also measured by a pair of photoelectric-diodes attached to the base of the test machine. With the data acquisition system, only load (the resistive force of the specimen) vs. time and the initial impact velocity (just prior to impact) can be measured directly. Using the equations of motion, energy absorbed by the specimen, velocity of impactor and deflection at the impact center were derived and recorded.

Figure 1 Schematic of the drop-weight impact test set-up

Two strain gauges were mounted on the impact side (i.e., the top surface in Figure 1) of each specimen. These strain gages were mounted at a distance of 25.4 mm from the center of the composite panel, where the impactor impacted the specimens. Figure 2 shows a schematic of the strain gage orientations and locations on the specimen. SG-1 measured the radial strain while SG-2 measured the hoop strain.

Post damage evaluation: A modular and expandable ultrasonic system was used to scan and conduct the damage evaluation for the impacted specimens. Scanning was performed using the through-transmission technique. Flat and focused 5MHz transducers (6.35mm in diameter) were used to scan these post-impacted composite specimens.

445

Figure 2 The locations of strain-gages on a composite specimen for drop-weight impact test

EXPERIMENTAL RESULTS

C-SCAN

BACK

FRONT

Reviewing the literature, it was found that many different factors tend to affect the damage mechanisms and damage patterns in composite materials. Some of these factors include lay-up configuration, laminate thickness, impactor size and shape, constituent properties, temperature, impact velocity and energy, etc. To the authors’ knowledge, no studies have been carried out on the effect of temperature on the impact response of thick section non hybrid GL, GR or hybrid GL/GR/GL and GR/GL/GR composites. Visual inspection showed that the GL specimens were the most resistant to impact damage. As shown in Figure 3, for the highest test temperature, i.e. 125 °C, horizontal and vertical front surface cracks were created in GL specimens. This is often referred to as micro-buckling, where the fibers on the top surface buckle due to compression on the upper surface. As the temperature decreases, it can be seen that the severity of micro buckling decreases i.e. the length of the front surface crack decreases with decreasing temperature. For all the test temperatures it can be seen that no back surface splitting occurred. From the C-scan images, it can also be seen that the damaged area slightly decreases with a decreasing test temperature.

125 °C

75 °C

R.T

−20 °C

−60 °C

Figure 3 Optical fractographs and C-scan of non-hybrid GL composite specimens after a 30J energy level drop-weight impact at various temperatures

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Impact tests were conducted on the hybrid GR/GL/GR which consists of graphite layers as the face sheets and a glass layer as the core material. In Figure 4, drop-weight tests conducted on the hybrid composite specimens showed horizontal and vertical front surface cracks for all specimens tested. The size of cracks decreased with decreasing temperature from 125 °C to R.T, but it was difficult to determine a proper trend of horizontal or vertical front surface cracks at the cryogenic temperatures i.e. −20°C and −60°C. Back surface splitting was not severe but can be seen only for R.T, −20°C and −60°C. The difference in stacking sequence results in some differences in the size of the generated horizontal and vertical front surface cracks as well as the severity of penetration and back surface splitting. Research was also done for the case where GR was the core and sandwiched by GL [5]. There it was shown that horizontal and vertical front surface cracks tend to decrease as the test temperature decreases from125 °C to −20 °C. At the lowest test temperature however, the cracks increased. Upon carefully examination of the specimens, it was seen that there was no back splitting on any of the samples after testing.

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Figure 4 Optical fractographs and C-scan of hybrid GR/GL/GR composite specimens level drop-weight impact at various temperatures

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Figure 5 Representative force-time history curve from drop-weight impact test

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The Dynatup 930-I data acquisition system measures only the initial velocity of the impactor and force vs. time directly. The remaining parameters, such as absorbed energy, velocity of impactor and deflection, are calculated using equations of motion. Important damage mechanisms can be studied by understanding the force time history curve from an impact test. Fig 5 shows a representative force-time history curve from a drop weight impact test. The initial peak on the curve represents the initiation of delamination. If the impact force is less than the initial peak value, then there will be no damage or delamination of the material and the response will be elastic, i.e. the force time history curve will almost be sinusoidal. In the event that the maximum force of impact is greater than the load bearing capabilities of the material, there will be fiber shear-out. In other words, the commencement of fiber shear-out indicates that the load bearing capacity of the laminate is reached, hence little resistance will be provided against higher impact energies.

Figure 6 Time histories of the impact forces of non-hybrid GL composites at −60°C, −20°C, R.T, 75°C, 125°C. In Figure 6 it can be seen that as temperature decreases, the initial peak of the time histories of impact force increases. This indicates that the force required to create a delamination increases with a decrease in temperature. The time it takes to reach this initial peak also increases with a reduction in test temperature. Also the maximum force increases as temperature decreases. Since the impact force is the force measured by the impactor, it is the resistance the impactor experiences when it is in contact with the specimen. Therefore, at low temperatures, the GL specimen is more resistant to damage and is therefore tougher.

Figure 7 Time histories of the impact forces of non-hybrid GR/GL/GR composites at −60°C, −20°C, R.T, 75°C, 125°C.

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Figure 7 shows the impact force time history of the hybrid composite specimen, i.e. GR/GL/GR. It can be seen that although the maximum force is almost the same at R.T and 75°C, their initial peaks are more distinct. At −60°C, −20°C, R.T for hybrid GR/GL/GR, not only is there a drop after the initial peak but a few other drops along the impact force history curve. It was realized that these were the specimens that had back splitting after impact.

Dynamic strain histories

Figure 8 Dynamic strain histories for GL impacted at various temperatures Figure 8 shows the strains obtained from two strain gages placed on a GL specimen impacted at the five test temperatures. Strains above the time axis are the radial strains while those below the time axis are the hoop strains. Strain gages were placed one inch away from the point of impact. In the figure, it can be seen that the absolute value of the hoop strains is higher than that for the radial strain at each temperature.

