Advanced Structured Materials Volume 15
Series Editors Andreas Öchsner, Johor, Malaysia Lucas Filipe Martins da Silva, Porto, Portugal Holm Altenbach, Halle, Germany
For further volumes: http://www.springer.com/series/8611
Holm Altenbach Victor A. Eremeyev Editors •
Shell-like Structures Non-classical Theories and Applications
123
Prof. Dr.-Ing.habil. Dr.h.c. Holm Altenbach Lehrstuhl Technische Mechanik Institut für Mechanik Fakultät für Maschinenbau Otto-von-Guericke-Universität Magdeburg Universitätsplatz 2 D-39106 Magdeburg Germany e-mail:
[email protected]
Prof. Dr. Victor A. Eremeyev Lehrstuhl Technische Mechanik Institut für Mechanik Fakultät für Maschinenbau Otto-von-Guericke-Universität Magdeburg Universitätsplatz 2 D-39106 Magdeburg Germany and South Federal University and South Science Centre of RASci Rostov on Don Russia e-mail:
[email protected]
ISSN 1869-8433 ISBN 978-3-642-21854-5 DOI 10.1007/978-3-642-21855-2
e-ISSN 1869-8441 e-ISBN 978-3-642-21855-2
Springer Heidelberg Dordrecht London New York Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
Shell-like structures are used in Civil and Aero-space engineering as basic structural elements. As a model of analysis are also used such structures in other branches, e.g. Mechanical engineering, but also in new branches like Medicine and Biology. New applications are primarily related to new materials—instead of steel or concrete, now one has to analyze laminates, foams, functionally graded materials, nanofilms, biological membranes, soft tissues, etc. The new trends in applications demand the improvements of the theoretical foundations of shell theory, since new effects must be taken into account. For example, in the case of small-size shell-like structures (thin films, multiwalled nanotubes) the surface effect plays an important role in the mechanical analysis of these structural elements. Scientific meetings like conferences or colloquia are regularly organized by civil engineers, mathematicians, etc. Within the EUROMECH Colloquium 527 Shell-like structures – Non-classical Theories and Applications, which will bring together specialists from different areas, various items related to the colloquium title are discussed by international experts. The forum was addressed to scientists and researchers from industries. The focus was related to the following problems: new theories (based on two-dimensional field equations but describing non-classical effects), mathematical methods, e.g. the asymptotical analysis, new constitutive equations (for materials like sandwiches, foams, biological membranes, etc. and which can be combined with the two-dimensional shell equations), complex structures (folded, branching and/or self-intersecting shell structures, etc.) and shell-like structures on different scales (thin- and nanofilms, nanotubes, and nanoparticles) or very thin structures (similar to membranes, but with bending stiffness). In addition, coupled effects, phase transitions in shells and refined shell thermodynamics are discussed. This book contains papers submitted before the above mentioned colloquium. The reason for publication on such early stage was that a lot of the tentative colloquium participants gave a positive response on our call to publish their contribution in advance. By this way each participant is informed in some details about the forthcoming presentations at the beginning of the conference, which v
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Preface
stimulates a deeper scientific discussion. Finally, more than 50 papers were received and after reviewing 48 were included in this volume. It should be mentioned that some very long submissions (over 50 pages) were rejected and shorter versions was prepared for publication in this volume. The Full-length contributions will be published in the SpringerBriefs (http://www.springer.com/briefs) in 2011. This book deals with various items of the theories of shells, plates, beams, etc. In addition, traditional and new applications are presented. Its contents is split into 8 parts: • • • • • • • •
Mathematical Problems Dynamics and Stability Nonlinear Models and Coupled Fields Numerical Analysis Engineering Design Micro- and Nanomechanical Applications Biomechanics FGM and Laminated Plates and Shells
This monograph is not only based on the authors’ contributions. The publication and the conference were supported by different organizations and you can find some personal acknowledgements at the end of various papers. Here we acknowledge our main supporters: • • • • • •
Martin-Luther-Universität Halle-Wittenberg, Leucorea (Lutherstadt Wittenberg), Deutsche Forschungsgemeinschaft, Deutscher Akademischer Austauschdienst, Japan Society for the Promotion of Science (ID No. RC 21115001), Springer Publisher.
Only a few persons will be listed here: Dr. Christoph Baumann (Springer) for solving all our publishing problems just in time, Mrs. Barbara Renner for checking the language of some papers, Ms. Anna Girchenko, for writing a lot of letters, etc., Mr. Andreas Kutschke for solving technical and other problems. Last but not least we have to thank Prof. Henry Petryk (representative of the EUROMECH) and the members of the scientific committee (Profs Ren´e de Borst, Elena A. Ivanova, Reinhold Kienzler, Gennady I. Mikhasev, Wojciech Pietraszkiewicz, Paolo Podio-Guidugli) for their support. Prof. Dr. Holm Altenbach Prof. Dr. Victor A. Eremeyev
Contents
Part I Mathematical Problems 1
Nonclassical Spatial Boundary Value Problems of Statics and Dynamics of Shells and the Asymptotic Method of Their Solution . . . Lenser Abgar Aghalovyan
3
2
Analytical Solution for the Bending of a Plate on a Functionally Graded Layer of Complex Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Sergey Aizikovich, Andrey Vasiliev, Igor Sevostianov, Irina Trubchik, Ludmila Evich and Elena Ambalova
3
Analysis of the Deformation of Multi-layered Orthotropic Cylindrical Elastic Shells Using the Direct Approach . . . . . . . . . . . . . . . 29 Mircea Bˆırsan and Holm Altenbach
4
Asymptotic Integration of One Narrow Plate Problem . . . . . . . . . . . . . . 53 Valentina O. Finiukova and Alexander M. Stolyar
5
On Cusped Shell-like Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 George Jaiani
6
Effect of the Tangential Loads on the Bending of Elastic Plates . . . . . . 75 Kristine L. Martirosyan
7
On the Convergence of an Iteration Method in Timoshenko’s Theory of Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Jemal Peradze
8
Mathematical Models of Micropolar Elastic Thin Shells . . . . . . . . . . . . 91 Samvel H. Sargsyan
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Part II Dynamics and Stability 9
Closed-Form Approximate Solution for the Postbuckling Behavior of Orthotropic Shallow Shells Under Axial Compression . . . . . . . . . . . . 103 Matthias Beerhorst, Michael Seibel and Christian Mittelstedt
10 Nonlinear Magnetoelastic Waves in a Plate . . . . . . . . . . . . . . . . . . . . . . . . 125 Vladimir I. Erofeev, Alexey O. Malkhanov, Aleksandr I. Zemlyanukhin and Vladimir M. Catson 11 Basic Concepts in the Stability Theory of Thin-Walled Structures . . . 135 Ardeshir Guran and Leonid P. Lebedev 12 High-Frequency Free Vibrations of Plates in the Reissner’s Type Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Elena A. Ivanova 13 On the Reconstruction of Inhomogeneous Initial Stresses in Plates . . . 165 Rostislav D. Nedin and Alexander O. Vatulyan 14 Dynamic Response of Pre-Stressed Spatially Curved ThinWalled Beams of Open Profile Impacted by a Falling Elastic Hemispherical-Nosed Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Yury A. Rossikhin and Marina V. Shitikova 15 On Stability of Elastic Rectangular Sandwich Plate Subject to Biaxial Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Denis Sheydakov Part III Nonlinear Models and Coupled Fields 16 On the Nonlinear Theory of Two-Phase Shells . . . . . . . . . . . . . . . . . . . . . 219 Victor A. Eremeyev and Wojciech Pietraszkiewicz 17 A Gradient-Enhanced Damage Model for Viscoplastic Thin-Shell Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 An Danh Nguyen, Marcus Stoffel and Dieter Weichert 18 On Constitutive Restrictions in the Resultant Thermomechanics of Shells with Interstitial Working . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Wojciech Pietraszkiewicz 19 Free Finite Rotations in Deformation of Thin Bodies . . . . . . . . . . . . . . . 261 Leonid I. Shkutin 20 On Universal Deformations of Nonlinear Isotropic Elastic Shells . . . . 279 Leonid M. Zubov
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Part IV Numerical Analysis 21 Application of Genetic Algorithms to the Shape Optimization of the Nonlinearly Elastic Corrugated Membranes . . . . . . . . . . . . . . . . . . . . . . 297 Mikhail Karyakin and Taisiya Sigaeva 22 Advances in Quadrilateral Shell Elements with Drilling Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Stephan Kugler, Peter A. Fotiu and Justin Murin 23 Invariant-Based Geometrically Nonlinear Formulation of a Triangular Finite Element of Laminated Shells . . . . . . . . . . . . . . . . . . . . 329 Stanislav V. Levyakov 24 Consistency Issues in Shell Elements for Geometrically Nonlinear Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Teodoro Merlini and Marco Morandini 25 An Algorithm for the Automatisation of Pseudo Reductions of PDE Systems Arising from the Uniform-Approximation Technique . . . . . . . 377 Patrick Schneider and Reinhold Kienzler 26 Recent Improvements in Hu-Washizu Shell Elements with Drilling Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Krzysztof Wi´sniewski and Ewa Turska Part V Engineering Design 27 Dynamic Analysis of Debonded Sandwich Plates with Flexible Core – Numerical Aspects and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 Vyacheslav N. Burlayenko and Tomasz Sadowski 28 On Elasto-Plastic Analysis of Thin Shells with Deformable Junctions 441 Jacek Chr´os´cielewski, Violetta Konopi´nska and Wojciech Pietraszkiewicz 29 Thermal Stress and Strain of Solar Cells in Photovoltaic Modules . . . 453 Ulrich Eitner, Sarah Kajari-Schr¨oder, Marc K¨ontges and Holm Altenbach 30 Computational Models of Laminated Glass Plate under Transverse Static Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 Ivelin V. Ivanov, Dimitar S. Velchev, Tomasz Sadowski and Marcin Kne´c 31 Unbending of Curved Tube by Internal Pressure . . . . . . . . . . . . . . . . . . . 491 Alexei M. Kolesnikov
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32 Characterization of Polymeric Interlayers in Laminated Glass Beams for Photovoltaic Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 Stefan-H. Schulze, Matthias Pander, Konstantin Naumenko, Anna Girchenko and Holm Altenbach 33 On the Determination of Edge Reinforcement Properties for Optimum Lightweight Design of Composite Stiffeners . . . . . . . . . . . . . . 507 Philipp Weißgraeber, Christian Mittelstedt and Wilfried Becker Part VI Micro- and Nanomechanical Applications 34 Evaluation of the Mechanical Parameters of Nanotubes by Means of Nonclassical Theories of Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 Svetlana M. Bauer, Andrei M. Ermakov, Stanislava V. Kashtanova and Nikita F. Morozov 35 Gauss-Codazzi Equations for Thin Films and Nanotubes Containing Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 Svyatoslav Derezin 36 Effective Mechanical Properties of Closed-Cell Foams Investigated with a Microstructural Model and Numerical Homogenisation . . . . . . 549 Nina-Carolin Fahlbusch and Wilfried Becker 37 What Shell Theory Fits Carbon Nanotubes? . . . . . . . . . . . . . . . . . . . . . . 561 Antonino Favata and Paolo Podio–Guidugli 38 A Variationally Consistent Derivation of Microcontinuum Theories . . 571 Johannes Meenen, Holm Altenbach, Victor Eremeyev and Konstantin Naumenko 39 Shell-Models for Multi- L ayer Carbon Nano- Particles . . . . . . . . . . . . . . 585 Melanie Todt, Franz G. Rammerstorfer, Markus A. Hartmann, Oskar Paris and Franz D. Fischer Part VII Biomechanics 40 Mechanics of Biological Membranes from Lattice Homogenization . . 605 Mohamed Assidi, Francisco Dos Reis and Jean Franc¸ois Ganghoffer 41 Biological and Synthetic Membranes: Modeling and Experimental Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 Rasa Kazakevi˘ci¯ut˙e-Makovska and Holger Steeb 42 Nonclassical Theories of Shells in Application to Soft Biological Tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 Eva B. Voronkova, Svetlana M. Bauer and Anders Eriksson
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Part VIII FGM and Laminated Plates and Shells 43 Axisymmetric Bending Analysis of Two Directional Functionally Graded Circular Plates Using Third Order Shear Deformation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 Reza Akbari Alashti and Hossein Rahbari 44 Stability Analysis of Functionally Graded Plates Subject to Thermal Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 Mokhtar Bouazza, A. Tounsi, E. A. Adda-Bedia and A. Megueni 45 A Best Theory Diagram for Metallic and Laminated Shells . . . . . . . . . 681 Erasmo Carrera, Maria Cinefra and Marco Petrolo 46 In-Plane Strain and Stress Fields in Theories of Shearable Laminated Plates Subject to Transverse Loads . . . . . . . . . . . . . . . . . . . . 699 Giovanni Formica, Marzio Lembo and Paolo Podio-Guidugli 47 On the Use of a New Concept of Sampling Surfaces in Shell Theory . . 715 Gennady M. Kulikov and Svetlana V. Plotnikova 48 Theory of Thin Adaptive Laminated Shells Based on Magnetorheological Materials and Its Application in Problems on Vibration Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727 Gennady I. Mikhasev, Marina G. Botogova and Evgeniya V. Korobko
Part I
Mathematical Problems
Chapter 1
Nonclassical Spatial Boundary Value Problems of Statics and Dynamics of Shells and the Asymptotic Method of Their Solution Lenser Abgar Aghalovyan
Abstract Classical and improved theories of shells consider only one class of boundary value problems - on the facial surfaces of the shell the values of the corresponding stress tensor components are given. If on the facial surfaces other conditions - displacement vector or mixed boundary conditions of theory of elasticity - are given, a nonclassical boundary value problem of the theory of shells arises, which is important in some application cases. It is proved that the Kirchhoff-Love hypotheses of the classical theory of shells are not applicable for the solution of this class of problems. It is shown that such problems can be solved by the asymptotic method of solution of singularly perturbed differential equations. Non-contradictory asymptotic orders of the stress tensor components and a displacement vector are established. The iteration processes for the determination of all sought values with the beforehand given asymptotic accuracy are built. As special cases, the solutions of static and dynamic problems, illustrating the effectiveness of the asymptotic method, are presented. Nonclassical problems related to free and forced vibrations of isotropic and anisotropic shells are considered. The amplitudes of forced vibrations are determined. For layered shells the efficiency of the method is shown. Keywords Asymptotic method · Nonclassical boundary value problems · Shell theory · Layered shell
1.1 Introduction The classical theory of plates and shells considers one class of boundary value problems only. It is assumed that on the facial surfaces of the plate or the shell the values of the corresponding stress tensor components (first boundary value problem of L. A. Aghalovyan (B) Institute of Mechanics of National Academy of Sciences of Armenia, Armenia e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 1, © Springer-Verlag Berlin Heidelberg 2011
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L.A. Aghalovyan
elasticity theory) are given. Improved theories of plates and shells (in the sense of Reissner, Ambartsumian or Timoshenko) are devoted to this case, too. With respect to applications, particularly, in soil mechanics or foundation engineering, seismology, geophysics, in interaction of thin and rigid bodies and others, the problems ifon the facial surfaces of the plate or the shell other conditions—displacement vector or mixed conditions of elasticity theory - are given, are important. It is proved that the Kirchhoff-Love hypotheses of the classical theory are not applicable for the solution of these classes of problems [1]. Taking into account that plates and shells are structural elements—one of their geometrical dimensions (thickness) significantly differs from the others. Hence, it is possible, to correlate the three-dimensional problem of elasticity theory to dimensionless coordinates and components of the displacement vector and to obtain a system of equations containing a small dimensionless geometrical parameter. This system of differential equations turned out to be singularly perturbed by the small parameter. For its solution the asymptotic method is found to be effective. The solution of similar equations and systems is the combination of the solutions of the internal problem (I int) and the boundary layer (Ib ) [7, 8] I = I int + Ib
(1.1)
The asymptotic theory of isotropic shells (the first boundary problem of elasticity theory) was established by Goldenveizer [6] among others and for anisotropic plates and shells, for example, in [1]. Below we show the effectiveness of the asymptotic method for the solution of static and dynamic problems of anisotropic homogeneous and multi-layered shells considering the second or the mixed boundary value problem.
1.2 Second and Mixed Static Boundary Value Problem of Anisotropic Thermoelastic Shells Let us have an anisotropic shell of thickness 2h covering the domain D = {(α, β, γ) : (α, β) ∈ D0 , −h ≤ γ ≤ h}, where D0 is the middle surface, related to the lines of curvature α, β, the rectilinear axis γ is directed perpendicularly to the middle surface. The given volume forces with the component F j (α, β, γ) and temperature effects, which are taken into account by the Duhamel-Neumann model, acting on the shell. The change of the temperature field is characterized by the function θ(α, β, γ) = T (α, β, γ) − T 0 (α, β, γ), which is considered to be known. Anisotropy is assumed to be general and is described by 21 elasticity parameters. It is required to find the solutions of the equations and correlations of the spatial problem of elasticity theory in the domain D under the boundary conditions uα (−h) = u− (α, β), uβ (−h) = v− (α, β), uγ (−h) = u−γ (α, β)
(1.2)
1 Nonclassical Spatial Boundary Value Problems of Statics and Dynamics of Shells
5
and in one of the variants of the boundary conditions at γ = h a) σαγ (h) = σ+αγ , (α, β), σγγ (h) = σ+γγ (α, β), b) uα (h) = u+ (α, β), uβ (h) = v+ (α, β), uγ (h) = w+ (α, β),
(1.3) (1.4)
c) uα (h) = u+ (α, β), uβ (h) = v+ (α, β),
(1.5)
d) σαγ (h) =
σ+αγ (α, β),
+
σγγ (h) = σ+γγ (α, β),
uγ (h) = w (α, β)
(1.6)
and under the conditions on the lateral surface of the shell, which are not specified, but they result in the boundary layer. It is considered that the functions σ+jk , u± , v± , w± belong to the class C n , ∀n. In order to decrease the calculations, we use the components of the nonsymmetric stress tensor τi j : ταα = (1 + γ/R2)σαα , τββ = (1 + γ/R1)σββ , τγγ = (1 + γ/R1)(1 + γ/R2)σγγ , ταγ = (1 + γ/R2)σαγ , τβγ = (1 + γ/R1)σβγ , (1 + γ/R1)ταβ = (1 + γ/R2)τβα
(1.7)
ταβ = (1 + γ/R2)σαβ , τβα = (1 + γ/R1)σαβ , (the symmetry condition),
where σ jk are the components of the symmetric tensor. The following equilibrium and constitutive equations for an anisotropic body, taking into account temperature deformations, in the chosen system of coordinates are valid: Equilibrium equations ∂ταγ 2ταγ 1 ∂ 1 ∂ (Bταα ) − kβ τββ + (Aτβα ) + kα ταβ + (1 + γ/R1) + + Fα∗ = 0, AB ∂α AB ∂β ∂γ R1 ∂τβγ 2τβγ 1 ∂ 1 ∂ (Bτββ ) − kα ταα + (Bταβ ) + kβ τβα + (1 + γ/R2) + + Fβ∗ = 0, AB ∂β AB ∂α ∂γ R 2 ∂τγγ ταα τββ 1 ∂ταγ 1 ∂τβγ − + + + + kβ ταγ + kα τβγ + F γ∗ = 0, ∂γ R1 R2 A ∂α B ∂β F ∗j = (1 + γ/R1)(1 + γ/R2)F j (α, β, γ), j = α, β, γ (1.8) Constitutive equations (state equations) uγ γ 1 ∂uα γ 1+ + k α uβ + = 1+ (a11 ταα + a15ταγ + a16 ταβ ) R2 A ∂α R1 R1 γ + 1+ (a12τββ + a14τβγ ) + a13τγγ + α11 δ11 (γ)θ, R2 uγ γ 1 ∂uβ γ 1+ + k β uα + = 1+ (a12 ταα + a25ταγ + a26 ταβ ) R1 B ∂β R2 R1 γ + 1+ (a22τββ + a24τβγ ) + a23τγγ + α22 δ11 (γ)θ, R2
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L.A. Aghalovyan
∂uγ γ δ11 (γ) = 1+ (a13 ταα + a35ταγ + a36 ταβ ) ∂γ R1 γ + 1+ (a23 τββ + a34 τβγ ) + a33τγγ + α33 δ11 (γ)θ, R2 γ 1 ∂uα γ 1 ∂uβ 1+ − kβ u β + 1 + − k α uα R1 B ∂β R2 A ∂α γ = 1+ (a16 ταα + a56ταγ + a66 ταβ ) + a36τγγ R1 γ + 1+ (a26 τββ + a46 τβγ ) + α12δ11 (γ)θ, R2 ∂uα γ uα 1 γ ∂uγ δ11 (γ) − 1+ + 1+ ∂γ R2 R1 A R2 ∂α γ = 1+ (a15 ταα + a55ταγ + a56 ταβ ) + a35τγγ R1 γ + 1+ (a25 τββ + a45 τβγ ) + α13δ11 (γ)θ, R2 ∂uβ γ uβ 1 γ ∂uγ δ11 (γ) − 1+ + 1+ ∂γ R 1 R2 B R1 ∂β γ = 1+ (a14 ταα + a45ταγ + a46 ταβ ) + a34τγγ R1 γ + 1+ (a24 τββ + a44 τβγ ) + α23δ11 (γ)θ, R2 1 1 γ2 δ11 (γ) = 1 + γ + + R1 R 2 R1 R2
(1.9)
1 ∂A 1 ∂B where kα = AB ∂β , kβ = AB ∂α are geodesic curvatures, A, B are the coefficients of the first quadratic form, R1 , R2 are the principal radii of the middle (coordinate) surface curvature of the shell, ai j are the elasticity parameters, αi j are the coefficients of the thermal expansion. In order to solve the stated boundary problems, in Eqs (1.8) and (1.9) we introduce dimensionless coordinates and displacements by formulae
α = Rξ, β = Rη, γ = εRζ = hζ, uα = Ru, uβ = Rv, uγ = Rw,
(1.10)
where R is the characteristic dimension of the shell (least of radii of curvature and linear dimensions of the middle surface), ε = h/R is the small parameter. As a result we have the system, which is singularly perturbed with respect to the parameter ε. Its solution has the form of (1.1). The solution of the internal problem will be sought in the form of I int = εqI +s I (s) ,
s = 0, N
(1.11)
1 Nonclassical Spatial Boundary Value Problems of Statics and Dynamics of Shells
7
The summation by integer values from 0 to N, N is the number of approximations. The values of indexes q I should be determined in that moment, that after substitution (1.11) into the transformed, according to (1.10), equations (1.8), (1.9) and equating in each equation the coefficients with similar degrees of ε, we can get a non-contradictory system for determining desired values I (s) . Such a system was first time presented in [1] qσ = −1 for stresses,
qu = 0 for all displacements
(1.12)
Note that the asymptotics (1.11), (1.12) essentially differ from the asymptotics in the first boundary-value problem (classical theory) [6] and specific conclusion arises. The contribution of the volume forces and the temperature field will be comparable with the contribution of exterior forces, if the corresponding volumes allow the decompositions F α = ε−2+s F α(s) ,
(α, β, γ);
θ = ε−1+s θ(s) ,
s = 0, N
(1.13)
Substituting (1.11)-(1.13) into the transformed system (1.8), (1.9) and solving again the obtained system, we get: (s) (s) τ(s) αγ (ξ, η, ζ) = ταγ0 (ξ, η) + ταγ∗ (ξ, η, ζ),
(α, β)
(s) (s) τ(s) γγ (ξ, η, ζ) = τγγ0 (ξ, η) + τγγ∗ (ξ, η, ζ), (s) (s) (s) (s) τ(s) αα (ξ, η, ζ) = A13 τγγ0 + A14 τβγ0 + A15 ταγ0 + ταα∗ (ξ, η, ζ),
(α, β; 1, 2)
(s) (s) (s) (s) τ(s) αβ (ξ, η, ζ) = A63 τγγ0 + A64 τβγ0 + A65 ταγ0 + ταβ∗ (ξ, η, ζ),
(1.14)
(s) (s) (s) (s) u(s) (ξ, η, ζ) = ζ(A53 τ(s) γγ0 + A54 τβγ0 + A55 ταγ0 ) + u0 (ξ, η) + u∗ (ξ, η, ζ), (s) (s) (s) (s) v(s) (ξ, η, ζ) = ζ(A43 τ(s) γγ0 + A44 τβγ0 + A45 ταγ0 ) + v0 (ξ, η) + v∗ (ξ, η, ζ), (s) (s) (s) (s) w(s) (ξ, η, ζ) = ζ(A33 τ(s) γγ0 + A34 τβγ0 + A35 ταγ0 ) + w0 (ξ, η) + w∗ (ξ, η, ζ) (s) (s) (s) (s) (s) In Eqs (1.14) the functions τ(s) αγ0 , τβγ0 , τγγ0 , u0 , v0 , w0 , which will be uniquely determined by satisfying of the boundary conditions (1.2)-(1.6), are unknown yet. The values with star for every s are known, if the previous approximations are determined, particularly, we have ζ 1 ∂ 1 ∂ τ(s) (ξ, η, ζ) = − (Bτ(s−1) (Aτ(s−1) αγ∗ αα ) + βα ) + AB ∂η 0 AB ∂ξ ⎛ ⎞ ⎜⎜⎜ ∂τ(s−1) ⎟ αγ (s−1) (s−1) (s) (s−1) ⎟ ⎜ +R(kα ταβ − kβ τββ ) + RFα + r1 ⎜⎝ζ + 2ταγ ⎟⎟⎟⎠ + ∂ζ (s−1) (s−2) 2 +ζR(r1 + r2 )F α + ζ r1 r2 RFα dζ, (α, β; ξ, η; r1 , r2 ; A, B) (1.15) ζ (s) (s) (s) (s) u(s) (a15 τ(s) ∗ = αα∗ + a25 τββ∗ + a35 τγγ∗ + a45 τβγ∗ + a55 ταγ∗ + 0
(s) a56 τ(s) αβ∗ + α13 θ −
1 ∂w(s−1) (s−1) + r1 u(s−1) + ζ(a15r1 τ(s−1) αα + a25 r2 τββ + A ∂ξ
8
L.A. Aghalovyan
(s−1) (s−1) (s−1) (s−1) ∂u +a45r2 τ(s−1) + a r τ + a r τ + (r + r ) α θ − − 1 2 13 55 1 αγ 56 1 αβ βγ ∂ζ 1 ∂w(s−2) ∂u(s−2) r2 + r1 r2 u(s−2) ) + ζ 2 r1 r2 α13 θ(s−2) − ) dζ, A ∂ξ ∂ζ (s) (s) (s) τ(s) αα∗ (ξ, η, ζ) = B11 P1 + B12 P2 + B16 P3 ,
Q
(m)
r1 = R/R1 , r2 = R/R2
≡ 0 at m < 0 (s)
There are analogous formulae for the rest Q∗ [1]. In Eqs (1.14), (1.15) B jk = (a jl akl − a jk all )/Δ,
B jk = Bk j
Bll = (a j j akk − a2jk )/Δ,
( j k l j; j, k, l = 1, 2, 6) Alm = −a1m Bl1 − a2m Bl2 − a6m Bl6 , Amn Anm Anm = an1 A1m + an2 A2m + an6 A6m + anm , Δ=
(1.16)
(n, m = 3, 4, 5)
a11 a22 a66 + 2a12a26 a16 − a11 a226 − a22 a216 − a66 a212
The solution of the interior problem (1.11), (1.12), (1.14) permits us to satisfy all the variants of the boundary conditions (1.2), (1.3)-(1.6). Satisfying the conditions (1.2) we get (s) (s) −(s) u(s) = (1 + ζ)(A53τ(s) (ξ, η) − γγ0 + A54 τβγ0 + A55 ταγ0 ) + u (s) −u(s) ∗ (ζ = −1) + u∗ (ξ, η, ζ); (u, v, w; A5k , A4k , A3k ; k = 3, 4, 5) u−(0) = u− /R, u−(s) = 0, s 0
(1.17)
Satisfying the conditions (1.3) we get (s) +(s) (s) τ(s) = τ+(s) αγ (ζ = 1), (α, β), τγγ0 = τγγ − τγγ∗ (ζ = 1), αγ0
(1.18)
where +(1) +(s) + + τ+(0) αγ = εσαγ , ταγ = εr2 σαγ , ταγ = 0, s > 1 +(1) + + τ+(0) γγ = εσγγ , τγγ = ε(r1 + r2 )σγγ
τ+(2) γγ
= εr1 r2 σ+γγ , τ+(s) γγ
(1.19)
= 0, s > 2
Thus, the solution of the interior problem, corresponding to boundary conditions (1.2), (1.3), will be determined by Eqs (1.11), (1.12), (1.14), (1.17)-(1.19). From here it follows, that unlike the conditions of the first boundary value problem, the solution of the internal problem is uniquely determined by means of the function values, entering into the boundary conditions on the facial surfaces γ = ±h. This solution, as a rule, will not satisfy the conditions on the lateral surface of the shell. Rising misunderstanding is removed with the help of the solution for the boundary layer (Ib ). The values of the boundary layer, when removing from the lateral surface into the interior of the shell, decrease exponentially.
1 Nonclassical Spatial Boundary Value Problems of Statics and Dynamics of Shells
9
In general case the solution for the boundary layer is built and conjugated with the solution of the internal problem with the described in [1] method. Satisfying the conditions (1.4) we have τ(s) = (B55Vα(s) + B54Vβ(s) + B53Vγ(s) )/Δ, αγ0 (s) (s) (s) τ(s) βγ0 = (B45 Vα + B44Vβ + B43 Vγ )/Δ,
(1.20)
τ(s) = (B35Vα(s) + B34Vβ(s) + B33Vγ(s) )/Δ, γγ0 where 1 (s) Vα(s) = [u+(s) − u−(s) + u(s) ∗ (ζ = −1) − u∗ (ζ = 1)], 2 u±(0) = u± /R, u±(s) = 0, s 0, (α, β, γ; u, v, w), B jk = A jk All − A jl Akl , Bll = A jk Ak j − A j j Akk , B jk Bk j , j k l j, j, k, l = 3, 4, 5,
(1.21)
Δ = A33 B33 + A34 A43 + A35 B53 The solution of the internal problem, corresponding to boundary conditions (1.2), (1.4), will be determined by Eqs (1.11), (1.12), (1.14), (1.17), (1.20). In the analoguous way conditions (1.2), (1.5); (1.2), (1.6) are satisfied.
1.3 Solutions of Illustrative Problems On the base of the derived in Sect. 1.2 formulae we get solutions of some applied problems. A) Let the exterior surface of the lying horizontally orthotropic cylindrical shell be rigidly fastened, and inside pressure of the constant intensity P acts. The weight influence of the shell will be taken into account, too. Let α be the length of the generator, β is the length of the arch directing the cylinder counted from vertical (Fig. 1.1). Then A = B = 1, F α = 0,
r1 = 0,
r2 = 1,
F β = ρg sin ϕ,
kα = kβ = 0,
F γ = −ρg cos ϕ,
1 = 0, R1 ϕ = β/R
R2 = R, (1.22)
and uα (γ = h) = uβ (γ = h) = uγ (γ = h) = 0 σγγ (γ = −h) = −P, σαγ (γ = −h) = σβγ (γ = −h) = 0
(1.23) (1.24)
will be boundary conditions, where R is the radius of the middle surface of the cylinder, ρ is the density. It is considered that the temperature field does not change.
10
L.A. Aghalovyan
Fig. 1.1 Cylindrical shell
Using Eqs (1.11)-(1.19) and restricting the asymptotic expansion to O(ε2 ) exactly, which is enough for practical applications, according to (1.22) and conditions (1.23), (1.24), we get the following solution of the internal problem: σαγ = σαβ = 0,
uα = 0,
R−h h+γ (h + γ)2 P− A23 P + (h + γ)ρg cosϕ + A23 ρg cos ϕ, R+γ R+γ 2(R + γ) (h + γ)2 σβγ = (h + γ)ρg sinϕ − (2 + A23)ρg sin ϕ, 2R a12 A33 h − γ a12 a44 (h − γ)2 σαα = A13 σγγ − (2P − (3h + γ)ρg cosϕ) + ρg cos ϕ, Δ1 R + γ 2Δ1 R + γ R+γ A33 a11 h − γ σββ = A23 σγγ + (2P − (3h + γ)ρg cosϕ) R Δ1 R a44a11 h − γ γ − ρg cos ϕ + A23 (P − (h + γ)ρg cosϕ), (1.25) 2Δ1 R R 1 ρg uβ = a44 (h − γ)2ρg sin ϕ + (h − γ)(a44(1 + A23)(γ2 + 4γh + 7h2) 2 6R +A33 (γ − h)(γ + 5h)) sinϕ, 1 a44 A23 uγ = A33 (h − γ)(2P − (3h + γ)ρg cosϕ) − (h − γ)3 ρg cos ϕ 2 6R 1 − A33 (h − γ)((γ + 3h − 4A23h)P + 2A23h2 ρg cos ϕ), 2R A13 = −(a22a13 − a12a23 )/Δ1 , A23 = −(a11a23 − a12a13 )/Δ1 , σγγ = −
A33 = a13 A13 + a23 A23 + a33 ,
Δ1 = a11 a22 − a212
If the cylindrical shell has a finite length, the solution (1.25) will be just beginning with the distances from the end-wall of the cylinder, equal to the zone of extension
1 Nonclassical Spatial Boundary Value Problems of Statics and Dynamics of Shells
11
of the boundary layer. The boundary layer may be studied with the described in [1] method. B) The exterior one is free from loadings, the volume forces are absent. The shell is under the action of a variable temperature field. For the simplicity of the calculation we shall assume that the temperature field changes along the thickness of the shell linearly (θ = aγ + b). Using the same formulae and being restricted to the exactness of O(ε2 ), we get the solution: γ+h C22 [a(γ − h) + 2b], R+γ γ+h σαα = C11 (aγ + β) + A13C22 [a(γ − h) + 2b] 2(R + h) a12 γ−h − C33 [a(γ + h) + 2b], 2Δ1 R+γ γ+h σββ = C22 (aγ + b) + A23C22 [a(γ − h) + 2b] 2R a12 γ−h + C33 [a(γ + h) + 2b], (1.26) 2Δ1 R 1 uα = α13 (γ − h)[a(γ + h) + 2b], 2 1 1 uβ = α23 (γ − h)[a(γ + h) + 2b] + α23 (γ − h)[a(γ − h)(γ + 2h) + 3b(γ + 3h)], 2 6R 1 b uγ = C33 (γ − h)[a(γ + h) + 2b] + A33C22 (γ − h)(γ + 3h) 2 2R b a 2 − A23C33 (γ − h) + (A33C22 − A23C33 )(γ − h)2(γ + 2h), 2R 6R C11 = (a12 α22 − a22 α11 )/Δ1 , C22 = (a12α11 − a11α22 )/Δ1 , C33 = A13 α11 + A23 α22 + α33 σαγ = σβγ = σαβ = 0,
σγγ =
The results obtained in Sects 1.2 and 1.3 may be extended to two-layered and multilayered shells. The asymptotics (1.11), (1.12) are reserved for each layer, and the number of the layer must be attached to the solution (1.14) in the form of the index. Satisfying the boundary conditions on the facial surfaces and conditions of full contact between the layers, all the desired displacement vector components and the stresses tensor for each layer of the shell are determined [1, 2]. As illustration of the above described we present one solution for the two-layered shell. C) We have a two-layered orthotropic cylindrical shell with layers thicknesses h1 , h2 , respectively. The surface γ = h1 is rigidly fastened, and on the interior surface γ = −h2 the pressure of the constant intensity P acts. It is considered that the volume forces are absent, and the temperature field does not change. Ascribing the numbers of the layers index i, i = 1, 2 to the solution (1.14) and fulfilling all the required procedure, with exactness O(ε2 ), ε = h/R, h = max(h1, h2 ) we have the solution
12
L.A. Aghalovyan (i) (i) σ(i) αγ = σβγ = σαβ = 0,
(i) u(i) α = uβ = 0,
i = 1, 2,
R − h2 P (2) P − (A(i) 23 γ + A23 h2 ) R + γ , R+γ P (i) (i) (2) (i) σαα = −A13 (R − h2 + h2 A23 + γA23 ) R+γ σ(i) γγ = −
+
a(i) 12 Δ(i) 1
(A(i) γ − h1 A(1) ) 33 33
P , R+γ
(1.27)
(i) (2) (i) σ(i) ββ = −A23 (R − h2 − γ + h2 A23 + γA23 )
(i) P a11 (i) P + (i) (A33 γ − h1 A(1) ) , 33 R R Δ 1
u(i) γ
=
(1) (2) P −(A(i) 33 γ − h1 A33 )(R − h2 + h2 A23 ) R
1 P h1 P − (A(1) h21 − γ2 A(i) ) − A(1) (A(i) γ − h1 A(1) ) , 33 33 33 23 23 2 R R Δ(i) = a(i) a(i) − (a(i) )2 11 22 12
1.4 Non-Classical Spatial Dynamic Problems for Shells The asymptotic method is effective for the investigation of forced and free vibrations of shells, as well. Below we consider two types of forced vibrations: 1. Forced vibrations caused by the conditions at γ = −h uα (−h) = u− (α, β) exp(iΩt), uγ (−h) = w− (α, β) exp(iΩt)
uβ (−h) = v− (α, β) exp(iΩt) (1.28)
Simulating, particularly, seismic actions on the shell, where Ω is the frequency of the forcing actions. At γ = h the conditions of free or rigidly fixed surface are given: ταγ (h) = τβγ (h) = τγγ (h) = 0 (1.29) or uα (h) = uβ (h) = uγ (h) = 0
(1.30)
2. Forced vibrations of the shell with rigidly fastened exterior surface under the action of pulsing loading uα (γ = −h) = 0, uβ (γ = −h) = 0, uγ (γ = −h) = 0 σ jγ (γ = h) = σ+jγ (α, β) exp(iΩt), j = α, β, γ
(1.31) (1.32)
In order to solve the formulated dynamic problems, in the dynamic equations of three-dimensional problem of elasticity theory, obtained from Eqs (1.8), (1.9), substituting in the equations of equilibrium (1.8) Fα∗ , Fβ∗ , F γ∗ into
1 Nonclassical Spatial Boundary Value Problems of Statics and Dynamics of Shells
γ γ ∂2 u j F j = −ρ 1 + 1+ , R1 R2 ∂t2
j = α, β, γ
13
(1.33)
We again pass to dimensionless coordinates and displacements (1.10) and the solution of the transformed system is sought in the form of Qαβ = Q jk (ξ, η, ζ) exp(iΩt),
(α, β, γ), j, k = 1, 2, 3,
(1.34)
where Qα,β is any of the stresses and displacements. As a result we again obtain a singularly perturbed system, the solution of which has the form of (1.1). The solution of the internal problem is sought in the form of −1+s (s) τint τ jk (ξ, η, ζ), j, k = 1, 2, 3; s = 0, N, jk (ξ, η, ζ) = ε
(1.35)
(u (ξ, η, ζ), v , w ) = ε (u (ξ, η, ζ), v , w ) int
int
int
s
(s)
(s)
(s)
Substituting (1.35) into the according to (1.10), (1.34) transformed equations, we get a recurrent system, from where all the stresses may be expressed through the displacements [4, 5]. For determining the displacements in the case of orthotropic shells, we have the equations ∂2 u(s) (s−1) + a55 Ω2∗ u(s) = Fu (u, v; a55, a44 ), ∂ζ 2 ∂2 w(s) Δ 2 (s) + Ω w = F w(s−1) , Ω2∗ = ρh2 Ω2 , 2 Δ12 ∗ ∂ζ
(1.36)
where F u(s−1) , Fw(s−1) are known functions [5], and Δ = a11 Δ23 + a13Δ2 + a12 Δ1 , Δi j = aii a j j − a2i j , i, j = 1, 2, 3, Δ1 = a13 a23 − a33 a12 , Δ2 = a12 a23 − a22a13
(1.37)
The solution of the system (1.36) is (s)
(s)
u(s) (ξ, β, γ) = C1 (ξ, η) sin δu ζ + C2 (ξ, η) cos δu ζ + u(s) (ξ, η, ζ), v(s) (ξ, β, γ) = C3(s) (ξ, η) sin δv ζ + C4(s) (ξ, η) cos δv ζ + v(s) (ξ, η, ζ), w (ξ, β, γ) (s)
(1.38)
= C5(s) (ξ, η) sin δw ζ + C6(s) (ξ, η) cos δw ζ + w(s) (ξ, η, ζ)
where u(s) , v(s) , w(s) are special solutions of equations (1.36). In dynamic problems it is necessary to build the solution for the boundary layer as well. It is possible to use the formulae, presented in [5]. Having expressions for stresses and displacements, without any difficulties boundary conditions (1.28) (1.32) are satisfied. It is possible to show that in case of problem (1.28), (1.29) the solution will be finite, if cos 2δu 0,
cos 2δv 0,
cos2δw 0
(1.39)
14
L.A. Aghalovyan
Those values Ω, for which conditions (1.39) or the analogous conditions for the rest of the boundary conditions are not fulfilled, coincide with the values of the frequencies of free vibrations [3]. With these values of frequencies Ω of the exterior action resonance will rise.
References 1. Aghalovyan, L.A.: Asymptotic Theory of Anisotropic Plates and Shells (in Russian). Moscow, Nauka (1997) 2. Aghalovyan, L.A., Gevorgyan, R.S.: Nonclassical Boundary-value Problems of Anisotropic Layered Beams, Plates and Shells. Publishing House of the National Academy of Sciences of Armenia. Yerevan (2005) 3. Agalovyan, L.A., Gulghazaryan, L.G.: Asymptotic solutions of non-classical boundary-value problems of the natural vibrations of orthotropic shells. J. Appl. Math. Mech. 70, 102–115 (2006) 4. Agalovyan, L.A., Gulghazaryan, L.G.: Forced vibrations of orthotropic shells: nonclassical boundary-value problems. Int. Appl. Mech. 45(8), 105–122 (2009) 5. Aghalovyan, L.A., Gevorgyan, R.S., Ghulghazaryan, L.G.: The asymptotic solutions of 3D Dynamic problems for orthotropic cylindrical and toroidal shells. Proc. National Academy of Sciences of Armenia. Mechanics. 63(1), 6–22 (2010) 6. Gol’denveizer, A.L.: The Theory of Thin Elastic Shells (in Russian). Nauka, Moscow (1976) 7. Lomov, S.A.: Introduction in General Theory of Singular Perturbations (in Russian). Nauka, Moscow (1981) 8. Nayfeh, A.H.: Perturbations Methods. Wiley, New York (1973)
Chapter 2
Analytical Solution for the Bending of a Plate on a Functionally Graded Layer of Complex Structure Sergey Aizikovich, Andrey Vasiliev, Igor Sevostianov, Irina Trubchik, Ludmila Evich and Elena Ambalova
Abstract The problem of interaction between an asymmetrically loaded thin circular plate and a supporting elastic foundation is reduced to the solution of system of the dual integral equations for the unknown normal contact pressure and differential equation of plate bending. The supporting medium is an isotropic elastic functionally graded layer of complex structure of constant thickness lying, with or without friction, on a homogeneous elastic half-space. In designing raft foundations, prediction of differential settlements and of bending moments induced in the foundation is of great importance. It is common practice to assume that the raft foundation behaves like a thin elastic plate. Development of analytical methods of the solution of such problems for a foundation with elastic properties varying in-depth is based on the numerical construction of transforms for the kernels of integral equations of the corresponding contact problems [1]. For the construction of the approximate analytical solution of the corresponding integral equations the bilateral asymptotic method is used [2]. This method is found to be effective for rigid, as well as for flexible plates, in contrast to the method of orthogonal polynomials, collocation method, or asymptotic methods (of small or large parameter). Representation of plate deflections in the form of series by their own oscillations [3] is used under corresponding boundary conditions. It is possible to reduce the specified problems to the solution of the systems of linear algebraic equations. We provide the analysis of influence of various laws of variation of the elastic properties of the coating on the distribution of contact pressures under the circular plate; its deflection; surface settlement of the functionally-graded layer outside of plate; S. Aizikovich (B) · A. Vasiliev · I. Trubchik · L. Evich · E. Ambalova Department of Scientific Research, Don State Technical University, Gagarin sqr., 1, Rostov-on-Don, 344000, Russia e-mail:
[email protected] I. Sevostianov Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 880003, USA e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 2, © Springer-Verlag Berlin Heidelberg 2011
15
16
S. Aizikovich et al.
and value of the radial and tangential moments in the thin circular plate. The developed method allows one to construct analytical solutions with prescribed accuracy and gives the opportunity to conduct multiparametric and qualitative analysis of the problem with minimal computation time. Keywords Plate · FGM foundation · Dual integral equations
Introduction For the behavior of the supporting soil, various models have been proposed and applied within the framework of linear elasticity. Well known of them are: Winkler springs, half-space continuum, and layered continuum. In this paper we discuss interaction of an axisymmetrically loaded thin circular Kirchhoff plate and functionally graded layer coupled with elastic half space. This elasticity problem models the actual problem of circular raft foundations which are used to support large cylindrical structures such as chimneys, silos, water tanks, etc. Many attempts have been made to treat them approximately using various techniques [4]– [15], since it is impossible to obtain closed form analytical solutions for interaction problem of this type. The aim of this paper is to develop an effective approximate analytical method for solution of this problem using integral transform. Present work assumes that a variation of Lame coefficients in the isotropic elastic functionallygraded layer has general form, i.e. they are arbitrarily continuous or piecewise continuous functions vertical coordinate. Friction between plate and the foundation is neglected. It is assumed that all deformations are elastic.
2.1 Interaction of a Thin Circular Plate with a Non-Homogeneous Half-Space. Statement and Formulation of the Problem An elastic layer of constant thickness H is lying, with or without friction, on a semiinfinite elastic base (Fig. 2.1).
Fig. 2.1 Axisymmetrically loaded circular plate on a functionally-graded halfspace
2 Analytical Solution of the Bending of Plates
17
A circular plate of radius R and constant thickness h governed by the KirchhoffLove thin plate theory rests on the surface z = 0 of the elastic layer. Under the action of an axisymmetrically distributed load of intensity p(r) and of the reactive normal contact pressure q(r) of the elastic layer, the plate is deflected with the deflection shape described by function w(r). Assuming no loss of contact at the interface between the plate and the elastic layer, the surface normal displacement of the elastic layer is also described by the function w(r). The surface z = 0 is free of tangential stresses while outside the contact region it is also free of normal stresses. It is required to determine the function w(r), the normal contact pressure distribution q(r) and the bending moments induced in the plate. The mathematical treatment of the stated axisymmetric interaction problem involves solution of two coupled boundary value problems, one for the bending of the thin circular plate with free-edge boundary conditions and the other for the stress and displacement fields in the foundation. Lam´e coefficients Λ and M of the half-space vary with depth as follows: Λ = Λ0 (z), M = M0 (z), − H z 0 Λ = Λ1 = Λ0 (−H), M = M1 = M0 (−H), − ∞ < z < −H
(2.1)
We use static equilibrium equations of the theory of elasticity written in the displacements in the case of axially symmetric deformation for the cylindrical coordinate system. It is required to determine the distribution of contact pressure q(r ) under the plate and the shape of the deflection w(r ). According to accepted assumptions, the considered problem is reduced to solution of the following system of equations Dw(r) = p(r) − q(r), 0 r 1 ∞
(2.2)
∞ Q(α)L(αλ)J0 (αr)dα = sλw(r), 0 r 1;
0
Q(α)J0 (αr)dα = 0, r > 1 (2.3) 0
where D is differential operator of the plate bending in cylindrical coordinate system, p(r) is external loading, and q(r) stays for contact pressures under the plate. Dual integral equation (2.3) establishes the connection between contact pressures and the bending profile of the plate lying on elastic functionally-graded coating of the elastic half-space. Here 1 Q(α) =
q(ρ)J0 (αρ)ρdρ; 0
q(r) = (2π)
−1
∞ Q(α)J0 (αr)αdα,
(2.4)
0
λ = H/R is dimensionless geometrical parameter, s = ΘR3 D−1 — parameter which describes deflection rigidity of the plate, D — bending rigidity of the plate Θ = 2M(z) (Λ(z) + M(z))/ (2M(z) + Λ(z)), r = r /R. Function w(r) should satisfy traction-free boundary conditions, 2
d w ν dw
d
+ = 0, (w) (2.5)
r=1 = 0, dr dr2 r dr r=1
18
S. Aizikovich et al.
where ν is the Poisson’s ratio of the plate, and — Laplace operator (written in cylindrical coordinate system). Construction of kernel’s transform L(u) in the general case of properties variation across the functionally-graded layer is considered in [16]. When the following conditions are satisfied: min Θ(z) c1 > 0,
z∈(−∞;0]
max Θ(z) c2 < ∞, lim Θ(z) = const z→−∞
z∈(−∞;0]
(2.6)
it can be demonstrated [17], that the kernel transform has the following properties: L(u) = A + B |u| + O(|u|2 ), u → 0; L(u) = 1 + C |u|−1 + O(|u|−2 ),
A = lim Θ(0)/Θ(z), z→−∞
u → ∞;
B, C = const.
(2.7) (2.8)
2.2 Construction of the Solution We will utilize the function of bending in the form of series expansion with respect to own bendings of a circular plate with a free boundary [3] w(r) =
∞
1 wm ϕm (r),
wm =
m=0
ϕ0 (r) =
√
2,
w(ρ)ϕm (ρ)ρdρ,
(2.9)
0
ϕm (r) = Am J0 (km r) − J1 (km )I1−1 (km )I0 (km r) ,
Values Am and km for m = 0, 1, . . . , 10 are given in [3]. Taking into account linearity of problem, the reactive normal contact pressure are represented as linear combination of special solutions with the same coefficients wm , as for bending function w(r) in (2.9): q(r) =
∞
wm qm (r),
0 r 1.
(2.10)
m=0
We will determine special solution qm (r) (m = 0, 1, . . .) from integral equation (2.3) similarly to [1], [16]. We assume that function L(u) in (2.3) belongs to the class ΠN (Σ M , SN, M ), if it has the form: ⎧N ⎪ 2 2 −2 2 2 −2 −1 ⎪ ⎪ ⎪ (α + ai λ )(α + bi λ ) ≡ LN (λα) ∈ ΠN , ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎨ M L(λα) = ⎪ ⎪ ck λ−1 | α | (α2 + dk2 λ−2 )−1 ≡ LΣM (λα) ∈ Σ M , ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ LN (λα) + LΣ (λα) ∈ SN, M . M
2 Analytical Solution of the Bending of Plates
19
Here ai , bi (i = 1, . . . , N), ck , dk (k = 1, . . . , M) — certain complex constants (ai − ak )(bi − bk ) 0, k i. It has been proved [1], that provided the function L(u) possesses properties (2.7), (2.8), it admits approximation by expressions of type L(λα) = LN (λα) + LΣ∞ (λα).
(2.11)
Using (2.11), eq. (2.3) can be written in terms of the operator in the form Π N q + Σ∞ q = f.
(2.12)
Definition 2.1. Condition A will be said to hold for equation (2.12) if when one can construct a closed solution for it by following [18]. We denote it by q N = Π−1 N f.
(2.13)
Condition A means that for functions f (x) in certain class W(c, d), there exists a function q(x) in some class V(c, d) such that (2.13) holds. It follows from representation (2.13), that q N V(c, d) m(ΠN ) f W(c, d) ,
m(ΠN ) = const.
(2.14)
Let us substitute in the right part of (2.3) m-eigenfunction ϕm (r) instead of w(r). The closed approximate solution of the equation (2.3) of type (2.13) has the following form: ⎡ ⎤ N −1 ⎥⎥⎥ √ −1 ⎢⎢⎢ −1 N 0 −1 q0 (r) = 2 2π λs ⎢⎢⎣⎢LN (0) 1 − r2 + Ci Ψ (r, ai λ )⎥⎥⎦⎥ , (2.15) i=0
⎡ ⎢⎢
⎢ N qm (r) = 2π−1 Am λs ⎢⎢⎢⎢⎣L−1 N (λkm ) f (r, km ) J1 (km )I1−1 (km )L−1 N (iλkm )Ψ(r, km ) +
−
N j=0
1 ch A sh Atdt Ψ(r, a) = √ −a ; √ t2 − r 2 1 − r2 r
⎤ ⎥⎥
⎥ Cim Ψ (r, iAi λ−1 )⎥⎥⎥⎥⎦ ,
(2.16) m = 1, 2, . . . ,
1 cos ε sin tεdt f (r, ε) = √ +ε . √ t2 − r 2 1 − r2 r
(2.17)
Constants Cim are determined from the following systems of linear algebraic equations N i=1
−1 Ci0 α(ai λ−1 , bk λ−1 ) + L−1 N (0)λbk = 0,
k = 1, 2, . . . , N,
(2.18)
20
S. Aizikovich et al. N Cim α(ai λ−1 , bk λ−1 )+β(bk λ−1 , km )−J1 (km )I1−1 (km )γ(bk λ−1 , km ) = 0, i=1
k = 1, 2, . . . , N;
(2.19)
m = 1, 2, . . . ,
α(a, b) = (a sh a + b ch a)(b2 − a2 )−1 , 2 2 −1 β(μ, b) = (b cosμ − μ sin μ)L−1 N (λμ)(b + μ ) , 2 2 −1 γ(k, b) = (k sh k + b ch k)L−1 N (iλk)(b − k ) . N In turn, contact pressures qm (r) can be presented as series (2.9):
N qm (r) =
∞
1 ymj ϕ j (r),
ymj
j=0
=
N qm (ρ)ϕ j (ρ)ρdρ.
(2.20)
0
Let’s write the conditions required to present f (r) in the form (2.20). We use the following definitions: Definition 2.2. We define C k (0, 1) as the space of functions having continuous derivatives on an interval [0, 1]. Definition 2.3. We define L2 (0, 1) as the space of functions f , defined on interval 0 r 1 , measurable on this interval by Lebesgue’s norm, and such, that 1 | f (r)|2 dr < ∞, 0
where integral is taken in the Lebesgue sense. Then, the following theorem takes place [19]. Theorem 2.1. Let function f ∈ C 4 (0, 1) satisfies boundary conditions (2.5). Then representation (2.9) on [0, 1] takes place, and a series converges on [0, 1] evenly. More general expansion and completeness theorem in L2 (0, 1) also takes place [19]. Theorem 2.2. If f ∈ L2 (0, 1), than equation (2.9) is satisfied, and it is understood in the sense that 1 k lim f − fm ϕm (r) = 0, fm = f (ρ)ϕm (ρ)ρdρ. (2.21) k→∞ 2 0 m=0 L (0,1)
Besides, Parseval’s equality takes place f 2L2 (0, 1) =
∞
fm2 .
(2.22)
m=0
We assume that function p(r) can be presented in the form of series expansion
2 Analytical Solution of the Bending of Plates ∞
p(r) =
21
pm ϕm (r),
1
pm =
p(ρ)ϕm (ρ)ρdρ.
(2.23)
0
m=0
Then, substituting decompositions (2.9), (2.10), (2.23) into (2.2), we obtain infinite system of linear algebraic equations for coefficients wm : −4 wm − akm
∞
−4 w j E mj = pm km ,
m = 0, 1, 2, . . . ;
a = −1,
(2.24)
j=0
⎡
⎤
N ⎢⎢⎢ ⎥⎥⎥ m −1 −1 m E mj = 2π−1A j λs ⎢⎢⎢⎣L−1 (λk )F −J (k )I (k )L (iλk )R + Cnm S nm ⎥⎥⎥⎦, j 1 m 1 m N j j N j n=0
F mj = Rmj =
1
0
S nm =
f (ρ, k j )ϕm (ρ)ρdρ = Am [x(k j , km ) − J1 (km )I1−1 (km )X(km, k j )],
(2.26)
Ψ (ρ, k j )ϕm (ρ)ρdρ = Am [X(k j , km ) − J1 (km )I1−1 (km )x(ik j , ikm )],
(2.27)
0 1
0
(2.25)
1
Ψ(ρ, an λ−1 )ϕm (ρ)ρdρ =Am [X(anλ−1, km )−J1 (km )I1−1(km )x(an λ−1, ikm )].
Here j = 1, 2, . . . ,
(2.28) m = 0, 1, 2, . . .;
x(a, b) = b−1 (cosa sin b+a2−1 (sin (a−b)(a−b)−1 −sin(a+b)(a+b)−1 )), X(a, b) = (a2 + b2 )−1 (a sh a cos b + b sinb ch a).
(2.29)
In particular, ⎡ ⎤ N ⎢⎢⎢ ⎥⎥⎥ √ −1 −1 −1 E0m = 2 2π−1 λs ⎢⎢⎢⎣ L−1 Cnm S nm ⎥⎥⎥⎦ , N (0)Am km (sin km − J1 (km )I1 (km ) sh km ) +2 n=1
⎡ N ⎤ ⎥⎥⎥ √ −1 ⎢⎢⎢ √ 0 0 0 ⎢ E0 = 2π λs ⎢⎢⎣2+ Cn S n ⎥⎥⎥⎦ , S n0 = 2(an λ−1 sh (an λ−1 )− ch (an λ−1 )).
(2.30) (2.31)
n=1
We have
∞
−4 2 |pm km | M
m=0
∞
|pm|2 < ∞
(2.32)
m=0
at the restrictions imposed on p(r). From estimation (2.14) and condition of equilibrium of plate, 2π1
2π1 q(r)rdrdϕ =
0 0
p(r)rdrdϕ 0 0
(2.33)
22
S. Aizikovich et al.
We have ∞ m=0
−8 km
∞
|E mj |2
j=0
∞ ∞
|E mj |2 M(ΠN , Σ∞ )
m=0 j=0
∞
|pm|2 < ∞.
(2.34)
m=0
It can be solved by reduction method [20] (by replacement with system from n equations with n unknowns): −4 wm − akm
n−1
−4 w j E mj = pm km ,
m = 0, 1, 2, . . ., n − 1,
a = −1,
(2.35)
j=0
Besides, if n is sufficiently large, system (2.35) is solvable and the convergence of approximated solutions to the exact one takes place. When coefficients wm (m = 0, 1, . . ., n) are determined for the fixed value, equation (2.35) yields the contact pressure n q(r) = wm qm (r), 0 r 1, (2.36) m=0
and plate deflections w(r) =
n
wm ϕm (r),
0 r 1.
(2.37)
m=0
Practical calculations require the values of radial Mr and tangential moments Mϕ in plate. Let us write out their expressions in terms of the plate deflections: 2 D d 2 w ν dw D d w 1 dw Mr = − 2 + , Mϕ = − 2 ν 2 + . (2.38) r dr R dr2 r dr R dr Using expressions for plate deflections (2.9), we receive: Mr = Mϕ =
M D
R2
m=0
M D
R2
2 wm Am km
2 wm Am km
m=0
where Vi (km r) = Ji (km r) + Bm Ii (km r),
ν−1 V1 (km r) + V0 (km r) , km r
1−ν V1 (km r) + νV0 (km r) , km r
i = 0, 1.
(2.39)
2 Analytical Solution of the Bending of Plates
23
2.3 Determination of the Surface Displacement Outside of the Plate Equation (2.3) has been derived as a result of the following representation of vertical displacement of the surface foundation: 1 f (r) = sλ
∞ Q(α)L(λα)J0 (αr)dα.
(2.40)
0
Below we derive analytical expression for function f (r) in (2.40), when L(α) ∈ ΠN , r > 1. For L(α) ∈ Π N function QN (α) is received in analytical form in the process of solving equation (2.3). Using formula (2.40), we receive expression for function f (r), r > 1, if L(α) ∈ ΠN we designate it as f N (r). According to (2.9), function QN (α) can be written as ∞ N QN (α) = w m Qm (α). (2.41) m=0
Corresponding expressions for functions fmN (r) are given by ⎧ ⎪ ⎪ ⎪ ⎨ fmN (r) = 2π−1 Am ⎪ G(r, km ) − BmG(r, ikm )+ ⎪ ⎪ ⎩ ⎡ ⎤⎫ N N ⎢⎢ ⎥⎥⎪ ⎪ ⎪ −1 ⎢⎢⎢ m −1 ⎥⎥⎥⎬ + Dn bn λ In ⎢⎢⎣Ψn (km )−BmΨn (ikm )+ C j γn (ia j λ )⎥⎥⎦⎪ , ⎪ ⎪ ⎭ n=1 j=1 r > 1,
m = 0, 1, 2, . . .,
2 −2 2 Ψn (a) = L−1 N (λα)γn (a), γn (a) = a sin a/(bn λ + a ),
r In = 1
exp[λ−1 bn (t − 1)] dt, √ r 2 − t2 Dn = (a2n b−2 n − 1)
Finally f N (r) =
M
1 0 N $ −b2n + a2i
i=1,in
−b2n + b2i
wm fmN (r),
r > 1,
where wm are the same magnitudes as those in (2.9).
(2.43)
cos at dt, √ r2 − t2
G(r, a) =
m=0
(2.42)
.
(2.44)
24
S. Aizikovich et al.
2.4 Numerical Examples For illustration, let us consider bending of a circular plate by uniformly distributed load of unitary intensity. The plate is impressed into a half-space; variation of the Young’s modulus with depth is described by E(z) = E 0 φ(z),
−1 z 0,
E(z) = E 0 φ(−1),
(2.45)
−∞ < z < −1,
and Poisson’s ratio of the foundation ν = 1/3, −∞ < z 0. Poisson’s ratio of plate νn = 0,15. For uniquely distributed load, the conditions of the Theorem 2.1 (p(r) √ =p= 1) are fulfilled, decomposition coefficients (2.10) are given by: p0 = p 2, pm = 1, m = 1, 2, . . .. Let’s consider the following types of non-homogeneity: 1) monotonic (power): a) increasing with depth φ1 (z) = 0,1 + z2ak ,
ak =
ln 0,1(k − 1) , 2 ln 0,5
k = 2,
b) decreasing with depth φ2 (z) = 1,1 − z2ak ,
ak =
ln(1, 1 − k/10) , 2 ln 0,5
k = 9,
2) non-monotonic φ3 (z) = 1,1 + sin(πz),
φ4 (z) = 0,1 − sin(πz).
Figures 2.2, 2.5 presents graphical diagrams for parameter χ(r) = q N (r)/q0 (r) characterizing distribution of normal contact pressure under the plate on non-homogeneous foundation in comparison with homogeneous q0 (r) (for E = E 0 φ(−1)) at various values of λ.
Fig. 2.2 Graphical diagrams of the distribution of normal contact pressures under the plate on non-homogeneous foundation in comparison with homogeneous one. a) s = 0, 1, b) s = 3
y 1.0 φ(y) 0.0 0.1 ϕ1 -1.0
6
τ0
16
8
8
4
2
5
0
16 2
4 2
τ0
0.5 a)
1
r
2
4
0
8 0.2
0.5
0.5 b)
1
r
2 Analytical Solution of the Bending of Plates
Fig. 2.3 Graphical diagrams of the distribution of normal contact pressures under the plate on non-homogeneous foundation in comparison with homogeneous one. a) s = 0, 1, b) s = 3
y 1.0 φ(y) 0.0 0.1 ϕ2 -1.0
Fig. 2.5 Graphical diagrams of the distribution of normal contact pressures under the plate on non-homogeneous foundation in comparison with homogeneous one. a) s = 0, 1, b) s = 3
τ0
6
τ0
8
16 8 4 0.2
16
4 2
5
8
0
2 0.2
τ0
1.25
0.5
1
r
2
0
τ0
0.2
1
0.5
8
1
0
4
0.5 b)
2
8
0.5
0
1
r
0
1.5
τ0 4 16 2
1 0
2
8
0
0.5 b)
0.5
0
0.5
a)
1
r
0
1
r
1
r
τ0 4 16
1
0.2
r
2
a)
y 1.0 0.0 0.1 φ(y) -0.5 ϕ4 -1.0
1
0.2
0.5 16 4
0.5
16
0.25
0.5
4 0.5
a)
y 1.0 0.0 0.1 φ(y) -0.5 ϕ3 -1.0
Fig. 2.4 Graphical diagrams of the distribution of normal contact pressures under the plate on non-homogeneous foundation in comparison with homogeneous one. a) s = 0, 1, b) s = 3
25
0.2
0
0.5
8
2
0.5 b)
The digits curves correspond to the values of λ for which the calculation was conducted. Values q N (r), q0 (r) are found with formula (2.13) at N = 10, M = 10. In fig. 2.2 χ(r) corresponds to φ1 (z), in fig. 2.3 — φ2 (z), in fig. 2.4 — φ3 (z), and in fig. 2.5 — φ4 (z). One might conclude then, that, in the case of monotonously decreasing Young’s modulus described by φ2 (z) the decrease of coefficient is observed for contact pressure as compare to its value for homogeneous half-space (fig. 2.5). In such case the formulation of the problem must be modified. The contact zone of plate with foundation can be determined from condition of vanish of contact pressures on the boundary of contact. The lift-off area extends at flexural rigidity increase of plate (fig. 2.5b). It is typical for non-monotonic variations φ3 (z) and φ4 (z) that in case when φ(z) increases with depth from surface (φ3 (z)), the increase of the value χ(r), is observed. In this case, plate separation from of the foundation does not take place.
26
S. Aizikovich et al. y 1.0 φ(y) 0.0 0.1 ϕ1 -1.0
Fig. 2.6 Relative settlement of the surface of the inhomogeneous half-space as compared to the homogeneous half-space. a) s = 0, 1, b) s = 3
2
4
Δ(r) 1
16 8
0.5
8
0.5
0.5
0.5
0.2
0
y 1.0 φ(y) 0.0 0.1 ϕ2 -1.0
0
r
4
2 a)
0
0
2
3
8
4 2
8
r
4
2
Δ(r)
1
0
2
2
16 8
4
1
0.5
0.2 0.5
2
2
2
4
r
0
0
0.5 b)
a) Δ(r)
2
0.5 0.2
1
2 4
0
2 a)
4
r
0
r
4
r
4 0.5 8
16
1
Δ(r)
1
8
0.5 0
r
Δ(r)
8
0.2 4
4
b)
16
1
y 1.0 0.0 0.1 φ(y) -0.5 ϕ4 -1.0
16
0.5 0.2
0.2
0
r
2
2 0.5
2
4
4
a)
0
4
Δ(r)
0
Fig. 2.8 Relative settlement of the surface of the inhomogeneous half-space as compared to the homogeneous one. a) s = 0, 1, b) s = 3
2
b)
16
0
y 1.0 0.0 0.1 φ(y) -0.5 ϕ3 -1.0
16
0.2
Δ(r)
10 5
Fig. 2.7 Relative settlement of the surface of the inhomogeneous half-space as compared to the homogeneous half-space. a) s = 0, 1, b) s = 3
Fig. 2.9 Relative settlement of the surface of the inhomogeneous half-space as compared to the homogeneous one. a) s = 0, 1, b) s = 3
Δ(r)
1
0
0.2 2
16
2 b)
If φ(z) decreases with depth from surface (φ4 (z)), the decrease of the value χ(r) is observed at approach from within to boundary of plate. The form of curves χ(r) depends on the values of λ, and s. One can see from Figs 2.2–2.5 that distribution of contact pressures essentially depends on the thickness of non-homogeneous layer, type of non-homogeneity, and rigidity of the plate. Figures 2.6–2.9 illustrate the surface displacement. ⎧ ⎪ ⎪ ⎨wN (r)/w0 (r), 0 r 1, Δ(r) = ⎪ ⎪ ⎩ fN (r)/ f0 (r), r > 1,
2 Analytical Solution of the Bending of Plates y 1.0 φ(y) 0.0 0.1 ϕ1 -1.0
0,0 -0,2
27
Mr0
-0,6
2
-0,01 0 M -0,02 r -0,03
Mφ0
0.2
-1,0
-1,4 1,0 0,6
16
0,0
16 4 0.5
0,6 1,0
0
-0,04 -0,05 1,0
2
8
0.5
Mφ0
0.2
0 0,2
0,4
1,0
b)
a)
Fig. 2.10 Distribution of the bending moments. a) s = 0, 1, b) s = 3 y 1.0 φ(y) 0.0 0.1 ϕ2 -1.0
12
0.5
Mr0
0.2
2
8
0
Mφ
1,0
0
Mφ
Mr0
0,2
16
4
0,4
0,0 0,4
0 a)
0,4
1,0
-0,2 1,0 0,6
0,6 1,0
0 b)
Fig. 2.11 Distribution of the bending moments. a) s = 0, 1, b) s = 3 y 1.0 0.0 0.1 φ(y) -0.5 ϕ3 -1.0
0
Mr
1,5
0
0.2
4
8
16
0,0
0,04
0.5
-1,5 -3,0 1,0
0,08
Mφ
0 a)
0,4
0.2 4
8
0.5
-0,04 1,0
Mφ0
16
0,0
2
0,4
Mr0
0,4
1,0
2
0 0,4 b)
1,0
Fig. 2.12 Distribution of the bending moments. a) s = 0, 1, b) s = 3 y 1.0 φ(y) 0.0 0.1 -0.5 ϕ4 -1.0
8
0
Mφ
0,0
4 2
Mr
-1,5
0.5
0,4
0 a)
0,4
1,0
-0,06 1,0
0
Mφ
8
0.2
-0,04
0.2
2
16
-0,02
-3,0 1,0
0
16
1,5 Mr0 0,0
0.5
4
0,4
0 0,4 b)
1,0
Fig. 2.13 Distribution of the bending moments. a) s = 0, 1, b) s = 3
of non-homogeneous half-space of type (2.45) in comparison with homogeneous w0 (r) (under the plate and outside of it) at various values of λ and s. Figures 2.10–2.13 illustrate behavior of dimensionless radial (Mr0 = Mr /M0r ) and tangential moments (Mϕ0 = Mϕ /M0ϕ ). Values Mr (M0r ), Mϕ (M0ϕ ) are calculated with formulas (2.39) at N = 10, M = 10.
28
S. Aizikovich et al.
Acknowledgements The work is performed with the financial support of RFBR (09-08-011410a, 10-08-01296-a, 10-08-00839-a, 10-08-90025-Bel-a, 11-08-91168-GFEN-a), GK No P 1107, GK No 02.740.11.0413, GK No 02.740.11.5193, AVCP 2.1.2/10063, NASA EPSCOR Grant GR0002488; Los Alamos National Labs MOU Grant GR0002842.
References 1. Aizikovich, S.M., Alexandrov, V.M., Kalker, J.J., Krenev, L.I., Trubchik, I.S. (2002) Analytical solution of the spherical indentation problem for a half-space with gradients with the depth elastic properties, Int. J. Solids Structures, 39 10, 2745–2772. 2. Aizikovich, S.M., Alexandrov, V.M., Trubchik, I.S. (2009) Bilateral asymptotic solution of one class of dual integral equations of the static contact problems for the foundations inhomogeneous in depth. Operator Theory: Advances and Applications, 39 3–17. 3. Tseitlin, A.I. The bending of a circular plate resting on a linearly deformable foundation. Izv. Akad. Nauk SSSR. MTT (1969)1 99–112. 4. Popov, G. Ya. Plates on a linearly elastic foundation (a survey). Prikl. Mekh. 8(3) (1971) – pp. 3–17. 5. Hooper, J. A. (1978) Foundation interaction analysis, in Developmenls in Soil Mechanics-l (Edited by R. C. Scott), Applied Science Publ., London, 149–213. 6. Selvadurai, A. P. S. (1979) Elastic Analysis of Soil-Foundation Interacrion: Developments in Geotechnical Engineering. Elsevier, Amsterdam, Vol. 17. 7. Gladwell, G.M.L. (1980) Contact Problems in the Classical Theory of Elasticity. 8. Ishkova, A. G. (1951) Bending of a circular plate on the elastic half-space under the action of a uniformly distributed axisymmetrical load. Uch. Zap. Mosk. Gos. Univ. 3, 202–225. (in Russian) 9. Palmov, V. A.(1960) The contact problem of a plate on an elastic foundation. PMM, Vol. 3, 416–422. 10. Barden, L. (1965) Contact pressures under circular slabs. Struct. Engineer, 5 (43), 153–154. 11. Brown, P. T. (1969) Numerical analyses of uniformly loaded circular rafts on deep elastic foundations. Geotechnique, Vol. 19, 399–404. 12. Brown, P. T. (1969) Numerical analyses of uniformly loaded circular rafts on elastic layers of finite depth. Geotechnique, Vol. 19, 301–306. 13. Chattopadhyay, R., Ghosh, A. (1969) Analysis of circular plates on semi-infinite elastic substrates. Indian J. Technol. 7, 312–319. 14. Selvadurai,A. P. S. (1979) The interaction between a uniformly loaded circular plate and an isotropic half-space: A variational approach. J. Strucr. Mech. 7(3), 231–246. 15. Selvadurai, A. P. S. (1980) Elastic contact between a flexible ircular plate and a transversely isotropic elastic halfspace. Int. J. Solids Struct. 16, 167–176. 16. Aizikovich, S.M. (1982) Asymptotic solutions of contact problems of the theory of elasticity for media inhomogeneous in depth. Prikl. Mat. Mekh. 46 (1), 148–158. 17. Aizikovich, S.M., Aleksandrov, V.M. (1982) On the properties of compliance functions corresponding to polylayered and continuously inhomogeneous half-space. Dokl Akad Nauk SSSR, 266 (1), 40–43 (English Transl. in Soviet. Phys. Dokl. 27). 18. Aleksandrov, V.M.(1973) The solution of a class of dual equations. Dokl. Akad. Nauk SSSR 210 (1), 55–58. 19. Coddington, E.A., Levinson, N. Theory of Ordinary Differential Equations. McGraw-Hill, New York, 1955. 20. Kantorovich, L.V., Akilov, G.P. Functional analysis, 2nd rev. ed., Nauka, Moscow, 1977.
Chapter 3
Analysis of the Deformation of Multi-layered Orthotropic Cylindrical Elastic Shells Using the Direct Approach Mircea Bˆırsan and Holm Altenbach
Abstract In this paper we analyze the deformation of cylindrical multi-layered elastic shells using the direct approach to shell theory. In this approach, the thin shelllike bodies are modeled as deformable surfaces with a triad of vectors (directors) attached to each point. This triad of directors rotates during deformation and describes the rotations of the thickness filament of the shell. We consider a general set of constitutive equations which can model orthotropic multi-layered shells. For this type of shells we investigate the equilibrium of thinwalled tubes (not necessarily circular) subjected to external body loads and to resultant forces and moments applied to the end edges. We present a general procedure to derive the analytical solution of this problem. We consider that the external body loads are given polynomials in the axial coordinate, which coefficients can be arbitrary functions of the circumferential coordinate. We illustrate our method in the case of circular cylindrical three-layered shells and obtain the solution in closed form. For isotropic shells, the solution is in agreement with classical known results. Keywords Direct approach · Multilayered shell · Reissner theory · Simple shell · Orthotropic material · Cylindrical shell
M. Bˆırsan (B) Department of Mathematics, “A.I. Cuza” University of Ias¸i, 700506 Ias¸i, Romania Faculty of Civil Engineering and Architecture, Lublin University of Technology, 20-618 Lublin, Poland e-mail:
[email protected] H. Altenbach Department of Engineering Sciences, Martin-Luther-University Halle-Wittenberg, 06099 Halle (Saale), Germany Faculty of Mechanical Engineering, Otto-von-Guericke-University, 39106 Magdeburg, Germany e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 3, © Springer-Verlag Berlin Heidelberg 2011
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M. Bˆırsan and H. Altenbach
3.1 Introduction The paper is concerned with the deformation of thin elastic shells modeled by the direct approach. We employ the theory of simple shells, which was introduced by Zhilin [17,18]. In this approach, the thin shell-like bodies are modeled as deformable surfaces with a triad of rigidly rotating directors attached to each point. The model has been successfully applied in the case of plates to describe the mechanical behavior of functionally graded, viscoelastic or porous plates in [2–4]. A mathematical study of the theory of simple shells is presented in [7]. We investigate the deformation of cylindrical shells with arbitrary shape of crosssection (not necessarily circular), and made of orthotropic materials. We consider a quite general set of constitutive equations which can describe various types of composite shells, such as multi-layered shells or reinforced shells. Moreover, our results can be applied also to non-homogeneous shells, since the constitutive coefficients are assumed to be arbitrary functions of the circumferential coordinate. In this framework, we consider the following mechanical problem: find the equilibrium of cylindrical shells under the action of resultant forces and moments applied to the end edges of the cylindrical shell, and the action of given body loads (body forces and couples). We present a general analytical solution procedure for this problem. Thus, the solution for the displacement and rotation fields is expressed in terms of the exact solutions to auxiliary boundary-value problems for ordinary differential equations. The idea of this method is taken from the three-dimensional linear elasticity: in this context, the deformation of solid cylinders made from Cosserat materials has been investigated in details, see e.g. [10]. Adapting the method of Ies¸an [11], we investigate the deformation of cylindrical surfaces with directors. We derive first the equilibrium solution of our problem for the case when the body loads are independent of the axial coordinate, but they depend on the circumferential coordinate of the surface. Then, we consider the more general case when the body loads are polynomials in the axial coordinate (of degree n) and we obtain a solution of our equilibrium problem by the mathematical induction procedure with respect to the degree n. We show that our solutions are in agreement with the classical results concerning the deformation of cylindrical elastic shells. Finally, we apply this general method to the case of a three-layered circular cylindrical shell, and determine the closed-form solution for the displacement and rotation fields.
3.2 Basic Equations for Shells. General Geometry We consider a direct approach to shells, in which each material point of the surface is connected with an orthonormal triad of vectors (directors). This model has been suggested first by the Cosserat brothers [9] and it has been developed by Zhilin in [17,18], who established the constitutive equations and presented several interesting
3 Analysis of the Deformation of Multi-Layered Orthotropic Cylindrical Elastic Shells
31
applications. This approach, also called the theory of simple shells, can also account thermal effects in shells [8]. Let S denote the actual configuration of the shell at time t, and Dk (k = 1, 2, 3) designate the 3 directors connected to each point. Then, the configuration S of the shell is determined by the functions {R(x, t), Dk (x, t)} , where R is the position vector of any point with curvilinear material coordinates x = (x1 , x2 ), see Fig. 3.1. The triad
S S0
d3 d 2 (x1 , x2 )
D3
D2 D1
d1
(x1 , x2 )
r
R O
Fig. 3.1 Reference configuration S0 and actual configuration S of the shell
of directors { Dk } remains orthonormal during deformation, so that Dk · Dm = δkm (the Kronecker symbol). The rotations of the triad { Dk } describe the rotations of the thickness filament of the shell during deformation. In the reference configuration S0 (at t = 0) we denote by r(x) = R(x, 0) the position vector and by dk (x) = Dk (x, 0) the directors. Then, the rotation tensor P is defined by P(x, t) = Dk (x, t) ⊗ dk (x). Here, the usual summation convention over repeated indices is employed, and the direct tensor notation in the sense of [12]. Let P˙ denote the material time derivative of P, and ∂α P = ∂∂P xα . We introduce the angular velocity ω as the axial vector of the antisymmetric tensor P˙ · PT (the superscript T denotes the transpose). Also, let Φ α be the axial vector of the antisymmetric tensor ˙ t) = ω (x, t)×P(x, t) and ∂α P(x, t) = Φ α (x, t)×P(x, t). Through∂α P · PT . We have P(x, out the paper, the Greek indices take the values {1, 2}, while the Latin indices range over the set {1, 2, 3} (unless otherwise specified). Let n denote the unit normal to the reference surface S0 , the covariant base vectors in the tangent plane are rα (x) = ∂α r(x), and the dual basis {rα (x)} is defined by rα · rβ = δαβ . We introduce the first fundamental tensor a, the second fundamental tensor b, and the alternator tensor c given by a = rα ⊗ rα ,
b = −rα ⊗ ∂α n,
c = −a × n = −n × a.
The nonlinear theory of simple shells is presented in [5, 18]. In this paper, we confine our attention to the linear theory and present the relevant equations.
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The infinitesimal displacement vector u and the infinitesimal rotation vector ϕ are introduced by the relations u = R−r
and
ϕ˙ = ω ,
∂ αϕ = Φ α .
ϕ × 1, where 1 is the unit tensor of Then, the rotation tensor is expressed as P = 1 +ϕ second order. Let us choose the reference director d3 (x) to coincide with the unit normal to S0 , i.e. d3 = n. We may consider that ϕ (x, t) · n(x) = 0, since the (drilling) rotations about the normal n as usual do not intervene in the governing equations in the linear theory of simple shells (exceptional cases, for example, folded structures, will be not discussed here). The vector fields u and ϕ can be decomposed as u = uα rα + u3 n , ϕ = ϕα rα , which shows that we have 5 independent kinematical scalar fields (three displacements ui and two rotations ϕα ). Thus, the theory of simple shells is in this sense a Reissner-type theory for shells. ϕ , κ = rα ⊗ ∂αϕ . Then, the Let us denote by e and κ the tensors e = rα ⊗ ∂α u + a ×ϕ strain tensors for simple shells are expressed by ε=
& 1% e · a + a · eT , γ = e · n = (rα ⊗ ∂α u) · n + c · ϕ , 2 1 k = κ · a + (e · ·c)b. 2
(3.1)
Here ε is a symmetric tensor describing the extensional and in-plane shear strains of the shell, γ is a vector which accounts for the transverse shear deformation, and k is a tensor for the bending and twist strains. For any subset P0 of the reference surface S0 , we designate by ν the external unit normal to the boundary curve ∂P0 which lies in the tangent plane, and by t and m the vectors of external force and moment acting on the boundary ∂P0 . Then, the force tensor T and the moment tensor M satisfy the relations of Cauchy type t = ν · T, m = ν · M . The equations of equilibrium have the form rα · ∂α T + F = 0,
rα · ∂α M + T× + L = 0.
(3.2)
Here the vectors F and L are the external forces and moments per unit area of the surface S0 . The term T× denotes the vector invariant (or ‘Gibbsian cross’) of the tensor T, see e.g. Appendix A in [13]. γ , k) stand for the inWe present next the constitutive equations. Let U = U(εε ,γ ternal energy function, which will be assumed to be a quadratic function of its arguments. This means that “eigenstresses” presented by linear terms and nonlinear elastic behavior are ignored. For general orthotropic shells, U can be expressed as ρU =
1 1 1 ε · ·C1 · ·εε + ε · ·C2 · ·k + k · ·C3 · ·k + γ · Γ · γ , 2 2 2
(3.3)
where ρ is the mass density in the reference configuration, Ci represent stiffness tensors of order 4, and Γ is a stiffness tensor of order 2 (the transverse shear
3 Analysis of the Deformation of Multi-Layered Orthotropic Cylindrical Elastic Shells
33
stiffness). These stiffness tensors satisfy certain symmetry conditions and they express the effective elastic properties and the thickness geometry of the shell. The functional form (3.3) can describe a large variety of shells, like for instance multilayered shells, as we will see later on. The stiffness tensors have been determined for various classes of shells and plates in [1–4, 18]. The constitutive equations for simple shells are 1 ∂(ρU) sym(T · a) ≡ (T · a + a · TT ) = = C1 · ·εε + C2 · ·k, 2 ∂εε ∂(ρU) ∂(ρU) N ≡ T·n = = Γ · γ, M= = (εε · ·C2 + C3 · ·k)T . ∂γγ ∂k
(3.4)
For shells of constant thickness made of non-polar materials, we have the following restrictions [18] TT · ·c + MT · ·b = 0,
M · n = 0.
(3.5)
We notice that the relation (3.5)1 gives the antisymmetric part of T · a . In view of the constitutive equations (3.4)1,2 , the force tensor T will be completely determined. The expressions of the constitutive tensors Ci and Γ are presented in [18], Sect. 3.10. Let us consider that the vectors rα are directed along the principal directions of the tensor b and the basis {rα } is orthogonal. We introduce the tensors a1 = a = r1 ⊗ r1 + r2 ⊗ r2 , a3 = c = r1 ⊗ r2 − r2 ⊗ r1 ,
a2 = r1 ⊗ r1 − r2 ⊗ r2 , a4 = r1 ⊗ r2 + r2 ⊗ r1 .
Then the fourth order tensors Ci and the second order tensor Γ are expressed by C1 = A11 a1 ⊗ a1 + A12 (a1 ⊗ a2 + a2 ⊗ a1 ) + A22a2 ⊗ a2 + A44 a4 ⊗ a4 , C2 = B13a1 ⊗ a3 + B14a1 ⊗ a4 + B23a2 ⊗ a3 + B24a2 ⊗ a4 + B41a4 ⊗ a1 + B42a4 ⊗ a2 , C3 = C11 a1 ⊗ a1 + C12 (a1 ⊗ a2 + a2 ⊗ a1 ) + C22a2 ⊗ a2 + C33 a3 ⊗ a3 + C34 (a3 ⊗ a4 + a4 ⊗ a3 ) + C44a4 ⊗ a4 , Γ = Γ1 a1 + Γ2 a2 ,
(3.6)
where the coefficients Ars , Brs , Crs and Γα are the effective stiffness moduli. As mentioned by Zhilin [18], Sect. 3.10, the formulas (3.6) have a wide range of applicability. By a suitable choice of the effective stiffness moduli, these equations can be used to describe the mechanical behavior of the following types of shells: transversal isotropic shells with constant thickness; orthotropic multi-layered shells (with symmetric or non-symmetric sequences of layers); reinforced shells; or shells with several inelastic layers. In this paper, we shall employ the relations (3.6), together with the constitutive equations (3.3)-(3.5), to investigate the deformation of orthotropic multi-layered cylindrical shells made of elastic materials.
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M. Bˆırsan and H. Altenbach
3.3 Cylindrical Shells with Arbitrary Cross-Section Shape. Formulation of the Problem In what follows we assume that the reference configuration S0 is a cylindrical surface (see Fig. 3.2). We refer the cylindrical surface to a Cartesian orthogonal frame Ox1 x2 x3 , such that the generators of S0 are parallel to the axis Ox3 . We denote the material surface coordinates by x1 = s, x2 = z, where the axial coordinate z is the distance to the plane x1 Ox2 , while the circumferential coordinate s is the arclength parameter along the cross-section curves z = const. Let ei be the unit vectors along the Oxi axes. Then, the reference surface S0 is characterized by the parametric equations r = r(s, z) = xα (s) eα + z e3 ,
s ∈ [0, s¯], z ∈ [0, z¯],
(3.7)
where x1 (s), x2 (s) are arbitrary functions which describe the shape of the cylinder’s cross-section. For our subsequent analysis we assume that the functions xα (s) are of class C 3 [0, s¯]. Let us designate by τ (s) the unit tangent vector to the cross-section curve, n(s) the unit normal vector to the surface, and r(s) the radius of curvature of the cross-section curve. Then, the fields introduced previously have the expressions r1 = r1 = τ = xα eα ,
r2 = r2 = e3 , n = eαβ xβ eα , c = eαβ rα ⊗ rβ , 1 a = τ ⊗ττ + e3 ⊗ e3 , b = − τ ⊗ττ, r(s) = [eαβ xα (s)xβ (s)]−1 , r(s)
(3.8)
where eαβ stands for the two-dimensional alternator (e12 = −e21 = 1, e11 = e22 = 0), and ( ) = d(ds) . Due to the identification r1 = τ , r2 = e3 we denote the components of the above tensors in the following manner u = u1τ + u2 e3 + u3 n, ϕ = −ϕ2τ + ϕ1 e3 , γ = γ1τ + γ2 e3 , ε = ε1τ ⊗ττ + ε2 e3 ⊗ e3 + ε12τ ⊗ e3 + ε21 e3 ⊗ττ, k = −δ1τ ⊗ττ + δ2 e3 ⊗ e3 + κ1τ ⊗ e3 − κ2 e3 ⊗ττ, T = T 1τ ⊗ττ + T 2 e3 ⊗ e3 + T 12τ ⊗ e3 + T 21e3 ⊗ττ + N1τ ⊗ n + N2 e3 ⊗ n, M = −M12τ ⊗ττ + M21 e3 ⊗ e3 + M1τ ⊗ e3 − M2 e3 ⊗ττ.
(3.9)
Cz¯ (s, z) e3 r Fig. 3.2 Reference configuration S0 of the cylindrical shell
O e1
τ n
e2 C0
3 Analysis of the Deformation of Multi-Layered Orthotropic Cylindrical Elastic Shells
35
In view of (3.1), (3.8) and (3.9), the components of the strain tensors are expressed by the geometrical equations 1 1 ε1 = u1,1 + u3 , ε2 = u2,2 , ε12 = ε21 = (u1,2 + u2,1 ), r 2 1 γ1 = u3,1 − u1 + ϕ1 , γ2 = u3,2 + ϕ2 , r 1 δ1 = ϕ2,1 + (u1,2 − u2,1 ), δ2 = ϕ1,2 , κ1 = ϕ1,1 , κ2 = ϕ2,2 . 2r Here we designate the partial derivatives in the usual manner: f,1 = ∂f ∂x2
∂f ∂z
∂f ∂x1
(3.10)
=
∂f ∂s
, f,2 =
= , for any function f . The equilibrium equations (3.2) can be put in the operator form P(v) = L , L = −(F i , Lα ), (3.11) ϕ ) is the field of displacements and rotations, L represents the external where v = (u,ϕ body loads acting on the shell, and the operator P = (P1 , ..., P5 ) is given by P1 (v) = T 1,1 (v) + T 21,2(v) + 1r N1 (v),
P2 (v) = T 12,1 (v) + T 2,2(v), 1 P3 (v) = N1,1 (v) + N2,2(v) − T 1 (v), r P4 (v) = M1,1 (v) + M21,2 (v) − N1(v), P5 (v) = M12,1 (v) + M2,2(v) − N2(v).
(3.12)
To write the constitutive equations in component form, we use the relations (3.3)(3.6) and (3.10)6,7 . Thus, we have T 1 =a1 ε1 + a3ε2 + b1 κ1 + b2 κ2 , T 2 = a2 ε2 + a3 ε1 + b3κ1 + b4 κ2 , 1 1 1 1 1 1 T 12 = b5 − c4 δ1 + b6 − c6 δ2 + a4 − b5 ε12 , N1 = d1 γ1 , 2 r 2 r 2 r (3.13) 1 1 1 1 1 1 T 21 = b5 + c4 δ1 + b6 + c6 δ2 + a4 + b5 ε12 , N2 = d2 γ2 , 2 r 2 r 2 r M1 = b1 ε1 + b3ε2 + c1 κ1 + c3 κ2 , M2 = b2 ε1 + b4 ε2 + c3 κ1 + c2 κ2 , M12 = b5 ε12 + c4 δ1 + c6 δ2 , M21 = b6 ε12 + c6 δ1 + c5 δ2 , where the coefficients ar (r = 1, ..., 4), b s , c s (s = 1, ..., 6) and dγ are expressed in terms of the effective elastic moduli Ars , Brs , Crs and Γα by a1 = A11 + 2A12 + A22 , a2 = A11 − 2A12 + A22 , a3 = A11 − A22 , a4 = 4A44 , b1 = −B13 − B23 + B14 + B24, b2 = −(B13 + B23 + B14 + B24), b3 = −B13 + B23 + B14 − B24, b4 = −B13 + B23 − B14 + B24, b5 = −2(B41 + B42), b6 = 2(B41 − B42), d1 = Γ1 + Γ2 , d2 = Γ1 − Γ2 , c1 = C33 − 2C34 + C44 , c2 = C33 + 2C34 + C44 , c3 = C33 − C44 , c4 = C11 + 2C12 + C22 , c5 = C11 − 2C12 + C22 , c6 = −C11 + C22 . (3.14) The general form of the constitutive equations (3.13), (3.14) is applicable for orthotropic multi-layered cylindrical shells. For each given material distribution in the layers, a suitable choice of the effective elastic moduli Ars , Brs , Crs and Γα need to
36
M. Bˆırsan and H. Altenbach
be made. In our work, we present a general solution procedure based on the form (3.13), (3.14) for the constitutive equations. Then, we apply this method to determine the deformation of 3-layered cylindrical shells. We mention that, in the general case (3.13), (3.14), the effective elastic moduli Ars , Brs , Crs and Γα are prescribed functions of the circumferential coordinate s, so that our approach can treat also cylindrical shells with circumferential nonhomogeneity. Let us formulate the mechanical problem which will be investigated: determine the equilibrium deformation of a multi-layered cylindrical shell under the action of body loads F, L, and subjected to given resultant forces and resultant moments acting on its end edges. In what follows, we consider closed cylindrical surfaces (i.e. thin-walled tubes), but we mention that the same method can be applied also for open cylindrical surfaces. We denote by C0 the end edge of the cylindrical surface S0 characterized by z = 0, and Cz¯ the end edge corresponding to z = z¯. Then, the necessary and sufficient conditions for the equilibrium of the cylindrical shell are F da + t dl = 0, S0 C0 ∪Cz¯ (3.15) (L + r × F) da + (m + r × t) dl = 0. S0
C0 ∪Cz¯
These relations express the overall balance of forces and couples applied to the ϕ), we introduce the cylindrical shell. For any displacement and rotation field v = (u,ϕ vector functionals F (v) and M (v) given by F (v) = t(v) dl, M (v) = [m(v) + r × t(v)] dl, (3.16) C0
C0
which represent the resultant force and the resultant moment about O, respectively, acting on the end edge C0 and corresponding to the field v. Then, the global boundary conditions on the end edge C0 have the form F (v) = F 0 ,
M (v) = M 0 ,
(3.17)
where F 0 and M 0 are the prescribed resultant force and moment acting on C0 . We mention that the global boundary conditions on the opposite end edge Cz¯ can be easily derived from relations (3.15), in view of (3.16) and (3.17). We denote the components of the vectors F and M by: F = Fi ei and M = Mi ei . Since we consider closed cylindrical shells, we impose that the displacement and ' ( ϕ (s, z) satisfy the continuity conditions for s = 0, s¯: rotation field v(s, z) = u(s, z),ϕ v(0, z) = v( s¯, z),
∂v ∂v (0, z) = ( s¯, z), ∂s ∂s
∂2 v ∂2 v (0, z) = 2 ( s¯, z), 2 ∂s ∂s
∀ z ∈ [0, z¯]. (3.18)
To recapitulate, the basic equations for our shell problem are: the geometrical relations (3.10), the equilibrium equations (3.11), (3.12), the constitutive equations
3 Analysis of the Deformation of Multi-Layered Orthotropic Cylindrical Elastic Shells
37
(3.13), the boundary conditions (3.16), (3.17) and the continuity conditions (3.18). As we can see, the data of our problem are the body loads F, L, the resultant force F 0 and the resultant moment M 0 acting on C0 , while the unknown is the displaceϕ). We denote this problem by Π(F F 0 ,M M0 , F, L). Let ment and rotation field v = (u,ϕ ϕ) of us designate by D(F, L) the set of all displacement and rotation fields v = (u,ϕ class C 1 (S¯ 0 ) ∩ C 2 (S0 ) which satisfy the equilibrium equations (3.11), (3.12), and the continuity conditions (3.18). We mention that the corresponding problem for solid cylinders has been extensively studied in the three-dimensional theory of elasticity (see e.g. [10] and the references given therein). In this context, if the body loads are absent (F = 0, L = F 0 ,M M 0 , 0, 0) is called the relaxed Saint-Venant problem. 0) then the problem Π(F Also, if the body loads F and L are independent of z (but they depend only on F 0 ,M M0 , F, L) is known as the Almansi-Michell problem. Moreover, if s) then Π(F F and L are polynomial functions of z (with coefficients depending on s) then F 0 ,M M 0 , F, L) is usually called the Almansi problem. Due to their importance Π(F in engineering, these problems have received considerable attention in the threedimensional elasticity theory. An interesting general method to investigate the deformation of solid cylinders has been presented in the book [10]. In our work we consider the corresponding problems for shells and adapt the method of Ies¸an [10] to study the deformation of thin-walled tubes. We remark first F 0 ,M M 0 , F, L) is not unique, due to the presence that the solution of the problem Π(F of global boundary conditions (3.17) on the end edges. However, by virtue of SaintVenant’s principle for shells, the difference between any two solutions is negligible, except possibly near the end edges. In the next sections we present a general method to construct solutions of these shell problems.