Figure 9 Dynamic strain histories for GR/GL/GR impacted at various temperatures Figures 9 show the strain histories for the hybrid composite. The absolute value of strain tends to increase with an increase in temperature. One reason for this is that as the temperature increases, its stiffness decreases, hence the material deforms more. This was evident from the tensile testing. It can also be seen that as the temperature increases, the initial slope of the strain history tends to get steeper or in other words, the strain rate increases with an increase in temperature. This can also be

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explained by the reduction in stiffness in the material with an increase in temperature. Due to reduction in stiffness, and increase in ductility, the material exhibits larger deformation at a fixed time as compared to a material with a higher stiffness.

Contact force vs. deflection In Figure 10 the contact force vs. deflection of the composite plate is plotted. This is sometimes referred to as the stiffness. During the loading portion, it can be seen that the material has a highly oscillating behavior. Also the initial peak of the force-deflection curve tends to decrease with an increase in temperature. This peak represents the beginning of delamination. It can also be seen that the initial slope of curve tend to decrease with an increase in temperature. We can therefore conclude by looking at Figure 9 that the stiffness of the material tends to increase as the test temperature decreases, and the maximum force increases with a decrease in temperature.

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(b) Figure 10 Contact force vs. deflection of (a) GL and (b) GR/GL/GR for the impact tests conducted at −60°C, −20°C, R.T, 75°C, 125°C

Impact energy Composite materials use various mechanisms to absorb energy when they are impacted. It is believed that the three major method of energy absorption by composites are delamination, fiber shear out and fiber breakage. When an impactor drops from a certain height, its potential energy is converted to kinetic energy. At the time of impact, this kinetic energy goes into deforming the specimen (elastically and plastically) while some of it gets dissipated by the internal mechanisms of the composite. There are also other ways in which this energy gets dissipated, such as friction, etc. At the time of maximum penetration or when the velocity of the impactor is zero, the corresponding energy is referred to as the impact energy. If the energy absorbed by the composite specimen is small, usually the impactor bounces back with rebound energy. In Figure 11, the energy history of the composites is shown at all the test temperatures. It can be seen that up to the point where the energy curve reaches its maximum peak, all the composite layups. As the temperature decreases, the slope of the energy history curve tends to increase. Looking only at the curves for high temperatures i.e. 75°C, 125°C, it can be seen that the absorbed

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energy for the 125°C test is more that for 75°C. In all cases the absorbed energy for −60°C, −20°C, R.T did not vary significant and was more than that for high temperatures

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(b) Figure 11 Energy-time histories of (a) GL and (b) GR/GL/GR for the impact tests conducted at −60°C, −20°C, R.T, 75°C, 125°C

Delamination In this research, since delamination occurs inside of the material and is difficult to see with the naked eye, two approaches were used to study delamination. The first method used was the Ultrasonic C-scan method. After scanning was completed, the damaged specimens were cut into halves to reveal the damaged cross- sectional area. In Figures 3 and 4 the C-scan images of the delaminated areas of the impacted specimens at all test temperatures are shown. Using these images the delamination area was measured in order to quantify the effect of temperature on it. It can be seen that for the GL composite, the delamination area is smaller and symmetric while that of the non hybrid is more asymmetric and larger. This is due to the bonding of different materials, in which the adhesion strength between the dissimilar materials is weaker and the propagation of delamination and failure is not uniform in all directions, thus giving rise to an unsymmetrical delamination pattern. The delamination area of glass tends to decrease with a decrease in temperature; this contradicts our intuition that the delamination area is expected to increase with a decrease in test temperature. Figure 12 shows the cross-sectional view of GL specimens at all the test temperatures after impact. At high temperature a darker center core can be seen throughout the thickness of the cross-section, and is due to compression of the material. As the temperature decreases, due to brittleness of the material, the compression effect is significantly smaller so from the figure it can be seen that this core fades away. It can also be seen that at the point of impact, the delamination area is very small but the delamination area tend to get larger further away from the point of impact.

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(e) Figure 12. Optical fractographs of sectioned composites after drop impact conducted at 30 J on GL: (a) 125°C, (b) 75°C, (c) R.T (d) −20°C, (e) −60°C

It is interesting to note that in Figure 12 the rate of increase of delamination from the top surface to the bottom surface is a lot greater for the low temperature tests and this rate decreases with an increase in temperature. This can be easily seen by the white area of the cross-section. In looking very carefully in Figure 3 at the C-scan results for the GL specimens and in Figure 12 at the cross-section revealing the damage area, our question as to why GL appears to have a decrease in delamination area as temperature decreases is answered. At high temperature the dark core at the center is the effect of compression and is not the actual delamination of the specimen. This can be verified by looking at the C-scan images where at high temperature a white core is seen surrounded by the delamination area which is darker. Therefore at high temperature although the delamination area is small, it appears to be large because it surrounds a large compressive core. This compression area along with the delamination surrounding it has also been reported in literature. At lower temperature, this compression effect is minimal and the entire area is delamination. This can be verified by the absence of the white core in the C-scan images and also the dark core in the cross-sectional images. Therefore the actual delamination area increases as temperature decreases but only appears at first to be the reverse. Figure 12 shows the cross-sectional view of the hybrid GR/GL/GR specimens. It can be seen that at R.T, −20°C and −60°C there is back splitting of the material. Usually, when a material is impacted, energy

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transferred by the impactor is dissipated by the damage mechanisms in the composite material. The first major form of energy dissipation is usually delamination. When delamination cannot occur fast enough to dissipate the energy, then the material seeks other methods to dissipate the energy. The two other major energy dissipation mechanisms in composites are fiber shear out and fiber fracture. In Figure 13, fiber shear out is seen, this is when material near the impactor cannot remain perpendicular to the impactor anymore so it start to shear out. The back splitting is due to fibers failing in tension since the back end of the specimens elongates to its fracture point due to the deflection of the specimen. It can also be seen clearly from the GL layer in Figure 13 that on the surface closer to the impactor the delamination is smaller and further away from the impactor the delamination area increases.