3.4 Solution Procedure An important role in our solution procedure will be played by the following result. ϕ) ∈ D(F, L) be a displacement and rotation field such that Theorem 3.1. Let v = (u,ϕ ' ∂F ∂L ( ∂v 1 ¯ 2 is of class C ( S ) ∩ C (S0 ). Then, we have ∂v 0 ∂z ∂z ∈ D ∂z , ∂z . Moreover, the resultant forces and moments acting on C0 , corresponding to the field ∂v ∂z , are given by F
∂v ∂z
=
C0
F dl,
M
∂v ∂z
= eαβ Fβ (v)eα +
C0
(L + r × F) dl.
(3.19)
Proof. From the linearity of the theory and the definition of the set D(F, L) it fol' ∂F ∂L ( lows that ∂v ∂z ∈ D ∂z , ∂z . Next, to prove the relations (3.19), we employ the equations (3.16), the equilibrium equations (3.11), (3.12), and the continuity conditions (3.18). These calculations are not detailed here. The preceding theorem has the following important consequence.
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M. Bˆırsan and H. Altenbach
Corollary 3.1. Assume that the external body loads F and L do not depend on the ϕ ) be a solution of the problem Π(F F 0 ,M M0 , F, L), axial coordinate z, and let v = (u,ϕ ∂v ∂v 1 ¯ 2 such that ∂z ∈ C (S0 ) ∩ C (S0 ). Then, the field ∂z is a solution of the problem G,Q Q, 0, 0), where the resultant force G and the resultant moment Q are expressed Π(G by G=
C0
F dl,
Q = eαβ Fβ0 eα +
C0
(L + r × F) dl.
(3.20)
∂L ∂v Proof. Indeed, in this case we have ∂F ∂z = 0, ∂z = 0, so that ∂z ∈ D(0, 0) by virtue of Theorem 3.1. The expressions of G and Q in (3.20) follow from the relations (3.19).
These results will be used in our subsequent developments. F 0 ,M M0 , F, L), it is useful to analyze first the In order to solve our problem Π(F solutions of this problem that depend only on s.
3.4.1 Solutions Depending Only on the Circumferential Coordinate Let us consider the equilibrium equations (3.11), (3.12), together with the continuity ϕ(s)) that depend only on s. conditions (3.18), and search for solutions v(s) = (u(s),ϕ More precisely, we consider the equations P(v(s)) = L(s)
for
s ∈ [0, s¯],
(3.21)
and the boundary conditions v (0) = v ( s¯),
v(0) = v( s¯),
(3.22)
where the body loads L(s) = −(Fi (s), Lα (s)) are assumed to depend only on s. Concerning the existence of solutions to the boundary-value problem (3.21), (3.22), we have the following result. Lemma 3.1. The boundary-value problem (3.21), (3.22) admits a solution v(s) = ϕ(s)) if and only if the external body loads F(s) and L(s) satisfy the following (u(s),ϕ conditions F dl = 0, (L + r × F) dl · e3 = 0. (3.23) C0
C0
Moreover, if the body loads F(s) and L(s) satisfy in addition the continuity conditions F(0) = F( s¯),
L(0) = L( s¯),
then the solution v(s) of the boundary-value problem (3.21), (3.22) verifies the continuity relation v (0) = v ( s¯). The proof of Lemma 3.1 is not difficult, but it necessitates lengthy calculations, and will be omitted. We only mention that the equilibrium equations (3.21) reduce to a
3 Analysis of the Deformation of Multi-Layered Orthotropic Cylindrical Elastic Shells
39
system of linear ordinary differential equations for the unknowns ui (s) and ϕα (s). Using the relations (3.12), the constitutive equations (3.13), and the boundary conditions (3.22), we are able to determine the solution of equations (3.21) in analytic form, provided the necessary and sufficient conditions (3.23) are satisfied. As a physical interpretation, the condition (3.23)1 expresses that the resultant force of the external body forces acting on C0 is zero. The condition (3.23)2 means that the axial component of the resultant moment about O of the external body moments and couples acting on C0 is vanishing. Since the problem (3.21), (3.22) is related only to the cross-section curve of the cylindrical shell and the fields do not depend on z, we call the boundary-value problem of type (3.21), (3.22) the cross-section plane problem. In view of the above results, we know how to determine the analytic solution of cross-section plane problems. In the following sections, we will have the opportunity to verify the conditions (3.23) and to solve several cross-section plane problems of the type (3.21), (3.22).
3.4.2 Basic Solutions to the Equilibrium Equations Suggested by the results concerning the deformation of the three-dimensional cylinϕ) of the equilibrium equations (in the ders [10], we search for the solutions v = (u,ϕ absence of the external body loads) such that ∂v ∂z is a rigid body displacement of the cylindrical shell. This type of solutions will be useful to construct the general F 0 ,M M0 , F, L). In other words, we look for displacement solutions of the problem Π(F ϕ(s, z)) that satisfy the equations and rotation fields v(s, z) = (u(s, z),ϕ P(v) = 0
with
v(0, z) = v( s¯, z),
v (0, z) = v ( s¯, z),
(3.24)
and
ϕ ∂u ∂ϕ ϕ∗ × r, = u∗ +ϕ = a · ϕ∗ , (3.25) ∂z ∂z where u∗ and ϕ ∗ are some arbitrary constant vectors. The relations (3.25) express ' ϕ( ∂ϕ the fact that ∂u ∂z , ∂z is a rigid body displacement and rotation field of the simple ϕ) has shell. Integrating the equations (3.25) with respect to z, we obtain that v = (u,ϕ the following form (up to a rigid displacement field) 1 u1 = − z2 (ωα xα ) + ω4 z(eαβ xα xβ ) + y1 (s), 2 u2 = z(ωα xα + ω3 ) + y2(s), 1 u3 = z2 (eαβ ωβ xα ) − ω4 z(xα xα ) + y3(s), 2 ϕ1 = ω4 z + ζ1 (s), ϕ2 = z(eαβ ωα xβ ) + ζ2 (s),
(3.26)
where ω1 , ..., ω4 are arbitrary constants, while yi (s) and ζα (s) are arbitrary functions of class C 2 [0, s¯]. We denote the components by y = y1τ + y2 e3 + y3 n , ζ = −ζ2τ + ζ1 e3
40
M. Bˆırsan and H. Altenbach
' ( ϕ) and by w(s) = y(s),ζζ (s) . If we regroup the terms containing ωk , the field v = (u,ϕ given by (3.26) can be written in the form v(s, z) = w(s) +
4
ωk vˆ(k) (s, z),
(3.27)
k=1
where we have denoted by vˆ(k) (s, z), k = 1, ..., 4, the fields 1 1 vˆ(α) = − z2 xατ + z xα e3 − z2 eαβ xβ n , −z eαβ xβτ , 2 ' 2 ( vˆ(3) = (z e3 , 0), vˆ(4) = z eαβ xα xβτ − z xα xα n , z e3 .
(3.28)
Imposing that v satisfy the equilibrium equation (3.24)1 from (3.27) we get P(w(s)) = −
4
ωk P(ˆv(k) (s, z)).
(3.29)
k=1
By virtue of the linearity, we separate (3.29) into 4 different boundary-value problems (k = 1, ..., 4): ' ( ' ( P w(k) (s) = −P vˆ(k) (s, z) with
w(k) (0) = w(k) ( s¯), w(k) (0) = w(k) ( s¯)
(3.30)
Since the fields v(k) are given by (3.28), it easy to show that (3.30) represents a crosssection plane problem of the type (3.21), (3.22). By a straightforward calculation we verify that the conditions (3.23) are satisfied in our case and thus, in view of Lemma 3.1, we can solve the problem (3.30) to find the unknown w(k) (s). In what follows, we consider that the field w(k) (s) have been determined in this way, for each k = 1, ..., 4. On the basis of (3.27), (3.29) and (3.30), we obtain v(s, z) =
4
ωk v(k) (s, z)
with v(k) (s, z) = w(k) (s) + vˆ (k) (s, z).
(3.31)
k=1
Let ω ˆ denote the quadruple ω ˆ = (ω1 , ω2 , ω3 , ω4 ). In order to indicate the dependence on the arbitrary constants ωk , we designate the field (3.31)1 by v{ω}, ˆ i.e. we put v{ω} ˆ =
4
ωk v(k) .
(3.32)
k=1
Since the fields vˆ(k) and w(k) are known, then the basic solutions v(k) are also known (k = 1, ..., 4). For any constants ωk , the displacement and rotation field v{ω} ˆ defined by (3.32) possesses the following important properties: i) v{ω} ˆ is a solution of the equilibrium equations with zero body loads, i.e. we have P(v{ω}) ˆ = 0;
(3.33)
3 Analysis of the Deformation of Multi-Layered Orthotropic Cylindrical Elastic Shells
41
ˆ ii) the field ∂v{∂zω} is a rigid body displacement and rotation field of the cylindrical shell; iii) the strain tensors corresponding to to the field v{ω} ˆ are independent of the axial coordinate; iv) we have Fα (v{ω}) ˆ = 0, α = 1, 2. (3.34)
We mention that these properties are analogous to the properties of the wellknown Saint-Venant’s solutions in three-dimensional elasticity. These solutions v{ω} ˆ will be employed in the next sections to express the soF 0 ,M M0 , F, L). We introduce the constants Drk , (r, k = lutions of our problem Π(F 1, ..., 4), defined by ' ( Dαk = eαβ Mβ v(k) ,
' ( D3k = −F3 v(k) ,
' ( D4k = −M3 v(k) ,
(3.35)
which are known quantities, since v(k) are known and the functionals F3 and Mi are given by (3.16). We observe that the field v{ω} ˆ does not depend on the external loads, but they depend only on the geometry of the cylindrical shell and the material properties. It can be calculated for every specific geometry and given material properties of shells. In Sects 3.5 and 3.6 we compute the field v{ω} ˆ for different types of circular cylindrical shells.
3.4.3 Cylindrical Shells Subjected to Terminal Resultant Forces and Moments F 0 ,M M0 , 0, 0), which consists in the equiLet us present a solution of the problem Π(F librium deformation of cylindrical shells under the action of resultant forces F 0 and resultant moments M 0 acting on the end edges, in the absence of body loads (F = 0, L = 0). The result is given in the next theorem. F 0 ,M M0 , 0, 0) concerning the deformation of Theorem 3.2. Consider the problem Π(F cylindrical shells due to terminal resultant forces and moments. Then, there exists a ϕ) of this problem having the form solution v = (u,ϕ z v= v{ω} ˆ dz + v{ˆη} + w(s), (3.36) 0
where ω ˆ = (ω1 , ω2 , ω3 , ω4 ) and ηˆ = (η1 , η2 , η3 , η4 ) are quadruples of constants and w(s) is a displacement and rotation field of class C 2 [0, s¯] which depends only on the circumferential coordinate s. Proof. Let us determine the constants ω, ˆ η, ˆ and the field w(s) such that the field v given by (3.36) satisfies v ∈ D(0, 0) and Fi (v) = Fi0 ,
Mi (v) = M0i .
(3.37)
42
M. Bˆırsan and H. Altenbach
First, we shall use the Corollary 3.1 to determine the constants ω. ˆ If we differentiate (3.36) with respect to z and take into account the property (ii) for the field v{ˆη}, then we obtain (up to a rigid body displacement field) ∂v = v{ω} ˆ . ∂z
(3.38)
On the other hand, from Corollary 3.1 we deduce that ∂v ∂z is a solution of the problem 0 Π(0, eαβ Fβ eα , 0, 0). Then, in view of (3.38) and (3.34), we have F3 (v{ω}) ˆ = 0,
Mα (v{ω}) ˆ = eαβ Fβ0 ,
M3 (v{ω}) ˆ = 0.
Using the notations (3.35), the last relations become ⎛ 4 ⎞ ⎜⎜⎜ ⎟⎟⎟ ' ( ⎜⎜⎜ Dkr ωr ⎟⎟⎟⎠ = − F10 , −F20 , 0, 0 . ⎝ r=1
(3.39)
k=1,...,4
The relations (3.39) represent a system of 4 algebraic linear equations for the constants ωk (k = 1, ..., 4). In what follows, we consider that ωk have been thus determined. Next, we apply the Lemma 3.1 to determine the field w(s). In view of (3.36), the equilibrium equations P(v) = 0 can be written as z P(w(s)) = −P v{ω} ˆ dz − P(v{ˆη}). 0
Taking into account (3.33), from the last equations we obtain the following boundary-value problem for the unknown w(s): z P(w(s)) = −P v{ω} ˆ dz with w(0) = w( s¯), w (0) = w ( s¯). (3.40) 0
') z
( ' ( ∂ We notice that ∂z P 0 v{ω}dz ˆ = P v{ω} ˆ = 0, in view of (3.33). Thus, the righthand side of (3.40)1 does not depend on z, so that (3.40) represents a cross-section plane problem of the type (3.21), (3.22). We apply the Lemma 3.1 for the crosssection plane problem (3.40) and we verify that the conditions (3.23) are satisfied in our case. Then, we can solve the problem (3.40) to obtain the field w(s). In what follows, we assume that w(s) has been determined. Finally, we compute the constants ηˆ by imposing the end edge conditions (3.37). If we denote by z
w(s, ˆ z) = w(s) +
v{ω}dz ˆ
0
for brevity, then from the conditions (3.37) we deduce F3 (v{ˆη}) = F30 − F3 (w), ˆ
Mi (v{ˆη}) = M0i − Mi (w), ˆ
3 Analysis of the Deformation of Multi-Layered Orthotropic Cylindrical Elastic Shells
or equivalently, using the notations (3.35), ⎛ 4 ⎞ ⎜⎜⎜ ⎟⎟⎟ % ' & ( ⎜⎜⎜ Dkr ηr ⎟⎟⎟⎠ = eαβ M0β − Mβ (w) ˆ , F3 (w) ˆ − F30 , M3 (w) ˆ − M03 . ⎝ r=1
43
(3.41)
k=1,...,4
Relations (3.41) form a system of 4 algebraic equations which permits the determination of the constants ηk (k = 1, ..., 4). The proof is complete. Remark. The proof of Theorem 3.2 describes the modality to construct effecF 0 ,M M0 , 0, 0). tively the solution of the form (3.36) to the problem Π(F
3.4.4 Deformation of Loaded Cylindrical Shells In this section we analyze the equilibrium of cylindrical shells when we have also F 0 ,M M0 , F, L). external body loads F and L acting on the shell, i.e the problem Π(F For the first stage, we assume that the external body loads F and L do not depend on the axial coordinate z, but they are arbitrary functions of s. In this case, the solution of our problem is given by the following result. F 0 ,M M0 , F, L) for the deformation of loaded Theorem 3.3. Consider the problem Π(F cylindrical shells in the case when the body loads F and L depend only on the ϕ) of the circumferential coordinate s. Then, this problem admits a solution v = (u,ϕ form z
v= 0
z
z
v{ω} ˆ dz dz +
0
ˆ + z w(s) + w(s), v{ˆη} dz + v{ξ} ˜
(3.42)
0
where ω, ˆ ηˆ and ξˆ are quadruples of constants and w(s), w(s) ˜ are displacement and rotation fields of class C 2 [0, s¯] which depend only on s. Proof. We follow the same idea as in the preceding proof. First, we determine the constants ω, ˆ ηˆ and the field w(s) by applying the Corollary 3.1. Indeed, by differentiating (3.42) with respect to z we find (up to a rigid body field) z ∂v = v{ω} ˆ dz + v{ˆη} + w(s). (3.43) ∂z 0 On the other hand, by virtue of Corollary 3.1, ∂v ∂z is a solution of the problem G,Q Q, 0, 0) where G and Q are given by (3.20). If we apply the Theorem 3.2 (and Π(G its proof) for the solution (3.43), we obtain that the constants ωr are given by the algebraic system ⎛ 4 ⎞ ⎜⎜⎜ ⎟⎟⎟ ' ( ⎜⎜⎜⎝ Dkr ωr ⎟⎟⎟⎠ = − G1 , −G2 , 0, 0 , (3.44) r=1
k=1,...,4
44
M. Bˆırsan and H. Altenbach
the field w(s) is the solution of the cross-section plane problem (3.40), while the constants ηr are determined by the equations ⎛ 4 ⎞ ⎜⎜⎜ ⎟⎟⎟ % ' & ( ⎜⎜⎜⎝ Dkr ηr ⎟⎟⎟⎠ = eαβ Qβ − Mβ (w) ˆ , F3 (w) ˆ − G3 , M3 (w) ˆ − Q3 . (3.45) r=1
k=1,...,4
Thus, we have found ω, ˆ ηˆ and w(s). Let us denote for brevity by z z z V{ω, ˆ η, ˆ w} = v{ω} ˆ dz dz + v{ˆη} dz + z w(s), 0
0
(3.46)
0
which is a known field. Then, the equilibrium equations P(v) = −(F, L) reduce to the following cross-section plane problem for the determination of the field w(s) ˜ ' ( P(w(s)) ˜ = −(F, L) − P V{ω, ˆ η, ˆ w}
with w(0) ˜ = w( ˜ s¯), w˜ (0) = w˜ ( s¯).
(3.47)
We can verify that the conditions (3.23) are satisfied in the case of the problem (3.47), so that we can compute its solution w(s). ˜ Finally, in order to determine the constants ξr (r = 1, ..., 4) we impose the end edge conditions of the form (3.37), with v given by (3.42). These conditions reduce to the following system of algebraic equations 4
% ' (& Dαr ξr = eαβ M0β − Mβ w˜ + V{ω, ˆ η, ˆ w} ,
r=1
4 r=1 4
' ( D3r ξr = F3 w˜ + V{ω, ˆ ηˆ , w} − F30 ,
(3.48)
' ( D4r ξr = M3 w˜ + V{ω, ˆ ηˆ , w} − M03 ,
r=1
which allows for the determination of the constants ξr . The proof is complete.
For the second part of this section, we consider the case when the external body loads are polynomials in the axial coordinate z of arbitrary degree n, i.e. F=
n k=0
F(k) (s) zk ,
L=
n
L(k) (s) zk ,
(3.49)
k=0
where the coefficients F(k) (s) and L(k) (s) are given functions of s. By virtue of (3.49) and the linearity of the theory, we can additively decompose F 0 ,M M0 , F, L) as the sum of the problems Π(F F 0 ,M M0 , F(0) , L(0) ) and our problem Π(F k k F 0 ,M M 0 , F(0) , L(0) ) Π(0, 0, F(k) z , L(k) z ), k = 1, ..., n. We remark that the problem Π(F has already been solved in Theorem 3.3. Let us find also a solution of the problem Π(0, 0, F(k) zk , L(k) zk ). To this aim, we employ the method of mathematical induction with respect to the degree k = 1, ..., n. In other words, we assume that we know a
3 Analysis of the Deformation of Multi-Layered Orthotropic Cylindrical Elastic Shells
45
solution of the problem Π(0, 0, F(k) zk , L(k) zk ) and we want to find a solution v to the problem Π(0, 0, F(k+1) zk+1 , L(k+1) zk+1 ). The procedure to compute the new solution v is given by the next result. Theorem 3.4. Assume that v∗ is a solution of the problem Π(0, 0, F(k+1) zk , L(k+1) zk ). ϕ) of the Then, the problem Π(0, 0, F(k+1) zk+1 , L(k+1) zk+1 ) admits a solution v = (u,ϕ form z
v = (k + 1)
v∗ dz + v{ω} ˆ + w∗ (s) ,
(3.50)
0
where ω ˆ = (ω1 , ω2 , ω3 , ω4 ), is a quadruple of constants and w(s), w∗ (s) is a field of displacement and rotation which depends only on s. Proof. Let us determine w∗ (s) and the constants ωk . Imposing that the field (3.50) satisfies the relation v ∈ D(F(k+1) zk+1 , L(k+1) zk+1 ), we derive the following boundaryvalue problem for the field w∗ (s) z 1 ∗ k+1 k+1 ∗ P(w (s)) = − (F(k+1) z , L(k+1) z ) − P v dz , (3.51) k+1 0 ∗ ∗ ∗ ∗ w (0) = w ( s¯), w (0) = w ( s¯). Since the conditions of Lemma 3.1 are satisfied, we can use the cross-section plane problem (3.51) to determine w∗ (s). Next, from the end edge boundary conditions Fi = 0, Mi = 0 we get the system of 4 algebraic equations ⎛ 4 ⎞ ⎜⎜⎜ ⎟⎟⎟ % & ⎜⎜⎝⎜ Dkr ωr ⎟⎟⎠⎟ = eβα Mβ (w), ¯ F3 (w), ¯ M3 (w) ¯ , (3.52) r=1
k=1,...,4
)z where we have denoted by w(s, ¯ z) = 0 v∗ dz + w∗ (s). From Eqs (3.52) we find the constants ωr and the proof is complete Remark. The proofs of Theorems 3.3 and 3.4 present the method to construct F 0 ,M M0 , F, L), when F and L are polynomials in z, the solution of the problem Π(F of arbitrary degree. In the general case when F and L are given (smooth enough) functions of (s, z) we can approximate them by polynomials in z of the form (3.49), F 0 ,M M0 , F, L). and then apply the solution procedure to solve the problem Π(F In the next sections we confine our attention to circular cylindrical shells and apply our solution procedure to different types of material.
3.5 Circular Cylindrical Shells Let us consider in this section circular cylindrical shells made of isotropic materials. In this simple case, we are able to obtain the solution in closed form and to compare it with the results obtained in previous approaches to shell problems.
46
M. Bˆırsan and H. Altenbach
The cross-section curves are circles of radius r0 , so that the parametric equations of the reference surface S0 is given by (3.7), with x1 (s) = r0 cos
s , r0
x2 (s) = r0 sin
s , r0
s ∈ [0, 2πr0 ].
(3.53)
In this case we have the relations xα = eβα
xβ , r0
xα = −
xα , r02
r(s) = r0 .
(3.54)
Let us denote by E the Young’s modulus, ν the Poisson’s ratio, μ the shear modulus of the isotropic material and by h the thickness of the shell. According to [18] Chapt. 3, in the case of isotropic shells the effective stiffness moduli introduced in (3.6) are given by Eh E h3 ν , A12 = 0, A22 = A44 = μh, B13 = , 2(1 − ν) 24r0 (1 − ν)2 E h3 E h3 E h3 B14 = − , B24 = B42 = 0, B41 = , B23 = , 24r0 (1 − ν) 48r0 (1 + ν) 24r0 (1 − ν2 ) h2 E h3 E h3 E h3 C11 = , C = , C = C = , 33 22 44 24(1 − ν) 24(1 + ν) 48r02 24(1 + ν) π2 C12 = C34 = 0, Γ1 = μ h Γ0 , Γ2 = 0 Γ0 = . 12 (3.55) 2 The coefficient Γ0 = π12 stands for the shear correction factor. If we denote by C, D, and γ the following quantities A11 =
C=
Eh , 1 − ν2
D=
E h3 , 12(1 − ν2)
γ=
1 h 2 , 48 r0
(3.56)
then in view of (3.13), (3.14) and (3.55), the constitutive equations in our case are D (κ1 + ν2 κ2 ), r0 (1 − ν) D T 2 = C(νε1 + ε2 ) − [νκ1 + (ν2 + ν − 1)κ2], r0 (1 − ν) * + D T 12 = (1 − ν) C(1 + γ)ε12 + [−(2 + γ)δ1 + γ δ2 ] , 4r0 , D T 21 = (1 − ν) C(1 − γ)ε12 + [γ δ1 + (2 − γ)δ2] , 4r0 π2 N1 = (μ h Γ0 )γ1 , N2 = (μ h Γ0 )γ2 Γ0 = , 12 D M1 = D(κ1 + νκ2 ) − (ε1 + νε2 ), r0 (1 − ν) D M2 = D(νκ1 + κ2 ) − [ν2 ε1 + (ν2 + ν − 1)ε2], r0 (1 − ν) T 1 = C(ε1 + νε2 ) −
3 Analysis of the Deformation of Multi-Layered Orthotropic Cylindrical Elastic Shells
(1 − ν)D 1 M12 = − ε12 + (1 + γ)δ1 + (1 − γ)δ2 , 2 r 0 (1 − ν)D 1 M21 = ε12 + (1 − γ)δ1 + (1 + γ)δ2 . 2 r0
47
(3.57)
F 0 ,M M0 , 0, 0), i.e. we consider Our aim is to find the solution of the problem Π(F circular cylindrical shells deformed by terminal resultant forces F 0 and moments M 0 . Let us determine first the basic solution field v{ω}. ˆ Following the procedure described in Sect. 3.4.2, we find that the displacement and rotation field v{ω} ˆ for isotropic circular cylindrical shells is given by u3 = −
u1 = 2
z2 (eαβ ωα xβ ) + ω4 r0 z, 2r 0
z + r0 F ωα xα − ν r0 ω3 , 2r0
u2 = z (ωα xα + ω3 ),
ϕ1 = ω4 z − Geαβ ωα xβ ,
ϕ2 =
where the coefficients F and G are constants specified by the relations 1−ν 1−ν 4γ 4γ ν2 Γ0 + 1 F + Γ0 + G = −ν + , 1−ν 1−ν 2 2 1−ν 4γ 1−ν 4γ ν2 Γ0 + F+ Γ0 + 4γ G = − . 2 1−ν 2 1−ν
z ωα xα , r0 (3.58)
(3.59)
Using the definitions (3.35) and (3.16) we compute next the coefficients Drk as follows 4γν2 4γ D11 = D22 = π r03 C (1 + Fν) + (G − F) + (4 − 5ν + 2ν3) , 1−ν 1−ν D33 = 2π r0 C(1 − ν2 ), D44 = π r03 C(1 − ν)(1 + 11γ + γ2), Dkr = 0 for k r. (3.60) We are now able to express the solution. According to the procedure presented in F 0 ,M M0 , 0, 0) in the form Sect. 3.4.3, we search for a solution v of the problem Π(F (3.36). Following the steps described in the proof of Theorem 3.2, we obtain the solution z3 z2 (eαβ ωα xβ ) + (eαβ ηα xβ ) + η4 r0 z, 2r0 6r0 1 u2 = z2 + Pr02 ωα xα + z (ηα xα + η3 ), 2 3 2 z z u3 = − + z r0 F ωα xα + − + r0 F ηα xα − ν r0 η3 , 6r0 2r0 2 z z ϕ1 = −zGeαβ ωα xβ − Geαβ ηα xβ + η4 z , ϕ2 = + r0 Q ωα xα + ηα xα , 2r0 r0 u1 =
(3.61)
where the constants ωα and ηk are expressed in terms of the resultants Fi0 and M0i by eαβ M0β F0 M0 F0 ω α = − α , ηα = , η3 = − 3 , η4 = − 3 . (3.62) D11 D11 D33 D44
48
M. Bˆırsan and H. Altenbach
The constant coefficients F and G are given by (3.59), while the constant coefficients P and Q are determined by the equations (1 + 3γ + γ2)P − 2γ(2 + γ)Q = K, 2γ(2 + γ)P − [Γ0 + 4γ(1 + γ)]Q = L,
(3.63)
with the notations . / 4γ K = 2 1 + νF + 1−ν (νG − ν2 − ν + 1) − (1 − ν)γ2G , ( 4γ ' 1−ν2 2 2 2 L = 1−ν 2 Γ0 F + 1−ν 2 G + ν F + ν + 2ν − 2 − 2(1 − ν)γ G.
(3.64)
From the above solution, we can see the geometrical significance of the overall strain measures ωα and ηk . Thus, η3 , ηα and η4 , respectively, represent measures of stretch, curvature and twist of the cylindrical shell considered as a beam, while ωα are global measures of strain appropriate to flexure. In what follows, let us consider the case of very thin shells (h r0 ), i.e. the limiting case when the parameter γ tends to zero: 1 % h &2 γ= → 0. (3.65) 48 r0 In view of (3.65), the solution (3.59)-(3.63) simplifies. Indeed, from (3.59), (3.60) and (3.63) we get D11 = π r03 E h, D33 = 2π r0 E h, D44 = 2π r03 μ h, F = −G = −ν, P = 2(1 − ν2), Q = ν(1−ν) 2 .
(3.66)
E where μ = 2(1+ν) is the shear modulus of the material. Then, the relations (3.62) between the overall strain measures ηk and the resultants F30 , M0i become
ηα =
eαβ M0β π r03 E h
,
η3 = −
F30 2π r0 E h
,
η4 = −
M03 2π r03 μh
.
(3.67)
The above relations (3.67) are in accordance with the results obtained previously for the deformation of thin cylindrical shells, presented in the classical books [16] Sect. 102, [15] Sect. 47, or in the papers [6,14]. Moreover, the displacement field ui given M0 , 0, 0) is in perfect by (3.61) for the extension-bending-torsion problem Π(F30 e3 ,M agreement with the corresponding results presented by Reissner and Tsai [14]. We can conclude that our solution is in agreement with previously known results from the literature, in the particular case of circular cylindrical thin shells made of isotropic materials. In addition, the solution procedure presented in Sect. 3.4 allows for the treatment of more complex structures, such as multi-layered orthotropic cylindrical shells (not necessarily circular). As an example, we study in the next section the deformation of a three-layered cylindrical shell.
3 Analysis of the Deformation of Multi-Layered Orthotropic Cylindrical Elastic Shells
49
3.6 Three-Layered Shells Let us present first the constitutive equations for three-layered shells with general geometry. We consider shells with a symmetric arrangement of the 3 layers (Fig. 3.3a): the two exterior layers have the same thickness h2 and they are made of the same isotropic material, while the inner layer has the thickness 2h1 and is made from a different isotropic material. Thus, the total thickness of the shell is 2h, where h = h1 + h2 .
h2 2h1 h2 (a)
(b)
Fig. 3.3 Three-layered shell: (a) General geometry; (b) Circular cylindrical shell
We denote by E 1 , ν1 and μ1 the Young’s modulus, Poisson’s ratio and shear modulus of the interior layer, respectively. The notations E2 , ν2 and μ2 will be used for the exterior layers. The constitutive equations for a three-layered isotropic shell have been presented in [18], Chapt. 5. In order to keep the calculations as simple as possible, we consider a simplified version of the constitutive equations, in which the tensor C2 = 0. More precisely, the effective stiffness moduli which appear in the constitutive equations (3.6) have the following expressions [18] E1 h1 E2 h2 A22 1−ν1 + 1−ν2 , E1 h31 E2 (h3 −h31 ) C33 = 3(1−ν1 ) + 3(1−ν2 ) ,
A11 =
C11 = C12 = C34 = 0,
= A44 =
E1 h1 1+ν1
E2 h2 + 1+ν , 2
C22 = C44 = Bkl = 0
E1 h31 3(1+ν1 )
(∀ k, l),
A12 = 0, +
E2 (h3 −h31 ) 3(1+ν2 )
(3.68)
Γ2 = 0.
For the transverse shear stiffness Γ1 , Zhilin [18] proposes the value ⎛ ⎞ 3 E 2 (h3 − h31 ) ⎟⎟⎟ λ2 λ2 ⎜⎜⎜⎜ E1 h1 ⎟⎟ , Γ1 = 2 C22 = 2 ⎜⎝ + 3(1 + ν2) ⎠ h h 3(1 + ν1) where λ is the smallest positive solution of the equation λh1 λh2 μ1 tan tan = . h h μ2
,
(3.69)
50
M. Bˆırsan and H. Altenbach
We notice that Γ (s) ≤ Γ1 ≤ Γ (p) ,
Γ (s) =
π2 μ1 h1 μ2 h2 , 6 4μ1 h1 + μ2 h2
Γ (p) =
π2 (μ1 h1 + μ2 h2 ). 6
The value Γ (p) corresponds to the connection in parallel, while Γ (s) represents the connection in series of the layers’ stiffness. It is useful to introduce the notations E1 h1 C1 = 2 1−ν 2 , 1
E2 h2 C2 = 2 1−ν 2 , 2
D1 =
3 2 E 1 h1 3 1−ν2 1
,
D2 =
3 3 2 E2 (h −h1 ) 3 1−ν22
.
(3.70)
In view of (3.68)–(3.70), the constitutive equations (3.13) for the case of threelayered cylindrical shells read T 1 = (C1 + C2 )ε1 + (C1 ν1 + C2 ν2 )ε2 , T 2 = (C1 ν1 + C2 ν2 )ε1 + (C1 + C2 )ε2 , T 12 = [C1 (1 − ν1) + C2 (1 − ν2 )]ε12 − 4r1 [D1 (1 − ν1 ) + D2 (1 − ν2)](δ1 + δ2), T 21 = [C1 (1 − ν1) + C2 (1 − ν2 )]ε12 + 4r1 [D1 (1 − ν1 ) + D2 (1 − ν2)](δ1 + δ2), λ2 λ2 N1 = 2h N2 = 2h 2 [D1 (1−ν1 ) + D2 (1−ν2 )]γ1 , 2 [D1 (1−ν1 ) + D2 (1−ν2 )]γ2 , M1 = (D1 +D2 )κ1 + (D1 ν1 +D2 ν2 )κ2 , M2 = (D1 ν1 +D2 ν2 )κ1 + (D1 +D2 )κ2 , M12 = M21 = 12 [D1 (1 − ν1 ) + D2 (1 − ν2 )](δ1 + δ2 ). (3.71) For the remaining of this section, we restrict our attention to circular cylindrical shells, three-layered (Fig. 3.3b). The geometry of the reference surface S0 is described by the equations (3.7), (3.53) and (3.54). We want to determine the equilibrium of this shell under the action of given body loads and arbitrary resultant forces F 0 ,M M0 , F, L). and moments applied to the end edges, i.e. to solve the problem Π(F To this aim, we apply the solution procedure presented in Sect. 3.4, and we determine first the basic solution v{ω}. ˆ For three-layered shells, we obtain the following expression for v{ω} ˆ z2 (eαβ ωα xβ ) + ω4 r0 z , u2 = z (ωα xα + ω3 ), 2r0 z2 ' ( u3 = − ωα xα + r0 Fωα xα + Hω3 , 2r0 z ϕ1 = ω4 z − G eαβ ωα xβ , ϕ2 = ω α x α , r0 u1 =
where the constants F and G are specified by the relations ( / λ2 ' λ2 . (C1 +C2 ) + 2h α ) F + 2h2 Dα (1−να ) G = −(Cα να ) , 2 Dα (1−ν 0 1 / ( λ2 . λ2 ' 1 2 Dα (1−να ) F + D1r+D + 2h 2 2 Dα (1−να ) G = r2 (Dα να ) , 2h2 0
0
(3.72)
(3.73)
3 Analysis of the Deformation of Multi-Layered Orthotropic Cylindrical Elastic Shells
51
and the constant H is given by H=−
C1 ν1 + C2 ν2 . C1 + C 2
(3.74)
The coefficients Drk in our case are computed on the basis of Eqs (3.35) and (3.16) ⎡ ⎤ ⎢ ( Dα ν α D1 + D2 ⎥⎥⎥⎥ 3 ⎢⎢⎢ ' D11 = D22 = π r0 ⎢⎣F Cα να − G 2 + (C1 + C2 ) + ⎥⎦ , r0 r02 (C1 + C2 )2 − (Cα να )2 (3.75) D33 = 2π r0 , Dkr = 0 for k r, ⎡ C1 + C2 ⎤ ⎢⎢' ( 9 ' (⎥⎥ D44 = π r03 ⎢⎢⎢⎣ Cα (1 − να) + 2 Dα (1 − να ) ⎥⎥⎥⎦ . 4r0 For the equilibrium problem that we want to solve, we take external body loads independent of the axial coordinate z. More precisely, we consider the body loads F and L of the following form F = F (a) e3 + F (p) n ,
L = 0,
(3.76)
where F (a) and F (p) are given constants. The load system (3.76) represents an uniform pressure force F (p) combined with a constant body force F (a) acting in the axial direction. F 0 ,M M0 , F, According to Theorem 3.3, we search for a solution of our problem Π(F L) in the form (3.42), where the field v{ω} ˆ is given by (3.72). Repeating the steps described in the proof of Theorem 3.3, we find after some calculations the following ϕ) form for the solution v = (u,ϕ z3 z2 (eαβ ηα xβ ) + (eαβ ξα xβ ) + ξ4 r0 z, 6r0 2r0 2 ( z ' u2 = ηα xα + η3 + z (ξα xα + ξ3 ) + 2Pr02 ηα xα , 2 r2 F (p) z3 z2 u3 = − ηα xα − ξα xα + z r0 (Fηα xα +Hη3 ) + r0 (Fξα xα +Hξ3 ) + 0 , 6r0 2r0 C1 +C2 ϕ1 = −zG (eαβ ηα xβ ) + ξ4 z − G (eαβ ξα xβ ) , z2 z ϕ2 = ηα xα + ξα xα + Q r0 ηα xα − H r0 η3 , 2r0 r0 (3.77) where the constants ηi are expressed in terms of the forces Fα0 and F (a) by the relations F0 2π r0 F (a) ηα = − α , η3 = − , (3.78) D11 D33 u1 =
and the constants ξk are given in terms of F30 , M0i and F (p) by ξα =
eαβ M0β D11
,
ξ3 = −
F30 D33
+
r0 F (p) (Cα να ) , (C1 +C2 )2 − (Cα να )2
ξ4 = −
M03 D44
.
(3.79)
52
M. Bˆırsan and H. Altenbach
The constants coefficients P and Q are specified by the equations Cα (1 − να ) F(Cα να ) + (C1 + C2 ) 1 + 4r02 P − Q = G + 4r02 , D (1 − ν ) Dα (1 − να ) α α ⎛ ⎞ ' ( ⎜⎜ λ2 r2 ⎟⎟ λ2 r 2 G Dα (1 + να ) − 2(D1 + D2 ) P − ⎜⎜⎜⎝1 + 2 0 ⎟⎟⎟⎠ Q = 2 0 F + . Dα (1 − να) h h
(3.80)
The solution (3.80) represents the displacement and rotation field for the deformed three-layered shell, corresponding to the given system of external loads. In the same way, we can apply the general solution procedure to solve various practical problems concerning multi-layered cylindrical shells. Acknowledgements For the first author M.B.: the research leading to these results has received funding from the European Union Seventh Framework Programme, FP7-REGPOT-2009-1 under grant agreement no:245479, and from the Polish Ministry of Science and Higher Education, grant no:1471-1/7,PR UE/2010/7; and sponsorship by the Alexander von Humboldt–Foundation.
References 1. Altenbach H (2000) An alternative determination of transverse shear stiffnesses for sandwich and laminated plates. Int J Solids Struct 37: 3503-3520 2. Altenbach H, Eremeyev VA (2008) Direct approach-based analysis of plates composed of functionally graded materials. Arch Appl Mech 78: 775-794 3. Altenbach H, Eremeyev VA (2009) On the bending of viscoelastic plates made of polymer foams. Acta Mech 204: 137-154 4. Altenbach H, Eremeyev VA (2010) On the effective stiffness of plates made of hyperelastic materials with initial stresses. Int J Non-Lin Mech 45: 976-981 5. Altenbach H, Zhilin PA (2004) The theory of simple elastic shells. In: Kienzler R, Altenbach H, Ott I (eds) Theories of Shells and Plates, Lecture Notes in Applied and Computational Mechanics 16: 1-12. Springer, Berlin 6. Berdichevsky V, Armanios E, Badir A (1992) Theory of anisotropic thin-walled closed-crosssection beams. Comp Eng 2: 411-432 7. Bˆırsan M, Altenbach H (2010) A mathematical study of the linear theory for orthotropic elastic simple shells. Math Meth Appl Sci 33: 1399-1413 8. Bˆırsan M, Altenbach H (2011) On the dynamical theory of thermoelastic simple shells. ZAMM, doi: 10.1002/zamm.201000057 9. Cosserat E, Cosserat F (1909) Th´eorie des corps d´eformables. Herman et Fils, Paris 10. Ies¸an D (2009) Classical and Generalized Models of Elastic Rods. Chapman & Hall / CRC Press, Boca Raton - London - New York 11. Ies¸an D (2011) Deformation of porous Cosserat elastic bars. Int J Solids Struct 48: 573-583 12. Lurie AI (2005) Theory of Elasticity. Springer, Berlin 13. Naumenko K, Altenbach H (2007) Modeling of Creep for Structural Analysis. SpringerVerlag, Berlin 14. Reissner E, Tsai WT (1972) Pure bending, stretching, and twisting of anisotropic cylindrical shells. J Appl Mech 39: 148-154 15. Sokolnikoff IS (1956) Mathematical Theory of Elasticity. McGraw-Hill, New York 16. Timoshenko S, Goodier JN (1951) Theory of Elasticity. McGraw-Hill, New York 17. Zhilin PA (1976) Mechanics of deformable directed surfaces. Int J Solids Struct 12: 635-648 18. Zhilin PA (2006) Applied Mechanics: Foundations of Shell Theory (in Russian). Politekhn Univ Publ, Sankt Petersburg
Chapter 4
Asymptotic Integration of One Narrow Plate Problem Valentina O. Finiukova and Alexander M. Stolyar
Abstract The derivation of a narrow plate model is carried out in the paper. It has been based on the method of asymptotic integration in connection with the boundary layer method and applied to one non-linear dynamic problem. It allows to reduce the solution of the given problem to a sequence of linear or nonlinear one-dimensional initial boundary-value problems (as a result of the first iteration process) and a sequence of two-dimensional linear problems (second order process) and get the solution of a given problem with high accuracy. The small parameter is equal to the relation of the lengths of adjacent sides of the plate. It has been obtained that the main term of the asymptotic expansion satisfies the known equation of a beam theory. Some examples are calculated in order to find the limits of asymptotics obtained. Keywords Asymptotic integration · Narrow plate
4.1 Introduction In this paper we consider one singularly disturbed initial-boundary value problem, which describes the dynamic behavior of so-called narrow rectangular plates with free longitudinal sides. Both the asymptotic integration and the boundary layer method are applied to this problem. The solution is constructed as power series in parameter δ which is a ratio of adjacent sides of the plate. Coefficients of the first series are obtained as a result of the first iteration process by its direct substitution in equations and boundary (and initial) conditions. The main terms of expansions satisfy the known linear or non-linear equations of lower dimension. During the first iteration process discrepancies appear. They are compensated during the V. O. Finiukova (B) · A. M. Stolyar Southern Federal University, Milchakova, 8a, 344090 Rostov-on-Don, Russia e-mail:
[email protected],
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 4, © Springer-Verlag Berlin Heidelberg 2011
53
54
V.O. Finiukova and A.M. Stolyar
second iteration process by boundary-layer functions which are the solutions of the corresponding generalized linear bi-harmonic equations for a semi-strip. The transfer from three-dimensional linear equations of elasticity theory to the two-dimensional one in the case of thin domains have been investigated in [1, 3, 14, 17] and in [2, 5, 12] – for non-linear problems. Non-linear problems of oscillations and dynamic stability of narrow cylindrical panel are considered in [8]. The beam type equations are derived on the base of Mindlin-Timoshenko plate equations in [6, 7]. The following problems are considered in [10]: 1. 2. 3. 4.
linear bending of the plate on non-linear elastic foundation; linear bending of orthotropic plate on linear foundation; linear oscillations of orthotropic plate on linear foundation; oscillations of orthotropic plate (von K´arm´an type equations [2]).
The transition from two-dimensional plate equations to one-dimensional beam equations takes place when longitudinal sides of the plate are free. Another boundary conditions are defined which allow such transition; some relationship between mechanical parameters are found which makes such transition impossible [10]. In this paper we develop the approach presented in [10] to consider the problem concerning oscillations of orthotropic rectangular plate on the linear foundation.
4.2 Oscillations of Narrow Orthotropic Rectangular Plate on Elastic Foundation. Governing Equations Let us consider the corresponding initial-boundary value problem, given in dimensionless form (here we use von K´arm´an plate equations [2]): c2 ∂4y w + 2c3δ2 ∂2y ∂2x w + δ4 ∂4x w + k1δ4 w + δ4∂2t w = δ2 L(w, Φ) + δ4 q(x, t),
(4.1)
c2 ∂4y Φ + 2c1 δ2 ∂2y ∂2x Φ + δ4 ∂4x Φ = −αc2 δ2 L(w, w), ∂ ∂ ∂ a [w, ∂t w]t=0 = 0, ∂ x = , ∂y = , ∂t = , δ = ∂x ∂y ∂t l
(4.2)
L(w, Φ) = ∂2x w∂2y Φ + ∂2y w∂2x Φ − 2∂ x ∂y w∂ x ∂y Φ, ∂2x Φ, ∂ x ∂y Φ, ∂2y w + ν1 δ2 ∂2x w, ∂3y w + c4 δ2 ∂y ∂2x w = 0, y=±1 w, ∂2x w, δ2 ∂2x Φ − ν2 ∂2y Φ, δ2 ∂3x Φ + c5 ∂ x ∂2y Φ =0
(4.3)
x=0, 1
(4.4) (4.5)
Besides we use known identity (written in dimensionless form as well) 1% 1 & ∂2y Φ − ν1 δ2 ∂2x Φ dx = αδ2 (∂ x w)2 dx 0
0
(4.6)
4 Asymptotic Integration of One Narrow Plate Problem
55
The introduced coefficients are connected with dimensionless values by the following relations: x1 = lx, c2 =
D2 , D1
Di =
x2 = ay, c3 =
D3 , D1
E i h3 , 12(1 − ν1ν2 )
W = wh,
q=
Ql4 , D1 h
c4 =
D3 + 2Dk , D2
τ = ct,
c2 =
c5 = 2c1 + ν2 ,
k1 =
Dk =
ρhl4 , D1
Kl4 , D1
Gh3 , 12
2c1 =
D3 = D1 ν2 + 2Dk ,
F = D1 Φ,
i = 1, 2,
E2 − 2ν2 , G
α = 6(1 − ν1ν2 ),
E1 ν2 = E 2 ν1 ,
where W and h are the displacement and the thickness of the plate, K is the foundation reaction coefficient, E i is the Young’s modulus, νi is the Poisson’s ratio, G is the shear modulus, ρ is the material density, F is the function of forces, Q(x1 , τ) is the transverse load, τ is the time coordinate. We suppose that our plate occupies the domain Ω: Ω = {(x1 , x2 ) : 0 ≤ x1 ≤ l, −a ≤ x2 ≤ a}. Equations (4.4) and (4.5) describe boundary conditions for free of forces and moments longitudinal sides of the plate (x2 = ±a) and simply supported edges x1 = 0, x1 = l. The parameter δ is a characteristic of the relative width of the plate. We suppose it to be small. It stands near several terms of Eqs (4.1), (4.2) and the boundary conditions (4.4), (4.5) (and identity (4.6)) and makes the problem (4.1)–(4.5) singularly disturbed.