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(e) Figure 13. Optical fractographs of sectioned composites after drop impact conducted at 30 J on GR/GL/GR: (a) 125°C, (b) 75°C, (c) R.T (d) −20°C, (e) −60°C.

CONCLUSIONS a) As the test temperature decreases, the peak force of the composite tends to increase slightly. b) For GL, the delamination area tends to decrease with a decrease in test temperature, the delamination pattern of GR/GL/GR was not clear. c) As the test temperature decreases, the strain rate of the material due to impact also tends to decrease. d) The first major form of energy dissipation in the composite material was delamination which was directly linked to the material peak force. Other dissipation methods seen were fiber shear out and fiber breakage. e) It can be seen that GL specimen is more impact resistant and hence stronger than the hybrid.

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REFERENCES [1] Budhoo, Y., Liaw, B., Delale, F., 2010 “Effect of temperature on the impact damage of composite materials”, Proceedings of the ASME 2010 International Mechanical Engineering Congress & Exposition IMECE2010 [2] Hirai Y., Hamada H. and Kim J.K., 1998, “Impact response of woven glass-fabric composites- II. Effect of temperature”, Composites Science and Technology, Vol. 58, Issue 1, pp. 119-128 [3] Im K.H, Cha C.S, Kim K.S, Yang I.Y., 2001, “Effect of temperature on impacts damages in CFRP composite laminates”. Composites Part B, Vol. 32, pp. 669–82. [4] Parlevliet P.P, Bersee H.E.N, and Beukers A, 2006, “Residual stresses in thermoplastic composites- A study of the literature-Part 1: Formation of residual stresses”, Applied Science and Manufacturing, Vol. 37, Issue 11, pp. 1847-1857 [5] Dlouhy I., Chlup Z., Boccaccini D. N., Atiq S. and Boccaccini A. R., 2003, “Fracture behavior of hybrid glass matrix composites: thermal ageing effects”, Composites Part A: Applied Science and Manufacturing, Vol 34, Issue 12, pp 11771185. [6] Sevkat E., Liaw B., Delale F., Raju B.B., 2009, “Drop-weight impact of plain-woven hybrid glass–graphite/toughened epoxy composites”, Composites Part A: Applied Science and Manufacturing, Vol. 40, Issue 8, pp. 1090-1110 [7] Jones M. R., Mechanics of Composite Materials, 1998, “Thermal and Mechanical Stress Analysis”, Edwards Brothers, Ann Arbor, MI. Chap. 4.5, [8] Naik N. K., Sekher C.Y. and Meduri S., 2000, “Damage in woven-fabric composites subjected to low-velocity impact”, Composites Science and Technology, Vol 60, Issue 5, pp 731-744

Dynamic Mode-II Characterization of A Woven Glass Composite

Wei-Yang Lu, Bo Song, Helena Jin Sandia National Laboratories, Livermore, CA 94551-0969, USA Delamination of laminated composites has been a reliability issue in applications. Fracture toughness is a critical indicator of resistance to such a delamination. Quasi-static experimental techniques determining fracture toughness have been well developed. For example, end notched flexure (ENF) technique has been commonly used to determine mode-II fracture toughness. However, dynamic fracture characterization of composites is much more challenging. Such immature dynamic techniques have resulted in conflictive and confusing description of rate effects on fracture toughness of composites. Currently dynamic mode-II characterization employs three-point bending with a Kolsky bar on an ENF specimen and uses the same analysis as quasi-static ENF technique. The quasi-static ENF analysis is applicable to dynamic experiments only when the forces at both ends of the ENF specimen are equilibrated. However, both the weak delamination strength of composites and the complicated loading structure of the three-point bending embedded to the Kolsky bar make it difficult to compare the forces at both ends of the ENF specimen. Polyvinylidene fluoride (PVDF) thin film force transducers have been introduced to directly measure the forces at both ends of the composite ENF specimen [1]. The results showed that the forces at both ends of the specimen were far from equilibration in conventional Kolsky bar experiments. This requires proper pulse shaping design in dynamic three-point bending experiments with Kolsky bar. In this study, we modified the Kolsky bar techniques with proper pulse shaping and applied PVDF force transducers directly on the contact surfaces of loading in dynamic mode-II characterization of a glass woven composite. Figure 1 shows the testing section of the Kolsky bar for dynamic three-point bending experiments. Three PVDF force transducers are directly attached to the specimen surfaces in contact with the front loading wedge and back span supports. The PVDF film used in this study is commercially available and has a thickness of 110 µm. The piezoelectric response of the PVDF has been carefully calibrated with Kolsky bar system within a large stress scale. The surfaces of the PVDF in contact with the loading wedge and spans were protected with a thin steel piece. By contrast, the other surfaces of the PVDFs were directly attached to the specimen surface in order to avoid additional inertia involved in the force measurement. A ø1/8×0.02” annealed copper disk was Fig. 1. Schematic of testing section attached to the impact surface of the incident bar as a pulse shaper to properly modify the profile of the incident pulse such that the force equilibrium on both ends of the ENF composite specimen can be achieved. Figure 2 shows the oscilloscope record from the strain gages on the incident bar. The modified incident pulse possesses a rise time of approximately 130 µs, which is 10 times longer than that in conventional Kolsky bar experiments. Since the composite exhibits low delamination strength, nearly all incident pulse is reflected back as the reflected pulse, as shown in Fig. 2. The PVDF at the impact wedge end is the first to sense the load on the specimen. This load produces transverse wave travelling towards both upper and lower ends of the specimen, which are sensed with the PVDFs attached to the back surface of the specimen. The PVDF signals are shown in Fig. 3. Figure 3 clearly shows the forces at both back span supports are equilibrated, while the sum of both is equal to the force at the front loading wedge after the first 113 µs. It takes the transverse wave approximately 30 µs to propagate from the central wedge to the end span supports, which corresponds to approximately 600 m/s of

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456 transverse wave speed. This is consistent with the DIC result previously conducted [2]. It takes another 83 µs to achieve force equilibrium. It is noted that the specimen has achieved force equilibrium prior to constant-speed loading. After the force is equilibrated, the forces on both sides of the specimen are built up simultaneously until the crack starts to propagate. Figure 3 also shows the speed history of the impact wedge. Figure 3 confirms that the specimen has achieved stress equilibrium before the impact speed reaches a constant of 5.6 m/s. In addition, the specimen failed (crack propagation) at the constant impact speed, which is important to study the rate effect. Figure 4 shows the shear strain on the specimen obtained from DIC analysis. The DIC images were taken with a Phantom V12.1 at a speed of 83,000 frames per second. Figure 4 indicates that the crack started to propagate at t = 253 µs, when the front force on the specimen reached a maximum (Fig. 3).

Fig. 2 Oscilloscope record of incident and reflected pulses

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457 where C = δ / P, which is the slope in the displacement-force curve; P is the force when the crack starts to propagate. The specimen geometry parameters used in Eq. (1) are defined in Fig. 5. Note that P = F1 = F2 + F3. For the specimen used in this study, t = 6.7 mm, w = 10 mm, L = 25 mm, a = 6.35 mm. C is calculated from Fig. 6, -7 C= 5.86×10 m/N, and the maximum force was also measured from Fig. 6, P=1922.3 N. The mode-II fracture 2 toughness wass then calculated as GIIC = 1227.1 J/m at the impact speed of 5.6 m/s. Following the same procedure but at different impact speeds, the rate effect on the mode-II fracture toughness of the composite is able to be determined.

Fig. 5 Specimen geometry

Fig. 6 Force-displacement curve

In summary, the quasi-static mode-II fracture toughness analysis can be applicable to dynamic experiments only after the testing conditions are validated. Pulse shaping needs to be properly designed such that the specimen can achieve stress equilibrium before the crack starts to propagate. The stress equilibrium needs to be carefully verified with appropriate measurements. Embedding PVDF thin film transducers has been demonstrated to be effective and straight-forward technique for stress equilibrium verification. In addition, proper pulse shaping facilitates a constant impact speed on the specimen, which is important to determine rate effect when incorporating quasi-static results.

ACKNOWLEDGEMENTS Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

REFERENCES 1. Lu, W.-Y., Song, B., and Jin, H., 2010, “Dynamic Mode-II Characterization Using SHPB Embedded with th PVDF,” In: Proceedings of the 7 Asian-Australasian Conference on Composite Materials (ASSM-7), Taipei, Taiwan, Novermber 15-18, 2010. 2. Song, B., Jin, H., and Lu, W.-Y., 2010, “Stress Wave Propagation in a Composite Beam Subjected to Transverse Impact,” In: Proceedings of IMPLAST 2010 – SEM 2010 Fall Meeting, Providence, RI, October 12-14, 2010. nd 3. Gibson, R. F., 2007, Principles of Composite Material Mechanics (2 Edition), CRC Press, New York.

Rate Dependent Material Properties of an OFHC copper Film

Jin Sung Kim1)· Hoon Huh*2) 1)

Railway Structure Research Department, Korea Railroad Research Institute, 360-1, Woram-dong, Uiwang-si, Gyeonggi-do 437-757, Republic of Korea 2) School of Mechanical, Aerospace and Systems Engineering, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon 305-701, Republic of Korea [email protected], [email protected] ABSTRACT The material properties of OFHC(Oxygen Free High thermal Conductivity) film with a thickness of 0.1 mm was evaluated at the strain-rates ranging from 0.001/s to 500/s using High-Speed Material Micro-Testing Machine(HSMMTM). The high strain-rate material properties of thin films are important especially for evaluation of structural reliability of micro-formed parts and MEMS products. The high strain-rate material testing methods of thin films, however, are not yet thoroughly established while testing methods of larger specimens for electronics, auto-body, train, ship and ocean structures has been well-established. For evaluation, a HSMMTM has been newly developed to conduct high-speed tensile tests of thin films. The machine developed has a capacity of sufficiently high tensile speed with an electromagnetic actuator, a novel gripping mechanism and an accurate load measurement system. The OFHC copper film shows high strain-rate sensitivity in terms of the flow stress, the fracture elongation and strain hardening. They increase as the tensile strain-rate increases. The quantitative comparison would provide material data at high strain-rates for design and analysis of micro-appliances and micro-equipments. INTRODUCTION Investigations of the mechanical properties of thin films for micro-parts and MEMS products have increased in researchers of these products over the past decades. In recent years, a variety of micro-parts, such as micro-levers, micro-connectors, microscrews and springs, have been developed. These parts require high reliability and good dimensional accuracy and productivity. The high strain-rate material properties of various materials for micro-parts and MEMS applications are required to design and evaluate the product quality and performance levels during the processes of high-speed forming and impact loading, the latter of which induces high strain-rate deformation of many different materials. Micro-forming processes generally demand a high production rate[1-2] which causes high strain-rate deformation as these processes proceed. Thus, the high strain-rate material properties of small-sized materials are required to analyze the high-speed micro-forming process accurately. Sharpe[3] pointed out two important factors in the testing of new materials on a microscale. First, the specimens must be similar in size to structural components. Secondly, the specimens must be produced by the same manufacturing processes used for the components. Therefore, a new material testing technique valid at high strain-rates is necessary for microscale specimens. Material testing results for various specimens[4-16] according to their cross-sectional areas and strain-rates are shown in Fig. 1. Material tests at various strain-rates have been performed by many researchers in accordance with the specimen size in relation to the size of the structural component. Material testing methods to obtain a stress–strain relationship generally differ depending on the strain-rate. General mechanical or hydraulic machines, such as the Instron UTM, are used to obtain the stress–strain relationship at a low strain-rate of less than 0.1/s. The material properties at intermediate strain-rates ranges ranging from 1/s to 1000/s were obtained using high-speed material testing machines with a servo-hydraulic type actuator[16-18]. A split-Hopkinson pressure bar apparatus, also known as a Kolsky bar, is a popular type of experimental equipment used for the identification of dynamic material characteristics at high strain-rates ranging from 103/s to 104/s[19]. For small-sized specimens, material tests using mechanical[20], piezoelectric[14, 21] or electro-magnetic actuators[9-12] are commonly conducted at quasi-static strain-rates of less than 10-2/s. The tensile testing method for thinfilm materials has become well established over the past few decades. However, high-speed material testing techniques for