4.3 First Iteration Process Following the approach of asymptotic integration method we construct solution of (4.1)–(4.5) as power series: w=
∞
w2k (x, y, t)δ2k ,
Φ=
k=0
∞
Φ2k (x, y, t)δ2k
(4.7)
k=0
Then we substitute (4.7) in (4.1)–(4.5) and gather coefficients standing near δ of the same degree. We find as a result of the first step: . / w0 = w0,0 (x, t), Φ0 = 0, w0,0 , ∂2x w0,0 = 0, w0,0 , ∂t w0,0 t=0 = 0 (4.8) x=0,1
Function w0,0 is undefined yet. Second step results in following relations: w2 = w2,0 (x, t) + w2,2 (x, t)y2 ,
w2 , ∂2x w2
x=0,1
=0
(4.9)
56
V.O. Finiukova and A.M. Stolyar
We may get from the first boundary condition in (4.9)
w2,0
x=0,1 = 0,
(4.10)
but we can not fulfill the second boundary condition in (4.9). The discrepancy will be eliminated using boundary layer functions further on. Besides we get formulas for Φ2 using (4.6) 1 d2 (t) = α 2
Φ2 = d2 (t)y , 2
1
' ( ∂ x w0,0 2 dx
0
On the third step we find the corresponding relations for definition of w4 and Φ4 , but some of them can not be solved because of discrepancies on the edges x = 0, x = 1. However, condition of compatibility of several equations, obtained on the first three steps of the first iteration process, allows to get the initial-boundary value problem for the main term of asymptotic expansion. We write it in dimension variables l% &2 E 1 h3 4 1 E1 h 2 ∂ x1 W + KW + ρh∂2τ W = ∂ x1 W ∂ x1 W dx1 + Q, 12 2 l 0 W, ∂2x1 W = 0, [W, ∂τ W]τ=0 = 0
(4.11)
x=0,l
Equation (4.11) with boundary and initial conditions describes oscillations of the beam on the linear foundation with simply supported edges. Continuing first iteration process we get the following relations for constructing series (4.7): k {w2k , Φ2k } = y2 j {w2k,2 j (x, t), Φ2k,2 j (x, t)} j=0
Functions w2k,2 j (where j 0) and Φ2k,2 j depend on the functions obtained on the previous steps of the first iteration process, functions w2k,0 satisfy the linear differential equations. For instance, one could get the following relations: Φ2,2 = d2 (t),
Φ4,0 = Φ4,4 =
&2 αν1 % 2 ∂ x w0,0 , 12
(1 − ν1ν2 )∂4x w2,0 + kw2,0 + ∂2t w2,0 − 2Φ2,2 ∂2x w2,0 1 1 +2α∂2x w0,0 w2,0 ∂2x w0,0 dx = q2 − 2Φ4,4 ∂2x w0,0 + 4 Φ4,4 dx, 0
(4.12)
0
where q2 denotes the known functions on the current step of the first iteration process.
4 Asymptotic Integration of One Narrow Plate Problem
57
4.4 Second Iteration Process In order to remove discrepancies which appear in first iteration process we use boundary layer functions. We construct the solution of the problem (4.1)–(4.5) as follows: ∞ w = δ2k {w2k (x, y, t) + u2k (ξ, y, t) + v2k (η, y, t)} , k=0
Φ=
∞
(4.13) δ {Φ2k (x, y, t) + ϕ2k (ξ, y, t) + ψ2k (η, y, t)} , 2k
k=0
where ξ = x/δ, η = (1 − x)/δ are the so-called “extended” variables, u2k , ϕ2k and v2k , ψ2k are the boundary layer functions, which are concentrated near the edges x = 0 and x = 1, respectively. After substituting (4.13) in (4.1)–(4.6) and taking into account the results of the first iteration process one can obtain the relations concerning the boundary layer functions. On the first steps of the second iteration process we get u0 = ϕ0 = 0; then it is easy to show that the boundary value problems for the functions u2k , ϕ2k and v2k , ψ2k are linear (k 0). Let us consider first non-trivial boundary value problems.
4.4.1 Solution of Generalized Bi-Harmonic Problem Function u4 satisfies the following equations c2 ∂4y u4 + 2c3 ∂2y ∂2ξ u4 + ∂4ξ u4 = 0,
u4 , ∂2ξ u4
(4.14)
= 0, ξ=1/δ ∂2y u4 + ν1 ∂2ξ u4 , ∂3y u4 + c4 ∂y ∂2ξ u4 = 0, y=±1
u4 ξ=0 = −w4 x=0 , ∂2ξ u4
ξ=0 = −∂2x w2
x=0
(4.15)
Here the variable t is a parameter. The form of the solution of the problem (4.14), (4.15) depends on relation between mechanical constants. If 2 G 1 ν2 1 = , √ E 1 2 ν1 1 + ν1 ν2 then the solution of the problem (4.14), (4.15) may be reduced to the case of isotropic plate/shell [8, 9]. If 2 G 1 ν2 1 ≺ , (4.16) √ E 1 2 ν1 1 + ν1 ν2
58
V.O. Finiukova and A.M. Stolyar
then the solution of problem (4.14), (4.15) is given by the following series1 : 0
u4 =
E k e−sk ξ F k (y) + 2 Re
∞
k=− j+1
E k e−sk ξ F k (y)
k=1
The so-called Papkovich functions Fk (y) satisfy the boundary value problems: c2 FkIV + 2c1 s2k F k + s4k Fk = 0,
[Fk + ν1 s2k F k , F k + c4 s2k F k ]y=±1 = 0,
(4.17)
where sk are the “reduced” roots of equation κ · sinhz − sin(ωz) = 0,
(4.18)
z = 2βsk , j is the number of its real roots; κ, ω and β are the constants, which are connected with mechanical parameters of the plate; one can obtain asymptotic expressions for real and imaginary parts of the complex roots of Eq. (4.18) as follows: 4πω ωπ 4k + 1 ln κ − k · − + O e 1+ω2 , 2 1 + ω2 1 + ω2 4πω π 4k + 1 ω ln κ − k Imz = · + + O e 1+ω2 2 1 + ω2 1 + ω2 Re z =
k→∞
(4.19)
Equations (4.19) are obtained in the case, when κ and ω are supposed to be fixed. Coefficients Ek are obtained using the identities [11, 15]
1% −1
& Fk Fn − ps2k s2n Fk F n dx = 0,
1% −1
k n,
& F k F¯ n − ps2k s¯2n F k F¯ n dx = 0,
p=
c1 , c2
for ∀ k, n,
(4.20)
(4.21)
where the overlined variables are the complex conjugate, as follows: Ek =
1
(F k f1 − s2k Fk f2 ) dy
−1 1 −1
(Fk )2 − s4k Fk2
(4.22) dy
In connection with the calculation of E k the problem of representation of two real functions f1 (y) and f2 (y) as series of Papkovich functions f1 (y) =
∞ k=− j+1
1
E k Fk (y),
f2 (y) =
∞
s2k E k Fk (y),
(4.23)
k=− j+1
If instead of inequality (4.16) the opposite consideration arises, the problem can be solved in a similar way.
4 Asymptotic Integration of One Narrow Plate Problem
59
has been considered [11]. Different problems of such kind are investigated in [4, 9, 13, 16] and so on including the problem of completeness of the system of Papkovich elementary solutions for bi-harmonic operator in a semi-strip. Let us give the conditions of probability of expansions (4.23) (theorem 1 was proved in [11], taking after Grinberg’s method [4]; proof of the theorem 2 is similar to the first one). Theorem 4.1. Let an even function f1 (y) be continuously differentiable and its second derivative be absolutely integrated on [0, 1]. Then the sum of the first series in (4.23) coincides with the function f1 (y) when y ∈ [0, 1) if the following relations take place 1 ν2 f1 (1) + f2 (y)dy = 0, (4.24) 0
c1 ν1 ν2 f1 (1) − 2ν1c4 f1 (1) + 2(2c3ν1 − c1 )
1
1
f1 (y) dy + c1ν1
0
f2 (y)y2 dy = 0
0
Theorem 4.2. Let an even function f2 (y) be continuously differentiable and its second derivative be absolutely integrated on [0, 1]. Let function f1 (y) meet the same requirements. Then the sum of the second series in (4.23) coincides with the function f2 (y) when y ∈ [0, 1) if the first relation (4.24) takes place. Equalities (4.24) may be called the conditions of convergence of the series (4.23). They have practical application as well. In the first iteration process we got a sequence of linear equations, which allows to obtain functions w2k,0 ; for instance function w2,0 satisfies Eq. (4.12). We have also two boundary conditions (4.10) for w2,0 . But we failed trying to get boundary values for ∂2x w2,0 . We needed the second iteration process to obtain them (functions −w4 | x=0 and −∂2x w2 | x=0 (see Eqs (4.15)) play the role of functions f1 (y) and f2 (y)). For that purpose we substitute −w4 | x=0 and −∂2x w2 | x=0 instead of f1 (y) and f2 (y) in the first relation (4.24) and taking into account the results of the first iteration process we get the missing boundary value
1 G 4 2 ∂ x w2,0 x=0 = ν1 1 + 8ν1 ∂ x w0,0
x=0 6 E1 Function v4 satisfies the boundary value problem similar to (4.14)–(4.15). Its solution allows to get the boundary value of ∂2x w2,0 on the edge x = 1.
4.4.2 One More Boundary Value Problem Function ϕ2 satisfies the following equations c2 ∂4y ϕ2 + 2c1 ∂2y ∂2ξ ϕ2 + ∂4ξ ϕ2 = 0, [Aϕ2 , Bϕ2 ]ξ=1/δ = 0,
∂2ξ ϕ2 , ∂y ∂ξ ϕ2
y=±1
= 0,
Aϕ2 |ξ=0 = 2ν2 d2 (t),
(4.25) Bϕ2 |ξ=0 = 0,
60
V.O. Finiukova and A.M. Stolyar
A = ∂2ξ − ν2 ∂2y ,
B = ∂2ξ + c5 ∂ξ ∂2y
The solution of the problem (4.25) is also constructed as a series of Papkovich functions and may be reduced to the case considered in [4].
4.5 Estimation of the Remainder of Series in the Linear Case If we investigate the linear model of an orthotropic plate [10] and consider the linear initial boundary value problem instead of the problem (4.1)–(4.5) one can estimate the remainder term of solution. Let the solution be given by the first series (4.13). We use the notation U 2n =
n
δ2k {w2k (x, y, t) + u2k (ξ, y, t) + v2k (η, y, t)}
(4.26)
k=0
and the following known definition. Definition 4.1. We call function U the asymptotic δ-approach to solution w in the domain Ω, if supΩ |w − U| → 0 when δ → 0. Besides if supΩ |w − U| = O(δn ) then we call function U the asymptotic approach to solution w in Ω with accuracy of δn order. Using this definition, one can establish the following theorem. Theorem 4.3. Let the function q(x, t), x ∈ [0, 1], has two continuous derivatives and ∂2x q(0, t) = ∂2x q(1, t) = 0. Then the function U4 is an asymptotic approach to solution w of the corresponding linear problem [10], i. e. w(x, y, t) = U 4 + O(δ6 ), when δ → 0.
4.6 Closing Remarks In this paper we use the method of asymptotic integration with boundary layer method for reducing of nonlinear plate initial boundary value problem to the sequence of initial boundary value problems of smaller dimension; the main term of asymptotic expansion satisfies the known non-linear integro-differential equation and junior terms are the solutions of linear beam oscillations problems. Boundary values, which we need for the definition of mentioned junior terms are calculated using the conditions of convergence of the series, which appear as the solutions of the corresponding boundary-layer problems. These solutions are the eliminators of discrepancies, which appear during the first iteration process. Remark 4.1. Let the longitudinal sides of the plate be not absolutely free, but the bending moment M(x1 , τ) be given there on the boundaries x2 = ±a. It means that instead of the third equality (4.4) we have the following condition:
4 Asymptotic Integration of One Narrow Plate Problem
∂2y w + ν1 δ2 ∂2x w
y=±1
= −δ2 m(x, t),
61
m(x, t) =
M(x1 , ct) 2 l D1 h
In that case one may also transit the plate problem to the problem of beam oscillations under reduced load Q(x1 , τ) + ν1 ∂2x1 M(x1 , τ). Remark 4.2. Let the third equality in (4.4) be replaced by [∂y w]y=±1 = 0. It means that free clamping of the longitudinal sides of the plate takes place. In that case one may also transit the plate problem to the problem of beam oscillations with reduced elasticity modulus. The corresponding equation is D1 ∂4x1 W + KW + ρh∂2τ W = Q(x1 , τ) Remark 4.3. Let the Young’s modulus be connected by the relation E1 /E 2 = O(δ2 ). It means that the rigidity of the material is much more in Ox2 direction than in longitudinal direction Ox1 . Asymptotic integration does not lead to any transition in that case. Remark 4.4. While carrying out the asymptotic integration of the plate equations, we need to obtain initial values of the first iteration process functions of course. It is a special part of the problem, we did not present it in this paper. We only note here that some matching conditions are necessary. For instance, if we keep the terms up to O(δ4 ) order in solution (4.13) of the problem, then the following equalities must take place q(1, 0) = ∂2x q(1, 0) = ∂t q(1, 0) = q(0, 0) = ∂2x q(0, 0) = ∂t q(0, 0) = 0 Remark 4.5. For the purpose of estimation of the boundaries of asymptotics obtained application, numerical algorithms of beams, plates and cylindrical panels oscillations and dynamical buckling have been developed on the base of finite difference method. Corresponding numerical analysis has been carried out and estimations defined for isotropic material [8, 10].
References 1. Aghalovyan, L. A.: On the boundary layer of the orthotropic plates. (in Russian). Izv. AN Arm. SSR. Mechanics. 26, 2, 27–43 (1973). 2. Ciarlet, P. G., Rabier, P.: Les equations de von K´arm´an. Springer-Verlag, Berlin-HeidelbergNew York (1980). 3. Gol’denweiser, A. L.: Theory of Thin Shells (in Russian). Nauka, Moscow (1976). 4. Grinberg, G. A.: On the method, suggested by P. F. Papkovich for solution of the plane problem of the theory of elasticity in rectangular domain and the problem of bending of rectangular thin plate with two fastened edges and its some generalizations (in Russian). Prikl. Mat. Mech. 17, 2, 211–228 (1953). 5. Kucherenko, V. V., Popov, V. A.: Asymptotics of solution of elasticity theory problems in thin domains (in Russian). Dokl. AN SSSR. 274, 1, 58–61 (1984). 6. Russell, D. L., White, L. W.: Formulation and Validation of Dynamical Models for Narrow Plate Motion. Applied Mathematics and Computation. 58, 103–141 (1993).
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7. Russell, D. L., White, L. W.: The lowed narrow plate model. Electronic J. Diff. Equations. 27, 1–19 (2000). 8. Srubshchik, L. S., Stolyar, A. M., Tsibulin, V. G.: Asymptotic integration of nonlinear equations of cylindrical panel vibrations. J. Appl. Math. Mech. 52, 4, 511–518 (1988). 9. Stolyar, A. M.: Asymptotic integration of the equation of vibrations of long rectangular plate (in Russian). Izv. Sev.-Kavk. Nauchn. Tsentra Vyssh. Shk., Estestv. Nauki, 4 (56), 46–50 (1986). 10. Stolyar, A. M.: Asymptotic analysis of the problems of statics and dynamics of narrow rectangular plates. (in Russian). Izv. Vyssh. Ucheb. Zaved., Sev.-Kavk. Region, Estestv. Nauki, Specvypusk Pseudo-differential Equations and Some Problems of Mathematical Physics. 107–111 (2005). 11. Stolyar, A. M.: Representation of two real functions as series of P. F. Papkovich functions in one problem of limiting transition (in Russian). Izv. Vyssh. Ucheb. Zaved., Sev.-Kavk. Region, Estestv. Nauki, Specvypusk Actual Problems of Mathematical Hydrodynamics. 203– 206 (2009). 12. Sugimoto, N.: Nonlinear theory for flexural motions of thin elastic plate. Trans. ASME. J. Appl. Mech. 48, 2, 377–390 (1981). 13. Ustinov, Yu. A., Yudovich, V. I.: On the completeness of a system of elementary solutions of the biharmonic equation in a semi-strip. J. Appl. Math. Mech. 37, 4, 665–674 (1973). 14. Ustinov, Yu. A.: Boundary problems and the problem of limiting transition from threedimensional problems of the theory of elasticity to two-dimensional problems for heterogeneous plates. Dr. habil. Thesis (in Russian). Moscow (1977). 15. Vasil’ev, V. V., Lur’e, S. A.: Plane problem of elasticity theory for orthotropic clamped strip (in Russian). Izv. AN SSSR. MTT. 5, 125–135 (1984). 16. Vorovich, I. I.: Some mathematical problems of the theory of plates and shells (in Russian). In Trans. of II All-Union Congress on Theoretical and Applied Mechanics, pp. 116–136. Moscow (1966). 17. Vorovich, I. I.: Some results and problems of the asymptotic theory of plates and shells. (in Russian). In Trans. of I All-Union school on theory and numerical methods of calculation of plates and shells, pp. 51–149. Tbilisi (1975).
Chapter 5
On Cusped Shell-like Structures George Jaiani
Abstract This paper is updated concise survey of results concerning elastic cusped shells, plates, and beams and cusped prismatic shell-fluid interaction problems. Keywords Cusped beams · Cusped plates · Cusped prismatic shells · Mathematical modeling · Linear elasticity
5.1 Introduction The present paper is an updated concise survey of investigations concerning elastic cusped (in other words, said differently, sharpened, cuspidate, cuspate, cuspidal, tapered) shell-like structures, namely, cusped (standard and prismatic) shells, plates, and beams, as well as cusped shell-like elastic body – fluid interaction problems. Under cusped shells (see, e.g., [1, 2]) we understand shells whose thickness vanishes either on a part or on the whole boundary of either the standard shell “middle” surface or prismatic shell projection. Beams are called cusped (see [3, 4]) if at least at one of the endpoints of their axis the cross-section area vanishes. Mathematically, one is lead to posing and solving, in the static case, certain boundary value problems (BVPs) for even order equations and systems of elliptic type with order degeneration, while in the dynamical case one faces initial boundary value problems (IBVPs) for even order equations and systems of hyperbolic type with order degeneration. In the study of cusped shell-like bodies, forces concentrated along cusped edges and at points may be encountered; moreover, it may happen that well-posed displacement boundary conditions (BCs) along cusped edges depends on their geometry and sharpness. At present, we can count on a sufficiently complete mathematical G. Jaiani (B) I. Vekua Institute of Applied Mathematics of Iv. Javakhishvili Tbilisi State University, 2 University St., 0186, Tbilisi, Georgia e-mail:
[email protected],
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 5, © Springer-Verlag Berlin Heidelberg 2011
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theory of elastic cusped prismatic shells and beams; however, the study of cusped standard shells of general form remains topical.
5.2 Geometry of Structures Under Consideration Investigations of cusped elastic prismatic shells, actually, takes its origin from the fifties of the last century, namely, in 1955 I. Vekua [1, 2] raised the problem of investigation of elastic cusped prismatic shells, whose thickness on the prismatic shell entire boundary or on its part vanishes. Such bodies, considered as 3D ones, may occupy 3D domains with, in general, non-Lipschitz boundaries. In practice, such cusped prismatic shells, in particular, cusped plates, and cusped beams (i.e., beams whose cross-sections’ area vanishes at least at one end of the beam) are often encountered in spatial structures with partly fixed edges, e.g., stadium ceilings, aircraft wings, submarine wings, etc., in machine-tool design, as in cutting-machines, planning-machines, in astronautics, turbines, and in many other application fields of engineering. (+)
x3=h(x1,x2) x3
Fig. 5.1 Prismatic shell of a constant thickness. ∂Ω is a Lipschitz surface
x2
(-)
x3=h(x1,x2)
x1
(+)
x3=h(x1,x2) (-)
x3
Fig. 5.2 Sharp cusped prismatic shell with a semicircle projection. ∂Ω is a Lipschitz surface
x3=h(x1,x2) x2 x1
G0 = g
0
Let Ox1 x2 x3 be an anticlockwise-oriented rectangular Cartesian frame of origin O. We conditionally assume the x3 -axis vertical. The elastic body is called a prismatic shell if it is bounded above and below by, respectively, the surfaces (so called face surfaces) (+)
(−)
x3 = h (x1 , x2 ) and x3 = h (x1 , x2 ),
5 On Cusped Shell-like Structures
65 (+)
x3=h(x1,x2)
g1
g2
x3
Fig. 5.3 Cusped plate with sharp γ1 and blunt γ2 edges, γ0 = γ1 ∪ γ2 . ∂Ω is a nonLipschitz boundary
x2 (-)
x1
x3=h(x1,x2)
(+)
x3=h(x1,x2)
Fig. 5.4 Blunt cusped plate with the edge γ0 . ∂Ω is a Lipschitz boundary
G0 = g
(-)
0
x3=h(x1, x2) =0
Fig. 5.5 Cross-sections of a prismatic (left) and a standard shell with the same midsurface
laterally by a cylindrical surface Γ of generatrix parallel to the x3 -axis. By Γ0 we denote a part of Γ which degenerates in a curve. In other words, the 3D elastic prismatic shell-like body occupies a bounded region Ω with boundary ∂Ω, which is defined as: * + (−) (+) Ω := (x1 , x2 , x3 ) ∈ R3 : (x1 , x2 ) ∈ ω, h (x1 , x2 ) < x3 < h (x1 , x2 ) , (5.1) where ω := ω ∪ ∂ω is the so-called projection of the prismatic shell Ω := Ω ∪ ∂Ω (see Figs 5.1-5.4); γ = ∂ω and ∂Ω denote boundaries of ω and Ω, respectively. * (+) (−) > 0 for (x1 , x2 ) ∈ ω, 2h(x1 , x2 ) := h (x1 , x2 ) − h (x1 , x2 ) ≥ 0 for (x1 , x2 ) ∈ ∂ω is the thickness of the prismatic shell Ω at the points (x1 , x2 ) ∈ ω ¯ = ω ∪ ∂ω. max{2h} is essentially less then characteristic dimensions of ω. Let (+)
(−)
2h(x1 , x2 ) := h (x1 , x2 ) + h (x1 , x2 ).
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In the symmetric case of the prismatic shells, i.e., when
(−)
h (x1 , x2 )
(+)
= − h (x1 , x2 ), i.e., 2h(x1 , x2 ) = 0, we have to do with plates of variable thickness 2h(x1 , x2 ) and a middle-plane ω (see Figs 5.3, 5.4). Prismatic shells are called cusped if a set γ0 , consisting of (x1 , x2 ) ∈ ∂ω for which 2h(x1 , x2 ) = 0, is not empty (see Figs 5.3, 5.4, 5.11–5.18). Distinctions between the prismatic shell of a constant thickness and the standard shell of a constant thickness are shown in Fig. 5.5. The lateral boundary of the standard shell is orthogonal to the “middle surface”, while the lateral boundary of the prismatic shell is orthogonal to the prismatic shell’s projection on x3 = 0. Fig. 5.6 Cusped beam with rectangular cross-sections. ∂Ω is a Lipschitz boundary
Fig. 5.7 A cusped beam with rectangular crosssections. ∂Ω is a Lipschitz boundary
Fig. 5.9 A conical wedge. ∂Ω is a Lipschitz boundary
Fig. 5.8 A cusped circular beam. ∂Ω is a Lipschitz boundary
Fig. 5.10 A cusped circular beam. ∂Ω is a non-Lipschitz boundary
3 (−) (+) If in (5.1) ω := (x1 , x2 ) ∈ R2 : 0 < x1 < L, h2 (x1 ) < x2 < h2 (x1 ), 0 ≤ 2h2 (x1 ) := 4 (+) (−) h2 (x1 ) − h2 (x1 ) L, L = const > 0 , the prismatic shell-like body Ω will become a b
beam-like body with a cross-section of arbitrary form. In particular, let a domain Ω of R3 occupied by an elastic beam be * (−) (+) Ωb := (x1 , x2 , x3 ) ∈ R3 : 0 < x1 < L, hi (x1 ) < xi < hi (x1 ), + (+) (−) 5 2hi (x1 ) := hi − hi ≥ 0, hi ∈ C([0, L]) C 2 (]0, L[), i = 2, 3, L = const > 0
and 2h3 and 2h2 be correspondingly the thickness and the width of the beam and their maxima be essentially less then the length L of the bar; the superscript “b” means beam, see Figs 5.11–5.18 showing at the cusp the angle ϕ between tangents
5 On Cusped Shell-like Structures (+)
67
(−)
T and T , where typical longitudinal (vertical and horizontal) sections of beams are given, and Figs 5.6–5.10. If at least one of the conditions 2hi (0) = 0 and 2hi (L) = 0, i = 1, 2, is fulfilled, a beam is called the cusped one. The last class of beams consists of beams with rectangular cross-sections which may degenerate in segments or points at the beams ends, see Figs 5.6, 5.7. (+)
T
(-)
x2
T
Fig. 5.12 ϕ = π
Fig. 5.11 ϕ = 0
(+)
(+)
T
(-)
T
Fig. 5.13 ϕ =
T
(-)
x2 π 2
Fig. 5.14
(+)
π 2
T
x2
(+)
Fig. 5.17 0 < ϕ < π
x2
T
π 2
(+)
T
T
T
(-)
Fig. 5.16 0 < ϕ <
Fig. 5.15 ϕ = 0
(-)
<ϕ<π (+)
T
T
(-)
x2
T
x2
(-)
T
x2
Fig. 5.18 Wedge, 0 < ϕ < π
5.3 Hierarchical Models In what follows Xi j and ei j are the stress and strain tensors, respectively, ui are the displacements, Φi are the volume force components, ρ is the density, λ and μ are the Lam´e constants, δi j is the Kronecker delta, subscripts preceded by a comma mean partial derivatives with respect to the corresponding variables. Moreover, repeated indices imply summation (Greek letters run from 1 to 2, and Latin letters run from 1 to 3, unless stated otherwise), bar under one of the repeated indices means that we do not sum. I. Vekua’s hierarchical models for elastic prismatic shells are the mathematical models, which are based on the multiplication of the basic equations of linear elasticity: motion equations, generalized Hooke’s law (isotropic case), kinematic relations by Legendre polynomials Pr (ax3 − b), where
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a(x1 , x2 ) :=
1 h(x1 , x2 ) , b(x1 , x2 ) := , h(x1 , x2 ) h(x1 , x2 ) (−)
(+)
and then integration with respect to x3 within the limits h (x1 , x2 ) and h (x1 , x2 ). By these calculations in Vekua’s first version on upper and lower face surfaces stressvectors are assumed as prescribed, while there the values of the displacements are calculated from their (displacements’) Fourier-Legendre series expansions on the (−) (+) segment x3 ∈ h (x1 , x2 ), h (x1 , x2 ) and vice versa in his second version. So, we get the equivalent infinite system of relations with respect to the so called l-th order moments %
h% & & Xi jl , ei jl , uil (x1 , x2 , t) := (−) Xi j , ei j , ui (x1 , x2 , x3 , t)Pl (ax3 − b) dx3 . (+)
h
Then, having followed the usual procedure used in the theory of elasticity, we get an equivalent infinite system with respect to the l-th order moments uil . After this if we assume that the moments whose subscripts, indicating moments’ order, are greater than N equal zero and consider only the first N + 1 equations in the obtained infinite system of equations with respect to the l-th order moments uil we obtain the N−th order approximation (hierarchical model) governing system with respect to N
N
u il (roughly speaking u il is an “approximate value” of uil ). In the N th approximation for cusped, in general, homogeneous elastic prismatic shells the governing system in displacements reads as [1]: N r N N N N μ h2r+1 v αr, j + h2r+1 v jr,α +λδα j h2r+1 v γr,γ + Bα jks hr+s+1 v ks ,α
,α
,α
s=r+1
⎡ ⎤⎥ r−1 N l ⎥ N r+l+1 N r+l+1 N r+s+1 N r ⎢⎢⎢⎢ ⎢ v γ l,γ + μh v il, j + v jl,i + v ks ⎥⎥⎥⎥⎦ + ail ⎣⎢λδi j h Bi jks h l=0
s=l+1
r
+hr X j = ρhr N
,α
(5.2)
N ∂2 hr+1 v
∂ t2
jr
, r = 0, N, j = 1, 2, 3,
q−1 (· · · ) ≡ 0, q
N
u
kr where v kr := hr+1 , k = 1, 2, 3, r = 0, N, are unknown so called weighted “moments” of
r
r
l
displacements, X j , ail , and Bi jks are known functions. Similar hierarchical models, when on the face surfaces displacements are prescribed can be found in [1]. The analogues system in (N3 , N2 ) approximation of hierarchical models for cusped, in general, prismatic beams are derived in [3] as follows
5 On Cusped Shell-like Structures
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N3 N2 3 i j (N3 ,N2 ) i j (N3 ,N2 ) 2n 2 +1 2n 3 +1 (N3 ,N2 ) v jn 3 n 2 , 1 + Λ j h2 h3 Rrs v irs, 1 + S rs v irs ,1
n + hn2 2 h33
n3 ,n2 X 0j
=
i=1 r=0 s=0
n2+1 n3 +1 (N3 ,N2 ) v j n 3 n2 h3 n2 n3 ∂h2 ρh2 h3 , 2 ∂t
j = 1, 2, 3; nk = 0, N k ,
(5.3)
(N3 ,N2 )
where v jn3 n2 are unknown so called weighted double “moments” of displacements, k = 2, 3, * λ + 2μ , j = 1, Λ j := μ, j = 2, 3, n3 ,n2
ij ij Rrs (x1 ) , S rs (x1 ) , and X 0j are known functions. In 2007 Chinchaladze and Jaiani constructed hierarchical models for shallow waters occupying non-Lipschitz, in general, prismatic domains within the scheme of small displacements linearized with respect to the rest state.
5.4 Cusped Shells and Plates The first correct work concerning classical bending of cusped elastic plates was done by Makhover and Mikhlin [4]. The bending equation of the Kirchhoff-Love plate of variable thickness has the form (see, e.g., [5]) (Dw,11 ),11 +(Dw,22 ),22 +ν(Dw,22 ),11 +ν (Dw,11 ),22 +2(1 − ν)(Dw,12 ),12 = f (x1 , x2 ), where w is a deflection, D is a flexural rigidity of the plate, f is the lateral load. They considered such a cusped plate with an adjacent to the axis x1 bounded domain ω lying in the half-plane x2 ≥ 0 and the flexural rigidity D(x1 , x2 ) satisfying D1 xk21 ≤ D(x1 , x2 ) ≤ D2 xk21 , x2 ≥ 0, D1 , D2 , k1 = const > 0.
(5.4)
Namely, it has been shown that for k1 < 2 the deflection can be prescribed on the cusped edge x2 = 0 of the plate, while its normal derivative can be prescribed for k1 < 1 there. Applying more natural spaces Jaiani has analyzed in which cases the cusped edge can be fixed (k1 < 1) or simply supported (k1 < 2). Moreover, he gave the correct formulation of all the reasonable principal BVPs. The corresponding BVPs are solved in the explicit (integral) form by Jaiani, when ω is the half-plane. In 1971 in the zero approximation of Vekua’s hierarchical models of shallow prismatic shells Khvoles represented the forth order degenerate operator acting on the Airy stress function as a product of two second order degenerate operators in the case when the plate thickness 2h is given by k
2h = h0 x22 , h0 = const > 0, k2 = const > 0, x2 ≥ 0.
(5.5)
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Since 1972 Jaiani investigated the tension-compression problem for such cusped prismatic shells and constructed effective solutions of BVPs in integrated stresses, when ω is the half-plane, half-strip, or angle. A qualitative part of Jaiani’s abovementioned results can be summarized as follows. Statement 5.1 Let n be the inward normal of the prismatic shell projection boundary. In the case of the tension-compression (N = 0) problem on the cusped edge, where (see Figs 5.15–5.18, which can be considered as cusped prismatic shells’ cross-sections) 0 ≤ ∂h ∂n < ∞ (in the case (5.5) this means k2 ≥ 1), which will be called a sharp cusped edge, the displacement vector can not be prescribed, while on the cusped edge, where (see Figs 5.11–5.14) ∂h ∂n = + ∞ (in the case (5.5) this means k2 < 1), which will be called a blunt cusped edge, the displacement vector can be prescribed. In the case of the classical bending problem when at a plate cusped edge ∂h = O(dk−1 ) as d → 0, k = const > 0, ∂n
(5.6)
where d is the distance between an interior reference point of the plate projection and the cusped edge, the edge can not be fixed if k ≥ 1/3, but it can be fixed if 0 < k < 1/3; it can not be freely supported if k ≥ 2/3 , and it can be freely supported if 0 < k < 2/3; it can be free or arbitrarily loaded by a shear force and a bending moment if k > 0. Note that in the case (5.5), the condition (5.6) implies that d2 = x2 and k = k2 = k1 /3. In other words, when a profile, i.e., cross-section (see Fig. 5.19) of the plate in a neighborhood of a plate boundary point P (blunt cusp): (i) lies in the green zone then all three main BVPs can be correctly posed; (ii) lies in the blue zone then the edge can be either simply supported or be free (or be arbitrarily loaded); (iii) lies in the red zone then such edge can be only free (or be arbitrarily loaded). The last from the above three assertions is also valid for the sharp cusp.
Fig. 5.19 Zones of setting of different BCs at cusped edges depending on geometry of tapering (sharpening)
5 On Cusped Shell-like Structures
71
Fig. 5.20 A cusped cylindrical shell
The cylindrical bending of a cylindrical shell has been considered by Tsiskarishvili and Khomasuridze when the thickness has the following form h = h0 sinκ ϕ, h0 , κ = const > 0, where ϕ ∈] 0, ϕ0 [ is the polar angle counted from the plane Oζ x (see Fig. 5.20). Let w be a radial component of the displacement vector, i.e., deflection, which obviously characterizes bending, and v be an angular component of the displacement vector, i.e., displacement in the middle surface (cylinder) orthogonal to the element of the cylinder which (i.e., displacement v) characterizes tension-compression of cylindrical shell. They proved (as foreseen by Jaiani) that deflection w can be given on the cusped edge only when κ ∈ 0, 23 , its first derivative can be given only when κ ∈ 0, 13 , but v can be given if the cusped edge is blunt, i.e., κ ∈] 0, 1 [, as in the zero approximation (compare with Statement 1). The same authors have considered the strength problem of a uniformly loaded 3D elastic structure consisting of a rotational cylindrical shell of constant thickness and two rotational cusped conical shells with a linearly changing thickness. Jaiani has proved that Statement 1 remains valid in the case of classical bending of orthotropic cusped plates. However, for general cusped shells and general anisotropic cusped plates, corresponding analysis is yet to be done. The classical bending of plates with the flexural rigidity (5.4) in energetic and in weighted Sobolev spaces has been studied by Jaiani [5]. In the energetic space some restrictions on the lateral load has been relaxed by Devdariani. In 2004 Jaiani has studied bending of an elastic cusped Kirchhoff-Love plate on an elastic foundation with a constant compliance. In 2002 in the case of the cylindrical bending Chinchaladze has explored the well-posedness of the Keldysh type and weighted BVPs for the Kirchhoff-Love plate with two cusped edges and studied general and harmonic vibrations of such plates are studied. She has shown that the setting of BCs depends on the geometry of sharpening of plate edges, while the setting of initial conditions is independent of it. In 2009 for geometrically non-linear cusped Timoshenko plate with flexural rigidity (5.4) Chinchaladze and Jaiani proved that at a cusped edge the bending moment and intersecting force can be prescribed only if 0 ≤ k1 < 2 and 0 ≤ k1 < 1, respectively (in contrast to the linear case, when these quantities can always be
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G. Jaiani
prescribed). Applying the function-analytic method developed by Fichera, in the N = 0 approximation%for general & form cusped prismatic shells Jaiani established that at blunt cusped edge ∂h = +∞ displacement vector components can be prescribed, ∂n % & ∂h while sharp cusped edge 0 ≤ ∂n < ∞ should be freed from BCs (the Keldysh type BVP for displacements). The last result concerning sharp cusped edges is true for the N-th approximation as well. In the case (5.5) the tension-compression system of the N = 1 approximation is investigated by Devdariani, Jaiani, Kharibegashvili, and Natroshvili in 2000. The existence and uniqueness of generalized solutions of BVPs with Dirichlet (for weighted zero-moments when κ2 < 1 and for weighted first moments when κ2 < 1/3) and Keldysh type (for weighted zero-moments when κ2 ≥ 1 and for weighted first moments when κ2 ≥ 1/3) BCs is proved in weighted Sobolev spaces. In [6] the vibration tension-compression system of the N = 1 approximation is studied when 2h(x1 , x2 ) ≥ hκ xκ2 , hκ = const > 0, κ = const ≥ 0, x2 ≥ 0. The method of investigation of hierarchical models based on the idea to get Korn’s type inequality for 2D models from the 3D Korn’s inequality for non-cusped domains and then use Lax-Milgram theorem belongs to Gordeziani. This method found its complete realization in [7]. The analogous approach is developed by Schwab [8]. This idea for cusped but Lipschitz 3D domains with corresponding modifications was successfully used in [9]. In the case when cusped prismatic shell occupies a Lipschitz 3D domain; on face surfaces stress vectors, while on the noncusped edge weighted moments of displacement vector components are given, with the help of variational methods, the existence and uniqueness theorems for the corresponding 2D BVPs are proved. The above method does not allow to consider BVPs when on the cusped edge either displacements or loads are prescribed. In the case under consideration the loads are concentrated along the cusped edge. In [10] the well-posedness of BVPs for elastic cusped plates (i.e., symmetric prismatic shells) in the N-th approximation N ≥ 0 of I.Vekua’s hierarchical models. Note, that for the r-th order moments Dirichlet and Keldysh BCs are correct 1 1 when κ < 2r+1 and κ ≥ 2r+1 (r=0,1,2,...N), respectively. In the zero approximation of Vekua’s hierarchical models of prismatic shells, when the thickness has the form (5.5) Jaiani has solved the Flamant, Cerutti and Carothers’ type problems in the explicit form. The purpose of [11] is to show that internal concentrated contact interactions are at times to be expected within the framework of Vekua’s hierarchical theory of prismatic shells [1]. The results can be summarized as follows. When equilibrium problems for cusped prismatic shells subject to concentrated loads are formulated within the framework of the zero approximation, (i) in both the Flamanttype problem and Cerruti-type problem, where the external concentrated force acts in direction x1 , force-like internal concentrated interactions do occur, while they do not occur in the Cerruti-type problem where the external concentrated force acts in direction x3 ; (ii) in the Carothers-type equilibrium problem, neither forcelike nor couple-like internal concentrated interactions occur. These results parallel closely those obtained, by Podio-Guidugli. He has considered the plane versions of Flamant’s, Cerruti’s, and Carothers’ problems.
5 On Cusped Shell-like Structures
73
5.5 Cusped Beams In 1879 G. Kirchhoff constructed solutions for the harmonic eigenvibration of beams having a form either of a wedge (see Fig. 5.18) or a conical wedge (see Fig. 5.9) by means of Bessel functions. In 1916, using the above exact solution for the wedge shaped beam, Timoshenko investigated the eigenvibration when cusped beam end is free while the non-cusped end is built-in. In 1980 Uzunov numerically solved the problem of bending of the cusped circular beam (see Fig. 5.8) on an elastic foundation with a constant compliance. The blunt cusped end is free and the noncusped end is clamped. In 1990–1995 the bending vibration of homogeneous Euler-Bernoulli cone beams and beams of continuously varying rectangular cross-sections, when one side (width) of the cross-section is constant, while the other side (thickness) has the form (5.5), where x2 is the axial coordinate measured from the cusped end, were considered by Naguleswaran. In these investigations the cusped end is always free. Shavlakadze has studied the contact problem for an unbounded elastic medium composed of two welded half-planes x1 > 0 and x1 < 0 having different elastic constants and strengthened on the semi-axis x2 > 0 by a cusped thin beam (as an inclusion) with a free cusped end. The well-possedness of BVPs in weighted displacements for (0,0) and (1,0) approximations are investigated by Jaiani. Setting of BCs in the general (N3 , N2 ) approximation is considered in [3]. Kharibegashvili and Jaiani have studied well-possedness of dynamical problems in (0,0) approximation with mixed BCs at beam ends when either integrated stresses or weighted displacement components are prescribed. In [4] for cusped Euler-Bernulli beams all the reasonable BVPs are solved in explicit (integral) forms. Moreover, the existence and uniqueness of weak (generalized) solutions to vibration problems in suitably chosen weighted Sobolev spaces are established. In [12] a dynamical problem in the (0,0) approximation of elastic cusped prismatic beams is investigated when stresses are applied at the lateral surfaces and the ends of the beam. Two types of cusped ends are considered when the beam cross-section turns into either a point or a straight line segment. Correspondingly, at the cusped end either a force concentrated at the point or forces concentrated along the straight line segment are applied. The existence and uniqueness theorems in appropriate weighted Sobolev spaces are proved.
5.6 Relations of 3D, 2D, and 1D Problems Paper [13] deals with the analysis of the physical and geometrical sense of N-th (N=0,1,. . . ) order moments and weighted moments of the stress tensor and the displacement vector in the theory of cusped prismatic shells. The peculiarities of the setting of BCs at cusped edges in terms of moments and weighted moments are analyzed. The relation of such BCs to the BCs of the 3D theory of elasticity is also discussed.
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5.7 Cusped Prismatic Shell-Fluid Interaction Problems Works of Chinchaladze and Jaiani are devoted to some solid-fluid interaction problems when the solid part is an elastic cusped plate. Namely, the adequate transmission conditions on the interface for the interaction problem, when for elastic part the classical Kirchhoff-Love model is used, are studied; the bending of KirchhoffLove plates with two cusped edges under action of either incompressible ideal or incompressible viscous fluids has been considered as well, in particular, harmonic vibration is studied. In 2005-2006 by Chinchaladze, Gilbert an interaction between an elastic prismatic shell and an incompressible fluid, when in the elastic part the N = 0 approximation of Vekua’s hierarchical models for a cusped elastic prismatic shell is used, is investigated. References to all the works of the scientists mentioned in the present paper can be found in the bibliographies of papers [4, 9, 14]. An expanded version of this exploratory survey, supplemented with a large bibliography, will be published in Springer Briefs series.
References 1. Vekua, I. N. (1955). On one method of calculating of prismatic shells. Trudy Tbilis. Mat. Inst., 21, 191-259 (in Russian). 2. Vekua, I. N. (1985). Shell Theory: General Methods of Construction. Pitman, Boston. 3. Jaiani, G. (2001). On a mathematical model of bars with variable rectangular cross-sections. ZAMM Z. Angew. Math. Mech., 81 (3), 147-173. 4. Jaiani, G. V. (2002). Theory of Cusped Euler-Bernoulli Beams and Kirchoff-Love Plates. Lect. Notes TICMI, 3. 5. Mikhlin, S. G. (1970). Variational Methods in Mathematical Physics. Nauka, Moscow. 6. Jaiani, G., Schulze, B.-W. (2007). Some Degenerate Elliptic Systems and Applications to Cusped Plates. Mathematische Nachrichten, 280 (4), 388-407. 7. Avalishvili, M., Gordeziani, D. (2003). Investigation of two-dimensional models of elastic prismatic shell. Georgian Math. J., 10 (1), 17-36. 8. Schwab, C. (1996). A-posteriori modelling error estimation for hierarchic Plate Models. Numer. Math., 74, 221-259. 9. Jaiani, G., Kharibegashvili, S., Natroshvili, D., Wendland, W.L. (2004). Two-dimensional Hierarchical Models for Prismatic Shells with Thickness Vanishing at the Boundary. J. Elasticity, 77 (2), 95-122. 10. Chinchaladze, N., Gilbert, R., Jaiani, G., Kharibegashvili, S., Natroshvili, D. (2008). Existence and uniqueness theorems for cusped prismatic shells in the N-th hierarchical model. Math. Methods Appl. Sci., 31(11), 1345-1367. 11. Chinchaladze, N., Jaiani, G., Maistrenko, B., Podio-Guidugli, P. (2011). Concentrated contact interactions in cuspidate prismatic shell-like bodies. Archive of Applied Mechanics, DOI: 10.1007/s00419-010-0496-6. 12. Chinchaladze, N., Gilbert, R., Jaiani, G., Kharibegashvili, S., Natroshvili, D. (2010). Cusped elastic beams under the action of stresses and concentrated forces. Appl. Anal. 89(5), 757774. 13. Jaiani, G. (2008). On Physical and Mathematical Moments and the Setting of Boundary Conditions for Cusped Prismatic Shells and Beams. Proceedings of the IUTAM Symposium on Relation of Shell, plate, Beam, and 3D Models, IUTAM Bookseries, 9, 133-146, Springer. 14. Jaiani, G.V. (1996). Elastic bodies with non-smooth boundaries–cusped plates and shells. Z. Angew. Math. Mech., 76, Suppl. 2, 117-120.
Chapter 6
Effect of the Tangential Loads on the Bending of Elastic Plates Kristine L. Martirosyan
Abstract In this work, the problem of bending of a semi-infinite plate under the action of tangential loads is considered on the base of the classical theory, theory of Reissner-Hencky-Mindlin in Vasilyev’s sense, and theory of Ambartsumyan. The comparisons between deflections, transverse shear forces and moments by firstorder and higher-order refined theories are performed. Keywords Bending · Tangential loads · Plate theories
6.1 Problem Formulation The problem of bending of plates in the presence of tangential loads is considered on the basis of the following three theories: classical theory of Kirchhoff (K) [1]; improved theory of first order - the theory of Reissner (R) [2]; improved theory of higher order - the theory of Ambartsumyan (A) [3]. Another approach of analogous problems investigation is suggested in [4]. We consider a semi-infinite plate with thickness 2h, on the front surfaces of which tangential loads are given. Rectangular Cartesian coordinate system (x, y, z) is chosen so that the middle plane of the plate coincides with the plane (Oxy). Let us assume that the rectangular plate occupies the region 0 ≤ x < ∞, 0 ≤ y ≤ b, −h ≤ z ≤ h (Fig. 6.1). It is supposed that on the front surfaces tangential loads are given: z=h:
σ33 = 0, σ31 = X + (x, y), σ32 = 0,
z = −h : σ33 = 0, σ31 = −X − (x, y), σ32 = 0
(6.1)
K. L. Martirosyan (B) National Academy of Sciences, Institute of Mechanics, Yerevan, Armenia e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 6, © Springer-Verlag Berlin Heidelberg 2011
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x
O
y
−X −
y
X+
z Fig. 6.1 Semi-infinite plate band under tangential loads.