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small-sized specimens are not yet established owing to the delicacy inherent in the testing methods and the difficulty in specimen handling. In contrast, high-speed material testing techniques for various materials of common sizes, such as conventional auto-body steel sheets, different types of copper, aluminum alloys and polymeric materials are well developed. The present paper suggests a novel high-speed material micro-testing technique at microscale and investigates the high strainrate material properties of an OFHC copper film. The material properties obtained are compared to those of a bulk OFHC copper sheet in terms of the rate-dependent flow stress curve, strain-rate sensitivity, strain hardening and fracture elongation. The tensile material properties are indispensible for analyzing large plastic deformations, such as deformation after an impact, of thin-walled structures by bending in which the primary deformation mechanism by bending consists of tension and compression across the thin sheets. The proposed high-speed material testing technique can support the computer-aided simulation of micro-forming processes, reliability assessments of micro-products and MEMS products in relation to the strain-rate.

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Strain rate [/sec] Fig. 1 Material testing results with respect to specimen cross-sectional area at various strain-rates EXPERIMENTAL TECHNIQUES The material for the micro-tensile tests in this study was cold-rolled OFHC copper with thickness of 0.1 mm. OFHC copper film was hot-rolled once initially. The number of cold rolling passes is two for OFHC copper film. OFHC copper film was annealed after the first cold rolling process and hard (H04) tempered in the annealing process. OFHC copper consists of 99.99% copper. The dimensions of the micro specimen used here are shown in Fig. 2. The gauge length is 1 mm and the width of the gauge section is 0.2 mm. The overall width is 2 mm and the total length of the specimen is 20 mm. One end of the micro specimen is longer than the other end of the micro specimen, as high-speed tensile tests require a sufficient acceleration distance. The micro specimens were fabricated by micro photo etching, which is a process used in microfabrication to remove parts selectively from a thin film or from a bulk substrate. A specimen pattern after micro photo etching is shown in Fig. 3. The micro specimens were prepared for the tensile tests by carefully removing the supporting beams from the micro specimen pattern. SEM images of a fabricated micro specimen confirm the accurate dimensions and fine surface quality, as shown in Fig. 4 (a) and (b). The gauge width measured was 200±1.8 μm in the entire gauge length. The several black spots visible in the SEM image are tiny dust spots; these have a negligible effect on the outcome of the tests.

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Fig. 4 Micro specimen witth a gauge widdth of 200 μm m machined byy micro photo etching: (a) full f view; (b) magnified vieew of a side wall off the gauge secction M Machine (HS SMMTM) waas developed to investigatte the high strain-rate matterial A High-Speeed Material Micro-Testing properties of micro couponns. The HSMM MTM consistts of a loadingg actuator, a gripper g and a load measureement system.. The SMMTM is shhown in Fig. 5. A linear gu uide block aliigns two stagees precisely, aand the facing g surfaces betw ween developed HS the two stagess are perfectlyy parallel. A XY X stage finallly adjusts the vertical and horizontal h alignment with a loading actuaator. The nominal strain-rate raange for the HSMMTM H is from 1/s to 1000/s with regard r to a m micro specimenn proposed with w a gauge length of 1 mm. Tennsile loading should s be appllied after the actuator a speedd reaches a deesignated consstant tensile sp peed. The actuator for the HSM MMTM shouldd have sufficiient acceleratiion that is higgher than thaat of a conven ntional high-sspeed material testinng machine, as a the dimensiions of the miicro specimen are much sm maller than those of a convenntional high-sspeed material testiing specimen. Moreover, it is importannt to achievee the loadingg speed quickkly. To satisfy y the accelerration performance, the actuator of o an electro-m magnetic lineaar motor was used u for the HSMMTM. H Thhe electro-maggnetic linear motor m employed is a servotube acctuator (STA22510 model off Copley Co.) with a maxim mum accelerattion of 580 m//s2, a load cappacity of 780 N andd a maximum speed of 4.2 m/s. m In order to verify the performance p o this apparaatus, the veloccity of the cylinder of was comparedd using an inpput command at velocities of o 1 mm/s to 1000 1 mm/s wiithout a payloaad, as shown in Fig. 6 (a) to o (d). In a severe caase, the velocity of the cylinnder reaches an a input comm mand velocity of 1000 mm/ss after approxximately 14 ms and a displacemennt of 6 mm duue to the acceleration time.