Concerning the displacements, by the theories (K), (R) and (A) the following respective assumptions are accepted [5]: • Kirchhoff theory (K)
∂W , ∂x ∂W U2 = V − z , ∂y U3 = W
(6.2)
U1 = U − zθ1 , U2 = V − zθ2 , U3 = W
(6.3)
∂W z z 1 + X1 + X2 + g(z)ϕ1 , ∂x 2G 2h G ∂W 1 U2 = V − z + g(z)ϕ2 , ∂y G U3 = W
(6.4)
U1 = U − z
• Reissner theory (R)
• Ambartsumyan theory (A) U1 = U − z
Here, U, V are displacements of the median plane; W is the deflection of the plate; ϕ1 , ϕ2 , θ1 , θ2 are functions independent of the coordinate z, G is the shear modulus, and X1 = X + − X − , X2 = X + + X − are the tangential loads. The function g(z) in Eqs (6.4) is defined as z2 g(z) = z 1 − 2 (6.5) 3h
6 Effect of the Tangential Loads on the Bending of Elastic Plates
77
The equations of bending of the plate have the form • by theory (K) Δ2 W =
h ∂X1 D ∂x
(6.6)
• by theory (R) ⎧ ∂θ1 ∂θ2 ⎪ ⎪ ⎪ ΔW − − = 0, ⎪ ⎪ ⎪ ∂x ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ∂θ1 ∂θ2 4Gh ∂W 2h ⎨ D Δθ1 + θ + + − θ1 = X1 , ⎪ ⎪ ⎪ ∂x ∂x ∂y 1 − ν ∂x 1 −ν ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ∂θ2 ∂θ1 4Gh ∂W ⎪ ⎪ ⎪ D Δθ2 + θ + + − θ2 = 0 ⎩ ∂y ∂y ∂x 1 − ν ∂y
(6.7)
• by theory (A) ⎧ ∂ϕ1 ∂ϕ2 3 ∂X1 ⎪ ⎪ ⎪ + =− , ⎪ ⎪ ⎪ ∂x ∂y 4 ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 8h3 ∂ ∂ϕ1 ∂ϕ2 4h 2h3 ∂2 X1 1 − ν ∂2 X1 ⎨ ∂ D ΔW − Δϕ +θ + + ϕ = + , 1 1 ⎪ ⎪ ⎪ ∂x 15 ∂x ∂x ∂y 3 3(1 − ν) ∂x2 2 ∂y2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ 8h3 ∂ ∂ϕ1 ∂ϕ2 4h h3 θ ∂2 X1 ⎪ ⎪ ⎪ D ΔW − Δϕ2 + θ + + ϕ2 = ⎪ ⎩ ∂y 15 ∂y ∂x ∂y 3 3 ∂x∂y (6.8) In Eqs (6.6) to (6.8), ν is the Poisson‘s ratio, D is the bending stiffness of the plate. θ and D have the following form θ=
1+ν , 1−ν
D=
4Gh3 3(1 − ν)
The applied loads are given as it follows: X1 = τ0 sin λy,
X2 = 0
at λ =
π b
(6.9)
The edges of the plate at y = 0, b are hinged; the corresponding boundary conditions are then: • by theory (K): W = 0,
∂W 2 = 0 at ∂y2
y = 0, b
(6.10)
• by theory (R): W = 0,
θ1 = 0,
∂θ2 = 0 at ∂y
y = 0, b
(6.11)
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• by theory (A): 5 ∂2 W 4 ∂ϕ2 W = 0, ϕ1 = − X1 , − =0 2 8 5G ∂y ∂y
at
y = 0, b
(6.12)
With X1 the only tangential loads applied on front surfaces (6.9). It should be noted, that in the case of X2 = 0, generalized plane stress has a trivial solution. By the theories (K), (R) and (A) equations in the particular case of one-dimensional problem under the conditions (6.10), (6.11) and (6.12) have solutions. Therefore, solutions of Eqs (6.6), (6.7) and (6.8) in the two-dimensional case by analogy with the approach of Nadai [1] we will search under the conditions • by theory (K):
lim W = 0
(6.13)
x→∞
• by theory (R):
lim W = 0,
x→∞
lim θ1 = −
x→∞
lim θ2 = 0,
τ0 , 2G(η2 + 1)
(6.14)
x→∞
where
hλ η= √ 3
• by theory (A):
lim W = 0,
x→∞
lim ϕ1 = −
x→∞
lim ϕ2 = 0,
5λ2 τ0 , 8χ2
(6.15)
x→∞
2
where χ=h
−1
( 5' 1 + ξ2 , 2
2 ξ=
2 λh 5
6.2 Solutions for Lightweight Restraint Conditions at x = 0 Let the edge x = 0 has the condition of lightweight restraint; the corresponding boundary conditions have the following form: • by the theory (K):
W = 0, ∂W = 0 at ∂x
x=0
(6.16)
6 Effect of the Tangential Loads on the Bending of Elastic Plates
79
• According to the theory (R) W = 0, θ1 = 0, ∂θ2 = 0 at ∂x
(6.17) x=0
• According to the theory (A) W = 0, ∂W 4 X1 − ϕ1 = ∂x 5G 2G ∂ϕ1 ∂ϕ2 5 ∂X1 − + = 0 at ∂y ∂x 8 ∂y
(6.18) x=0
The expressions for deflections will have the following form: • by the theory (K):
W=0
• by the theory (R): W=
(6.19)
e−λx xτ0 sin λy 2G(1 + 2η2(1 + θ))
(6.20)
e−λx xτ0 sin λy, 2G(1 + β)
(6.21)
• by the theory (A): W= where
4ξ2 1−ν From Eqs (6.20) and (6.21), it is obvious that the deflections by the theories (R) and (A) are identical when h2 λ2 1. In addition, we can calculate the estimated value for the moment M1 (0, y) and the forces N1 (0, y), N2 (x, 0). It is obtained, that the moments M1 (0, y) by the theories (R) and (A) coincide when hλ 1. The forces N1 (0, y) computed by the theories (R) and (A) coincide when h2 λ2 1. The expressions for the forces N2 (x, 0) when h2 λ2 1 are of the form β=
• by theory (R)
• by theory (A)
8ηe−λx N2 (x, 0) = −ηhτ0 e−(xα/h) − ; 1−ν
(6.22)
5ξhτ0 −xχ 4ξe−λx N2 (x, 0) = − e − ; 6 1−ν
(6.23)
When hλ 1 in corner point (x = 0, y = 0), we find that the ratio of the force N √2 (0, 0) calculated by the theory (R) to that calculated by the theory (A) is equal to 6/5.
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6.3 Solutions for Straitened Sliding Contact Conditions at x = 0 At the edge x = 0 the straitened sliding contact conditions are of the form • by the theory (K)
∂W = 0, ∂x ∂W = 0, X1 (6.24) ∂y 3 ∂ W h = 3 D ∂x Thus, according to Kirchhoff’s theory for the function of the deflection, three boundary conditions are obtained instead of the required two. When satisfying the condition of self-conjugacy of the problem [6], there are two variants of boundary conditions here: either the first two conditions are satisfied for W, which would be consistent with the task of anchorage, or the first and third conditions are satisfied, which coincides with the problem for sliding contact. Ambiguity of the formulation of the problem with the boundary conditions by the theory of Kirchhoff shows evidence of the need for applying refined theories. [7]. • by the theory (R) ∂W = 0, (6.25) ∂x θ1 = θ2 = 0 • by the theory (A)
∂W 1 + X1 = 0, ∂x 10G ∂W 4 − ϕ2 = 0, ∂y 5G
(6.26)
3 ϕ1 = − X1 4 Satisfying the boundary conditions (6.25) and (6.26), we obtain the expressions for deflections: • by the theory (R): W=
e−λx (1 + λx)τ0 % & sin λy; 2Gλη 1 + η2 1 − 2η2 (1 + θ) + 2η 1 + η2 (1 + θ)
• by the theory (A): % % & % && e−λx γλx 1 + ξ2 + β 6 + γ + ξ2 + γξ2 + 5λx τ0 sin λy W= % & , 10Gλβ 1 + ξ2 ξ + γ 1 + ξ2
(6.27)
(6.28)
6 Effect of the Tangential Loads on the Bending of Elastic Plates
where
81
⎛ ⎞ ⎟⎟⎟ 4ξ2 ⎜⎜⎜⎜ ξ ⎟⎟⎠ γ= ⎜⎝1 − 1−ν 1 + ξ2
From Eqs (6.27) and (6.28), it is obvious that the deflections by the theories (R) and (A), when h2 λ2 1, have the form: • by the theory (R): √ −λx 3e (1 + λx)τ0 sin λy W= , (6.29) 4η 2Ghλ2 (1 + 1−ν ) • by the theory (A): W=
3e−λx (1 + λx)τ0 sin λy √ 4ξ 10Ghλ2 (1 + 1−ν )
(6.30)
When hλ 1, we obtain that the ratio of the deflection, calculated by the theory √ (R), to the magnitude of the deflection, calculated by the theory (A) is equal to 5/6. Calculated values for the force N1 (0, y) by the theories (R) and (A) coincide and are identically equal to zero. When hλ 1, the ratio of the moment, calculated by √ the theory (R), to the magnitude of the moment, calculated by the theory (A) is 5/6. When hλ 1 in corner point x = 0, y = 0, we obtain that the ratio of the force, N2 (0, 0), calculated by √ the theory (R), to the magnitude of cutting force, calculated by the theory (A) is 6/5.
6.4 Conclusions The bending of a semi-infinite plate under the action of tangential loads has been studied analytically using three theories: the classical theory of Kirchhoff (K), the first order shear deformation theory of Reissner (R), and the refined higher-order theory of Ambartsumyan (A). It is found that the first theory is not satisfactory in general, while the solutions of the last two theories partly coincide for particular thickness ratios. These closed-form solutions can be helpful for numerical solutions assessment.
References 1. Timoshenko S., Woinowsky-Kriger S. Theory of Plates and Shells. McGraw Hill, New York, 1985. 2. Vasilyev, V. V.. Classical theory of plates: history and modern analysis. Russian Sc. Acad. Publishers, MTT, 1998, No 3, 46-58 (in Russian) 3. Ambartsumyan S.A. Theory of Anisotropic Plates (Strength, Stability, and Vibrations). TECHNOMIC Co, Stamford, 1970. 4. Kirakosyan R.M. On the improved theory cylindrical ortotropic plates of variable thickness. Proc. NAS of Armenia, 47. N. 5-6, 1994 (in Russian).
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5. Belybekyan M.V. On the equations of the theory of plates, taking into account the transverse shear. In: Problems of the mechanics of thin deformable bodies, Yerevan. Gitutyun (Ed.), NAS RA, 2002, p.67-88, (in Russian). 6. Belubekyan V.M, Belubekyan M.V. On the boundary conditions of plate theory. Proc. NAS of Armenia, V. 52. N. 2, 1999, p.11-21, (in Russian). 7. Belubekyan M.V. A paradox of the theory of Kirchhoff’s boundary conditions. In the coll. The study of modern scientific problems in high schools. Pub. Armenia, Yerevan, 2000, v.2. p. 134-138.
Chapter 7
On the Convergence of an Iteration Method in Timoshenko’s Theory of Plates Jemal Peradze
Abstract The boundary value problem for a Timoshenko system of differential equations describing the static behavior of a plate is considered. Two sought functions are expressed through the third one for which an integro-differential equation with the Dirichlet condition on the boundary is written. The application of the Galerkin method to the obtained problem leads to a nonlinear system of algebraic equations which is solved by iteration. The condition of iteration process convergence is established and its rate is estimated. Keywords Timoshenko equation · Galerkin method · Jacobi iteration · Convergence of the iteration method
7.1 Problem Formulation The nonlinear Timoshenko model for plates and shells is important from the theoretical and engineering standpoints. Mathematical analysis of the corresponding system of equations is connected with a big difficulty because of the complex structure inherent in these equations. I. Vorovich [7] regarded the nonlinear system of Timoshenko equations as one of the unsolved problems of the mathematical theory of plates and shells in terms of solvability and construction of approximate solution algorithms. As far as we know, presently this problem remains far from being solved. In this context, the study of simplified one-dimensional variants of the Timoshenko system will help get a better insight into the nature inherent in these models and will make it easier to proceed to the investigation of the two-dimensional case. J. Peradze (B) Tbilisi State University, 2, University Str., Tbilisi 0186, Georgia & Georgian Technical University, 77, M. Kostava Str., Tbilisi 0174, Georgia e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 7, © Springer-Verlag Berlin Heidelberg 2011
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J. Peradze
Here we consider the problem of convergence and accuracy of the iteration process of solution of the one-dimensional axisymmetric boundary value problem for the Timoshenko plate. All constants participating in the obtained convergence conditions and the method error estimate are expressed explicitly through the initial data of the problem, which, in the case of practical application of the method, enables us to define in advance the number of iteration steps and to calculate the upper bound of the error. Let us consider the nonlinear system of differential equations 1 % 2 & w + p(x) = 0, 2 ( Eh ' Eh 1 2 k02 w + ψ + u + w w + q(x) = 0, 2(1 + ν) 2 1 − ν2 u +
' ( h2 ψ − k02 w + ψ = 0; 6(1 − ν)
(7.1)
0 < x < 1,
with the boundary conditions u(0) = u(1) = 0,
w(0) = w(1) = 0,
ψ (0) = ψ (1) = 0.
(7.2)
Here u = u(x), w = w(x) and ψ = ψ(x) are the functions to be defined, p(x) and q(x) are the given functions, ν, E, h and k0 are the given constants, 0 < ν < 12 . System (7.1) is obtained from the system of Timoshenko equations for a shell presented in [7, p. 42], on the one hand, by throwing away the variables y and t and assuming that k x = ky = 0, and, on the other hand, by preserving the terms with cubic nonlinearity. Thus system (7.1) characterizes the static state of the plate under the action of axially symmetric load. Note that this system can also be obtained from the system of Timoshenko equations for a plate from [2, p. 24].
7.2 Reduction of the Problem Let us return to problem (7.1),(7.2). Using the first and the third equation of system (7.1) and the corresponding boundary conditions from (7.2), the functions u(x) and ψ(x) can be obtained through the function w(x) 1 x 1 x 2 w (x) dx − w (ξ) dξ + x (1 − ξ)p(ξ) dξ − (x − ξ)p(ξ) dξ, 2 0 0 0 0 x 1 σ ψ(x) = − cosh σ(x − 1) cosh σξw (ξ) dξ + coshσx cosh σ(ξ − 1)w (ξ) dξ , sinh σ 0 x (7.3) where k0 σ= 6(1 − ν) . h x u(x) = 2
1
2
7 On the Convergence of an Iteration Method in Timoshenko’s Theory of Plates
85
Using (7.3) in the second equation of system (7.1), we come the equation for the function w(x) 1 x Eh 1 − ν 2 1 1 2 k + w dx + (1 − x)p(x) dx − p(ξ) dξ w − p(x)w 2 0 2 0 1 − ν2 0 0 x 3Ek04 1 − ν − sinhσ(x − 1) cosh σξ w (ξ) dξ h sinhσ 1 + ν 0 1 + sinh σx cosh σ(ξ − 1)w (ξ) dξ + q(x) = 0, (7.4) x
to which we add the boundary condition from (7.2) w(0) = w(1) = 0.
(7.5)
After solving problem (7.4), (7.5), we substitute w(x) in (7.3) and find other unknown functions u(x) and ψ(x).
7.3 The Numerical Algorithm Let us consider a numerical algorithm of the solution of problem (7.4), (7.5). The approximation of w(x) with respect to the spatial variable is written as the finite sum wn (x) =
n 1 wni sin iπx, iπ i=1
(7.6)
where, in case we use the Galerkin method [4], the coefficients wni satisfy the nonlinear system of equations
p1i + p2 +
n 1 w2n j wni + p3i j wn j + qi = 0, i j=1 j=1
n
i = 1, 2, . . ., n.
(7.7)
Here the following notation is used p1i =
3 + (hπi) 2 1 x
1 2k02 (1−ν)
j p3i j = 8 − i qi
1
=−
0
8(1 − ν2 ) Ehπ
0 1
,
p2 = 4
1
(1 − x)p(x) dx,
0
1 1 p(ξ) dξ sin iπx sin jπx d + p(x) sin iπx cos jπx dx , iπ 0
q(x) sin iπx dx. 0
(7.8)
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J. Peradze
Note that to obtain the i-th equation of system (7.7), in the left-hand part of (7.4) we should replace w(x) by wn (x), multiply the obtained expression by iπ1 sin iπx and, after that, using the relations * 1 0, i j, sin iπx sin jπx dx = 1 0 2 , i = j, eax eax sin bx dx = 2 2 (a sin bx − b cosbx), a +b eax eax cos bx dx = 2 2 (a cosbx + b sin bx), a +b integrate it over x from 0 to 1 and equate the result to zero. We will solve system (7.7) by using the iteration method
p1i + p2 + w2ni,k+1 +
n j=1 ji
n 1 w2n j,k wni,k+1 +p3ii wni,k+1 + p3i j wn j,k + qi = 0, (7.9) i j=1 ji
where wni,k+l is the (k + l)-th approximation of wni , l = 0, 1, and k = 0, 1, . . . , i = 1, 2, . . ., n,. This method is nothing else but the Jacobi iteration process [5]. After defining wni,k , i = 1, 2, . . ., n, we use them in formula (7.6) instead of wni and, as a result, find the approximation of the function w(x), which, when used in (7.3) gives the approximation for the functions u(x) and ψ(x). Note that each i-th equation of system (7.9) is a cubic equation for wni,k+1 . Therefore, using the Cardano formula [1], wni,k+1 can be written in the explicit form wni,k+1 = σi,1 − σi,2 , where
k = 0, 1, . . .,
1 ⎡ ⎛ 2 ⎞1 ⎤ ⎢⎢⎢ 3 ⎟ 2 ⎥⎥ 3 ⎜ r s ⎥ si ⎜ ⎟ ⎥ ⎢ σi,l = ⎢⎢⎢⎢(−1)l + ⎜⎜⎜⎝ i + i ⎟⎟⎟⎠ ⎥⎥⎥⎥ , ⎣ 2 4 27 ⎦
ri = p1i + p2 + p3ii +
n
w2n j,k ,
j=1 ji
si =
i = 1, 2, . . ., n,
l = 1, 2,
n 1 qi + p3i j wn j,k . i
(7.10)
(7.11)
(7.12)
j=1 ji
7.4 The Jacobi Matrix Let us rewrite system (7.10) in the form ' ( wni,k+1 = ϕi wn1,k , wn2,k , . . . , wnn,k
(7.13)
7 On the Convergence of an Iteration Method in Timoshenko’s Theory of Plates
87
and consider the Jacobi matrix
∂ϕi J= ∂wn j,k
n .
(7.14)
i, j=1
By virtue of (7.10)–(7.13) we conclude that the diagonal elements of the matrix J are equal to zero. As to the nondiagonal elements, i j, we have for them ⎡ ⎤ ⎛ 2 ⎞− 1 ⎥⎥ ⎢ 2 ⎜⎜⎜ si ri3 ⎟⎟⎟ 2 si ⎥⎥ ∂ϕi 1 1 ⎢⎢⎢⎢ 2 l 2 ⎢⎢ p3i j + (−1) ⎜⎜⎝ + ⎟⎟⎠ =− p3i j + ri wn j,k ⎥⎥⎥⎥ . (7.15) ⎦ ∂wn j,k 6 4 27 2 9 σ2 ⎢⎣ i,l
l=1
By (7.11) ri σi,1 · σi,2 = , 3
σ3i,2 − σ3i,1
⎛ 2 ⎞1 ⎜⎜⎜ si ri3 ⎟⎟⎟ 2 σ3i,1 + σ3i,2 . ⎜⎜⎝ + ⎟⎟⎠ = 4 27 2
= si ,
(7.16)
Using these relations in (7.15) we get ∂ϕi 1 1 2 1 = − p3i j 2 ri + wn j,k si 4 . ri 2 2 ∂wn j,k 6 σi,1 − 3 + σi,2 3 σi,1 + ( 3 ) + σ4i,2
(7.17)
Taking into account the first equality of (7.16), we find that σ2i,1 −
ri 1 + σ2i,2 ≥ ri , 3 3
σ4i,1 + σ4i,2 ≥
2 2 r . 9 i
(7.18)
By (7.17) and (7.18) we write
∂ϕi
≤ φ + φ , 1i j 2i j
∂wn j,k
(7.19)
where 1 1 φ1i j = |p3i j | , 2 |ri |
φ2i j = 2|wn j,k | |si |
1 . ri2
(7.20)
Let us estimate either of φli j , l = 1, 2. Suppose that the initial data of the problem – the function p(x) and the constants ν, E, h and k0 – are such that p1i > |p2 + p3ii | .
(7.21)
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As follows from (7.8), (7.21) implies the fulfillment of the relation ⎛ ⎜⎜⎜ ⎜⎜⎝
⎞−1 1
1 3 ⎟⎟⎟⎟
+ > 4 (1 − x)p(x) dx ⎟ ⎠
2k02 (1 − ν) (hiπ)2 0 1 x
4 1 −8 p(ξ) dξ sin2 iπx dx + p(x) sin 2iπx dx
, iπ 0 0 0 i = 1, 2, . . ., n.
(7.22)
In that case, by virtue of (7.20) and (7.11) we have φ1i j ≤
1 |p3i j | , 2ci
(7.23)
where ci = p1i − |p2 + p3ii | . x 1√ Further, using (7.20), (7.21), (7.12) and the fact that max c+x 2 = 2 c , x > 0, c > 0, x we come to a conclusion that ⎛ ⎞ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟ ⎜⎜⎜ 12 12 ⎟⎟⎟⎟ n n ⎜ ⎟⎟ |wn j,k | |si | |wn j,k | ⎜⎜ 1 |qi | 1 ⎜⎜⎜ φ2i j = 2 ≤2 + p23il w2nl,k ⎟⎟⎟⎟ n 2 ⎟⎟⎟ |ri | ri | ci + wn j,k ⎜⎜⎜ i ci c + w2 l=1 l=1 ⎜⎜⎜ ⎟⎟⎟ i nl,k ⎜⎝ ⎟⎠ l i l i l=1 li
1 1 |qi | 1 ≤ √ + √ ci i ci 2 ci
n
p23il
12
.
(7.24)
l=1 li
7.5 The Iteration Method Convergence and Error Let us introduce the vector and matrix norms by means of the expressions and max
n
1≤ j≤n i=1
n i=1
|vi |
|mi j | for v = (vi )ni=1 and M = (mi j )ni, j=1 . Now we will formulate the main
result. Assume that the norm of the matrix J is smaller than a number Δ, 0 < Δ < 1. According to (7.14), (7.19), (7.23) and (7.24), this is equivalent to the requirement 2 n n n 1 1 1 1 |qi | max |p3i j | + 2 √ + p23i j < Δ. 2 1≤ j≤n ci 2 i=1 ci i ci 1
i=1 i j
j=1 ji
(7.25)
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Then, by virtue of Banach’s contraction principle [3] there exists a unique solution wni , i = 1, 2, . . ., n, of system (7.7) to which the sequence of approximations wni,k of the iteration method converges as k → ∞, whereas the error method decreases with a geometrical progression rate n
Δk |wni,0 − wni,1 |, 1 − Δ i=1 n
|wni,k − wni | ≤
i=1
k = 0, 1, . . . .
7.6 An Alternative Form of Requirements To conclude, we replace conditions (7.22) and (7.25) by more rigid but easily verifiable conditions. For this, we require the fulfillment of p1i > |p2 | + |p3ii | instead of (7.21), then apply (7.8) and the following relations obtained by means of the Cauchy–Bunyakowski inequality
1
≤ 1 p , (1 − x)p(x) dx
√ 0 0 3
1 x
p(ξ) dξ sin iπx sin jπx dx
≤
0
0
1 √ p0 , i j, 2 2
√ 3 p(ξ) dξ sin2 iπx dx
≤ p0 , 4 0 0
1
≤ 1 p , i j, p(x) sin iπx cos jπx dx
2 0 0
1
1
p(x) sin iπx cos iπx dx
≤ √ p0 ,
0 2 2
1
x
) 1 2 1 where p0 = 0 p2 (x) dx . As a result, instead of requirement (7.22) we will have ⎛ ⎜⎜ c = ⎜⎜⎜⎝
⎞−1 ⎛ √ ⎞ ⎜⎜⎜ 5 1 6 ⎟⎟⎟⎟ 2 ⎟⎟⎟ ⎜ ⎟⎠ p0 > 0, + − + ⎟ ⎝√ ⎠ π k02 (1 − ν) (hπ)2 3
(7.26)
and (7.25) will be replaced by the condition 1 2c
n √ 1 1 4 2 1 + √ + n + 2(n − 1)n p0 + √ max |qi | < Δ. π i c 1≤i≤n 2 π i=1
(7.27)
It is not difficult to verify that conditions (7.22) and (7.25) as well as (7.26) and (7.27) are fulfilled for sufficiently small functions p(x) and q(x).
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References 1. Birkhoff, G., Mac Lane, S.: A brief Survey of Modern Algebra. Second edition. The Macmillan Co., New York; Collier-Macmillan Ltd., London (1965) 2. Lagnese, J., Lions, J.-L.: Modelling analysis and control of thin plates. Recherches en Mathematiques Appliquees [Research in Applied Mathematics], 6. Masson, Paris (1988) 3. Lebedev, L. P., Vorovich, I. I.: Functional Analysis in Mechanics. Springer Monographs in Mathematics. Springer, New York (2003) 4. Marchuk, G. I.: Methods of Numerical Mathematics. Second edition. Transl. from the Russian by Arthur A. Brown. Applications of Mathematics, 2. Springer, New York–Berlin (1982) 5. Ortega, J. M., Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables. Reprint of the 1970 original. Classics in Applied Mathematics, 30. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2000) 6. Vol’mir, A. S.: Nonlinear Dynamics of Plates and Shells (in Russian). Nauka, Moscow (1972) 7. Vorovich, I. I.: Nonlinear Theory of Shallow Shells. Applied Mathematical Sciences, 133. Springer, New York (1999).
Chapter 8
Mathematical Models of Micropolar Elastic Thin Shells Samvel H. Sargsyan
Abstract In the present paper on the basis of mathematical (asymptotic) confirmed hypotheses method, depending on the values of dimensionless physical parameters, mathematical models of micropolar elastic thin shells with free rotation, with constrained rotation, with “small transverse stiffness” are constructed. Transverse shear and related to them strains are completely taken into account in the suggested theories of micropolar shells. On the basis of these models specific problems of strength and dynamics of micropolar elastic thin shells, plates and bars can be studied. Keywords Micropolar thin shells · Free and constrained rotations · Asymptotic methods
8.1 Introduction One of the actual problems of the micropolar (asymmetric, moment) theory of elasticity is the construction of general applied theories of shells, plates and bars [1, 2]. The main problem of the general theory of micropolar elasticity of thin bars, plates and shells is the approximate, but adequate reduction of the three-dimensional or two-dimensional equations of the theory of micropolar elasticity to applied twodimensional or one-dimensional equations. In [6] the asymptotic theory of micropolar elastic thin shells is constructed for a thin domain. It should be noted that the construction of the models of micropolar elastic thin shells by the asymptotic theory is connected with rather difficult and unwieldy mathematical calculations. From this point of view in this paper the method of hypotheses is applied for the construction of the models of micropolar elastic thin shells. The hypotheses are formulated on the S. H. Sargsyan (B) National Academy of Sciences of the Republic of Armenia, Gyumri State Pedagogical Institute, Armenia e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 8, © Springer-Verlag Berlin Heidelberg 2011
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basis of the qualitative results of the asymptotic integration of boundary problems of in the three-dimensional micropolar theory of elasticity for thin shell domain. It should be noted that the suggested models of micropolar elastic thin shells based on the asymptotic theory and on the mentioned method of hypotheses coincide. Depending on the values of the physical dimensionless parameters, the models of micropolar elastic thin shells with free rotation, with constrained rotation, with “small transverse stiffness” are established. The transverse shear and the related strains are completely taken into account.
8.2 Problem Statement An isotropic shell of constant thickness 2h is considered as a three-dimensional elastic micropolar body. We proceed from the fundamental equations of the spatial static problem of linear micropolar theory of elasticity with independent fields of displacements and rotations [4]: • Equilibrium equations ∇m σmn = 0, • Elasticity relations *
∇m μmn + enmk σmk = 0
(8.1)
σmn = (μ + α)γmn + (μ − α)γnm + λγkk δnm μmn = (γ + ε)κmn + (γ − ε)κnm + βκkk δnm
(8.2)
γmn = ∇m Vn − ekmn ωk ,
(8.3)
• Geometric relations κmn = ∇m ωn .
Here σ, ˆ μˆ are the tensors of the force and the moment stresses, γˆ , κˆ are the strain tensor and the bending-torsion tensor, V, ω are the vectors of displacement and free rotation, λ, μ, α, β, γ, ε are the elastic parameters of the micropolar material. The indices m, n, k take the values 1, 2, 3. We will use orthogonal curvilinear coordinates αk , adopted in the theory of shells. Appropriate boundary conditions will be attached to the governing equations (8.1) - (8.3) of three-dimensional micropolar theory of elasticity. On the surfaces α3 = ±h of the shell force and moment stresses are given (i = 1, 2): σ3i = ±q±i , σ33 = ±q±3 , μ3i = ±m±i , μ33 = ±m±3 ,
(8.4)
On the boundary Σ of the shell boundary conditions in force and moment stresses or displacements and rotations or in mixed form can be prescribed. It is assumed that the shell thickness is small compared with typical radii of curvature of the shell’s middle surface, and hence, we will proceed from the following
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basic concept: in the static case the general stress-strain state (SSS) of a thin threedimensional body, forming a shell, is composed of the internal SSS, covering the shell, and boundary layers, localizing near the surface of the shell edge Σ. The construction of the general applied two-dimensional theory of micropolar elastic thin shells is closely connected with the formulation of the internal problem. During the determination of both the internal and the external SSS of the shell [6], the values of the physical parameters of the shell material play an important role. From this point of view the following dimensionless physical parameters are introduced: μ , α
R2 μ , β
R2 μ , γ
R2 μ , ε
(8.5)
where R, which is a characteristic radius of curvature of the middle surface of the shell, is a scale factor.
8.3 Model of Micropolar Elastic Thin Shells with Independent Fields of Displacements and Rotations The case when the dimensionless physical parameters (8.5) take the following values μ R2 μ R2 μ R2 μ ∼ 1, ∼ 1, ∼ 1, ∼1 (8.6) α β γ ε is considered. Taking into account the qualitative results of the asymptotic solution of equations (8.1)–(8.3) with the above mentioned boundary conditions, for the case (8.6) in [6] a theory of micropolar elastic thin shells with independent rotations based on the following, rather general assumptions (hypotheses) is deduced: a) During the deformation initially straight and normal to the median surface fibers are rotated freely in space as a whole rigid body, without changing its length and without remaining perpendicular to the deformed middle surface. The hypothesis is mathematically written as follows: Vi = ui (α1 , α2 ) + α3 Ψi (α1 , α2 ), V3 = w(α1 , α2 ), ωi = Ωi (α1 , α2 ), ω3 = Ω3 (α1 , α2 ) + α3 ι(α1 , α2 )
(8.7)
Thus, normal to the middle surface displacement and tangential rotations are constant functions along the shell thickness, and the tangential displacements and normal rotation are changing by a linear law. It should be noted that from the point of view of displacements the hypothesis (8.7) are similar to Timoshenko’s kinematic hypothesis in the classical theory of elastic shells [5]. The hypothesis (8.7) is called Timoshenko’s generalized kinematic hypothesis in the micropolar theory of shells. b) In the generalized Hooke’s law (8.2) the force stress σ33 can be neglected in comparison to the force stresses σii .
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c) Using the assumption of thin-walled shells, it is assumed 1+
α3 ≈1 Ri
d) In computing the strains, bending-torsion, force and moment stresses, first for the force stresses σ3i and moment stress μ33 we will take: 0
σ3i = σ3i (α1 , α2 ),
0
μ33 = μ33 (α1 , α2 )
(8.8)
After determining of the mentioned quantities, the values of σ3i and μ33 will be finally defined by addition of summands to the corresponding values (8.8), obtained by integration of the first two or the sixth equilibrium equations of (8.1). The condition will be required for the summands that quantities averaged through the shell thickness are equal to zero. The basic system of equations and boundary conditions of the general theory of micropolar elastic thin shells with independent fields of displacements and rotations, constructed with the help of hypotheses a) - d), is the following: Equilibrium equations: 1 ∂T ii 1 ∂A j 1 ∂S ji 1 ∂Ai Ni3 + (T ii − T j j ) + + (S ji + S i j ) + = −(q+i + q−i ), Ai ∂αi Ai A j ∂αi A j ∂α j Ai A j ∂α j Ri 1 ∂Mii 1 ∂A j 1 ∂H ji 1 ∂Ai + (Mii − M j j )+ + (H ji + Hi j ) − N3i = −h(q+i − q−i ), Ai ∂αi Ai A j ∂αi A j ∂α j Ai A j ∂α j T 11 T 22 1 ∂(A2 N13 ) ∂(A1 N23 ) + − + = q+3 + q−3 , R1 R 2 A1 A2 ∂α1 ∂α2 (8.9) 1 ∂Lii 1 ∂A j 1 ∂L ji 1 ∂Ai Li3 + (Lii − L j j ) + + (L ji + Li j ) + Ai ∂αi Ai A j ∂αi A j ∂α j Ai A j ∂α j Ri + − +(−1) j (N j3 − N3 j ) = −(m + m ), i i L11 L22 1 ∂(A2 L13 ) ∂(A1 L23 ) (8.10) + − + − (S 12 − S 21 ) = (m+3 + m−3 ), R1 R2 A1 A2 ∂α1 ∂α2 1 ∂(A2 Λ13 ) ∂(A1 Λ23 ) L33 − + − (H12 − H21 ) = h(m+3 − m−3 ). A1 A2 ∂α1 ∂α2 Elasticity relations: 2Eh [Γii + νΓ j j ], S i j = 2h[(μ + α)Γi j + (μ − α)Γ ji], 1 − ν2 3 2Eh 2h3 (8.11) Mii = [Kii + νK j j ], Hi j = [(μ + α)Ki j + (μ − α)K ji ], 2 3 3(1 − ν ) Ni3 = 2h(μ + α)Γi3 + 2h(μ − α)Γ3i, N3i = 2h(μ + α)Γ3i + 2h(μ − α)Γi3,
T ii =
8 Mathematical Models of Micropolar Elastic Thin Shells
Lii = 2h
4γ(β + γ) 2γβ β κii + κjj + L33 , β + 2γ β + 2γ β + 2γ
95
Li j = 2h[(γ + ε)κi j + (γ − ε)κ ji ],
4γε γ − ε m+i − m−i L33 = 2h[(β + 2γ)ι + β(κ11 + κ22 )], Li3 = 2h κi3 + , γ+ε γ+ε 2 + − 2h3 4γε γ − ε mi + mi Λi3 = li3 + . 3 γ+ε γ + ε 2h
(8.12)
Geometric relations: 1 ∂ui 1 ∂Ai w 1 ∂u j 1 ∂Ai + uj + , Γi j = − ui − (−1) jΩ3 , Ai ∂αi Ai A j ∂α j Ri Ai ∂αi Ai A j ∂α j 1 ∂Ψi 1 ∂Ai 1 ∂Ψ j 1 ∂Ai Kii = + Ψ j, Ki j = − Ψi − (−1) j ι, Ai ∂αi Ai A j ∂α j Ai ∂αi Ai A j ∂α j Γ3i = Ψi − (−1) j Ω j , Γi3 = −ϑi + (−1) jΩ j , (8.13) 1 ∂w ui 1 ∂Ωi 1 ∂Ai Ω3 ϑi = − + , κii = + Ω2 + , Ai ∂αi Ri Ai ∂αi Ai A j ∂α j Ri 1 ∂Ω j 1 ∂Ai 1 ∂Ω3 Ωi (8.14) κi j = − Ωi , κi3 = − , Ai ∂αi Ai A j ∂α j Ai ∂αi Ri 1 ∂ι li3 = . Ai ∂αi Γii =
Boundary conditions (for α1 = const) ∗ or u = u∗ , ∗ or w = w∗ , T 11 = T 11 N13 = N13 1 1 ∗ or u = u∗ , ∗ ∗ S 12 = S 12 M = M 2 11 2 11 or K11 = K11 , ∗ or K = K ∗ , H12 = H12 12 12 ∗ , L11 = L∗11 or κ11 = κ11 ∗ ∗ , L13 = L13 or κ13 = κ13
∗ , L12 = L∗12 or κ12 = κ12 ∗ ∗ Λ13 = Λ13 or l13 = l13
(8.15)
(8.16)
In the model (8.9)–(8.16) of micropolar elastic shells the transverse shear and the related deformations are completely taken into account. One obtains a system of 52 equations for 52 unknown functions (i, j = 1, 2, i j): ui , w, Ψi , Ωi , Ω3 , ι, ϑi , T ii , S i j , Ni3 , N3i , Mii , Hi j , Lii , Li j , L33 , Li3 , Λi3 , Γii , Γi j , Γi3 , Γ3i , Kii , Ki j , κii , κi j , κi3 , li3 . The system of differential equations (8.9)–(8.14) is of 18-th order with 9 boundary conditions (8.16) on each of the contours of the shell’s middle surface Γ. Choosing appropriately Lam´e’s parameters Ai the basic relations of the statics of micropolar cylindrical shells and shells of revolution will be obtained from the general equations (8.9)–(8.14) of the shells theory. From the system of equations (8.9)–(8.14) and boundary conditions (8.15) and (8.16) of micropolar shells, when α = 0, the boundary problem of the classical theory of shells will be separated on the basis of Timoshenko’s hypothesis [5] (with some differences related to the static hypothesis d)). If in the model (8.9)–(8.16) the transverse shear is neglected: Γi3 + Γ3i = 0
or
Ψi = ϑi
(8.17)
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a model of micropolar elastic thin shells with independent rotations will be obtained, where instead of Timoshenko’s generalized kinematic hypothesis Kirchhoff-Love’s generalized kinematic hypothesis in the case of micropolar shell will be accepted (so, we take the formula (8.7) taking into account (8.17)). The system of equations of this model of micropolar shells is the following: the equilibrium equations (8.9) and (8.10); elasticity relations from (8.11) and (8.12) for quantities T ii , S i j , Mii , Hi j , Lii , Li j , L33 , Li3 , Λi3 , where formula Ni3 − N3i = 4hα(Γi3 − Γ3i )
(8.18)
must be added; geometric relations from (8.13) and (8.14) for quantities Γii , Γi j , κii , κi j , κi3 ,li3 , where formulas Kii =
1 ∂ϑi 1 ∂Ai + ϑ j, Ai ∂αi Ai A j ∂α j ϑi = −
1 ∂w ui + , Ai ∂αi Ri
Ki j =
1 ∂ϑ j 1 ∂Ai − ϑi − (−1) j ι, Ai ∂αi Ai A j ∂α j
Γi3 − Γ3i = 2[−ϑi + (−1) j Ω j ]
(8.19)
must be added. The main equations of micropolar elastic thin shells of the model with independent rotation, obtained by Kirchhoff-Love’s generalized kinematic hypotheses, must be solved together with the boundary conditions of the classical theory of elastic shells [7] and conditions (8.16). When α = 0, from the mentioned main system of equations and boundary conditions of micropolar elastic thin shells both the general equations and the boundary conditions of the classical theory of elastic shells will be separated on the basis of the Kirchhoff-Love’s hypotheses.
8.4 Model of Micropolar Elastic Thin Shells with Constrained Rotation The case when the dimensionless physical parameters (8.5) have the following values: R2 μ R2 μ R2 μ α μ, ∼ 1, ∼ 1, ∼1 (8.20) β γ ε is considered. The asymptotic analysis of the posed boundary problem [6] for system of equations (8.1)–(8.3) in the case of (8.20) shows that the asymptotic approximations of the vector of rotation ω are connected with the approximations of the vector of displacement V as in the classical theory of elasticity: 1 ω = rotV 2
(8.21)
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It means that in this case the suggested applied two-dimensional theory of micropolar shells is in the frame of micropolar theory with constrained rotation (or Cosserat pseudo continuum [3]). It should be noted that the general three-dimensional micropolar theory of elasticity with constrained rotation has some special features, for example: a) in this model the elastic body is characterized by four elastic constants λ, μ (or E, ν) and γ, ε (or l, η); b) from the six boundary conditions (8.4) remain five [3]: it is impossible to prescribe arbitrary boundary conditions for the moment stress μ33 . On the basis of the results of the asymptotic method of integration of the posed boundary problem for systems of equations (8.1)–(8.3), when the dimensionless physical parameters (8.5) take the values (8.20) [6], the following assumptions (hypotheses) can be applied for the construction of the general applied two-dimensional theory of micropolar shells with constrained rotation: 1. these are assumptions a)–d) of the previous section (but in this case the assumption d) must be referred only to the force stresses σ3i ); 2. the condition of constrained rotation (8.21) The basic system of equations of general applied two-dimensional theory of micropolar elastic thin shells with constrained rotation, in which the transverse shear and the related deformations are completely taken into account, is expressed as follows: equilibrium equations (8.9) and (8.10); elasticity relations from (8.11) and (8.12) for quantities T ii , Mii , Li j , Li3 , Λi3 , where formulas S 12 + S 21 = 4μh(Γ12 + Γ21), Ni3 + N3i = 4μh(Γi3 + Γ3i ), 3 H12 + H21 = 2h3 2μ(K12 + K21 ), Lii = 4γhκii
(8.22)
must be added; geometric relations from (8.13) and (8.14) for quantities Γii , Kii , Γi3 , Γ3i , κii , κi j , κi3 , li3 , ϑi , where formulas Γi j =
1 ∂u j 1 ∂A j 1 ∂Ψ j 1 ∂Ai − ui , Ki j = − Ψi , Ai ∂αi Ai A j ∂α j Ai ∂αi Ai A j ∂α j 1 Ωi = −(−1)i (Ψ j + ϑ j ), Ω3 = (Γ12 − Γ21 ), 2 1 ι = (K12 − K21 ) 2
(8.23)
must be added. Boundary conditions (8.15) and (8.16) should be added to the system of equations of micropolar elastic shells with constrained rotation. The system of equations of the theory of micropolar shells with constrained rotation has 18th order with nine boundary conditions (8.15) and (8.16) on each edge of the middle surface Γ. This system contains 51 equations for 51 unknown functions: T ii , Mii , S i j , Ni3 , N3i , Hi j , Lii , Li j , Li3 , Λi3 , Γii , Kii , Γi j , Ki j , Γi3 , Γ3i , κii , κi j , κi3 , li3 , ui , w, Ψi , ϑi , Ωi , Ω3 , ι. If transverse shear is neglected in this system of equations, i.e. if formulas (8.17) are valid, the model of micropolar elastic shells with constrained rotation will be
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obtained, when Kirchhoff-Love’s generalized hypothesis is valid instead of Timoshenko’s generalized kinematic hypothesis. Main equations and boundary conditions of this model of micropolar shells with constrained rotation are the followings: equilibrium equations (8.9) and (8.10), where Ni3 are neglected; elasticity relations from (8.11) and (8.12) for quantities T ii , Mii , Li j , Li3 , Λi3 , elasticity relations from (8.22) for quantities S 12 + S 21 , H12 + H21 , Lii ; geometric relations from (8.13) and (8.14) for quantities Γii , κii , κi j , κi3 , li3 , ϑi , geometric relation from (8.19) for quantity Kii , geometric relations from (8.23) for quantities Ω3 , ι and formulas Γi j =
1 ∂u j 1 ∂Ai − ui , Ai ∂αi Ai A j ∂α j
Ki j =
1 ∂ϑ j 1 ∂Ai − ϑi , Ai ∂αi Ai A j ∂α j
(8.24)
Ωi = −(−1)iϑ j . The boundary conditions for this model of micropolar elastic thin shells, when α1 = const, are the followings: ∗ T 11 = T 11 or u1 = u∗1 ,
S 12 +
H12 − L11 ∗ = S 12 or u2 = u∗2 , R2
∗ ∗ M11 + L12 = M11 or K11 = K11 ,
N13 +
1 ∂(H12 − L11 ) ∗ = N13 or w = w∗ , A2 ∂α2
(8.25)
∗ L13 = L∗13 or κ13 = κ13 ,
Λ13 = Λ∗13 or l13 = l∗13 .
8.5 Model of Micropolar Shells with “Small Transverse Stiffness” The following case for the physical dimensionless parameters (8.5) is considered: α ∼ μ,
R2 α R2 α R2 α 1, 1, 1 β γ ε
(8.26)
On the basis of asymptotic analysis of the formulated boundary problem [6] for the system of equations (8.1)–(8.3) in a thin three-dimensional domain and considering (8.26) the following asymptotically confirmed assumptions (hypotheses) can be formulated:
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• the assumptions a)–d) of the section three; • σi j −σ ji , σi3 −σ3i can be neglected in the moment equations of equilibrium (8.1), but taking into consideration the values of physical dimensionless parameters (8.26) (α is a small quantity for the given R) these terms are kept in the physical relations (8.2). It must be noted that this model of micropolar elastic shells, which is obtained on the basis of the accepted hypotheses, is called model with ”small shear stiffness” taking into consideration the fact that the physical parameter α is a shear modulus like the classical sher modulus μ. In this model the ”moment part” of the problem is separated as an independent boundary problem. The main system of equations and boundary conditions of the theory of micropolar elastic thin shells with ”small shear stiffness” with consideration of transverse shear is the following: • the ”moment part” of the problem: equilibrium equations (8.10) without (S 12 − S 21 ), (Ni3 − N3i ), (H12 − H21 ); elasticity relations (8.12); geometric relations (8.14); boundary conditions (8.16); • the ”force part” of the problem: equilibrium equations (8.9); elasticity relations (8.11); geometric relations (8.13); boundary conditions (8.15). The governing system of equations and boundary conditions of the theory of micropolar elastic thin shells with ”small transverse stiffness” without transverse shear is the following: • the ”moment part” of the problem: equilibrium equations (8.10); elasticity relations (8.12); geometric relations (8.14); boundary conditions (8.16); • the ”force part” of the problem: equilibrium equations (8.9); elasticity relations from (8.11) (except the last one) and elasticity relation (8.18); geometric relations from (8.13), (8.14) (except the last five relations of (8.13)) and geometric relations (8.19); boundary conditions of Kirchhoff-Love’s classical theory of elastic shells and conditions (8.16). In the model of micropolar shells with ”small transverse stiffness” (with or without transverse shear) the following fact is important: when the ”moment part” of the problem has zero solution (i.e. when the above mentioned corresponding equations and boundary conditions are homogeneous) the ”force part” of the problem doesn’t coincide with the classical models of elastic shells, as in the physical relations (8.11) members with physical constant α are present.
Conclusion General applied two-dimensional theory of micropolar elastic isotropic shells is constructed, when transverse shear and related to them strains are completely taken into consideration. Depending on the values of dimensionless physical parameters, theories of micropolar elastic thin shells with free fields of displacements and rotations,
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with constrained rotation, with “small shear stiffness” are constructed. General applied two-dimensional theory of micropolar thin plates and general applied onedimensional theory of micropolar thin bars can be obtained as private cases of the theory of shells.
References 1. Altenbach, J., Altenbach, H., Eremeyev, V. A.: On generalized Cosserat-type theories of plates and shells: A short review and bibliography. Arch. Appl. Mech. 80(1), 73–92 (2010) 2. Altenbach, H., Eremeyev, V.A.: On the linear theory of micropolar plates. ZAMM 89(4), 242–256 (2009) 3. Koiter, W.T.: Couple-Stresses in the Theory of Elasticity. Proc. Koninkl. Neterland. Akad. Wetensh. Pt. I-II. 67, 17–44 (1964) 4. Palmov, V.A.: Fundamental equations of the theory of asymmetric elasticity. Applied Mathematics and Mechanics. 28(3), 496–505 (1964) 5. Pelech, B.L.: Theory of Shells with Finite Transverse Shear Stiffness (in Russian). Naukowa Dumka, Kiev (1973) 6. Sargsyan, S.H.: General theory of elastic thin shells on the basis of asymmetrical theory of elasticity (in Russian). Doklady NAS RA. 108(4), 309–319 (2008) 7. Timoshenko, S., Woinowsky-Krieger, S.: Theory of Plates and Shells. McGraw Hill, New York (1959)
Part II
Dynamics and Stability
Chapter 9
Closed-Form Approximate Solution for the Postbuckling Behavior of Orthotropic Shallow Shells Under Axial Compression Matthias Beerhorst, Michael Seibel and Christian Mittelstedt
Abstract The current paper deals with a closed-form approximate solution for the postbuckling behavior of an unstiffened, singly-curved, orthotropic shell. As loading condition the case of uniform axial compression is treated. Concerning the boundary conditions all edges are supposed to be simply supported. Additionally, geometrical imperfections in form of an initial deflection of the shell can be accounted for. Choosing rather simple shape functions for the deflection a closed-form expression for the Airy stress function is obtained from the compatibility condition. As the equilibrium condition cannot be satisfied exactly the solution procedures of Galerkin as well as Ritz are employed to obtain an approximate solution. The resulting expressions from these procedures again allow for a closed-form solution of the load-deflection-relationship. After the force and the amplitude are known all other state variables such as stresses and displacements can be evaluated in a closed-form manner. Due to the rather simple formulation of the deflection shape the algorithm is limited to cases where the qualitative shape of the deflection does not change significantly. On the other hand the very high computational efficiency of the described solution procedure makes it ideally suited for use in the field of optimization and preliminary design, if the applied load does not exceed the linear buckling load too much. Keywords Postbuckling · Orthotropic shells · Closed-form solution · Analytical
M. Beerhorst (B) · M. Seibel HAW Hamburg, Berliner Tor 9, D-20099 Hamburg, Germany e-mail:
[email protected],
[email protected] C. Mittelstedt ELAN GmbH, Team Method and Tools, Karnapp 25, D-21079 Hamburg, Germany e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 9, © Springer-Verlag Berlin Heidelberg 2011
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9.1 Introduction Especially in those fields of engineering where minimum weight is of primary importance composite materials find an ever increasing use. In the case of thinwalled structural members stability problems occur if the structure is loaded in compression or shear. For analysis purposes structural members are often decomposed into relative simple geometries like plates and shells. Because of this the calculation of the stability behavior of plates and shells has been the subject of scientific research for over a hundred years. An important quantity in the design is the linear buckling load which describes the load at which the undeformed structure may abruptly undergo a deformation into its normal direction. However, since the linear buckling load is often significantly below the maximum load capacity of the structure, the so called post buckling behavior is also of special interest. For detailed analysis of the postbuckling behavior various iterative procedures including global approaches using Ritz- or Galerkin-methods (e.g. Shin et al. [15]), semi-discretizing approaches like the finite-strip method (e.g. Zou and Qiao [18]) or the discrete finiteelement-method (e.g. Reddy [13]) have been applied. The focus of the present work lies on the development of a solution procedure which employs a global approach in combination with only few shape functions instead of whole series in order to allow for a closed-form solution. As it has been shown lately for example by Diaconu and Weaver ( [3], [4]) who examined the case of an compressively loaded infinitely long plate as well as Mittelstedt and Schr¨oder [12] who dealt with a plate of finite length, closed-form-type solutions offer the merit of a very high computational efficiency while delivering quite satisfying results if the structure is not loaded far beyond the linear buckling load. In the present work the approach developed in [12] is extended to the geometry of singly-curved, shallow shells (see Fig. 9.1). Among the various shell theories 0
Nx
0
Nx
Fig. 9.1 Singly curved, shallow shell under uniaxial compression
existing (Donnell [5], Sanders-Koiter [14], [8], Love [10] and Fl¨ugge [6]) the one of Donnell which is valid for thin shells with small curvature was selected for the present work. Comparisons of the different theories can, inter alia, be found in the works of Simitses et al. [16] and Jaunky and Knight [7]. Beginning with the presentation of the governing equations related to this problem non dimensional parameters are introduced in order to transform the derived
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expressions in a generic manner. With the aid of a simple function of known shape but unknown amplitude for the deflection and a function of the same shape and known amplitude for the initial deflection, the compatibility requirement yields a closed-form expression for the Airy stress function. Employing the solution procedures of Galerkin as well as Ritz, the equilibrium condition is satisfied in an integral sense resulting in approximate expressions for the load-deflection-dependency. As a closure the resulting load-deflection-curves, buckling mode shapes and average strains are compared to an approach which uses full series expansions for the deflections as well as the stress function.
9.2 Governing Equations 9.2.1 Classical Formulations The geometry and coordinate systems of the shell which is dealt with in the present paper are visualized in Fig. 9.2. The shell has the length a, mid-surface arc length b, thickness t and mid-surface radius R. Since the shell in the present work is assumed
b a x, z, y, y= R
R
Fig. 9.2 Geometry and coordinate systems
to be thin-walled and of small curvature the material law which describes the relationship between forces and moments on the one hand and strains and curvatures on the other hand may be described by using the classical laminate plate theory [9]:
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⎧ 0⎫ ⎡ ⎪ ⎢⎢⎢A11 Nx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢⎢⎢⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢⎢⎢A12 Ny0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢⎢⎢ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎨ Nxy ⎪ ⎬ ⎢⎢⎢⎢A16 = ⎢⎢⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢⎢⎢ B11 ⎪ ⎪ M 0x ⎪ ⎪ ⎪ ⎪ ⎪ ⎢⎢⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢⎢⎢ B 0 ⎪ ⎪ ⎪ M ⎪ ⎪ y⎪ ⎢⎢ 12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ M 0 ⎭ ⎢⎣ B 16 xy
A12 A16 B11 B12 A22 A26 B12 B22 A26 A66 B16 B26 B12 B16 D11 D12 B22 B26 D12 D22 B26 B66 D16 D26
⎤⎧ 0 ⎫ B16 ⎥⎥⎥ ⎪ εx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎥⎥⎥ ⎪ ⎪ 0⎪ ⎪ ⎪ ⎪ B26 ⎥⎥⎥ ⎪ ε ⎪ ⎪ y ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎥⎥⎥ ⎪ ⎪ ⎪ 0 ⎪ ⎪ B66 ⎥⎥⎥ ⎪ ε ⎨ xy ⎪ ⎬ ⎥⎥⎥ ⎪ . ⎪ ⎪ ⎪ 0 ⎪ ⎪ D16 ⎥⎥⎥⎥ ⎪ κ ⎪ ⎪ x⎪ ⎪ ⎪ ⎪ ⎥⎥ ⎪ ⎪ ⎪ ⎪ ⎪ D26 ⎥⎥⎥⎥⎥ ⎪ ⎪ κy0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎦ ⎪ ⎪ ⎪ ⎩ 0 D κ ⎭ 66
(9.1)
xy
Gathering terms in sub-vectors and sub-matrices yields: ⎧ ⎫ 0⎪ ⎪ ⎪ ⎨N ⎪ ⎬ = ⎪ ⎪ ⎪ ⎩M0⎪ ⎭
⎡ ⎤⎧ ⎫ 0⎪ ⎢⎢⎢A B ⎥⎥⎥ ⎪ ⎪ ⎨ε ⎪ ⎬ ⎢⎢⎢⎣ ⎥⎥⎥⎦ ⎪ ⎪ ⎪. 0⎪ ⎩ BD κ ⎭
(9.2)
Herein a column vector is indicated by one underline and a matrix by two underlines. The partially inverted form of (9.1) reads: ⎤⎧ ⎫ ⎧ ⎫ ⎡ ⎢ 6 6 ⎪ B ⎥⎥⎥⎥ ⎪ ⎪ ⎪ ⎪N 0 ⎪ ⎪ ⎨ ε0 ⎪ ⎬ ⎢⎢⎢⎢ A ⎬ ⎥⎥⎥ ⎨ = (9.3) ⎢ ⎪ ⎪ ⎪ ⎪ T ⎢ ⎪ ⎪ ⎪ ⎪, 0 ⎦ ⎩ M 0 ⎭ ⎣−6 ⎩ 6 B D κ ⎭ with: 6 = A−1 , A
6 B = −A−1 B,
6 = D − B A−1 B. D
(9.4)
The strain distribution in the shell can be calculated from the in-plane strains and the curvatures of the reference surface according to the following kinematic relations: ε x = ε0x + zκ0x ,
εy = ε0y + zκy0 ,
ε xy = ε0xy + zκ0xy .
(9.5)
Herein the curvatures are identical to those of the linear plate theory and read: κ0x = −
∂2 w , ∂x2
κy0 = −
∂2 w , ∂y2
κ0xy = −2
∂2 w . ∂x∂y
(9.6)
In contrast to this the membrane strains have additional nonlinear components consisting of derivatives of the deflection which has been separated into a part with constant amplitude describing the initial deflection w0 and a load dependent variable part w: 2 ∂u 1 ∂w ∂w ∂w0 + + , ∂x 2 ∂x ∂x ∂x 2 ∂v 1 ∂w ∂w ∂w0 w 0 εy = + + + , ∂y 2 ∂y ∂y ∂y R ∂u ∂v ∂w ∂w ∂w ∂w0 ∂w ∂w0 ε0xy = + + + + . ∂y ∂x ∂x ∂y ∂x ∂y ∂y ∂x ε0x =
(9.7) (9.8) (9.9)
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These equations are identical to those proposed for the flat plate with initial deflection by Marguerre [11] with an additional term wR in ε0y describing the influence of the radius. Since it is assumed that the shell is stress free in the initial state only the variable deflection w and not w0 is accounted for in this term. Furthermore the initial deflection w0 is assumed to be small in comparison to w in the later loading stages so that terms including w0 in second order are neglected. From the six equilibrium conditions concerning the forces and moments the following three equations can be derived ( [16]): ∂Nx ∂Nxy + = 0, ∂x ∂y
(9.10)
∂Nxy ∂Ny + = 0, ∂x ∂y
(9.11)
∂2 M xy ∂2 My ∂2 M x ∂2 (w + w0 ) + 2 + + N x ∂x∂y ∂x2 ∂y2 ∂x2 2 2 ∂ (w + w0 ) ∂ (w + w0) Ny + 2N xy + Ny − + p = 0. ∂x∂y R ∂y2
(9.12)
Wherein p denotes pressure acting normal to the shell surface. By introducing the Airy stress function (9.13) as given below, conditions (9.10) and (9.11) derived from equilibrium are identically fulfilled. Nx0 =
∂2 Ψ , ∂y2
Ny0 =
∂2 Ψ , ∂x2
0 Nxy =−
∂2 Ψ ∂x∂y
(9.13)
Apart from equilibrium the compatibility condition of the in-plane strains (9.14) is taken into account. 2 0 2 0 ∂2 ε0x ∂ εy ∂ ε xy + − =0 (9.14) ∂x∂y ∂y2 ∂x2 Using the partially inverted material law (9.3) in conjunction with the Airy stress function (9.13) the moments can be described as: ∂2 ψ ∂2 ψ ∂2 ψ ∂2 w ∂2 w ∂2 w − B + B − D − D − 2D , 21 11 12 61 16 ∂x∂y ∂x∂y ∂y2 ∂x2 ∂x2 ∂y2 ∂2 ψ ∂2 ψ ∂2 ψ ∂2 w ∂2 w ∂2 w My0 = −B12 2 − B22 2 + B62 − D12 2 − D22 2 − 2D26 , ∂x∂y ∂x∂y ∂y ∂x ∂x ∂y ∂2 ψ ∂2 ψ ∂2 ψ ∂2 w ∂2 w ∂2 w M 0xy = −B16 2 − B26 2 + B66 − D16 2 − D26 2 − 2D66 . (9.15) ∂x∂y ∂x∂y ∂y ∂x ∂x ∂y M 0x = −B11
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Inserting the above equations into the third equilibrium-equation (9.12) yields: ∂4 ψ ∂4 ψ ∂4 ψ ∂4 ψ + (B11 + B22 − 2B66) 2 2 + B12 4 + (2B26 − B61) 3 4 ∂x ∂x ∂y ∂y ∂x ∂y 4 4 4 ∂ ψ ∂ w ∂ w ∂4 w (D ) + (2B16 − B62) + D + 2 + 2D + D 11 12 22 66 ∂x∂y3 ∂x4 ∂x2 ∂y2 ∂y4 2 4 4 2 2 ∂ w ∂ w ∂ ψ ∂ w ∂ w0 +4D16 3 + 4D26 2 2 − 2 + ∂x ∂y ∂x ∂y ∂y ∂x2 ∂x2 2 2 2 2 2 2 ∂ ψ ∂ w ∂ w0 ∂ ψ ∂ w ∂ w0 1 +2 + − 2 + − + p = 0. (9.16) ∂x∂y ∂x∂y ∂x∂y ∂x ∂y2 ∂y2 R B21
The membrane strains formulated according to the partially inverted material law (9.3) read: ∂2 ψ ∂2 ψ ∂2 ψ ∂2 w ∂2 w ∂2 w − A + A − B − B − 2B , 12 11 12 16 16 ∂x∂y ∂x∂y ∂y2 ∂x2 ∂x2 ∂y2 ∂2 ψ ∂2 ψ ∂2 ψ ∂2 w ∂2 w ∂2 w ε0y = −A12 2 − A22 2 − A26 − B21 2 − B22 2 − B26 , ∂x∂y ∂x∂y ∂y ∂x ∂x ∂y ∂2 ψ ∂2 ψ ∂2 ψ ∂2 w ∂2 w ∂2 w ε0xy = A16 2 + A26 2 − A66 − B61 2 − B62 2 − 2B66 . ∂x∂y ∂x∂y ∂y ∂x ∂x ∂y ε0x = −A11
(9.17) (9.18) (9.19)
Inserting these expressions into the compatibility-equation (9.14) results in: ∂4 ψ ∂4 ψ ∂4 ψ ∂4 ψ (2A ) + A + + A − 2A 11 12 66 16 ∂x4 ∂y4 ∂x2 ∂y2 ∂x∂y3 4 4 4 ∂ ψ ∂ w ∂ w ∂4 w −A26 3 − B21 4 − B12 4 + (2B66 − B11 − B22) 2 2 ∂x ∂y ∂x ∂y ∂x ∂y ∂4 w ∂4 w + (B62 − 2B16) + (B61 − 2B26) 3 = 0. ∂x∂y3 ∂x ∂y A22
(9.20)
Another formulation of the compatibility expression is obtained by inserting the membrane strains resulting from the kinematic relations (9.5) into (9.14):
∂2 w ∂x∂y
2
∂2 w ∂2 w ∂2 w ∂2 w0 ∂2 w ∂2 w0 − − 2 ∂x2 ∂y2 ∂x2 ∂y2 ∂y ∂x2 ∂2 w ∂2 w0 1 ∂2 w +2 + = 0. ∂x∂y ∂x∂y R ∂x2 −
(9.21)
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Now both representations of the compatibility condition are equated to achieve a relation between the Airy stress function and the deflection functions: ∂4 ψ ∂4 ψ ∂4 ψ ∂4 ψ ∂4 ψ + A11 4 + (2A12 + A66) 2 2 − 2A16 − A26 3 4 3 ∂x ∂y ∂x ∂y ∂x∂y ∂x ∂y 4 4 4 ∂ w ∂ w ∂ w ∂4 w −B21 4 − B12 4 + (2B66 − B11 − B22) 2 2 + (B62 − 2B16) ∂x ∂y ∂x ∂y ∂x∂y3 2 ∂4 w ∂2 w ∂2 w ∂2 w ∂2 w ∂2 w0 + (B61 − 2B26) 3 = − 2 2− 2 ∂x∂y ∂x ∂y ∂x ∂y ∂x ∂y2 ∂2 w ∂2 w0 ∂2 w ∂2 w0 1 ∂2 w − 2 +2 + . (9.22) 2 ∂x∂y ∂x∂y R ∂x2 ∂y ∂x A22
The further attempts to solve the problem will mainly rely on finding a solution of the equilibrium condition in the form of (9.16) and the compatibility condition (9.22).
9.2.2 Non-Dimensional Quantities In order to get a more general insight into the influence of geometry and laminate parameters it is convenient to introduce non dimensional quantities (cf. [4]). The coordinates are related to the length and width of the shell as: y η= . b
x ξ= , a
(9.23)
Additionally a dimensionless radius R and pressure p are defined: R=R
7 AD , b2
p= p
a2 b 2 . 8 7 D 611 D 622 AD
Herein the following abbreviation was used: 8 4 7 611 A 622 D 611 D 622 . AD = A
(9.24)
(9.25)
Nondimensionalization of the stress function ψ and the potential of the system Π is achieved by using relations (9.26). ψ= 8
ψ 611 D 622 D
,
Π =Π
ab . 8 72 D 611 D 622 AD
(9.26)
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The displacements become: u=
ua 2
7 AD
,
v=
vb 2
7 AD
,
w=
w , 7 AD
w0 =
w0 . 7 AD
(9.27)
Further abbreviations proposed by Diaconu and Weaver [4] are: 9 9 611 622 a 4 A b2 6 B21 a 4 D αA = , αB = 2 , αD = , b A 622 7 b D 611 a AD βA =
612 + A 666 2A , 8 6 6 2 A11 A22
γA = 8 4
δA = 8 4
616 A 63 A 6 A 11 22 626 A 611 A 63 A 22
ηA = 8
μB = θB =
6 B11 , 7 AD b6 B61 a AD 7
612 A 611 A 622 A
ρB = ,
ζB =
,
,
6 B22 , 7 AD b 26 B26 a AD 7
βB =
6 B11 + 6 B22 − 2 6 B66 , 7 AD
612 + 2D 666 D βD = 8 , 6 6 D11 D22
γB =
b 26 B26 − 6 B61 , a 7 AD
γD = 8
δB =
a 26 B16 − 6 B62 , b 7 AD
δD = 8
ηB =
a2 6 B12 , 2 7 b AD
ηD = 8
ωB = ,
k1 =
6 B66 , 7 AD a2 b 2 , √ 7 D11 D22 AD
4
4
616 D 63 D 6 D 11 22 626 D 611 D 63 D 22 612 D 611 D 622 D
,
,
,
a6 B16 a6 B62 , τB = , b AD 7 b AD 7 a 2 b2 k2 = . (9.28) 72 AD
νB =
A non dimensional formulation of the equilibrium (9.16) is now obtained by using the non dimensional quantities above and multiplying the resulting expression with k1 from (9.28): αB
∂4 ψ ∂4 ψ ∂4 ψ ∂4 ψ ∂4 ψ 1 ∂4 w ∂4 w + β + η + γ + δ + + 2β B B B B D ∂ξ4 ∂ξ2 ∂η2 ∂η4 ∂ξ3 ∂η ∂ξ∂η3 α2D ∂ξ4 ∂ξ2 ∂η2 ∂4 w γ D ∂4 w ∂4 w ∂2 ψ ∂2 w ∂2 w0 + α2D 4 + 4 + 4α δ − + D D αD ∂ξ3 ∂η ∂η ∂ξ∂η3 ∂η2 ∂ξ2 ∂ξ2 ∂ 2 ψ ∂ 2 w ∂ 2 w0 ∂2 ψ ∂2 w ∂2 w0 1 +2 + − 2 + − + p = 0. (9.29) ∂ξ∂η ∂ξ∂η ∂ξ∂η ∂ξ ∂η2 ∂η2 R
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The procedure for transforming the compatibility equation (9.22) works analogously but this time the multiplication factor is k2 from (9.28): 1 ∂4 ψ ∂4 ψ ∂4 ψ ∂4 ψ δA ∂4 ψ + α2A 4 + 2βA 2 2 − 2αAγA −2 2 4 3 αA ∂ξ3 ∂η ∂η ∂ξ ∂η ∂ξ∂η αA ∂ξ ∂4 w ∂4 w ∂4 w ∂4 w ∂4 w − η − β − δ − γ B B B B ∂ξ4 ∂η4 ∂ξ2 ∂η2 ∂ξ∂η3 ∂ξ3 ∂η 2 2 ∂ w ∂2 w ∂2 w ∂2 w ∂2 w0 = − 2 2− 2 ∂ξ∂η ∂ξ ∂η ∂ξ ∂η2 2 2 2 2 ∂ w ∂ w0 ∂ w ∂ w0 1 ∂2 w − 2 +2 + . ∂ξ∂η ∂ξ∂η R ∂ξ2 ∂η ∂ξ2 −αB
(9.30)
The relation of loads and Airy stress function is also transformed as: 0
N x π2 =
∂2 ψ , ∂η2
0
N y π2 =
∂2 ψ , ∂ξ2
0
N xy π2 = −
∂2 ψ , ∂ξ∂η
(9.31)
where: 0
Nx =
Nx0 b2 , 8 611 D 622 π2 D
0
Ny =
Ny0 a2 , 8 611 D 622 π2 D
0
N xy =
0 ab Nxy . 8 611 D 622 π2 D
(9.32)
In the special case of symmetric, orthotropic laminates the absence of many coupling terms leads to a drastic simplification of the expressions above. More precisely the non dimensional parameters γA , δA , γD , δ D and all abbreviation parameters of (9.28) with subindex B vanish. The equilibrium condition (9.29) then simplifies to: 4 1 ∂4 w ∂4 w ∂2 ψ ∂2 w ∂2 w0 2 ∂ w + 2βD 2 2 + αD 4 − 2 + ∂ξ ∂η ∂η ∂η ∂ξ2 ∂ξ2 α2D ∂ξ4 ∂ 2 ψ ∂ 2 w ∂ 2 w0 ∂2 ψ ∂2 w ∂2 w0 1 +2 + − 2 + − + p = 0, (9.33) ∂ξ∂η ∂ξ∂η ∂ξ∂η ∂ξ ∂η2 ∂η2 R and the compatibility condition reads: 2 2 4 1 ∂4 ψ ∂4 ψ ∂ w ∂2 w ∂2 w 2∂ ψ + α + 2β = − A A ∂ξ∂η ∂η4 ∂ξ2 ∂η2 ∂ξ2 ∂η2 α2A ∂ξ4 ∂2 w ∂2 w0 ∂2 w ∂2 w0 ∂2 w ∂2 w0 1 ∂2 w − 2 − + 2 + . ∂ξ∂η ∂ξ∂η R ∂ξ2 ∂ξ ∂η2 ∂η2 ∂ξ2
(9.34)
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9.2.3 In-Plane Displacements In the field of structural design not only the deflection of the shell but also the in-plane displacements are of interest. The partial derivatives of the in-plane displacements as shown below in (9.35) to (9.37) are calculated from the kinematic relationships (9.5). 2 ∂u 1 ∂w ∂w ∂w0 = ε0x − − ∂x 2 ∂x ∂x ∂x 2 ∂v 1 ∂w ∂w ∂w0 w = ε0y − − − ∂y 2 ∂y ∂y ∂y R ∂u ∂v ∂w ∂w ∂w ∂w0 ∂w ∂w0 + = ε0xy − − − ∂y ∂x ∂x ∂y ∂x ∂y ∂y ∂x
(9.35) (9.36) (9.37)
Inserting the membrane strains of (9.17) to (9.19) into (9.35) to (9.37) and performing the transformation yields: ∂u ∂2 ψ ∂2 ψ ∂w ∂w0 = α2A 2 + ηA 2 − , ∂ξ ∂ξ ∂ξ ∂η ∂ξ 2 ∂v ∂2 ψ 1 ∂2 ψ 1 ∂w ∂w ∂w0 w = ηA 2 + 2 2 − − − , ∂η 2 ∂η ∂η ∂η R ∂η αA ∂ξ ∂u ∂v ∂2 ψ ∂w ∂w ∂w ∂w0 ∂w ∂w0 + = 2(ηA − βA ) − − − , ∂η ∂ξ ∂ξ∂η ∂ξ ∂η ∂ξ ∂η ∂η ∂ξ
(9.38) (9.39) (9.40)
in case of an symmetric, orthotropic laminate. Often the overall change in length and width is of special interest, which is calculated by integration over the length and width as: 1 ∂2 ψ ∂2 ψ ∂w ∂w0 Δa = α2A 2 + ηA 2 − dξ, (9.41) ∂ξ ∂ξ ∂η ∂ξ 0 and:
⎞ 2 1 ⎛⎜ ∂w ∂w0 w ⎟⎟⎟⎟ ⎜⎜⎜ ∂2 ψ 1 ∂2 ψ 1 ∂w Δb = ⎝⎜ηA ∂η2 + 2 ∂ξ2 − 2 ∂η − ∂η ∂η − ⎟⎠ dη. αA R 0
(9.42)
If these values are related to the nondimensionalized initial length a and width b of the shell, the average strains become: εa =
Δa Δa = , a a
εb =
Δb b
=
Δb . b
(9.43)
The nondimensionalized length a and width b may be calculated from (9.27), which describes the displacements, by replacing the characters u with a and v with b.
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9.3 Closed-Form Solution 9.3.1 Boundary Conditions The edges of the shell are assumed to be simply supported, which demands vanishing normal displacements and bending moments along the edges. w (ξ = 0, η) = 0,
M η (ξ = 0, η) = 0,
(9.44)
w (ξ = 1, η) = 0,
M η (ξ = 1, η) = 0,
(9.45)
w (ξ, η = 0) = 0,
M ξ (ξ, η = 0) = 0,
(9.46)
w (ξ, η = 1) = 0,
M ξ (ξ, η = 1) = 0.
(9.47)
Regarding the in-plane displacements no specific boundary conditions are assumed, so the edges can freely move in-plane.
9.3.2 Linear Buckling Loads For the validation of the results later on the linear buckling loads of the plate and the shell will be referred to. For brevity only the resulting expressions are cited here, details can for example be found in [17]. The linear buckling load of an orthotropic plate with symmetric lay-up is: 0
Nx =
αD m + m αD
2 + 2 (βD − 1),
and for the shell: ⎡ ⎢⎢⎢ 2 n 2 0 2 ⎢⎢⎢⎢ m 1 N x = n ⎢⎢ + 2βD + αD + 2 % ⎢⎣ n αD m (πn)4 R m
1
& % 1 2 n n αA + 2βA + m α A
(9.48)
⎤ ⎥⎥⎥ ⎥⎥⎥ &2 ⎥⎥⎥⎥ . (9.49) ⎦
Herein the numbers of halfwaves m and n which lead to the lowest buckling load are the relevant ones.
9.3.3 Compatibility Condition - Stress Function As a first step towards the solution of the postbuckling problem the compatibility condition (9.34) is used to obtain a closed-form solution of the stress function. As this partial differential equation depends on the stress function on the one hand and
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the deflection on the other hand the shape of the deflection is assumed a priori to reduce the equation to a partial differential equation of only one unknown function. Products of sine-functions which fulfill the boundary conditions given in (9.44) – (9.47) are chosen as shape functions for the variable deflection w and the initial deflection w0 : w = A sin(mπξ) sin(nπη),
(9.50)
w0 = B sin(mπξ) sin(nπη).
(9.51)
Herein A and B are the unknown and known deflection amplitudes, respectively. The number of buckling halfwaves is expressed through m in longitudinal and n in circumferential direction. Inserting the shape functions into the compatibility condition (9.34) results in: 1 ∂4 ψ ∂4 ψ ∂4 ψ m2 + α2A 4 + 2βA 2 2 = −π2 A sin(mπξ) sin(nπη) 2 4 ∂η ∂ξ ∂η αA ∂ξ R % & 4 2 2 2 +π m n A(A + 2B) cos (mπξ) + cos2 (nπη) − 1 .
(9.52)
Since the above partial differential equation is of linear nature the overall solution of the stress function can be found by superposition of a homogeneous solution and the particular solutions: ψ = ψh + ψ p1 + ψ p2 . (9.53) The homogeneous solution of the problem corresponds to the situation of a plate with zero deflection. According to [12] it can be obtained via integration while neglecting the integration constants: 0 ψh = N x π2 dη dξ. (9.54) After integration the homogeneous solution reads: 0
ψh =
N x π2 η2 . 2
(9.55)
From the right-hand side of the compatibility condition (9.52) it can be observed that the particular solution should consist of a product of sine-functions and a sum of squares of cosine-functions. So the first and second particular solution are assumed to be of the form: ψ p1 = C sin(mπξ) sin(nπη),
(9.56)
ψ p2 = D cos (mπξ) + E cos (nπη).
(9.57)
2
2
Herein the first particular solution describes the influence of the radius, whereas the second is identical to the solution for a flat plate. Subsequently inserting the
9 Approximate Solution for the Postbuckling Behavior of Orthotropic Shells
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particular solutions into the compatibility condition, the unknowns C, D and E can be calculated by comparison of coefficients:
C= Rπ2 D=
−A m2 α2A
4
+ α2A mn 2 + 2βA n2
α2A n2 A(A + 2B) 16m2
,
E=
,
m2 A(A + 2B) . 16α2A n2
(9.58)
9.3.4 Equilibrium Condition - Load-Deflection-Relation After a closed form solution for the stress function was determined by using the compatibility condition, the relationship between the applied load and the corresponding amplitude of the deflection has to be found. This can be done be using the equilibrium condition (9.33). Because this equation does not allow for a closedform solution two approximation approaches are employed. First the minimization of the potential energy of the system as suggested by Ritz is outlined. According to Ritz the stationary value of potential energy is found when the derivations of the total potential of the system with respect to the variables vanish: ∂Π = 0. ∂Ai
(9.59)
The total potential of the system is the sum of the internal and the external potential: Π = Π i + Π e. In the present case the inner energy of the shell reads: % & 1 Πi = σx ε x + σy εy + σ xy γ xy dV. 2 V
(9.60)
(9.61)
Inserting the strains according to the kinematics (9.5) and performing integration over the thickness yields: 1 Πi = 2
a b% 0
0
& 0 0 N x0 ε0x + Ny0 ε0y + N xy γ xy + M 0x κ0x + My0 κy0 + M 0xy κ0xy dydx.
(9.62)
Now the laminate forces and moments are replaced by the relations (9.13) and (9.15) and the strains and curvatures are replaced by (9.17)–(9.19) and (9.6):
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a b
1 Πi = 2
0
0
2 2 2 2 ∂ ψ 6 612 ∂ ψ ∂ ψ A11 + 2 A ∂y2 ∂x21 ∂y2 2 2 2 2 622 ∂ ψ + A 666 ∂ ψ +A 2 ∂x1 ∂y ∂x ⎛ ⎞2 2 2 2 ⎜ ⎟ 611 ⎜⎜⎜⎜⎝ ∂ w ⎟⎟⎟⎟⎠ + 2 D 612 ∂ w ∂ w +D ∂x2 ∂y2 ∂x21 2 2 2 2 622 ∂ w + 4 D 666 ∂ w +D dydx. ∂xdx2 ∂y2
(9.63)
Equation (9.64) shows the internal energy after transformation. 1 Πi = 2
0
1
1 2 2 ∂ ψ ∂2 ψ ∂2 ψ α2A + 2η A ∂η2 ∂ξ2 ∂η2 0 2 2 2 2 1 ∂ ψ ∂ ψ + 2 + 2 (βA − ηA ) 2 ∂ξ∂η αA ∂ξ 2 1 ∂2 w ∂2 w ∂2 w + 2 + 2η D ∂ξ2 ∂η2 αD ∂ξ2 2 2 2 2 ∂ w ∂ w (β ) + α2D + 2 − η dηdξ. D D 2 ∂ξ∂η ∂η
(9.64)
The potential of the external load is identical to the external work performed by this load: 1 1 0 du Πe = − N x dηdξ. (9.65) dξ 0 0 With the derivative of the displacement u according to (9.38) the external potential becomes: 1 1 ∂2 ψ ∂2 ψ ∂w ∂w0 0 Πe = − N x α2A 2 + ηA 2 − dηdξ. (9.66) ∂ξ ∂ξ ∂η ∂ξ 0 0 In the present case the only unknown variable is the deflection amplitude A of the variable deflection w, so only condition (9.67) below has to be fulfilled to find the state of stationary potential energy. ∂Π = 0. ∂A
(9.67)
The above condition leads to an expression which contains third order terms of A 0 and linear terms of the external load N x . Now two options exist for finding the relation between load and amplitude.
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• The first one, which may be called load controlled, is to set the value of the load and then calculate the amplitude, which is the maximum real root of the polynomial of third order for A. • The second one, which will be referred as amplitude controlled, is to solve (9.67) 0 for N x . From the resulting expression the load can be directly calculated for a given amplitude. The advantage of the first method is that setting the range of the load might be more intuitive as the engineer is often interested in the behavior of the structure up to a certain load level, for example two times the linear buckling load. On the other hand the second approach can also handle situations in which the absolute value of the amplitude decreases with increasing load, for example when dealing with cylindrical shells. Another way to approximately satisfy the equilibrium condition is to apply the Galerkin-procedure. This method is based on the main idea to eliminate the weighted residuum of the differential equation in an integral sense. For the present case this leads to: 1 1 . / R w (ξ, η) W(ξ, η)dηdξ = 0, (9.68) 0
0
with the first factor being the residuum (cf. (9.34)): 4 . / 1 ∂4 w ∂4 w ∂2 ψ ∂2 w ∂2 w0 2 ∂ w R w (ξ, η) = 2 + 2βD 2 2 + αD 4 − 2 + ∂ξ ∂η ∂η ∂η ∂ξ2 ∂ξ2 αD ∂ξ4 ∂2 ψ ∂2 w ∂2 w0 ∂2 ψ ∂2 w ∂2 w0 1 +2 + − 2 + − . ∂ξ∂η ∂ξ∂η ∂ξ∂η ∂ξ ∂η2 ∂η2 R
(9.69)
The weighting function is identical to the deflection function with the amplitude set to unity: W(ξ, η) = sin(mπξ) sin(nπη). (9.70) After inserting the deflection functions (9.50) and (9.51) and the stress function (9.53) into the residuum-equation (9.69) the integrals of (9.68) can be evaluated. The result is an expression which contains third order terms of A and linear terms of 0 the external load N x . In order to obtain the load-amplitude-relationship either loadcontrolling or amplitude-controlling as described above for the Ritz-method may be employed.
9.4 Results and Discussion In order to establish a basis for estimating the accuracy and applicability of the proposed closed-form solution it is compared to a solution procedure which uses Fourier series expansions for the deflections:
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w= w0 =
M N
Amn sin (mπξ) sin (nπη),
(9.71)
Bmn sin (mπξ) sin (nπη),
(9.72)
m=1 n=1 M N m=1 n=1
as well as the particular parts of stress function: ψ p1 = ψ p2 =
2M 2N m=0 n=0 2M 2N
Cmn sin(mπξ) sin(nπη),
(9.73)
Dmn cos(mπξ) cos(nπη),
(9.74)
m=0 n=0
similar as proposed for example by [2] and [1]. Because the algorithm based on this idea is capable of analyzing unsymmetrical lay-ups it will be referenced as PBUS (Postbuckling of unsymmetric shells). Analogously the present method will be referred to as PBOS (Postbuckling of orthotropic shells). Additionally the linear buckling loads of the flat plate as well as the shallow shell as given in (9.48) and (9.49) will be used as reference. The load-deflection curves do also include the postbuckling deflection of a flat plate, which is achieved by setting the value for the radius to infinity. As in-plane coordinates for the load-deflection-history the point with the maximum deflection obtained from the eigenvalue analysis which lies nearest to the center of the shell is chosen. For the further analyses the material properties are taken from a CFRP T300/913C UD-prepreg: E11 = 93000 MPa, E22 = 7300 MPa, G12 = 4700 MPa, ν12 = 0.3. In all examples the thickness of each ply is tPly = 0.25 mm and the orientation of the eight plies in degree is [90/90/0/0] s. The numbers of buckling halfwaves m and n which are chosen for the present analysis are determined from the eigenvalue analysis according to (9.49). The amplitude B of the initial deformation is set to 10.4% of the laminate thickness. The current example deals with a shell of the length a = 500 mm, width b = 200 mm and radius R = 2000 mm. Performing the eigenvalue analysis reveals that the 0 linear buckling load is N x = −3.8 and the corresponding eigenmode consists of one halfwave in circumferential direction and four halfwaves in longitudinal direction. Carrying out a convergence study with PBUS showed that using 6 x 6 terms in this case is more than sufficient. In Fig. 9.3 the load-deflection-history is shown for the coordinates ξ = 5/8 and η = 0.5. It can be seen that the deflection of the plate increases most rapidly with increasing load and that the results of PBOS and PBUS are in excellent agreement. As expected the results of the shell show a slower increasing amplitude which indicates a stiffer structural behavior. The results of the Galerkin and Ritz approach are identical. On first glance the resulting curve from the PBUS-tool which uses a Fourier expansion of 6 x 6 terms for the deflection function
9 Approximate Solution for the Postbuckling Behavior of Orthotropic Shells
-12 -10
Nx0
-8 -6 -4
PBOS Shell PBOS Plate Plate Linear Shell Linear PBUS PBUS (Mean) PBUS Plate
-2 0
0
1
2
3
4
w+w0
5
6
7
8
Fig. 9.3 Deflection history for point at ξ = 5/8 and η = 0.5
8 6 4
w+w0
2 0
-2 -4 -6 -8
PBUS PBOS
-10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0
Fig. 9.4 Deflection along ξ in the middle of the shell (η = 0.5) at N x = −12
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1.0 0.8 0.6 0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
Fig. 9.5 Contour plot of initial deflection
1.0 0.8 0.6 0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0 0
Fig. 9.6 Contour plot of deflection in the deep postbuckling region (N x = −12)
9 Approximate Solution for the Postbuckling Behavior of Orthotropic Shells
-0.065 -0.066
a [%]
-0.067 -0.068 PBUS PBOS
-0.069 -0.070 -0.071 -0.072 -0.073 0.0
0.2
0.4
0.6
0.8
1.0
0
Fig. 9.7 Average strain εa at N x = −6
0.06 PBUS PBOS
0.04
b [%]
0.02 0.00 -0.02 -0.04 -0.06 0.0
0
0.2
Fig. 9.8 Average strain εb at N x = −6
0.4
0.6
0.8
1.0
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0 -2
Nx0
-4 -6 -8 -10 -12 -0.20
PBUS PBOS
-0.15
-0.10
-0.05
a [%]
0.00
0.05
Fig. 9.9 Strain history of the average strain εa in the middle of the shell (η = 0.5)
might be surprising because it shows a smaller deflection and thus a stiffer behavior than the present approach. This effect can be explained by the fact that the absolute value of the buckling peaks is not the same in the PBUS-analysis which can be seen in Fig. 9.4. Unlike in the case of a flat plate, due to the influence of the radius, the absolute values of the buckling peaks pointing inward the shell and thus in negative ζ-direction are greater than the ones pointing in the outward direction. Because the present method has only one degree of freedom with respect to the amplitude of the deflection it is not able to capture this effect and despite that returns an amplitude which is more like the arithmetic mean of the absolute values of the buckling peaks of PBUS. This mean value of the deflection is also visualized in Fig. 9.3 and is in good agreement with deflection calculated with PBOS. Another aspect which comes into play is the shape of the buckling modes at different load levels. At the stage of very low load levels the mode shape is identical to the linear eigenmode and can be described by a single product of two sine-functions. A contour plot (Fig. 9.5) shows elliptical contour lines at this point. If the load is increased the crest of the buckling wave flattens which leads to mode shapes which are often described as bath tub shaped and are visualized in Fig. 9.6. A comparison of the average strains in longitudinal and transversal direction is 0 given in Figs 9.7 and 9.8, respectively . With an amount of N x = −6 the load level is roughly 1.5 times the linear buckling load which is an important design load in aircraft design, as limit load is often set to the linear buckling load and ultimate load is 150% of limit load. Figure 9.7 shows that the average strain εa is qualitatively
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similar while being constant for the present approximation and only slightly varying in case of PBUS. The relative deviation amounts to approximately 10%. For the average strain εb depicted in Fig. 9.8 the qualitatively behavior is similar again, but the deviation at the peaks is relatively high and amounts to about −30%. In Fig. 9.9 the average longitudinal strain at the middle of the shell (η = 0.5) is given in dependence of the applied load. As it could be expect from Fig. 9.7 the agreement between the two methods is satisfying, while the structural behavior simulated by PBOS is stiffer.
9.5 Conclusions In the present contribution an algorithm of high computational efficiency for the estimation of the postbuckling behavior of shallow orthotropic shells was presented. As it can be seen from the comparison with more detailed analysis methods the results are in excellent agreement for the flat plate. With decreasing radius the irregular deflection pattern of the shell is only simulated in an averaging sense by the present method. While the average strains in longitudinal direction are approximated with satisfying accuracy up to a load level of 1.5 times the linear buckling load, the strains in circumferential direction show large deviations in the peak regions of the deflection. Returning results in milliseconds while being capable of tracing the main structural behavior up to the early postbuckling region, the developed algorithm is well suited for the fields of preliminary design and optimization.
References 1. B¨urmann, P.: A semi-analytical model for the post-buckling analysis of stringer- and framestiffened cylindrical panels under combined loading. Ph.D. thesis, Technische Universit¨at Braunschweig (2006) 2. Byklum, E.: Ultimate strength analysis of stiffened steel and aluminium panels using semianalytical methods. Ph.D. thesis, NTNU Trondheim (2002) 3. Diaconu, C.G., Weaver, P.M.: Approximate solution and optimum design of compressionloaded, postbuckled laminated composite plates. AIAA Journal 43(4) (2005) 4. Diaconu, C.G., Weaver, P.M.: Postbuckling of long unsymmetrically laminated composite plates under axial compression. International Journal of Solids and Structures 43, 6978–6997 (2006) 5. Donnell, L.H.: Stability of thin-walled tubes under torsion. NACA Report 479 (1933) 6. Fl¨ugge, W.: Stresses in shells, 2nd edn. Springer, Berlin (1973) 7. Jaunky, N., Knight, N.F.: An assessment of shell theories for buckling of circular cylindrical laminated composite panels loaded in axial compression. International Journal of Solids and Structures 36, 3799–3820 (1999) 8. Koiter, W.T.: A consistent first approximation in general theory of thin elastic shells. the theory of thin elastic shells. In: Proceedings IUTAM Symposium, pp. 12–33. Delft (1959) 9. Kollar, L., Springer, G.: Mechanics of composite structures. Cambridge University Press, Cambridge (2003)
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10. Love, A.: A treatise on the mathematical theory of elasticity, 4th edn. Dover Publication, New York (1927) 11. Marguerre, K.: Die mittragende breite der gedr¨uckten platte. Luftfahrtsforschung 14 3, 121– 128 (1937) 12. Mittelstedt, C., Schr¨oder, K.-U.: Postbuckling of compressively loaded imperfect composite plates: closed-form approximate solutions. International Journal of Structural Stability and Dynamics 10, 761–778 (2010) 13. Reddy, J.: Mechanics of laminated composite plates and shells, 2nd edn. CRC PRESS, Boca Raton et al. (2004) 14. Sanders, J.: An improved first-approximation theory for thin shells. NASA-TR-R-24 (1959) 15. Shin, D.K., Griffin, O.H., G¨urdal, Z.: Postbuckling response of laminated plates under uniaxial compression. International Journal of Non-Linear Mechanics 28, 95–115 (1993) 16. Simitses, G.J., Shaw, D., Sheinman, I.: Stability of cylindrical shells, by various nonlinear shell theories. Zeitschrift f¨ur Angewandte Mathematik und Mechanik 65, 159–166 (1985) 17. Wiedemann, J.: Leichtbau - Elemente und Konstruktion, 3rd edn. Klassiker der Technik. Springer, Berlin et al. (2007) 18. Zou, G., Qiao, P.: Higher-Order finite strip method for postbuckling analysis of imperfect composite plates. Journal of Engineering Mechanics (2002)
Chapter 10
Nonlinear Magnetoelastic Waves in a Plate Vladimir I. Erofeev, Alexey O. Malkhanov, Aleksandr I. Zemlyanukhin and Vladimir M. Catson
Abstract We consider an elastic plate made of conductive material in an external magnetic field. It is shown that the distribution of intense waves in such a system can be described by nonlinear evolution equations, combining the well-known Khokhlov–Zabolotskaya–Kuznetsov and Kodomtsev–Petviashvili model equations. The features of the propagation of two-dimensional nonlinear magnetoelastic waves are analytically and numerically analyzed. Keywords Magnetoelasticity · Elastic waves · Magnetic field · Plate
10.1 Introduction As a scientific direction magnetoelasticity formed in the mid-50th of the XXth century at the intersection of solid mechanics, electrodynamics and acoustics. The first works were initiated by the problems of geophysics—the need to describe the wave dynamics of the deep layers of the Earth taking into account its electrical conductivity and interaction with the geomagnetic field [1]. Since that time, the dynamic V. I. Erofeev (B) Nizhny Novgorod Branch of Blagonravov Mechanical Engineering Research Institute of RAS, Belinsky St. 85, 603024, Nizhny Novgorod & Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia e-mail:
[email protected] A. O. Malkhanov Nizhny Novgorod Branch of Blagonravov Mechanical Engineering Research Institute RAS, Belinsky St. 85, 603024, Nizhny Novgorod, Russia e-mail:
[email protected] A. I. Zemlyanukhin · V. M. Catson Saratov State Technical University, Saratov, Politekhnicheskaja St. 77, 410054 Saratov, Russia e-mail:
[email protected],
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 10, © Springer-Verlag Berlin Heidelberg 2011
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processes in the interaction of electromagnetic and strain fields are being actively studied. This is associated with numerous physical, technical and technological applications. Among these problems is the problem of strength of structures and mechanisms operating in conditions of strong magnetic fields. The inclusion of fields of different physical nature in mechanical systems opens new possibilities for the development of engineering and technology. Magnetoelasticity effects are manifested or in strong magnetic fields created when the load can significantly affect the wave and dissipative properties of the medium [2], or in thin bodies: rods, plates, shells [3]. The infinite magnetoelastic medium is characterized by the anisotropy of the properties which are induced by the magnetic field. The magnetic field in a medium with finite conductivity leads to an additional mechanism of dissipation. These properties of magnetoelastic systems open new possibilities for practical applications. In this paper we study the propagation of nonlinear magnetoelastic waves in a plate. Particular attention is paid to the formation of two-dimensional spatially localized waves. The problem of the magnetic field effect on the localization of the strain wave considered earlier in [4] for the one-dimensional case.