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Fig. 6 Displacement curves with respect to the tensile velocity without payload: (a) 1 mm/s; (b) 10 mm/s; (c) 100 mm/s; (d) 1000 mm/s The gripping mechanism is a slack adapter type, as shown in Fig. 7. In order to achieve a constant velocity during the tests, a special jig was designed to move some distance without loading a specimen and then seize the specimen instantly at the designated speed. A wedge-type clipper grips one end of the micro specimen and the clipper is then inserted into the moving grip, as shown in Fig. 7 (a), (b) and (c). The loading cylinder moves smoothly toward the fixed jig, and the other end of the micro specimen is gripped on the fixed jig, as shown in Fig. 7 (d) and (e). Finally, the actuator moves as much as the acceleration distance in the opposite direction of the tensile direction. The micro high-speed tensile tests start from this initial position. The grip faces are flat and are fastened with M2 bolts. The contact angle between the moving jig and the clipper is 60° to relieve the impact force caused by the high-speed collision. A load measurement system for the HSMMTM should be carefully designed, as it is one of the most important parts in determining the quality of the stress–strain curves. In a general case, as the strain-rate increases, the load does not transmit with a uniform distribution to the specimen, the jig and the load cell. The load acquired from the load cell subsequently begins to oscillate because the inertia and the stress wave deform parts of the equipment. This phenomenon is termed load ringing[16]. The load ringing phenomenon can be reduced by increasing the natural frequency of the fixed jig or by measuring the load from the specimen directly. For the HSMMTM, the natural frequency of the fixed jig should be increased, rather than the load being measured from the specimen directly, as the size of the micro specimen is too small to attach strain gages onto it. The load cell used in this step is a piezoelectric-type dynamic force sensor, PCB 201B02, with a maximum load capacity of 444.8 N and an upper frequency limit of 90 kHz. Generally, the load ringing characteristics of a jig are enhanced as the mass of the jig decreases and as the stiffness of the jig increases. By removing unnecessary parts from the grip structure and after shortening the length of the jig, the natural frequency of the proposed fixed jig becomes 25,500 Hz, which is nearly four times the natural frequency of a conventional high-speed material testing machine[22]. The load cell was calibrated seven times at the loading speed of 160 N/s in the static UTM. The scale factor was 100.7±0.5 mV/N. The nonlinearity of the load cell is about 1% in the full scale. The strain along a specimen was measured by a non-contact digital image processing technique[23]. Incremental deformation images with a resolution of 144×480 pixels were taken by a high-speed camera, a Phantom V9.1 model, at a frame rate up to 10,000 frames/s. The lens set used was an AF Micro Nikkor 60 mm F2.8D set with extension tubes of 65 mm. The axial strain was manually measured by analyzing images via image processing, which traces particular spots on a specimen. This is shown in Fig. 8. Two particular points in the reduced section were selected as measuring points, and the axial distance between these two points became the initial gauge length, which was approximately 80% of the length of the reduced section. The axial strain is determined by dividing the gauge length increment by the initial gauge length. The measurement error estimated is about 0.7% due to the resolution of the images from the high-speed camera.

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Fig. 7 Gripping procedure for HSMMTM: (a) locate a micro specimen on the clipper; (b) fasten the micro specimen to the clipper; (c) insert the clipper with the micro specimen into the moving jig; (d) locate the actuator to initial loading point; (e) fasten the other end of the micro specimen to the fixed jig; (f) Move the actuator to the initial position as much as the acceleration distance

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Fig. 9 Load and axial strain measured after the onset of the tensile loading for the designated strain-rate: (a) 1/s; (b) 10/s; (c) 100/s The slope of the axial strain with respect to time, which is the nominal strain-rate, should be kept constant during the tensile test. The axial strain changes during high-speed tensile tests are plotted in Fig. 9 (a) to (c). The axial strain begins to increase after some elapsed time due to a subsequent error caused by a sudden increase in the load, especially for a strain-rate range lower than 10/s. The strain-rates are kept nearly constant after some elapsed time for a strain-rate range that exceeds 10/s, while the strain-rate increases to 3/s for a designated strain-rate of 1/s. The load ringing phenomenon arises at a strain-rate of 100/s, but the load ringing frequency is 25,500 Hz and the amplitude of oscillation is negligible compared to the measured load response. Stress–strain curves were obtained by synchronizing the axial strain measured via digital image processing and the load from the load cell. It is difficult to synchronize these two measurements at high strain-rates since those are measured from different devices. The strain was tracked by analyzing the camera images. And the strain data points are interpolated to match the number of load data points. Therefore, the error from this synchronization was estimated as the relative frame rate of the high-speed camera used in the process. The frame rate of the camera was 10,000 frames/s and the incremental time was 0.1 msec. The maximum synchronization error is estimated to be a strain of 0.01 at a strain-rate of 100/s. RATE-DEPENDENT MATERIAL PROPERTIES OF OFHC COPPER FILM Accurate tensile properties of thin films were obtained at high strain-rates from the developed HSMMTM. The OFHC copper film was tested using a quasi-static testing machine and a high-speed material testing machine at various strain-rates. A micro tester [13] and the developed HSMMTM were used for the tensile tests. The reduced section of a micro specimen had a length of 1 mm and a width of 0.2 mm. The rate-dependent stress–strain curves of the OFHC copper film are shown in Fig. 10 (a). Tests were conducted twice for each condition, and three times if the results were not reproducible. The stress–strain