10.2 Equations of Magnetoelasticity In magnetoelasticity, the influence of the magnetic field on the deformation field is described employing the Lorentz forces Fm = ρe E + j × B,
(10.1)
which are introduced into equations of motion of an elastic body ρ
∂2 u = (λ + μ) grad div u + μ Δu + Fnonlinear + Fm ∂t2
(10.2)
Here E is the intensity of the magnetic field; j is the vector of the electric current density, B is the magnetic induction vector; ρe is the volumetric density of the electric charges; u is the displacement vector; λ, μ are the Lam´e constants; ρ is the density of the material; t is the time. The force Fnonlinear includes elements which result from the consideration of elastic nonlinearity. If only the quadratic nonlinearity is taken into account, then the components of the vector can be represented through the gradients of displacements as follows % &' ( F i = μ + A4 ul,kk ul,i + ui,l ul,kk + 2ul,k ui,lk % &' ( (10.3) + λ + μ + A4 + B ul,k ul,ik + ui,l uk,lk + (λ + B) ul,l ui,kk % &' ( A + (B + 2C) ul,l uk,ki + λ + μ + 4 + B ul,k ul,ki + ui,l uk,lk
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Here A, B, C are the Landau constants, index after comma means differentiation with respect to the corresponding coordinate, repeating indices mean summation. From the Maxwell equations, one can obtain equations for the electric and magnetic inductions (D) and (B), respectively, ∂D ∂B ∂u c2 = rotH − j, = rot ×B + ΔB, (10.4) ∂t ∂t ∂t 4πσ which should be added by the electromagnetic equations of state j = σE,
D = εe E,
B = μe H
(10.5)
These equations should be solved together with Eqs (10.1) and (10.2). Here H is intensity of the magnetic field, σ is the conductivity, εe is the permittivity and μe is the magnetic conductivity, c is the speed of light in vacuum. In magnetoelasticity, both the biasing current (∂D/∂t = 0) and the density of free electric charges (ρe = 0) are neglected. Due to this, equations of magnetoelasticity can be written as follows ∂2 u 1 (rot H × H) , = (λ + μ) grad div u + μΔu + Fnonlinear + 4π ∂t2 ∂H ∂u c2 = rot ×H + ΔH. ∂t ∂t 4πσ
ρ
(10.6)
We assume that the external constant magnetic field with the intensity H0 is transverse to the direction of wave propagation. The total magnetic field consists of its permanent value and the perturbations that result from the interaction with the strain H = H0 n + h,
(10.7)
where h is a small disturbance of the magnetic field, n is the normal vector. Further, we consider the propagation of longitudinal waves in nonlinear elastic homogeneous plate in an external magnetic field (Fig. 10.1). In this case a small perturbation and the total magnetic field vectors are as follows: % & % & h = h x , h y , h z , H = h x , h y , H0 + h z . (10.8) As is well known [3], the idea of bringing three-dimensional equations of elasticity theory to two-dimensional equations of equilibrium or dynamics of the plates is to express the quantities describing the stress-strain state at any point of the body, through the new values, which vary along the medial surface of the plate. The transition from an infinite number of degrees of freedom in the normal direction x⊥ = (x2 , x3 ) to a finite number of degrees of freedom (finite number of modes) is performed by approximating the displacement with polynomials. It is usually in the powers of the transverse coordinates, and as a small parameter serves the
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Fig. 10.1 Plate in a magnetic field
relative thickness of the plate 2kz h, where h is the half-thickness of the plate, kz is the normal component of the wave vector. If ωh < cl , (cl – propagation velocity of longitudinal elastic waves in an infinite medium, cl = (λ + 2μ)/ρ) the following approximation of the displacement is valid [6] λz u1 = u (x, y, t) , u2 = v (x, y, t) , u3 = − divV, (10.9) λ + 2μ where u, v –⎛ denote the displacement of the middle of the plate; x = x1 ; y = x2 ; ⎞ ⎜⎜⎜ u ⎟⎟⎟ z = x3 ; V = ⎜⎜⎜⎝ ⎟⎟⎟⎠. v Magnetoelasticity equations for a plate undergoing a longitudinal vibration can be written as follows 4 2 ∂2 u 2 ∂2 u % 2 2 & ∂2 v λ2 h2 ∂ u ∂4 v 2∂ u − c p 2 − cl − cτ −c − + ∂x∂y τ ∂y2 3 (λ + 2μ)2 ∂x2 ∂t2 ∂x∂y∂t2 ∂t2 ∂x 2 2 2 4 4 4 4 λ cτ h ∂ u ∂ u ∂ v ∂ v 1 ∂hz + + 2 2+ 3 + + H0 2 ∂x4 3 4πρ ∂x ⎤ ∂x ∂y ∂x∂y 3 (λ + 2μ) ⎡ ∂x ∂y 2 ⎢⎢⎢ ∂ ∂u ∂v ∂u ∂v ⎥⎥⎥ ∂u ∂2 u ∂ ∂u ∂v ⎢ ⎥⎥ = α1 + α2 ⎢⎣2 + + + + ∂x ∂x2 ∂y ∂y ∂x ∂x ∂y ∂x ∂y ∂x ⎦ ∂hy ∂h x ∂v ∂2 v ∂v ∂2 v 1 ∂hz + α3 + α4 − h + h − ; z y ∂y ∂x∂y ∂x ∂x2 4πρ ∂x ∂x ∂y 4 (10.10) 2 ∂2 v 2 ∂2 v % 2 2 & ∂2 u λ2 h 2 ∂ v ∂4 u 2∂ v − c p 2 − cl − cτ −c − + ∂y∂x τ ∂x2 3 (λ + 2μ)2 ∂y2 ∂t2 ∂y∂x∂t2 ∂t2 ∂y λ2 c2τ h2 ∂4 v ∂4 v ∂4 u ∂4 u 1 ∂hz + + + + + H0 2 ∂y4 2 ∂x2 3 ∂x 3 4πρ ∂y ⎤ ∂y ∂y ∂y∂x 3 (λ + 2μ) ⎡ 2 2 ⎢ ∂v ∂ v ∂ ∂v ∂u ⎥⎥⎥⎥ ⎢⎢⎢ ∂ ∂v ∂u ∂v ∂u = α1 + α 2 + + + + ⎢ ⎥ 2⎣ ∂y ∂y2 ∂x ∂x ∂y ∂y ∂x ∂y ∂x ∂y ⎦ ∂hy ∂h x ∂u ∂2 u ∂u ∂2 u 1 ∂hz + α3 + α4 − hz − hx − ; 2 ∂x ∂y∂x ∂y ∂y 4πρ ∂y ∂x ∂y
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∂h x ∂ ∂u ∂v c2 ∂2 h x ∂2 h x = hy − h x + + 2 ; ∂t ∂y ∂t ∂t 4πσ ⎛∂x2 ∂y ⎞ ∂hy ∂ ∂u ∂v c2 ⎜⎜⎜ ∂2 hy ∂2 hy ⎟⎟⎟ ⎜⎝ =− hy − h x + + 2 ⎟⎠ ; 2 ∂t ∂x ∂t ∂t 4πσ ∂y ∂x ∂hz ∂ ∂u ∂ ∂v c2 ∂2 hz ∂2 hz (H0 + hz ) (H0 + hz ) =− − + + . ∂t ∂x ∂t ∂y ∂t 4πσ ∂x2 ∂y2 2 0 % &2 1 Here c p = cl 1 − c2l − 2c2τ /c4l is the velocity of the longitudinal wave extending in the plate in the absence of geometric dispersion and interaction with the magnetic field; cτ is the propagation velocity of elastic shear waves in an infinite medium; cτ = μ/ρ.
10.3 Beam of Longitudinal Waves Propagation Consider the propagation of the beam longitudinal waves along the x axis, restricting the area in which the parameters of nonlinearity, dispersion and diffraction of the same order (∼ ε). Suppose that the beam of waves is a limited, slightly divergent and close to a plane wave. Let us introduce the ”ray” coordinates: √ ξ = x − c˜ t; η = εx; χ = εy, (10.11) and new functions u = u; v =
√ √ √ εv; hz = hz ; h x = εh x ; hy = εhy .
(10.12)
The choice of variables in the form (10.11) and replacement (10.12) reflect the fact that, by virtue of nonlinearity, dispersion and diffraction divergence, the displacements and the components of the %magnetic field varies both along the beam propa√ & gation direction (∼ ε), and across ∼ ε . Substituting (10.11), (10.12) into (10.10) and holding down the only terms with order of ε not higher than the first, we get the evolutionary equation for axial (longitudinal) strain U = ∂u/∂ξ: ∂ ∂U ∂U ∂3 U ∂2 U ∂2 U + αU + β 3 − δ 2 + γ 2 = 0, ∂ξ ∂η ∂ξ ∂ξ ∂ξ ∂χ where α=
3−
% c &2 A
cl
% &2 % &2 c cτ 6λ2 − (λ+2μ) + cpl g 2 1 − cl % &2 % &2 − 1, c 2 2 cpl + ccAl
(10.13)
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λ2 h2 β= 6 (λ + 2μ)2
% c p &2 % c &2 % c &2 + cAl − cτl cl % c &2 % &2 , 2 cpl + ccAl
c2 8% & % & , 2 cp 2 8πσc2l + ccAl cl % &2
δ=
% &2 % &2 c 1 + ccA − ccτ 2 − cp l l l γ = % &2 % &2 % &2 % &2 % &2 , cp cp cA cA 2 2 cl + cl + cl − ccτl cl g=
2A + 6B + 2C , ρc2l
c˜ =
8 c2p + c2A ;
8 : here cA = H02 4πρ is the Alfven wave velocity. Equation (10.13) generalizes the well known model equations of Khokhlov– Zabolotskaya–Kuznetsov and Kadomtsev–Petviashvili, which follow from (10.13), when β = 0 and δ = 0 respectively [7]. To obtain an analytical solution “by simplest equations method” [8] we take as a simple Riccati equation Yz = −Y 2 + aY + b, (10.14) which has a solution in the following form √ 1 4b + a2 1 Y(z) = a + tanh 4b + a2 (z + C2 ) , 2 2 2
(10.15)
where a and b define the later of the over determined system of equations. Equation (10.13) is of the order of singularity, which is equal to two, so the solution of the equation will be sought in the form of: U (z) = A0 + A1 Y + A2 Y 2 + B1
Y z
Y
+ B2
Y 2 z
Y
,
(10.16)
here z = k0 ξ + k1 χ + k2η, Ai , Bi – arbitrary constants. In view of (10.14) we obtain: U (z) = (A2 + B2 ) Y 2 + (A1 − B1 − 2aB2) Y + A0 + aB1+ % & bB1 + 2abB2 B2 b2 +B2 a2 − 2b + + 2 Y Y
(10.17)
Substituting (10.17) into (10.13) and equating terms with the same degree of Yz , we obtain a system of equations, because of the cumbersome nature of which we show here only the first two equations: %
&4 1 αB22 + 6βB2 2 %
= 0,
& 14βB2 α + αB1 B2 + 2βB1 + 2aαB22 + 2δB2 b3 = 0 ...
(10.18)
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Solution of the system is as follows: A2 = −
12β 1δ , a = − , A0 = A1 = B1 = B2 = b = 0. α 5β
Then the exact solution of Eq. (10.13) takes the form of a shock wave: 2 3 δ2 k0 ξ + k1 χ + k2 η U (ξ, χ, η) = − −1 + tanh , 25 αβ 2
(10.19)
(10.20)
5βγ k2
6 δ where is k0 = 15 βδ , k2 = − 125 + δ 1 , k1 is an arbitrary constant. The requirement β2 of positive definiteness of the velocity of the wave (k2 > 0) imposes a restriction on the k1 coefficient: 6 δ4 k12 > . (10.21) 625 γβ3 3
During the verification of the solution, (10.20) served as an initial condition in numerical simulations. Calculations were based on the pseudospectral semi-implicit scheme [9] with parameters: 256 × 64 – the dimension of the grid; Δξ = 0.25 – step length along the ξ axis, Δχ = 0.25 – step length along the χ axis, Δη = 0.003 – step length along the η axis. The evolution of the perturbation is shown in Fig. 10.2. The shock front propagates without disturbances in the positive direction of ξ, which indicates the reliability of the resulting analytical solutions. Also in the numerical experiment was considered another form of initial conditions: 2 ξ − 32 U0 (ξ, χ) = 12sech sech (χ−8) (10.22) 4 The dome of the perturbation (Fig. 10.3) is moving forward on ξ, while spreading to the sides of χ, the latter effect dominates. With the passage of “time” η is closer to the borders of the perturbation amplitude grows to a certain limit (U0 = 7.1), constantly spreading outward and moving forward, which leads to cross-structures
Fig. 10.2 Transmission of the shock wave (α = −1, β = 1, γ = 1, δ = 1)
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Fig. 10.3 Initial disturbance for the numeric experiment
Fig. 10.4 Result of the evolution of initial disturbance with dissipation (η = 5)
Fig. 10.5 Result of the evolution of initial disturbance without dissipation (η = 5)
(Fig. 10.4 and 10.5). The height of the cross in the maximum point equals to two (U 0 = 2) (Fig. 10.4). Figure 10.5 shows the evolution of the initial disturbance in the equation of Kadomtsev–Petviashvili. From a comparison of cross-structures in Fig. 10.4 and 10.5 that the evolution of the initial momentum for both equations is qualitatively the same character – the spreading of the sides and the formation of cruciform
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Fig. 10.6 Dependence of the beam amplitude on the external magnetic field intensity
Fig. 10.7 Dependence of the beam widths along axes on the external magnetic field intensity
structures. The only difference is decreasing amplitude and increasing blurring of the perturbation due to dissipation (δ 0). Let us trace the change in amplitude and width of the wave beam along each coordinate, depending on the external magnetic field. Defining the arbitrary constants as follows: √ 7 δ2 1 δ3 k1 = , k2 = (10.23) 25 γβ3 125 β2 we obtain expressions for the amplitude and width of the beam along axes: 3 δ2 10β 250β2 50 γβ3 A=− , Δξ = , Δη = , Δχ = √ (10.24) 25 αβ δ δ3 7 δ2 For condensed matter in magnetic fields with strength up to 10 Tl Alfven wave velocity is less than the propagation velocity of longitudinal waves [5], so changing the parameters investigated in the 0 ≤ cA/cl < 1. Practical interest are the same values in the interval 0 ≤ cA/cl < 0.3. Figure 10.6 shows the variation of the amplitude of the wave beam. It is seen that with increasing external magnetic field amplitude decreases. At the same time, the width of the beam along each of axes increases with increasing magnetic field (Fig. 10.7).
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10.4 Conclusion As a result, analytical studies and numerical simulations demonstrated the possibility of formation of intense space-localized magnetoelastic waves in a plate. The dependencies of wave parameters (amplitude, velocity, width) on the external magnetic field, showing that using a magnetic field can control the characteristics of localized waves. Acknowledgements The work has been supported by the Russian Foundation for Basic Researchers, grant No 09–08–00827 and No 09–08–00188.
References 1. Knopoff, L.: The interaction between elastic waves motion and a magnetic field in an electric conductor. J. Geophys. 60, 441–456 (1955) 2. Bagdasarian, G.E., Danoyan, Z.N.: Electromagnetoelastic Waves. Yerevan State University, Yerevan (2006) (in Russian) 3. Ambartsumian, S.A., Bagdasarian, G.E., Belubekyan, M.V.: Magnetoelasticity of Thin Shells and Plates. Nauka, Moscow (1977) (in Russian) 4. Erofeyev, V.I., Malkhanov, A.O.: Localized magnetoelastic waves formation. Int. Rev. of Mech. Eng. 4, 581–585 (2010) 5. Erofeyev, V.I.: Wave Processes in Solids with Microstructure. World Scientific, Singapore (2003) 6. Erofeyev, V.I., Potapov, A.I., Soldatov, I.N.: Nonlinear Waves in Elastic Bodies with Spatial Dispersion. Gorky Univ. Publ., Gorky (1986) (in Russian) 7. Dodd, R.K., Eilbek, J.C., Gibbon J.D., Morris, H.C.: Solitons and Nonlinear Wave Equations. Academic Press, London (1984) 8. Kydryashov, N.A.: Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos, Solitons and Fractals. 24, 1217–1231 (2005) 9. Press, W.H., Teukolsky, S.L., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C. The Art of Scientific Somputing. Cambrige University Press, Cambridge (1992)
Chapter 11
Basic Concepts in the Stability Theory of Thin-Walled Structures Ardeshir Guran and Leonid P. Lebedev
Abstract Numerous specialized books and papers have been written about the subject of stability in mechanics. Most of these concentrate on methods for obtaining critical values of certain parameters and typically contain algorithms and graphs generated for describing important but very specific problems. In the present paper we take a step back and discuss the truly central notions regarding mechanical stability. Our intention is to treat the required concepts on a fairly elementary level, while simultaneously offering a bit of useful historical perspective. We also attempt to explain some common discrepancies between theoretical and practical results. Keywords Stability · Thin-wall structure · Shell · Plate
11.1 Introduction The terms “stable” and “stability” are somewhat overused. In everyday life we call various processes stable when they exhibit steady, periodic, or closely self-similar behaviors with respect to changes in certain parameters such as those of loading. The term “loss of stability” is associated with qualitative changes in processes or state behaviors. In this way, the notion of stability is similar to the mathematical notion of continuous dependence on time or some other parameter. Effects related to loss of stability are given special designations like “phase transition” in physics. Each natural science (biology, geology, chemistry) employs its own terms for loss A. Guran (B) Institute of Structronics, Julius Raab Straße 10, A-4040 Linz, Austria & Abteilung Robotik, Johannes Kepler University, Altenberger Straße 69 A-4040 Linz, Austria e-mail:
[email protected] L. P. Lebedev Universidad Nacional de Colombia, Cr. 45, # 2685, Bogot´a D.C., Colombia e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 11, © Springer-Verlag Berlin Heidelberg 2011
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of stability with regard to its particular objects of study. The same can be said for the various branches of engineering. Because valid mathematical descriptions take the form of precise statements involving equations or inequalities, however, we must lay down more precise explanations for the terms mentioned above. No universal definition, acceptable in all circumstances, is available. We shall consider only some aspects of stability theory in mechanics. In particular, we shall confine ourselves to the theory of thin-walled structures. These structures play important roles in engineering practice. Structural stability is often assumed in civil engineering design. On the other hand, instabilities are easily produced and can severely damage a structure. Buckling under load represents a typical example of loss of stability for thin-walled structures such as plates and shells. The buckling of a compressed column is a phenomenon with which every engineering student should be familiar. It is, of course, easily replicated using an ordinary plastic or metal ruler, and was first studied mathematically by Leonhard Euler (1701–1783) some 250 years ago. Although buckling can be catastrophic when it occurs on the body of a submarine, the same phenomenon provides the operating principle for the ubiquitous thermal switch. So the requirements of engineering practice have necessitated further development of stability theory by both engineers and mathematicians. Scientific notions seldom develop in a simple or straightforward manner. Stability has been studied for many years, and we still do not fully understand effects as simple as the buckling of a spherical cap under pressure. The major stability criteria and associated mathematical apparatus remain topics of active research interest in mechanics. Since mechanics depends heavily on the tools of calculus, its manner of approach to stability considerations was inherited largely from that subject. Only so much guidance could be borrowed directly from mathematics, however; without the incorporation of a clear engineering-level understanding based on practical considerations, it would have been impossible to construct sensible tools for predicting loss of stability. Meanwhile, mathematicians can only process the information that engineers have supplied. Furthermore, this processing can occur only with the benefit of tools that were constructed throughout the long history of mathematics — much of which preceded the advent of modern engineering practice. The stimulus provided to mathematics has been considerable though. Throughout the history of science we see a wonderful symbiosis between mathematics and real life. We shall restrict our discussion to the main ideas and methods associated with stability (and instability) in one of the oldest of the natural sciences: mechanics. In particular, we will concentrate on applications within the theory of elasticity (cf., also [2, 3, 20, 30]).
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11.2 Some Historical Remarks The use of mathematical tools in mechanics is so intensive that it is possible to regard mechanics as a branch of mathematics.1 But the modern era has seen at least some divergence between the modes of investigation used in the two subjects. In particular, the development of mechanics is not tied to a strictly axiomatic approach; rather, the changing requirements of actual engineering practice continually come into play as we noted above. Here we would like to discuss those stability-related notions that have evolved within mechanics. It was from these notions that certain portions of mathematics (e.g., stability theory for differential equations, branching of solutions to nonlinear equations in Hilbert spaces) were able to draw significant developmental inspiration. The first serious advancement in the understanding of stability for elastic structures was made by Euler [9] during his work on axially compressed rods called elastica. In the framework of statics, the stability problem can be treated only via nonlinear equations. Euler considered the case for which the degree of compression present at the rod ends is increased in proportion to a parameter. Until some “critical value” of the parameter, now known as Euler’s value, is reached, the rod must remain straight. Beyond this value the rod can assume two other symmetric equilibrium shapes. The closer the loading parameter is to Euler’s value, the closer the buckled shape stays to rectilinear. Thus, at the critical value itself we observe three possible solutions; furthermore, the dependence on the loading parameter is continuous (with respect to some norm). Such a point is called a point of branching. In experiments one commonly observes a buckled shape, hence Euler reasonably supposed that his critical point represents the least axial compression above which the straight shape is unstable. The total energy for the buckled state is less than that for the straight shape. Later work by Lagrange (1736–1813) [35] and Dirichlet (1805–1859) [8] paved the way for judging stability of the buckled state under a compression that exceeds Euler’s critical value. In view of the continuous dependence of both the stress and displacement fields on the load parameter near the critical point, it makes sense to invoke the tools of calculus. Suggesting itself here is an approach based on linearization of the problem about the solution corresponding to the straight rod shape. Euler was thus able to pose a linear boundary value problem having nontrivial solutions at certain critical values of the load parameter, the smallest of which was his original critical value. In this way Euler introduced an early eigenvalue problem in the theory of ordinary differential equations. His method was subsequently extended to other, more complex problems of elastic stability, but remained substantially the same. In engineering practice Euler’s critical points, which are found for certain equilibrium states when we know the main state of a structure, are associated with loss of stability. This is confirmed by experiment: as a rule, when we increase a load well above its critical value the structure in question will fail. In abstract equations, however, branching 1
We might mention that in Russian universities it is common for divisions of mechanics to thrive within mathematics departments.
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solutions need not deviate greatly from the main branch, so it is possible that a branching point could be neglected. We may be able to calculate Euler’s critical value then, but the situation reminds us of that for the critical values of an ordinary function in calculus: we set the first derivative to zero as a necessary condition in order to identify candidates for maxima and minima, but must do additional work in order to clarify the real nature of these points. It is clear that branching tools alone can tell us nothing about the stability or instability of a structural state. In engineering practice we are interested not only in the behavior of a structure in its ideal equilibrium state, but also under random external loads that vary (slowly or rapidly) with time. It is common for designers to rely on the fact that the state of a good stable structure will deviate only slightly from equilibrium when subject to light excitation. Unfortunately the picture is very different for states that lie close to instability. So it is not surprising that a study of unstable states should be based on dynamical considerations. Classical mechanics is divided two areas — statics and dynamics — and the resulting methodological subdivision was inherited by the theory of elasticity. Euler’s linearization method pertains to statics. We shall now consider the stability problem from the dynamical point of view. One of the oldest and most interesting stability problems is that for the motion of the Earth around the Sun, taking into account the influence of the Moon and other planets. A simple approximation, and one still not fully treated, is the so-called three-body problem. So we consider the stability question for a system of three point masses moving purely under their mutual gravitational forces. We wish to learn whether the Earth’s trajectory, which has been watched closely throughout history, will remain the same forever under this model. This means we must consider semiinfinite time intervals and other abstract notions that violate our understanding of the processes by which the Sun is slowly dying. Nonetheless, this approximate model could tell us something about the Earth’s future. Lagrange proposed a definition for stability of motion, relevant in cases where small changes in external conditions cause only finite changes in the motion of a system. His notion was appropriate for certain types of motion in mechanics. The final form of definition of dynamic stability was developed by A.M. Lyapunov (1857–1918) [42, 43]. It is the stability with respect to the initial data of the motion. This type of the definition is determined by existence of resonance frequencies in elastic structures. The resonance is wellknown in bridge design, since it was quickly found that a comparatively small group of soldiers marching across a bridge could cause a collapse (whereas the bridge could survive a much larger static load). The resonances of linear and nonlinear oscillating systems are well-known. Their existence predetermined that the main definitions for stability and instability in dynamics would differ significantly from those used in statics. Let us consider why the phenomenon of resonance brought other notions into dynamic stability. A simple but instructive mechanical system consists of a point mass m, attached to a spring having stiffness coefficient k, and that oscillates under a harmonic force F = f cos ωt. The equation of motion is m x(t) ¨ + kx(t) = f cos ω0 t.
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√
If f = 0, we observe simple harmonic oscillations at frequency ω0 = k/m. It seems reasonable to consider the stability of such oscillations, using the same idea as for statics: if the system is stable, then small perturbations in the external forces should bring small perturbations to the motion. Can we apply this successfully to our present system? When ω ω0 , a small f will give rise to small perturbations in amplitude. When ω comes close to ω0 , however, these perturbations will begin to grow, and when ω = ω0 the solution will contain a term proportional to t cosω0 t. This term will grow infinitely with time, regardless of the magnitude of f . The same thing occurs with any elastic structure in dynamics: there exist resonant frequencies at which an arbitrarily small external force can excite growing oscillations. This effect can be explained in terms of energy that accumulates in the system during each cycle of the oscillation. Of course, we rarely see oscillations driven toward infinite amplitude! In fact, when a certain amplitude is reached the linear model will no longer apply; furthermore, in any real system friction will extract energy from the system. For many elastic systems, frictional forces reduce forced resonances; this is why we could use dynamics tools to study stability. If we had a rigorous understanding of friction, we might be able to consider the problem of dynamic stability in the same way as for statics (i.e., through the change of state of a system under the perturbation of forces or other system parameters). Unfortunately this is not the case: no good universal model exists to describe the myriad real effects covered under the term “friction.” Models of a very restricted nature, when coupled with the equations of elasticity, yield complex boundary value problems that were considered intractable until very recently (with the advent of digital computation). So the ideas of dynamic stability were developed around overly-idealized elastic structures, leading ultimately to a dead end. Progress could continue only through consideration of particular types of stability of motion. For the three-body problem, for example, one can consider stability of motion under small perturbations in the initial conditions. This kind of stability is called Lyapunov stability, after A.M. Lyapunov (1857–1918) who developed its principal definitions and methods. Lyapunov’s notions spread quickly throughout dynamics and are now considered the keys to understanding stability in a great many circumstances. Lyapunov stability is applicable to elastic structures under various types of loading. In many cases static stability of some states of elastic structures is confirmed within the framework of Lyapunov’s ideas, and so much of Euler’s theory of stability was eventually confirmed for many elastic structures. This was first discovered by Movchan [45, 46] for some particular elastic systems under potential forces. However, the stability of systems under non-potential loads can be studied only on a dynamical basis.
11.3 Notions and Definitions for Stability in the Theory of Thin-Walled Structures So what steps must we take in order to study stability or instability of a given body, process, or state?
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1. We must construct a model adequate for predicting loss of stability. We must establish what we mean by stability, and present our selected notion in mathematical form. 2. We must establish necessary conditions for stability and instability of a process or state of a body, and develop methods for checking these. This includes tests for stability and instability of solutions to boundary value problems. 3. We must establish sufficient conditions for stability and instability. In addition, we must provide some means for practical selection of stable and unstable processes or states. Finally, we must furnish guidelines for theoretical and experimental studies that can generate reliable results in the area. The term “loss of stability” has acquired its meaning gradually, and through several mechanisms: (a) engineering intuition and qualitative observations made on the behavior of structures; (b) the influence of existing mathematical tools for researching the behavior of structures; (c) experimental procedures that ultimately confirmed the applicability of available methods. All three steps are equally important. Typically, however, textbooks confine themselves to Steps 2 and 3 (and only that portion of the latter which can be mathematically formalized). The remainder would require the kind of “fuzzy” explanation that is normally excluded from textbooks. Let us touch on a few aspects of this. Adequate models. One example of a stability problem in mechanics is the famous problem of turbulence for the flow of a liquid through a pipe. At some speed, the highly-organized laminar flow pattern is replaced by the seemingly chaotic pattern characteristic of turbulent flow. The latter is not as efficient for transportation of the liquid, and is therefore more expensive. At this time there is no clear confirmation that the Navier–Stokes model for a viscous liquid can adequately describe turbulence or how it arises, and so it seems that Step 1 is crucial for the theory of stability of liquid motion (possibly much more so than the techniques of Step 2). The situation is much better for an elastic ruler under axial compression. Precise experiments show that Euler’s elastica model is appropriate for studying stability of a compressed straight ruler, and his linearization technique seems appropriate for finding critical loads for elastic structures. However, we can be in doubt whether the elastica model for studying stability applies to all cases involving real solids. For example, we may consider a buckled spot on a tin can produced by fingertip pressure; since the spot does not automatically disappear upon removal of the pressure, it is not so clear whether we can exploit Euler’s theory to describe this situation within the framework of elasticity theory. (The reader should understand that a simple tin can is prestressed by its mode of production; some portions are in plastic states, so the question whether elastic theory would apply is not simple.) Next, restricting ourselves to elastic structures, we note that linear models of 3-D bodies, beams, plates, and shells cannot reveal anything about stability. Indeed, if as is common in stability theory for elastic structures, we increase the load in proportion to some parameter, then the solution to a linear problem will also be proportional to this parameter. There will be no indication of any loss of stability. So we must employ nonlinear models for static studies of instability effects.
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However, various nonlinear equations can be formulated for the same real object. They differ in both their degrees of accuracy and their possibilities for predicting certain effects. We must select the simplest one that can still describe the effects we wish to study. This stage of investigation is quite informal, since we need to evaluate the possibilities of a model without the benefit of calculations. Stable and unstable states and processes. Sometimes it is easy to distinguish between stable and unstable states visually, but not to quantify this difference in mathematical terms. Such is the case with turbulent flow. Here, the only way to determine stability is to use the dynamics of the process under disturbances in some adjustable parameters. However, the independent time variable adds another dimension to the problem and this can increase the difficulties associated with numerical solution. For this reason, practitioners use static tools whenever possible. Euler’s method of finding critical points. The buckling of a ruler under axial compression gives us a good example of when we can avoid considering process dynamics. We recall what happens in this example. Until some critical value of compression is reached, the ruler maintains its straight shape; any deflection produced by a small normal force disappears as soon as the force is removed. When the critical value is exceeded, the straight state becomes unstable; any transverse force will deflect the ruler into a curved state. This change occurs dynamically and the resulting curvature can be significant. But for compression just above Euler’s critical value, the change in shape happens relatively slowly and could be regarded as quasistatic. This is why we can apply static tools in this case. Here the critical point is a point of branching of solutions, and can be found through the application of functional analytic methods. Since the picture of the loss of stability for a ruler was extended for application to some shapes of plates and shells, we should devote special attention to the location of critical points in given parameter ranges for certain main states of an object. The method developed by Euler will yield only critical points on some main branch of a solution. The next question is whether these correspond to loads at which loss of stability can occur. So we consider how to find a critical point on a branch of solutions which depends on some set of parameters. A boundary value problem for an elastic object can be written as an equation G(x, ω) = 0
(11.1)
in an unknown variable x ∈ X, where X is a Banach space. The nonlinear operator G takes values in a Banach space Z. The parameter ω, characterizing the load on the object or some other significant variable, is an element of another Banach space Ω. Its values are usually considered to be proportional to a numerical parameter λ, and we shall proceed with this understanding. For stability, we check a branch of solutions defined by the equation x = x(λ). In these terms, equation (11.1) takes the form F(x, λ) = 0. To get the equation for the branching point we introduce a new unknown y by x = x(λ) + y and try to find the values λk of λ where there arise new branches of solutions y = y(λ) to the equation F(x(λ) + y(λ), λ) = 0. In this, we assume that the solutions depend continuously on λ, and so at a critical point
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y(λk ) = 0. Linearizing the equation F(x(λ) + y(λ), λ) = 0 with respect to y at the point λ, we get F(x(λ) + y, λ) = B(x(λ), λ)y + N(x(λ), y, λ) = 0,
(11.2)
where B is a linear operator in y, and N, the nonlinear part of F, is small with respect to y near the branching point λ; that is, N(x(λ), y, λ) = o ( y ). The operator B(x(λ), λ)y is the Gˆateaux derivative of F at the point (x(λ), λ). Consider the problem B(x(λ), λ)y = 0, (11.3) supposing that B(x(λ), λ) is a bounded linear operator with respect to y that depends on a parameter λ. This equation has a trivial solution for any λ. For an axially compressed ruler, this zero solution corresponds to the straight ruler shape. Let us find values of λ for which this equation has nontrivial solutions. For many (but not all) problems of mechanics, this is possible only for a discrete set of values for λ which are called critical values or eigenvalues. The corresponding nontrivial solutions are called eigenfunctions or eigensolutions. Usually we consider loads that increase in proportion to λ, and so there is a smallest critical value that we call Euler’s critical value. For a ruler, it can be shown that this is the branching point for solutions, and that for λ less than Euler’s value there are no nontrivial solutions y for the complete nonlinear problem. The linear branching equation cannot give us the amplitude of the branching part of the solution. So special techniques were developed to find after-critical solutions (we will not pause to consider this). We only note that for the problem of buckling of a plate having general shape and compressed in its midplane, the results mentioned above for the ruler have been generalized (Vorovich [55–57]): there is an Euler value for compression below which there are no buckled equilibrium shapes, and from this critical load we see branching of solutions. The existence of Euler’s critical load has also been established for momentless solutions for shallow shells. It turned out that for some shells, however, there exist other types of solutions under loads “less” than the critical value. When there is a stable nontrivial state for a shell that is not subjected to loads, the shell is called non-rigid (Vorovich [55–57]). Such solutions cannot be found through direct application of the methods of branching theory. Thus we know that for thin-walled structures there is a smallest critical point of the loading parameter λ from which solution branches emanate. We must still ask which of the branches correspond to real shapes of the structure, and hence which of the solutions are stable. It is clear that we cannot answer these questions by using only static tools, but need to bring in dynamic considerations. Energy method. In classical mechanics, results on stability have been established — in large part by Lagrange and Dirichlet. They demonstrated that for an equilibrium state of a system of mass points under conservative load, at which the total potential energy assumes its minimum value, the trajectories beginning from some small neighborhood of the equilibrium point with sufficiently small velocity remain in some small neighborhood of that point. Thus this equilibrium can be considered stable.
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Although elastic systems under potential loads belong to another class of systems, it was thought that the same idea could be applied. This happened because stability results were obtained for linear problems of elasticity that were quite close to those for classical mechanics. So the idea that minimum total energy corresponds to a stable equilibrium solution took on a life of its own. This permits us to avoid detailed considerations of the dynamics of various motions. When each of the solutions is stable by the energy criterion, we need to select which of them is more probable. The branch for which the total potential energy of the solutions is less is regarded as more probable. For a long time this viewpoint on stability was supported by direct experimentation with rods and some plates, and by attempts at calculating the dynamics of certain problems. The tools of the calculus of variations and functional analysis allowed us to justify this method, extending the Lagrange–Dirichlet theorem on stability of finite-dimensional systems to some problems of nonlinear elasticity, including those for plates and shells. The minimality of the total potential energy (for conservative loads) at an equilibrium point can signify stability of equilibrium, whereas its non-minimality can signify instability. There may be systems for which we do not see branching of solutions, but we can still use the energy method to find points at which loss of stability occurs. So the energy method is not a simple supplementation of Euler’s linearization method. It is worth repeating that the energy method of testing an equilibrium solution for stability is done in the static mode, but its roots lie in the dynamics of processes. Lyapunov’s definition of stability. The most sensible of the tests for stability of some system equilibrium or motion involves checking what happens with the system over time after some disturbance in the load or other variable parameter. We saw that even for a linear elastic system, because of resonance effects, we cannot consider stability with respect to disturbances in forces. This brings us to a special stability test developed by Lyapunov. Although it is not appropriate for all types of problems, the associated ideas are versatile and can be used in many circumstances. Lyapunov introduced a few main definitions to describe the stability of a solution to a system of ordinary differential equations (cf., Merkin [44]). These could be seen as defining continuity of this solution with respect to the initial data (which could be regarded as the system input) when the solution is considered as a whole on a semi-infinite time interval. Namely, he studied the stability of a solution y = y(t) ∈ IRn of a Cauchy problem y (t) = F(y(t),t),
y(t0 ) = y0
(11.4)
with a sufficiently smooth function F. The main definition states that the solution is stable if for any ε > 0 there is a δ = δ(ε) > 0 such that for any ξ that satisfies ξ < δ the solution yξ (t) to this equation with initial value yξ (t0 ) = y0 + ξ satisfies the inequality yξ (t) − y(t) < ε, where the norm is a standard norm in IRn . If this definition is not fulfilled — that is, if for some ε we cannot find a corresponding δ with the needed properties — then the solution is called unstable (in the sense of Lyapunov).
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Let us consider a solution to the equation as an element of the normed space C(IRn ; t0 , ∞) of functions that are continuous on [t0 , ∞) , and treat the problem as the result of the action of an operator from the set of initial conditions y0 ∈ IRn to the set of solutions y = y(t) ∈ IRn ; t0 , ∞). Now the definition of a stable solution in the sense of Lyapunov coincides with the way in which we define the continuity of an operator at a point in functional analysis. In addition, Lyapunov introduced the notion of asymptotic stability of a solution to the above equation. A solution y = y(t) is asymptotically stable if it is stable in the above meaning and yξ (t) − y(t) → 0 as t → ∞. In the above terms, this means that the difference yξ (t) − y(t) belongs to the subspace of C(IRn ; t0 , ∞) whose elements x(t) satisfy x(t) → 0 as t → ∞. Note that we could introduce a special requirement on the order of convergence to zero of yξ (t) − y(t) as t → ∞, and could proceed to consider various types of asymptotic stability such as exponential stability, etc. So we see that Lyapunov’s stability represents nothing more than a special notion of operator continuity. Our next issue is the practical determination of stability or instability of a solution. Here Lyapunov used two longstanding ideas from mechanics. The first one was to linearize the equation with respect to the difference x(t) = yξ (t) − y(t): that is, to consider the equation for the difference x (t) = F(y(t) + x(t),t)−F(y(t),t),
(11.5)
and linearize its right-hand side with respect to x(t). For the linearized equation Lyapunov found some conditions to insure that its solution remains small if the initial values are small. Together with the conditions under which the behavior of the solution to the linearized equation determines stability or instability of the solution of the initial nonlinear equation, these are the main parts of what is called Lyapunov’s first method for the study of stability. Lyapunov’s second method generalizes an idea contained in the proof of the Lagrange–Dirichlet stability criterion. The criterion was based on the energy conservation theorem for a system of mass points. Lyapunov proposed that we consider functions of a special type that can play the role of energy in this proof, and demonstrated how this may be done practically. Both methods turned out to be applicable to a variety of problems. An extension of Lyapunov’s notions to elastic structures presents some difficulty, because we must now deal with equations involving partial derivatives. All functions and vectors appearing in the setup for the stability problem take values in the space IRn , where, as the reader is aware, all norms are equivalent. With solutions to boundary value problems for spatial elastic structures, we encounter functions in several variables that describe the fields of displacements, strains, and stresses in the body. In this case, over a set of functions, we can introduce various norms which are not equivalent. This means we can consider different types of stability that are not equivalent when introducing different functional spaces in which we consider the continuity of solutions to corresponding problems. In the theory of shells, one of the most attractive (from the viewpoint of the justification of stability results) types of norm to use for this purpose is one defined by the quadratic portion of the elastic
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energy. Of course, other types of norms can be used for such purposes as well (see Atanacovich–Guran [3]). We have been discussing stability with respect to disturbances in the initial data. We can also consider a load disturbance, provided it has small amplitude and finite duration or finite amplitude and small duration. Such a disturbance can do work and thereby introduce additional small energy into a mechanical system. After the disturbance has ceased, its effects can be incorporated into and considered under the heading of a disturbance in the “initial data” (with respect to a new initial time). We can even consider load disturbances of infinite duration, provided that the total energy added by these is small. The three notions of stability discussed above should not be regarded as mutually conflicting. We should understand that Lyapunov’s dynamic definition of stability is the most universal. Because of the restrictions imposed by the static approach, the two other methods can show stability in the situation when in dynamics the solution is unstable. There are problems for which static approaches regularly give erroneous results, so we must clearly understand the range of applicability of each approach. Stability of Periodic Solution by Means of Poincar´e map [52]. Let us consider a periodic system described by a nonlinear system of ordinary differential equations y = g(y, t),
(11.6)
where y ∈ IRm , and g(y, t) is a periodic function of time t with period T > 0, that is g(y, t) = g(y, t + T ) for any y and t. We assume that g(y, t) is a smooth function of y and a continuous function of t, ensuring existence and uniqueness of a solution with the initial condition y0 = y(t0 ) on the semi-infinite time interval t > t0 . A solution y(t) of the system (11.6) is said to be periodic if y(t) = y(t + T ) (11.7) for any instant t. It is sufficient to check condition (11.7) at an instant t = t0 . By uniqueness of the solution y(t) and periodicity of g(y, t), if y(t) = y(t0 ) = y(t0 + T ) then (11.7) holds for any t. A periodic solution can be stable, asymptotically stable, or unstable. Let us consider a map f (y), which transfers a point y = y(0) in the state space to the point f (y) = y(T ), where y(t) is a solution of the system (11.6). The map f (y) is called a Poincar´e map. The Poincar´e map describes the dynamics of the system over the period T . Successive application of the Poincar´e map on a point y yields f k (y) = f ( f (. . . f (y) . . .)) = y(kT ), ;<=> k times
which is the value of the solution y(t) at t = kT . The point y0 is called stationary for the map f (y) if f (y0 ) = y0 .
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The relation of the Poincar´e map to the periodic system (11.6) is given by the following theorems. Theorem 1. A point y0 is stationary for the Poincar´e map f (y) if and only if the solution y(t) of the system (11.6) with the initial condition y(0) = y0 is periodic with period T . The Poincar´e map determines the discrete dynamical system y = f 0 (y) → f 1 (y) → . . . → f k (y) → . . .
(11.8)
where k represents discrete (integer) time. The stationary point y0 remains unchanged under the action of the Poincar´e map. Points y that are close to y0 can approach or move off the stationary point under multiple action of the map f (y). Such behavior determines stability properties of the stationary point y0 for the discrete dynamical system (11.8). Definition. The stationary point y0 of the map f (y) is said to be stable if for any ε > 0 there exists δ > 0 such that f k (y) − f k (y0 ) < ε whenever y − y0 < δ for any integer k > 0. If, in addition, f k (y) − f k (y0 ) → 0 as k → ∞ then the stationary point is said to be asymptotically stable. Although the Poincar´e map f (y) contains only partial information regarding the system (11.6) (using this map we cannot predict the state of the system at instant t kT ), it is sufficient to make a decision on the stability of a periodic solution to (11.6). Theorem 2. A periodic solution y(t) of the system (11.6) is stable (asymptotically stable) if and only if the stationary point y0 = y(0) of the Poincar´e map f (y) is stable (asymptotically stable). Let y˜ (t) be a periodic solution of (11.6). Introducing the vector x(t) = y(t) − y˜ (t) describing the deviation from the periodic solution and using (11.6), we obtain x = y − y˜ = g(˜y(t) + x(t), t) − g(˜y(t), t)). Assuming that x is small and neglecting the higher order terms, we find x = G(t)x where G(t) =
∂g
∂y y=˜y(t)
(11.9)
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is the m× m Jacobian matrix of the vector function g(y, t) evaluated at y = y˜ (t). As the functions g(y, t) and y˜ (t) are periodic with period T , the matrix G(t) is periodic with period T as well. The linear periodical system (11.9) represents the linearization of the system (11.6) near the periodic solution y˜ (t). Analogously, we perform the linearization of the Poincar´e map y → f (y) near the stationary point y˜ 0 = y˜ (0) as x → Fx, where x = y − y˜ 0 and F=
∂ f
∂y y=˜y0
is the m × m Jacobian matrix of the mapping f (x) evaluated at the stationary point x˜ .
11.4 Why Do We See Discrepancies between Theoretical and Experimental Results on the Stability of Thin-Walled Structures? Precise experiments with buckling of spherical caps under pressure, as well with other types of shells, demonstrate good agreement with theory. But in engineering practice we see that actual critical values can differ significantly from predicted values. Why does this happen? First, conditions in practice are quite far from those of precise experimentation. What does this mean? In short, it could be said that the discrepancy in values is produced by inappropriate modeling for studying these effects. We must remember that in calculations we deal not with real objects but with their mathematical models. These can offer good descriptions in some circumstances and poor descriptions in others. Our experience with the design and real-life performance of structures is evidence of the reliability of many mathematical models. But no model is perfect under all circumstances. As a rule, the strength of a structure is described by a linear model which is crude but sufficient for the purpose. The problem of stability of a solution is more fragile. First of all, the picture of buckling for each real structure has some peculiarities that may or may not be taken into account by a model. Many results on stability of thin-walled structures are obtained by Euler’s method of linearization described above. Critical points thus found are quite sensitive to many things such as imperfections in the shape of shells, their boundary conditions, and even to the technology of production of a shell. Imperfections result in actual values significantly below the predicted ones. So much research effort is devoted to the question of how we should account for imperfections in order to obtain more realistic numerical results. Ideal models turn out to be inappropriate for finding critical loads for stability of some objects, and researchers seek more appropriate ones. Models need some robustness with respect to the critical loads they determine; small disturbances in the shape of objects, in loads, or in supports should give small deviations in critical load values. We would like to discuss why this
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remains a challenging problem for engineers and mathematicians, and what should be taken into account by a good model of a thin-walled object in a study of its stability. The points we will consider play different roles in the overall effect. The order in which we consider them does not indicate their relative importance. Imperfections. Various nonlinear models are used to describe shells in practice. In each model there is an assumed distribution of strains along the normal to the shell’s midsurface. Additional assumptions result in a boundary value problem having a relatively simple (but still cumbersome) structure. The model of a shell is an approximation to a 3-D model of the real shell; here we should understand that, as with any approximation, it takes on only some main features of the more general model. Hence it may produce results that do not agree with those produced by the more accurate model. The reasons are as follows. First, the shape of the midsurface differs a bit from its intended shape. Small deviations, if taken into account, bring new terms into the equations. This can change the critical values calculated by Euler’s method, and certain values can simply disappear. Another problem is the thickness of the shell. It enters the coefficients of the higher-order derivative terms, and therefore can have a strong influence on the critical point values. However, the imperfections we are discussing affect solutions not only in the framework of a two-dimensional shell model; more serious are their effects on the distribution of strains along the normal to the midsurface. Irregularities in the shape of a shell face frequently introduce complicated 3 − D internal stress distributions that cannot be well-approximated by a 2-D approximation. Stress concentration can lead to plasticity effects near such irregularities. It is clear that these effects can contribute to significant discrepancies between theoretical and practical values for critical loads. Stochastic studies on the influence of imperfections are important for the analysis of buckling of thin-walled structures (see Guran and Ahmadi [22], Palassopoulos [48], and Guran [11]). Prestressing and inelasticity. We have mentioned that a material can demonstrate inelastic — in particular, plastic — properties because of defects in the shape of a shell. This can happen at sufficiently high loads. It can also result from certain production technologies, which leave behind internal stresses. Existing technological methods significantly reduce the level of strain introduced by cutting, pressing, or other stages of production, but cannot eliminate such strains completely. Unfortunately we can seldom determine these strain fields without destroying the object, hence it is difficult to account for them. Since the dependence of shell deformation on load increment is not linear, we encounter significant indeterminacy in trying to find critical loads while trying to account for prestressing of the shell (Guran [12, 49]). Boundary conditions. When solving boundary value problems for shells and plates, we use ideal boundary conditions such as Dirichlet’s conditions. However, real clamping can be non-rigid. From the viewpoint of the strained state of a structure, the difference between solutions to problems with ideal and non-ideal boundary conditions may be negligible if the pliability of the support is low. But as to the critical values of the load it may be significant. In experiments with preserving membranes in the Laboratory of Shells at Rostov State University, it was found that
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even traces of oil on the clamped area of a membrane affects the critical buckling pressure quite significantly, making results on critical pressure practically random. In the same experiments with thin-walled preserving caps, it was seen that normally the buckled spot on the membrane starts to develop near the edge. At the clamped edge there are some unusual conditions because of the technology of cap production: along the bent curve we see a deep crack, and it seems the material in this area is in a plastic state. It is clear that much of this happens to other types of clamped shells and plates, and so a proper type of boundary condition can play a crucial role in finding critical loads for real objects. Temperature effects. Normally thermoeffects are not taken into account when finding critical loads. However, their influence can be crucial in some circumstances. Let us consider the problem of asymmetric buckling of a spherical thin-walled cap. During testing and in real conditions, the increase in pressure is not infinitely slow; this means that the process of loading a cap is not completely isothermal — it is intermediate between isothermal and isentropic. Any change in strain is supplemented by a small change in temperature that depends on the change in strain. This means that inside a cap whose deformation state under pressure is not homogeneous, nor in different points, nor along the normals to the midsurface, there arise heat flows because of which the strain state changes slightly with time. Thus the process of loading the cap, even if it is quite slow and can be considered as quasistatic, depends on time. Such light changes in strain should be neglected when we consider the strain state. But in the theory of stability we consider more fragile characteristics: the critical point and the subsequent development of the deformation under the pressure increase. Since for a cap, near the first critical point where there arise branches of asymmetric solutions, the symmetric solution is also stable, it is possible to imagine (as far as we know nobody has tried to study this effect) that in some regime of the load increase, this non-robust critical point (which is not a point when we consider a time-dependent problem) can be “circumvented” so that the asymmetric branches do not appear. We suppose that such behavior of a thin-walled structure is quite probable in the neighborhood of critical points that do not possess the property of robustness. This direction in stability theory deserves further study.
11.5 Conclusion Stability theory is an interesting and important field of applied mathematics, having numerous applications in the natural sciences as well as in aerospace, naval, mechanical, civil, and electrical engineering. Stability theory was always important in astronomy and celestial mechanics, and during the last decade it has been applied to the study of processes in chemistry, biology, economics, and the social sciences. In this keynote article we presented the central notions of the stability theory of mechanical systems. We also attempted to explain some common discrepancies between theoretical and experimental results in the stability of thin-walled mechanical structures. Thousands of papers have been devoted to the topics mentioned above.
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The interested reader can consult [1–7, 10, 13–21, 23–34, 47, 50, 51, 53–55, 57] for more information. Theoretical analysis of these problems requires advanced tools of functional analysis, some of which can be found in [2, 30, 36–41, 58–60]. We hope this article will assist readers not only in finding stable solutions that can provide for safe operation of structures, machines, and devices made of thin-walled elements in engineering, but also in developing stable and predictable processes in the technical, natural, and social sciences. Acknowledgements The first author is indebted to his collaborators Professors Teodor Atanakovic (University of Novi Sad), Vladimir Beletskii (Moscow State University), Iliya Blekhman (IPM, Saint Petersburg), Horst Leipholz (University of Waterloo), Raymond Plaut (Virginia Polytechnic Institute), Karl Popp (Technical University of Hannover), Richard Rand (Cornell University), Fred Rimrott (University of Toronto), John Roorda (University of Waterloo), Alexander Seyranian (Moscow State Lomonsov University), Bez Tabarrok (University of Victoria), and Peter Tovstik (Saint Petersburg State University) for their contributions to his understanding of the Stability theory of mechanical systems. This work is prepared to honor Professor Felix Leonidovich Chernousko (Director, Institute for Problems in Mechanics of the Russian Academy of Sciences in Moscow) on his birthday; may he have excellent health, a long life, much happiness, and many more years of creative work in science.