464 curves obtained here confirmed that the flow stress and the strain hardening increase as the strain-rate increases. The yield stress at a strain-rate of 100/s was 247.7 MPa, which is 19.3% higher than that at a strain-rate of 0.001/s, 288.6 MPa. Fig. 10 (b) illustrates the variation of the strain-rate sensitivity, indicating the variation of the flow stress with respect to the strainrate with variation of the strain. The curve marked with square symbols denotes the initial yield stress curve; the subsequent curves denote the flow stress curves according to the corresponding plastic strain. The interval between the symbols under the same strain-rate indicates the amount of strain hardening. For the OFHC copper film in Fig. 10 (b), the yield stress at a strainrate of 100/s is 19.3% higher than that at a strain-rate of 0.001/s while the flow stress at a plastic strain of 0.075 at a strainrate of 100/s is 26.4% higher than that at a strain-rate of 0.001/s. Thus, the strain hardening becomes larger as the strain-rate increases. This phenomenon was explained by Follansbee and Kocks [26] and Tong et al. [27]. Tong et al. demonstrated that this strain-rate dependence of strain hardening in copper arises due to the increasing difficulty of dislocations to overcome obstacles at high strain-rates, leading to an increased multiplication rate and to an increase in the strain-rate sensitivity of strain hardening. Fig. 10 (c) shows the fracture elongation of the OFHC copper film with respect to the logarithmic scale of the strain-rate. The fracture elongation increases as the strain-rate increases from 0.001 to 0.1/s and then decreases slightly at a strain-rate of 1/s before increasing again to 100/s. Rate-dependent fracture elongations of conventional steel sheets also show similar tendencies in that ductility of TRIP600 and DP600 steels does not decrease as the strain-rates increase [28-30]. The fracture elongation of the OFHC copper film at a strain-rate of 0.001/s is 11.4% while the fracture elongation at a strain-rate of 100/s is 23.4%, which is an increase of 105.3% compared to the quasi-static case. Strain hardening has critical effects on the fracture elongation. The final shapes of micro specimens were shown in Fig. 11. 400

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Fig. 11 Micro specimens after the fracture CONCLUSION A High-Speed Material Micro-Testing Machine (HSMMTM) was newly developed for the high-speed material testing of thin films at strain-rates ranging from 1/s to 500/s. The proposed HSMMTM utilizes a high-performance electro-magnetic servo actuator. A fixed grip with a ring-type load cell reduces the load ringing phenomenon effectively. A slack adapter type gripping mechanism was designed to seize the micro specimen instantly after the designated tensile speed was reached.

465 Micro specimens with a gauge length of 1 mm were fabricated by a micro photo etching technique, showing a good surface quality along with good dimensional accuracy. The rate-dependent material properties of an OFHC copper film with a thickness of 0.1 mm were evaluated at strain-rates that ranged from 1/s to 500/s using the developed HSMMTM. The stressstrain curves of the OFHC copper film show positive strain-rate sensitivity and the strain hardening gradually increases as the strain-rate increase. The strain-hardening increase with the increase in the strain-rate increases the fracture elongation with respect to the strain-rate. Experimental results of an OFHC copper film provide rate-dependent flow stress curves and fracture elongation values, which are indispensable mechanical properties that can be used to analyze the micro-forming process and can be used as part of a structural analysis of micro-parts in conjunction with a numerical simulation. REFERENCES [1] F. Vollertsen, et al., "State of the art in micro forming and investigations into micro deep drawing", Journal of Materials Processing Technology, Vol. 151, pp. 70-79, 2004. [2] U. Engel and R. Eckstein, "Microforming - from basic research to its realization", Journal of Materials Processing Technology, Vol. 125, pp. 35-44, 2002. [3] W. N. Sharpe, "Murray lecture - Tensile testing at the micrometer scale: Opportunities in experimental mechanics", Experimental Mechanics, Vol. 43, pp. 228-237, 2003. [4] S. Cheng, et al., "Tensile properties of in situ consolidated nanocrystalline Cu", Acta Materialia, Vol. 53, pp. 1521-1533, 2005. [5] R. Schwaiger, et al., "Some critical experiments on the strain-rate sensitivity of nanocrystalline nickel", Acta Materialia, Vol. 51, pp. 5159-5172, 2003. [6] C. D. Gu, et al., "Experimental and modelling investigations on strain-rate sensitivity of an electrodeposited 20nm grain sized Ni", Journal of Physics D-Applied Physics, Vol. 40, pp. 7440-7446, 2007. [7] F. Dalla Torre, et al., "Nanocrystalline electrodeposited Ni: microstructure and tensile properties", Acta Materialia, Vol. 50, pp. 3957-3970, 2002. [8] Z. H. Jiang, et al., "Strain-rate sensitivity of a nanocrystalline Cu synthesized by electric brush plating", Applied Physics Letters, Vol. 88, pp. -, 2006. [9] R. D. Emery and G. L. Povirk, "Tensile behavior of free-standing gold films. Part I. Coarse-grained films", Acta Materialia, Vol. 51, pp. 2067-2078, 2003. [10] R. D. Emery and G. L. Povirk, "Tensile behavior of free-standing gold films. Part II. Fine-grained films", Acta Materialia, Vol. 51, pp. 2079-2087, 2003. [11] Y. H. Huh, et al., "Application of micro-ESPI technique for measurement of micro-tensile properties", Advances in Nondestructive Evaluation, Pt 1-3, Vol. 270-273, pp. 744-749, 2004. [12] Y. H. Huh, et al., "Measurement of continuous micro-tensile strain using micro-ESPI technique", Advances in Fracture and Strength, Pts 1- 4, Vol. 297-300, pp. 53-58, 2005. [13] J. Lou, et al., "An investigation of the effects of thickness on mechanical properties of LIGA nickel MEMS structures", Journal of Materials Science, Vol. 38, pp. 4129-4135, 2003. [14] S. J. Lee, et al., "Measurement of Young's modulus and Poisson's ratio for thin Au films using a visual image tracing system", Current Applied Physics, Vol. 9, pp. S75-S78, 2009. [15] Y. M. Wang, et al., "Microsample tensile testing of nanocrystalline copper", Scripta Materialia, Vol. 48, pp. 1581-1586, 2003. [16] H. Huh, et al., "High Speed Tensile Test of Steel Sheets for the Stress-Strain Curve at the Intermediate Strain-rate", International Journal of Automotive Technology, Vol. 10, pp. 195-204, 2009. [17] L. D. Oosterkamp, et al., "High strain-rate properties of selected aluminium alloys", Materials Science and Engineering a-Structural Materials Properties Microstructure and Processing, Vol. 278, pp. 225-235, 2000. [18] K. Miura, et al., "High strain-rate deformation of high strength sheet steels for automotive parts", SAE Paper No.980952, 1998. [19] H. Kolsky, Stress waves in solids. New York: Dover Pubns, 2003. [20] W. N. Sharpe, et al., "A new technique for measuring the mechanical properties of thin films", Journal of Microelectromechanical Systems, Vol. 6, pp. 193-199, 1997. [21] N. N. Nemeth, et al., "Fabrication and probabilistic fracture strength prediction of high-aspect-ratio single crystal silicon carbide microspecimens with stress concentration", Thin Solid Films, Vol. 515, pp. 3283-3290, 2007. [22] W. Wang, et al., "An analytical model for assessing strain-rate sensitivity of unidirectional composite laminates", Composite Structures, Vol. 69, pp. 45-54, 2005. [23] W. N. Sharpe, et al., "Strain measurements of silicon dioxide microspecimens by digital imaging processing", Experimental Mechanics, Vol. 47, pp. 649-658, 2007.

Zirconium: Probing the Role of Texture using Dynamic-Tensile-Extrusion C.P. Trujilloa, J.P. Escobedo-Diaza, G.T. Gray IIIa, E.K. Cerretaa, D.T. Martineza a

Material Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Extended Abstract The effect of high strain-rate and high strains on mechanical behavior has been observed primarily in

isotropic, cubic materials. The behavior of low-symmetry, textured, materials is not as well understood. To examine the high strain and high strain-rate response of structural metals, a Dynamic Tensile Extrusion technique has been developed at Los Alamos National Laboratory. In this study, high-purity zirconium bullets were accelerated up to velocities of 615m/s and extruded through a high-strength steel die. The Zr bullets were fired at 23°C and 250C. Specimens were sectioned in two orthogonal directions from the as received plate: (1) loading direction aligned in plane (IP) to the rolling direction and (2) loading direction through thickness (TT) of the plate. A combination of in-situ and ex-situ characterization techniques has been used to study the response of Zr under this dynamic loading condition. Not only are these the first experiments of their kind performed at temperatures higher than room temperature, but high-speed imaging and for the first time, PDV (Photonic Doppler Velocimetry) has been employed to capture the time and velocity of the evolved deformation through the die. In the Dynamic Tensile Extrusion technique, an 81° die is rigidly affixed to the end of a 0.300 caliber barrel [1]. The die is ~ 7 ½ times the reduction in cross-sectional area. At velocities of several hundred meters per second, this dynamic event produces an oblique shock match of several GPa of pressure and associated shear. This strongly integrated test results in a wide range of strain rates and stress states within the test specimen. However, a significant component to the stress state is tension, allowing us to investigate dynamic tensile failure modes. Using PDV in a novel application, we can generate valuable data sets that will provide validation and assist in developing fracture models. This diagnostics technique relies on an optical probe and a 1550nm wavelength laser (components of the PDV system). With this system, reflected light, from the front surface of the accelerating and extruding sample, can be captured by the probe as it travels through the barrel and then into the die. However, for this to be possible, the probe must be precisely aligned along the axis of the T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_65, © The Society for Experimental Mechanics, Inc. 2011

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barrel and the die. In Fig. 1, a high speed image, taken at 2µs interframe times, are captured as the extruded sample is approaching the optical probe of the PDV system. Using this diagnostic, we can capture the samples transit time through the die as well as the velocity profile during this transition. The die transit time is directly utilized to inform the continuum level of processing time of the material in the die so constitutive models can further evaluate the measured behavior. Computationally, the projectile is allowed to "slip" thru the die via slide lines or interfacial slip and shear - both of which are approximations. If the transit time is accurately captured in the modeling, it is likely that the calculated pressures, strains, stresses, etc. realized by the projectile as a function of time may be more reasonable approximations. To validate this further, predictions of the ultimate segmentation of the extruded specimen, shown in Fig.2, can be compared with tested and soft recover specimens. As is evident in Fig. 1, we witness the tensile instability in the necking of the sample before impacting the probe. This leads to segments of the extruded specimen but also results in a “tensile pullback”. This is captured for the first time through the time and distance measurements of the PDV, Fig. 3. From soft recovered extruded specimens, post mortem characterization could be performed and correlations based on the influence of texture, temperature and test velocity on ductility could be made. We found that the In Plane samples consistently show larger elongations than Through Thickness specimens, by about 25%. Elongations were measured based on the total length of reassembled, soft recovered, extruded pieces. Reassembly of fully extruded segments was based on the high speed camera images that captured the in-situ damage process. For all tests, a segment of the dynamically loaded specimen remained in the die. For many of these tests, this piece was removed and microstructural analysis was performed. This analysis included: optically microscopy (OM), scanning electron microscopy (SEM), and electron back scattered diffraction (EBSD). In the In Plane cases, texture evolution due to the dynamic extrusion process was not as significant as that observed in the TT case. This correlated well with differences in observed twinning. More tensile twinning was observed in the TT specimens, as may be expected for Zr loaded in this orientation in a stress state dominated by tensile loads [2]. This difference in twinning is expected to generate differences in the development of plastic instabilities during deformation and is likely linked to observed difference in elongation between IP and TT specimens. Quantitative examination of the influence of temperature, texture, and extrusion velocity will be presented as well as PDV findings on transit times through the die.

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