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14. A. Guran, “Controlling the Chaos Using Fuzzy Estimation in a Gyrostat Satellite”, in Recent Progress in Controlling Chaos, M. A. F. Sanjuan and C. Grebogi (Eds), World Scientific, Singapore–New Jersey–London–Hong Kong, 2010. 15. A. Guran, “Buckling of a column as a benchmark nonlinear control design”, in The 3rd international IEEE scientific conference on physics and Control, J. Kurths, A. Fradkov, G. Chen (Eds), University of Potsdam, Germany, 2007. 16. A. Guran, “Ernst Mach and Peter Salcher: The development of nonlinear wave mechanics during the period 1850-1950”, in International Sysposium Peter Salcher and Ernst Mach: A Successful Teamwork, Rijeka, Croatia, 23-25 September, 2004. 17. A. Guran et al, “Oscillations, Bifurcations and Chaos in Ziegler’s Pendulum with Eccentric Load in a gravitational field”, in Proceedings of the 3rd International Conference on Complex Systems and Applications(ICCSA 2009), University of Le Havre Le Havre, Normandy, France, 2009. 18. A. Guran et al, On the Stability of a Spinning Viscoelastic Column, Mechanics Based Design of Structures and Machines 19(4), 437–455 (1991). 19. A. Guran et al, Studies in Spatial Motion of a Gyro on an Elastic Foundation, Mechanics Based Design of Structures and Machines An International Journal 21(2), 185–199 185–199 (1993). 20. A. Guran and T. M.Atanackovic, Lecture Notes on Theory of Elasticity (in Russian) (St. Petersburg University Press Petersburg, 2002). 21. A. Guran and T. M. Atanackovic, Fluid conveying pipe with shear and compressibility, Eur. J. Mech., A/Solids 17, 121–137 (1997)). 22. A. Guran and G. Ahmadi, “Postbuckling Behavior and Imperfection Sensivity of an Elastic Compressible Rod whose Flexural Resistance Changes with Load”, in Second International Conference on Application of Mathematics in Technical and Natural Sciences, Sozopol, Bulgaria, 2010 (to appear). 23. A. Guran, A. Bajaj, N. Perkins, G.DEluterio, and C. Pierre, Stability of Gyroscopic Systems (World Scientific, Singapore–New Jersey–London–Hong Kong, 1999). 24. A. Guran and M. Khoshnood, “Control of an Inverted Double Pendulum with Eccentric Fluid Load by Means of Computed Torque Method”, in Proceedings of EUROMECH Colloquium 515 Advanced Applications and Perspectives of Multibody System Dynamics, Blagoevgrad, 2010, pp. 36–37. 25. A. Guran, L. Sperling (Guest Editors), Wissenschaftliche Zeitschrift fur Grundlagen und Anwendungen der Technischen Mechanik, 24, Special issue In memoriam Friedrich P.J. Rimrott (2004). 26. A. Guran and I. G. Tadjbakhsh, “A mechanical actuator to suppress vibration”, in Proc. Smart Structures and Materials (SPIE), Smart Structures and Intelligent Systems, W. Hagood (Ed) 1917, 1993, pp. 1113–1114. 27. A. Guran and B. Yousefghahari, “Biomechanical study of spine”, in XIII Mediterranean Congress of Rheumatology; Clinical and Experimental Rheumatology 27, 726, 2009. 28. A. Guran and B. Yousefghahari, “A nonlinear Multibody System Dynamics Model to Predict the Impact Response of Human Chest during Cardiopulmonary Resuscitations”, in Proceedings of EUROMECH Colloquium 515 Advanced Applications and Perspectives of Multibody System Dynamics, Blagoevgrad, 2010, pp. 70–71. 29. A. Guran and B. Yousefghahari, “Shape-control of carbon nanotubes by means of nanofluidic actuators”, in Proc. Workshop on Biomedical applications of functionalized carbon nanotubes, Schloss Eckberg, Dresden, Germany, 2010. 30. R. J.Knops and E. W. Wilkens, Theory of Elastic Stability, Handbuch der Physik. Bd. VIa/3 (Springer-Verlag, Berlin, 1973). 31. W. T. Koiter, On the stability of elastic equilibrium, (Dissertation, Delft, 1945) in Trans. NASA, 10:833, 1967. 32. W. T. Koiter, “Elastic stability and postbuckling behavior”, in Proc. Symp. Non-Lrnear Probl., R.E. Langer, ed., Univ. of Wisconsin Press, 1963, pp. 257-275. 33. W. T. Koiter, “The non-linear buckling problem of complete sphercal shells under uniform external pressure”, Proc. Konik, Ned. Akad. Wetensch. Ser B, 72:40, 1969.
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34. W. T. Koiter, Current trends in the theory of buckling,Buckling Structures, 1-16 (1976). 35. J. L. Lagrange, M´ecanique Analutique” (Courier, Paris, 1788). 36. L. P. Lebedev and M. J. Cloud, The Calculus of Variations, Optimal Control, and Functional Analysis (World Scientific, Singapore–New Jersey–London–Hong Kong, 2003). 37. L. P. Lebedev and M. J. Cloud, Tensor Analysis (World Scientific, Singapore–New Jersey– London–Hong Kong, 2003). 38. L. P. Lebedev and M.,J. Cloud, Introduction to Mathematical Elasticity (World Scientific, Singapore–New Jersey–London–Hong Kong, 2009). 39. L. P. Lebedev, M. J. Cloud, and V. A. Eremeyev, Tensor Analysis with Applications in Mechanics (World Scientific, Singapore–New Jersey–London–Hong Kong, 2010). 40. L. P. Lebedev and I. I. Vorovich, Functional Analysis in Mechanics (Springer, New York, 2002). 41. L. P. Lebedev, I. I. Vorovich, and G. M. L. Gladwell, Functional Analysis: Applications in Mechanics and Inverse Problems, (Kluwer Academic Publishers, 1996 (2/e 2002)). 42. A. M. Lyapunov, On the stability of ellipsoidal figures of equilibrium of a rotating fluid (Master’s thesis (in Russian), Sankt-Petersbourgue, Akademy of Sciences, 109 pp, 1884). 43. A. M. Lyapunov, The General Problem of Motion Stability (in Russian) (Doctoral dissertation, Kharkov, Russia, 250 pp, 1892). 44. D. R. Merkin, Introduction to Stability of Motion (Springer-Verlag, New York, 1996). 45. A. A. Movchan, On the direct Liapunov method in the stability problems for elastic systems (in Russian), Prikl. Mat. Mekh. 23(3) (1959). 46. A. A. Movchan, On stability of motion of continuous bodies. Lagrange’s theorem and its inverse (in Russian), Inzhenernyj sbornik, 29, (1960). 47. J. F. Olesen and E. Byskov, Accurate detemination of asymptotic postbuckling stresses by the finite element method, Computer and Structures 15, 157-163 (1982). 48. G. V. Palassopoulos, “Effect of stochastic imperfections on the buckling strength of imperfectionsensitive structures”, in Fifth World Congress on Computational Mechanics, Vienna, Austria, 2002. 49. R. H. Plaut and A. Guran, Buckling of plates with stiffening elastically restrained edges, Journal of Engineering Mechanics–ASCE 120(2), 408–411 (1994). 50. J. Roorda, Stability of structures with small imperfections, Journal of Engineering Mechanics– ASCE, 87-106 (1965), . 51. J. Roorda and A. H. Chilver, Frame buckling: an illustration of the perturbation technique, Int.J.Journal Non-Linear Mechanics 5, 235-246 (1970). 52. A. P. Seyranian and A. A. Mailybeav, Multiparameter Stability Theory with Mechanical Applications (World Scientific, Singapore–New Jersey–London–Hong Kong, 2003). 53. B. Tabarrok and F. P. J. Rimrott, Variational Methods and Complementary Formulations in Dynamics (Kluwer Academic Publishers, Netherland, 1994). 54. P. E. Tovstik and A. L. Smirnov, Asymptotic Methods in the Buckling Theory of Elastic Shells (World Scientific, Singapore–New Jersey–London–Hong Kong, 2001). 55. I. I. Vorovich, Some question of shell stability “in the large”, Dokl.Akad.Nauk SSSR 122, 37–40 (1958). 56. I. I. Vorovich, The problem of non-uniqueness and stability in the non-linear mechanics of continuum mechanics, in Applied Mechanics, Proc. Thirteenth Intern. Congr. Theor. Appl. Mech., (Springer, Berlin, 340–357 (1973). 57. I. I. Vorovich, Nonlinear Theory of Shallow Shells, (Springer-Verlag, New York, 1999, translation from Russian edition by Nauka, Moscow, 1989). 58. I. I. Vorovich and L. P. Lebedev, Some issues of continuum mechanics and mathematical problems in the theory of thin-walled structures, International Applied Mechanics (Prikladnaya Mekhanika) 38(4), 387–398 (2002). 59. V. A. Yakubovich, G. A. Leonov,and A. Kh. Gelig, Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities (World Scientific, Singapore–New Jersey–London– Hong Kong, 2004). 60. V. Yurkevich and A. Guran, Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities (Book Reviews), IEEE Transaction on Automatic Control 50(4), 542–543 (2005).
Chapter 12
High-Frequency Free Vibrations of Plates in the Reissner’s Type Theory Elena A. Ivanova
Abstract The classic plate theory by Kirchhoff allows to accurately describe the processes slowly varying by time. To solve the problems of the plate vibrations in the case of the external loads quickly varying by time the Reissner’s type plate theory should be used. The Reissner’s type plate theory includes three eigenfrequency spectra: one low-frequency spectrum whose asymptotic order is O(h), and two high-frequency spectra whose asymptotic order is O(h−1 ). Solving the problems of plate vibrations under the action of the quickly varying by time loads it is necessary to take into account vibrations with eigenfrequencies from the highfrequency spectra. That is why the problem of plate free vibrations with eigenfrequencies whose asymptotic order is O(h−1 ) is interesting and practically important. In this paper asymptotic analysis of the equations of the Reissner’s type plate theory for high-frequency free vibrations is carried out and the approximate equations of plate vibrations with eigenfrequencies of the asymptotic order O(h−1 ) are proposed. Asymptotic analysis of the equations of plate free vibrations shows that behavior of the functions defining the stress-strain state of the plate for high-frequency free vibrations differs from it for low-frequency free vibrations. For high-frequency free vibrations the solution includes the functions which quickly vary along the space coordinates but which are not the boundary layer type functions. Because of that using of the exact equations of the Reissner’s type theory in numerical procedures is difficult. Approximate equations of high-frequency free vibrations of plates independent of quickly varying along the space coordinates functions are formulated in this paper. These equations describe vibrations with eigenfrequencies from the high-frequency spectra only, like the classic plate theory describes vibrations with eigenfrequencies from the low-frequency spectrum only. Keywords Reissner plate · High-frequency vibration E. A. Ivanova (B) Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, Bolshoy pr. V.O., 61, 199178, Saint Petersburg, Russia e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 12, © Springer-Verlag Berlin Heidelberg 2011
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12.1 Introduction It is known that the transverse shear deformations [1–4] and the inertia of rotation [5,6] must be taken into account in some problems of forced vibrations of plates, in particular, for the plate vibrations under impact and other rapidly time-varying loads, for the thick plates and for the composite laminates. [7–10]. Therefore, the solution of the problem of eigenfrequencies and modes of plate vibrations for the Reissner theory [11, 12] is of great importance. The problem of law-frequency free vibrations of the Reissner plate is studied in detail. It is well known that the solution of the problem includes the slowly varying functions of the space coordinates and the rapidly varying functions of boundary-layer type [13]. In contrast to lawfrequency free vibrations, the problem of high-frequency free vibrations of plate has been studied insufficiently. As far as we know, the problem was studied only in the paper [14] where the equations of free plate vibrations are obtained on the basis of three-dimensional elasticity by a variational-asymptotic method for various frequency spectra. In contrast to [14], we obtain the approximate equations of high-frequency free plate vibrations starting from the asymptotic analysis of the equations of the Reissner plate theory. We show that the functions that describe the stress-strain state of the plate for the law-frequency and high-frequency vibrations, have quite a different character of varying with respect to the space coordinates. For highfrequency free vibrations the solutions include rapidly varying functions as well as for low-frequency ones, but for the high-frequency case these functions are not the boundary-layer type functions. They deeply penetrate into the plate domain. In contrast to [14], our attempt revises the problem of high-frequency vibrations so as to make it more convenient for numerical implementation and to cover all possible types of boundary conditions. We also note that the given equations for high-frequency vibrations differ from the equations in [14].
12.2 Summary of the Basic Equations of Free Vibrations of Reissner’s Plate Let us consider the problem of vibrations of a plate with taking account of the inertia of rotation, and of the transverse shear deformation. The deflection w, the vector Ψ of rotation angles, the vector N of shear forces, and the moment tensor M are related to the displacements and stresses in three-dimensional elasticity as follows [15, 16]: hw = N=
h/2 −h/2
h/2
−h/2
u · n dz ,
a · τ · n dz ,
h Ψ= 3
M=
h/2 −h/2
h/2 −h/2
a · τ · az dz ,
u z dz (12.1) a = E − nn.
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Here u and τ are are the displacement vector and the stress tensor in the threedimensional theory, h is the plate thickness, n is the vector of the unit normal to the plate plane, E is the unit tensor. The theory of plate with taking account of the inertia of rotation, and of the transverse shear deformation includes the following equations. The equations of the motion are ∇ · N + ρhP = ρhw¨ ,
∇·M−N =
1 3¨ ρh Ψ, 12
(12.2)
where P(x, y, t) is the external load, ρ is the mass density. Constitutive equations take the form . / M = D (1 − μ)æ + μ tr æ a .
N = GhΓγ ,
(12.3)
Here γ is the transverse shear deformation vector, æ is the bending–twisting tensor, D = Eh3 /[12(1 −μ2)] is the bending stiffness, E is the Young modulus, μ is Poisson’s ratio, GhΓ is the shear stiffness, Γ is the coefficient of transverse shear, G = E/[2(1+ μ)] is the shear modulus. Geometric relations are γ = ∇w + Ψ,
æ=
& 1% ∇Ψ + ∇ΨT . 2
(12.4)
The kinematic boundary conditions acquire the form w|c = w∗ ,
ν · Ψ|c = Ψν∗ ,
τ · Ψ|c = Ψτ∗ .
(12.5)
The force boundary conditions can be written as follows ν · N|c = Nν∗ ,
ν · M · ν|c = Mν∗ ,
ν · M · τ|c = Mτ∗ .
(12.6)
Here ν and τ the unit outward normal vector and the unit tangent vector to the plate contour, respectively; the vectors ν, τ, and n, are assumed to form a right-handed system; Ψν∗ and Ψτ∗ are angles of rotation about the tangent vector and the normal vector to the plate contour, respectively; Nν∗ is the lateral force, Mν∗ is bending moment, Mτ∗ is the torque. Introducing potentials Φ and F we reduce the equations of the plate theory to the more convenient form [17]: 2 3 .... ρh3 2 ¨ ¨ + ρ h Φ + ρh P = 0, D ΔΔΦ + ρhΦ − 1+ ΔΦ (12.7) 12 Γ(1 − μ) 12GΓ 12Γ ρ F − F¨ = 0. (12.8) G h2 The quantities characterizing the stress-strain state of a plate are expressed in terms of the potentials Φ and F by the formulas ΔF −
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w = −Φ +
h2 ρh2 ¨ ΔΦ − Φ, 6Γ(1 − μ) 12GΓ
Ψ = ∇Φ + ∇F × n,
ρh3 ¨ ∇Φ + GhΓ∇F × n, 12 ( 1−μ ' M = D (1 − μ)∇∇Φ + μΔΦ a + ∇∇F × n − n × ∇∇F . 2 N = D ∇ΔΦ −
(12.9)
It is known [18] that in the Reissner’s type theory of plates there are three spectra of eigenfrequencies, which satisfy the following asymptotic estimates: 9 ω(2) i
=
(1) 2 (1) ω(1) i = hω1i + h ω2i + ... 9
12GΓ + ω(2) + ... 0i ρh2
ω(3) i =
12GΓ + ω(3) + ... 0i ρh2
(12.10)
where the first spectrum in Eqs (12.10) describes the low frequency bending vibrations, whereas the second and the third spectra in (12.10) characterize the high frequency shear and bending vibrations.
12.3 Asymptotic Analysis of the Equations of Reissner’s Plate Theory For the free high-frequency vibrations, the functions Φ and F substantially differ from those for the low-frequency vibrations or static bending. The function F varies slowly with respect to the spatial coordinates and is not of boundary-layer type. This is due to the fact that the leading terms in the first and in the second components in Eq. (12.8) cancel each other. For the approximate statement of the problem, Eq. (12.8) remains the same. A very important feature of the high-frequency vibrations is that the asymptotic orders of the functions F and Φ are the same: F ∼ Φ . The penetrating potential Φ for the high-frequency vibrations has quite a different structure than in the low-frequency case; namely, along with functions, slowly varying with respect to the spatial coordinates, it includes a rapidly varying function, which was lacking in the preceding cases. Let us denote this rapidly varying function by ϕ, and retain the notation Φ for the slowly varying component as well as for the penetrating potential itself. This seems convenient, since there is an asymptotic relation ϕ ∼ h2 Φ . The function ϕ seemingly need not be taken into account since it is relatively small. However, this is not the case, and the function ϕ may exert influence on the leading terms of some characteristics of the stress-strain state, which can depend not only on the penetrating potential, but also on its derivatives of order ≤ 3. Let us proceed from Eq. (12.7) to the approximate equations for the components Φ and ϕ. We suppose that for the functions Φ, and ϕ the following asymptotic estimates hold
12 High-Frequency Free Vibrations of Plates in the Reissner’s Type Theory
12GΓ F¨ = − + O(1) F, ρh2 ∂F ∂F ∼ ∼ F, ∂x ∂y
¨ = − 12GΓ + O(1) Φ , Φ ρh2
∂Φ ∂Φ ∼ ∼ Φ, ∂x ∂y
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12GΓ ϕ¨ = − + O(1) ϕ, ρh2
∂ϕ ∂ϕ 1 ∼ ∼ ϕ, ∂x ∂y h
ϕ ∼ h2 Φ ∼ h2 F.
12.4 Approximate Formulation of the Problem of High-Frequency Free Vibrations On substituting the expression for the penetrating potential Φ + ϕ into Eq. (12.7) and by retaining only the leading terms, we obtain Gh A(Φ) + D B(ϕ) = 0, % 2 & 12Γ ρ ¨ A(Φ) = Γ + ΔΦ − 2 Φ − Φ, 1−μ G h 12 % Γ(1 − μ) & B(ϕ) = Δ Δϕ + 2 1 + ϕ . 2 h
(12.11)
Note that the first equation in (12.11) contains an obvious contradiction. On the one hand, A(Φ) is a slowly varying function, since it depends on Φ, and B(ϕ) is a rapidly varying function, since it depends on ϕ. On the other hand, according to the first equation in (12.11), the functions A(Φ) and B(ϕ) are proportional. Thus, A(Φ) and B(ϕ) are slowly varying and rapidly varying simultaneously, that is, they are both zero. Hence, the first equation in (12.11) represents the two equations A(Φ) = 0,
B(ϕ) = 0.
(12.12)
The first equation in (12.12) has the form % 2 & 12Γ ρ ¨ Γ+ ΔΦ − 2 Φ − Φ = 0. 1−μ G h
(12.13)
Equation (12.13) permits us to find the leading term of the slowly varying part of the penetrating potential. It should be noted that Eq. (12.13) for Φ coincides with Eq. (12.8) for F to within a constant coefficient of the first summand, and the behavior of Φ is similar to that of F in the case of high-frequency vibrations. The second equation in (12.12) is Δz(ϕ) = 0,
z(ϕ) = Δϕ +
12 % Γ(1 − μ) & 1+ ϕ. 2 2 h
(12.14)
The solution of the equation Δz = 0 is a slowly varying function. Since z(ϕ) varies rapidly, all solution except for zero are excluded: z ≡ 0 . Then the equation for ϕ acquires the form
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12 % Γ(1 − μ) & 1 + ϕ = 0. (12.15) 2 h2 Equation (12.15) allows one to find the leading term of the rapidly varying part of the penetrating potential. Note that in contrast to the low-frequency vibrations, the rapidly varying function for the high-frequency vibrations is not of boundary-layer type, but penetrates into the entire plate domain. For the high-frequency vibrations, the characteristics of stress-strain state of the plate have the following asymptotic representations: Δϕ +
% & h2 2 ΔΦ − 1 + ϕ, 12 Γ(1 − μ) ' ( Ψ = ∇Φ + ∇F × n , N = GhΓ ∇Φ + ∇F × n , w=−
(12.16)
( 1−μ ' M = D (1 − μ)∇∇(Φ + ϕ) + μΔ(Φ + ϕ) a+ ∇∇F × n − n × ∇∇F . 2 Before proceeding to the statement of the boundary conditions, let us focus our attention on the following important property of ϕ. Consider a boundary condition depending on ϕ, for example, w|c = 0. Obviously, ϕ cannot vary rapidly on the boundary, since the other components in the boundary condition vary slowly. A similar conclusion holds for any boundary condition that depends on ϕ. In addition, ϕ is generally nonzero on the plate contour, since otherwise it would obviously be zero in the entire plate domain. The most natural conclusion follows: on the plate boundary, the function ϕ loses the property of being rapidly varying and becomes a slowly varying function along the contour. We point out that ϕ varies slowly only on the plate boundary; it rapidly varies in every direction in the interior of the domain arbitrarily close to the boundary. Thus, the tangent derivative of ϕ on the plate contour has the same asymptotic order as ϕ: ∂ϕ/∂τ|c ∼ ϕ. The kinematic boundary conditions acquire the form
2 % h 2 &
∂Φ ∂F
∂Φ ∂F
− ΔΦ − 1 + ϕ = 0, + = 0, −
= 0. 12 Γ(1 − μ) c ∂ν ∂τ c ∂τ ∂ν c (12.17) The force boundary conditions are written as follows: ∂Φ ∂F
GhΓ +
= 0, ∂ν ∂τ c
∂2 F % 1 ∂Φ ∂2 Φ & 12 % Γ(1 − μ) &
D(1 − μ) − + +D ΔΦ − 2 1 + ϕ = 0, (12.18)
c ∂ν∂τ R ∂ν ∂τ2 2 h
∂2 Φ % 1 ∂F ∂2 F & % ρh3 &
D(1 − μ) + + 2 − F¨ + GhΓ F
= 0.
c ∂ν∂τ R ∂ν ∂τ 12 The physical meaning of the boundary conditions (12.17) and (12.18) is the same as that of conditions (12.5) and (12.6), respectively. Equations (12.8), (12.13), and (12.15) supplemented with the boundary conditions (12.17) and (12.18) form an
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approximate statement of the problem of free high-frequency vibrations, which permits us to find the leading terms of the characteristics of the stress-strain state and the eigenfrequencies with a relativeerror O(h4 ). Recall that the main terms of the eigenfrequencies are equivalent to 12GΓ/(ρh2) .
12.5 Formulation of the Problem of High-Frequency Free Vibrations of the Reissner’s Plate without Taking Account of Rapidly Varying Function As was noted in the preceding, ϕ is a rapidly varying function and penetrates into the entire plate domain. Hence, the cited statement for the high-frequency vibrations is practically invalid in numerical implementations. Is it possible to revise this problem without taking account of ϕ? Before answering this question, let us note the following: the vector Ψ of rotation angles and the vector N of shearing forces, which do not depend on ϕ (see Eqs (12.16)), are two orders of magnitude larger than the deflection w and the moment tensor M respectively. Thus allows one to claim, that the stress-strain state is mainly characterized by the vector of rotation angles and the vector of shear forces, whereas the deflection and the moment tensor play a less important role. Therefore, the statement of this problem without the function ϕ is consistent in principle. Thus, let us eliminate Eq. (12.15) from the system of equations for the highfrequency vibrations. Then the order of the system with respect to the spatia] derivatives is reduced from six to four. Hence, the three boundary conditions in the original statement must be replaced by two boundary conditions independent of ϕ. One of the original three boundary conditions is the third condition in (12.17) or in (12.18). Since these conditions do not depend on ϕ, they are retained. Two other conditions are replaced by one condition according to the following rule: • the second condition in (12.17) and the first condition in (12.18) are equivalent, and if they are given simultaneously, there is no need to choose one of them (note that the function ϕ is identically zero for these boundary conditions); • if the first and the second conditions in (12.17) are given, the second condition must be retained, since it is independent of ϕ (whereas the first condition depends on ϕ); • if the first and the second conditions in (12.18) are given, the first condition must be retained, since it is independent of ϕ (whereas the second condition depends on ϕ); • since the first condition in (12.17) and the second condition in (12.13) depend on ϕ they can be replaced by the combination
∂2 F % 1 ∂Φ ∂2 Φ & % ρh3 & ¨ + GhΓ Φ D(1 − μ) − + 2 + Φ ∂ν∂τ R ∂ν ∂τ 12
= 0. c
(12.19)
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The physical meaning of the boundary conditions without the function ϕ is the following. The third condition in (12.17) means that the angle Ψτ of rotation about the normal to the plate contour is zero. The second condition in (12.17) (and the equivalent first condition in (12.18)) indicates that the angle Ψν of rotation about the tangent to the plate contour is zero. The third condition in (12.18) means that the torque Mτ is zero. Condition (12.19) indicates that the reduced bending moment Mν − GhΓw is zero.
12.6 Asymptotic and Numerical Analysis of High-Frequency Free Vibrations of Rectangular Plates Free vibrations of rectangular plates with frequencies belonging to high-frequency spectra are studied. The results predicted by the exact Reissner’s type theory are compared with those predicted by an approximate theory of high-frequency free vibrations which takes into account only functions slowly varying with respect to the spatial coordinates. It is well known that in solving some dynamical problems of plates, in particular, problems on forced vibrations under the action of impact loadings, one cannot ignore high-frequency vibrations which are associated with the inertia of rotation and the transverse shear deformation. As pointed out above, for high-frequency vibrations, the solution contains functions rapidly varying with respect to spatial coordinates and penetrating into the entire domain of the plate. The presence of such functions makes the exact equations of the Reissner theory practically unsuitable for the numerical analysis of the problems. Above an approximate statement of the problem on high-frequency free vibrations of a plate was suggested; only functions that vary slowly with respect to the spatial coordinates are taken into account. The asymptotic accuracy of this statement is O(h) compared with unity in determining eigenfrequencies and O(h4 ) in determining eigenfrequencies. This difference in the accuracy is accounted for by the fact that the leading terms of asymptotic expansions for all eigenfrequencies coincide and are known, whereas the approximate theory defines the first correcting term in the asymptotic expansions for the eigenfrequencies. Of course, the asymptotic accuracy of a theory is an important characteristic. However, to assess an asymptotic theory from the viewpoint of its practical significance, the actual accuracy of the theory is of importance rather than its asymptotic accuracy. (By the actual accuracy we mean the relative difference of the value of a quantity predicted by the approximate theory and the value of that quantity predicted by the exact theory for the given value of the small parameter.) In what follows we deal with the analysis of the actual accuracy of the approximate theory of highfrequency vibrations which was suggested above. The purpose on this research is to determine the area of applicability of the theory. The investigation is exemplified by problems having exact analytical solution, which allows us to rule out practically any errors of calculations. Now we consider rectangular plates two opposite sides of which are hinged. Let us consider a plate
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occupying a domain −a ≤ x ≤ a, −b ≤ y ≤ b. The constrained hinged support conditions are assumed to be satisfied at the sides y = ±b: w|c = 0,
Mν |c = 0,
Ψτ |c = 0.
(12.20)
The boundary conditions at the sides x = ±a can be arbitrary. We study vibrations symmetric with respect to the axes x = 0 and y = 0. The eigenforms satisfying the differential equations (12.7) and (12.8) and the boundary conditions (12.20) at y = ±b have the form Φn (x, y) = [C1n cos (λ1n x) + C2n cos (λ2n x)] cos (μn y), Fn (x, y) = C3n sin (δn x) sin (μn y), √ √ (12.21) μn = (2n − 1)π/(2b), λ1n = An − Bn , λ2n = An + Bn , δn = ρω2n /G − 12Γ/h2 − μ2n , An = [1 + Γ(1 − μ)/2]ρω2n/(2GΓ) − μ2n , Bn = ρh/D + ([1 − Γ(1 − μ)/2]ρω2n/(2GΓ))2. By satisfying the boundary conditions at x = ±a one reduces the problem to solving a system of homogeneous algebraic equations for the coefficients C1n ,C2n ,C3n . By equating the determinant of this system to zero, one obtains an equation for determining eigenfrequencies. We considered all types of boundary conditions possible in the Reissner’s type theory and obtained the frequency equations for each of them. Let us discuss the solution of the problem according to the approximate theory of high-frequency free vibrations. It can be readily shown that the eigenforms satisfying the differential equations (12.8) and (12.13) and the constrained hinged support conditions (12.20) at y = ±b have the form Φn (x, y) = C1n cos (λ1n x) cos (μn y), Fn (x, y) = C3n sin (δn x) sin (μn y), 2 2 ρ ω0n ρ 12GΓ λ1n = − μ2n , δn = ω0n − μ2n , ω0n = ω2n − . G Γ + 2/(1 − μ) G ρh2 (12.22) The asymptotic analysis shows that the frequency equations and the eigenforms predicted by tho asymptotic theory in question follow from the exact frequency equations and the exact eigenforms within an O(h) asymptotic error for all types of boundary conditions (by exact frequency equations and exact eigenforms we mean those obtained by the Reissner’s type theory). The numerical analysis was carried out for the problem discussed above. Computations were performed for plates of dimensions a = b = 1 m and thicknesses h = 0, 1 m and h = 0, 04 m with the elastic constants E = 2, 1 · 1011 Pa, μ = 0, 25, Γ = 5/6, and ρ = 7, 951 · 103 kg/m3 . The key results can be summarized as follows. The first 10 eigenfrequencies of high-frequency spectra are found. The calculations were performed by the exact
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theory and by the approximate theory for all types of boundary conditions possible in the Reissner’s type theory. The actual errors of the approximate theory are found. The calculations were performed for all types of boundary conditions and the plate thicknesses 0, 1 m and 0, 04 m. For the case in which the free edge conditions were imposed at the sides x = ±a, more detailed investigation was carried out. The first 10 eigenfrequencies for plates of thickness 0, 2 m, 0, 3 m, 0, 4 m, and 0, 5 m and the corresponding actual errors are calculated. The eigenforms corresponding to the first 10 eigenfrequencies are determined. The calculations were performed by the exact and approximate theories for all types of boundary conditions and the plate thickness 0, 1 m. In the case of the free edge conditions, the eigenforms are found for the plate of thickness 0, 04 m as well. For the eigenforms in calculation of which the approximate theory leads to the largest errors, the graphs of the rotation angles Ψx and Ψy versus the coordinate x are constructed. Let us point out some general features characteristic of high-frequency spectra and the approximation of these spectra by the approximate theory. The frequencies of the bending spectrum are higher than those of the shear spectrum. (Calculation show that for all types of boundary conditions at x = ±a, only two frequencies of the first ten belong to the bending spectrum.) The accuracy for the frequencies belonging to the bending spectrum is, as a rule, less than that for the frequencies belonging to the shear spectrum. This is quite natural, since in the approximate theory, the equation responsible for the shear vibrations is exact, whereas the equation responsible for the bending vibrations is approximate. While the approximate theories when applied to low-frequency vibrations yield higher values for the eigenfrequencies compared with the exact ones, this is not the case for high-frequency vibrations. For high-frequency vibrations, no monotonic increase in the relative error with the mode number is observed either. Of course, lower frequencies are, on the average, predicted more accurately than higher frequencies. However, for high frequency vibrations, the situation in which a frequency with a larger mode number is predicted more accurately than many frequencies with smaller mode numbers is usual. Let us briefly dwell on the results of calculation of eigenforms. We carried out the investigation of the accuracy provided by the approximate theory for eigenforms as follows. For all types of boundary conditions, analytical expressions for the potentials F(x, y) and Φ(x, y) were obtained according to the exact and approximate theories. Then the eigenforms represented by F(x, y) were compared with those represented by Φ(x, y). As a result, it was established that most of eigenforms predicted by the approximate theory virtually coincide with those predicted by the exact Reissner’s type theory. The errors turned out to be noticeable only in the case of the free edge conditions at x = ±a. Therefore, the subsequent discussion pertains just to this type of boundary condition (h = 0, 1 m). It has been established that the eigenforms predicted by the approximate theory and by the exact theory are in quite good agreement, which allows us to conclude that the approximate theory provides high accuracy in predicting eigenforms as well. This statement is valid for the overwhelming majority of eigenforms. Nevertheless, in exceptional cases, the difference in eigenforms predicted by the exact and approximate theories can be
12 High-Frequency Free Vibrations of Plates in the Reissner’s Type Theory
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large. It should be emphasized that such cases are encountered rather rarely: among eigenforms found in the course of the present investigation (10 eigenforms were calculated for each of the eight types of boundary conditions), the exact and the approximate theories were established to disagree only in one case.
12.7 Discussion of the Physical Meaning of Obtained Results From the physical viewpoint, it seems obvious that the high-frequency vibrations are produced by the shear phenomena. The presented asymptotic estimates confirm this assertion. Indeed, for the low-frequency vibrations the vector Ψ = −∇w + γ of rotation angles is actually determined by the deflection w, and the vector γ of transverse shear deformation represents the unimportant refinements: γ ∼ hΨ in the vicinity of the boundary and γ ∼ h2 Ψ inside the domain. For the high-frequency vibrations the situation is quite opposite: the vector Ψ of rotation angles practically coincides with the vector of transverse shear deformation γ, and the deflection w adds unimportant refinements:Ψ ∼ γ, ∇w ∼ hΨ. Since the nature of the high-frequency shear and high-frequency bending vibrations is the same (in particular, this is confirmed by the coincidence of leading eigenfrequency terms in the shear and bending spectra), the equations for both vibrations naturally seem to be similar. This is just the case in the suggested statement of problem for the high-frequency vibrations: Eq. (12.13) for the bending vibrations practically coincides with Eq. (12.8) for the shear vibrations. Thus, if we assume similarity of the bending and shear vibrations, then the order of the system with respect to the space coordinates is reduced. This means that yet another function is not included. Since we already have the equations for the shear and bending vibrations, the equation for this function is likely to be the static equation. In the presented statement, Eq. (12.15) for ϕ just does not contain the time derivatives. From the physical considerations, it is obvious that this function characterizes the bending phenomena and has no importance in the problem of highfrequency vibrations. This assertion is supported by the fact that this function rapidly varies with respect to the space coordinates, as we can see from Eq. (12.15), and penetrates into the entire plate domain. Would this function be of primary importance for high-frequency vibrations, we should adopt that the plate theory does not apply to this problem. Thus, the possibility of stating the problem without rapidly varying functions seems to be natural from the physical viewpoint. The boundary conditions in the proposed statement are also worth saying a few words. First, it must be noted that there are four possible boundary conditions, and two of them are kinematic: Ψν |c = 0 (the angle of rotation about the tangent) and Ψτ |c = 0 (the angle of rotation about the normal); the other two are the force conditions: Mν∗
c = 0 (reduced bending moment) and Mτ |c = 0 (torque). Note that the meaning of the kinematic and the force conditions slightly change in proceeding from the original statement to the approximate statement in the Reissner’s type theory. For the approximate statement, the conditions Ψν |c = 0 and
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Nν |c = 0 are equivalent, since N = GhΓΨ , and the force condition Nν |c = 0 becomes kinematic. The kinematic condition w|c = 0 disappears in the approximate state ment, and the deflection appears in the condition Mν∗
c = 0 , since Mν∗ = Mν −GhΓw , that is, it becomes part of the force boundary condition. A certain symmetry for the problem of high-frequency vibrations is not restricted by the similarity of Eqs (12.8) and (12.13). The similarity also occurs for the boundary conditions. In fact, the con dition Ψν |c = 0 yields Ψτ |c = 0 , and the condition Mν∗
c = 0 yields Mτ |c = 0 by the substitution Φ → F, F → −Φ. Such symmetry also supports the cited statement (rather aesthetically than physically).
References 1. Reissner, E.: On the theory of bending of elastic plates. J. Math. Phys. 23(1944)4, 184–191 2. Reissner, E.: The effects of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12(1945)2, 69–77 3. Reissner, E.: On transverse bending of plates, including the effects of transverse shear deformation. Int. J. Solids. Struct. 11(1975)5, 569–573 4. Reissner, E.: On the theory of transverse bending of elastic plates. Int. J. Solids. Struct. 12(1976)8, 545–554 5. Mindlin, R. D.: Influence of rotatory inertia and shear on flexural motion of isotropic, elastic plates. J. Appl. Mech. 18(1951)1, 31–38 6. Shen, R. W.: Reissner-Mindlin plate theory for elastodynamics. J. Appl. Math. 3, 179–189 (2004) 7. Reissner, E.: On bending of elastic plates. Quart. Appl. Math. 51, 55–68 (1947) 8. Reissner, E.: A note on bending of plates, including the effects of transverse shearing and normal strains. ZAMP. 32(1981)6, 764–767 9. Wenbin, Yu: Mathematical construction of a Reissner-Mindlin plate theory for composite laminates. Int. J. Solids Struct. 42(2005)26, 6680–6699 10. Batista, M.: Refined Mindlin-Reissner theory of forced vibrations of shear deformable plates. Engineering Structures. 33(2011)1, 265–272 11. Mindlin, R. D., Schacknow A., Deresiewicz H.: Flexural vibrations of rectangular plates. J. Appl. Mech. 23(1956)3, 430–436 12. Shen, H. S., Yang, J., Zhang, L.: Free and forced vibrations of Reissner–Mindlin plates with free edges resting on elastic foundation. J. Sound Vibr. 244(2001)2, 299–320 13. Kuznetsov, E. N.: Edge effect in the bending inextensible plates. J. Appl. Mech. 49(1982)3, 649–651 14. Berdichevskii, V. L.: A high-frequency long-wave vibrations of plates. Dokl. AN SSSR. 235(1977)6, 1319–1322 15. Altenbach, H., Zhilin, P. A.: The general theory of elastic simple shells. Advances in Mechanics. 11(1988)4, 107–148 16. Zhilin, P. A.: On the Poison and Kirchhoff theory of plates from the modern point of view. Izv. RAN. Mekhanika Tverdogo Tela (Mechanics of Solids). (1992)3, 18–64 17. Bolotin, V. V. (ed): Vibrations in Engineering (in Russian). Vol. 1, Mashinostroenie, Moscow (1978) 18. Zhilin, P. A., Il’icheva, T. P.: Vibration spectra and modes of rectangular parallelepiped, obtained on the basis of three-dimensional elasticity and plate theory. Izv. AN SSSR. Mekhanika Tverdogo Tela (Mechanics of Solids). (1980)2, 94–103
Chapter 13
On the Reconstruction of Inhomogeneous Initial Stresses in Plates Rostislav D. Nedin and Alexander O. Vatulyan
Abstract The direct problem formulation of the steady-state vibration of a thin elastic isotropic plate with non-homogeneous prestress field is described using the variational principle of Lagrange. The solving calculations of the model direct problems were made using finite element method. The investigation of the prestress level influence on the frequency response functions of the plate points was made. The operator relation of the inverse problem is constructed, which binds together the unknown prestress components and the given displacement field functions in the frequency set. The series of computation experiments on the identification of the smooth laws of the uniaxial pretension in the rectangular plate was conducted. On the base of these experiments, the conclusion about the efficiency of the proposed scheme of solving the important inverse problems class was drawn. The preferable frequency ranges are given for that the reconstruction of the initial stresses is most efficient. Keywords Thin plates · Inhomogeneous initial stresses · Inverse problem · Identification
13.1 Introduction The solid equilibrium problems in the presence of the initial stress state have been attracted attention of researchers for a long time. Stresses that exist in the solid without an application of any external (force or thermal) actions are termed initial stresses (also prestresses, residual stresses). Such stresses are often the results of welding, toughening, heat treatment and other manufacturing processes [1,3–5,10]. R. D. Nedin (B) · A. O. Vatulyan Southern Federal University, Faculty of Mathematics, Mechanics and Computer Sciences, ul. Milchakova 8a, 344090, Rostov-on-Don, Russia e-mail:
[email protected],
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 13, © Springer-Verlag Berlin Heidelberg 2011
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Mostly the homogeneous initial stress state model is used, at that it is enough to measure the elastic waves velocities for the estimation of such stresses [14]. At the same time the initial stress state is often essentially non-homogeneous and it depends on coordinates, especially about stress concentrators — cavities, insertions, cracks, weld seams neighborhoods. The opportunity of non-homogeneous prestress state identification can be used in extremely claiming nondestructive method in the fields of building, mechanical engineering, oil-and-gas sphere, biomechanics, modeling of manufacturing methods of the composite and functionally-gradient materials production. One of the most efficient nondestructive methods of non-homogeneous prestress state estimation is acoustical method. In the present paper the acoustical method is used for the theoretical research of the non-homogeneous prestress state determination opportunity. Even in the limits of the linearized model, the solving of the identification problem of the non-homogeneous prestress state can be realized only on basis of the solving the inverse elasticity theory problem with variable characteristics, which is non-linear and ill-posed problem. At that even for the influence estimation calculations, the elasticity theory problems with variable characteristics must be solved, and that is possible only with use of computing techniques, for example the finite element method (FEM) [16]. The principal difficulty of identification problems research is the complicated construction procedure of operator equation, which connects the unknown and given functions. And that conditioned by the variability of the differential operators coefficients and by the impossibility to construct the general presentations of corresponding elastic theory problems solutions in an explicit form [15]. In the present paper the direct problem formulation of the steady-state vibration of a thin elastic isotropic plate with non-homogeneous prestress field is described. In the case of homogeneous prestress state, such formulation was given in [12, 13]. On the basis of linearization method the inverse problem was reduced to the Fredholm integral equation of the first kind with continuous kernel, which was solved using Tikhonov’s regularization method [11, 17]. The solving calculations of the model problems were made using FEM in the package FreeFem++ [6, 22]; the series of computation experiment on the identification of the smooth laws of the uniaxial pretension in the rectangular plate was conducted; the preferable frequency ranges are given for that the reconstruction of the initial stresses is most efficient.
13.2 Formulation of Out-of-Plane Vibrations of the Thin Prestressed Plate Consider a thin elastic isotropic plate occupying the volume V with the boundary ∂V of arbitrary section outline and thickness 2h. Denote a plane section in the middle level by S . Regard ∂S = l = lu ∪ lσ ; the plate is fixed rigidly on the lateral surface part lu × [−h, h] and it is free from loads on lσ × [−h, h] (Fig. 13.1).
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Fig. 13.1 On the left: plate’s image in Cartesian coordinates. On the right: plate’s section of area S in the middle level (i.e. x3 = 0)
The stationary vibrations conditions of the plate are caused by periodic body q iωt forces with intensity 2h e , directed along the axis x3 . The linearized vibration equations are T i j , j +ρω2 ui + F i = 0, (13.1) where the components of the non-symmetric Piola tensor are [5, 10] T i j = σi j + ui,m σ0m j ,
(13.2)
the components of the symmetric tensor σi j submit to the Hooke’s law [9] σi j = λ∗ δi j uk,k + 2μ#i j ,
(13.3)
2λμ where λ∗ = λ+2μ — Lame parameter for two-dimensional stress state [11], μ — shear modulus (these coefficients are constants), σ0m j — the components of the symmetric prestress tensor, submitting the equilibrium equations:
σ0i j , j = 0.
(13.4)
Consider small deformations, regarding the Cauchy relations: & 1% ui , j +u j ,i . 2
(13.5)
T i j n j |∂V = Pi = 0.
(13.6)
#i j = Suppose no boundary loads:
In this problem let’s consider existing of 2-dimensional non-homogeneous prestess field in the plate, which is described by three components of prestress tensor σ011 , σ012 , σ022 being the functions of the coordinates x1 , x2 . We take Kirchhoff– Love hypotheses for thin plates, i.e. we neglect the stress component σ33 and all the components of the displacement vector depend on one function w = w(x1 , x2 ): u1 = −x3 w,1 ;
u2 = −x3 w,2 ;
u3 = w.
(13.7)
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Therefore the nonzero components of the stress tensor become . / σ11 = −x3 (λ∗ + 2μ)w,11 +λ∗ w,22 , σ12 = −2μx3 w,12 , . / σ22 = −x3 λ∗ w,11 +(λ∗ + 2μ)w,22 .
(13.8) (13.9) (13.10)
13.3 Derivation of the Motion Equation and the Boundary Conditions Let us apply the variational principle of Lagrange to the general formulation of the vibration problem of 3-dimensional solid, taking into account (13.2)–(13.10) [2, 7, 8, 19]. Consider the motion equations (13.1) T i j , j +ρω2 ui + F i = 0. Multiplying these equations by displacement variations δui and integrating by volume V, we have % & T i j , j δui + ρω2 ui δui + Fi δui dV = 0, V
or
% & (T i j δui ), j −T i j (δui ), j +ρω2 ui δui + F i δui dV = 0. V
Applying the Gauss–Ostrogradski formula, we obtain: T i j n j δui d(∂V) − T i j δui , j dV + ρω2 ui δui dV + Fi δui dV = 0. ∂V
V
V
V
By lack of external forces (13.6), the first integral vanishes. Therefore − T i j δui , j dV + ρω2 ui δui dV + F i δui dV = 0. V
V
(13.11)
V
Regarding V = S × [−h, h] and using (13.7)–(13.10), let’s integrate all terms in (13.11) by x3 . For that re-arrange elements of integration in every integral. The q body force vector has nonzero component F 3 = 2h . Hence V
1 F i δui dV = 2h
V
1 qδu3dV = 2h
h
1 · dx3
−h
qδwdS =
S
qδwdS . S
(13.12)
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169
The displacement variations are δu1 = −x3 δw,1 ;
δu2 = −x3 δw,2 ;
δu3 = δw.
(13.13)
The nonzero T i j expressions are T 11 = σ11 + σ011 u1 ,1 +σ012 u1 ,2 ,
(13.14)
= σ12 + σ012 u1 ,1 +σ022 u1 ,2 , = σ12 + σ011 u2 ,1 +σ012 u2 ,2 , = σ22 + σ012 u2 ,1 +σ022 u2 ,2 , = σ011 u3 ,1 +σ012 u3 ,2 , = σ012 u3 ,1 +σ022 u3 ,2 .
(13.15)
T 12 T 21 T 22 T 31 T 32
(13.16) (13.17) (13.18) (13.19)
Taking into consideration (13.13)-(13.19), let us integrate the equality (13.11) by coordinate x3 and use the Green’s formula. After simplification we have: ? 2 2 2 − h3 T˜ αβ δw,α nβ d(∂S ) + h3 T˜ αβ ,β δw,α dS + h3 ρω2 wα δw,α dS + 3 3 3 ∂S
S
+ 2h
ρω2 wδwdS − 2h
S
S
σ0αβ w,β δw,α dS +
S
qδwdS = 0, S
introducing σ˜ 11 = (λ∗ + 2μ)w,11 +λ∗ w,22 , σ˜ 12 = 2μw,12 , σ˜ 22 = λ∗ w,11 +(λ∗ + 2μ)w,22 , T˜ 11 = (λ∗ + 2μ)w,11 +λ∗ w,22 +σ011 w,11 +σ012 w,12 , T˜ 12 = 2μw,12 +σ0 w,11 +σ0 w,12 , 12
22
T˜ 21 = 2μw,12 +σ011 w,12 +σ012 w,22 , T˜ 22 = λ∗ w,11 +(λ∗ + 2μ)w,22 +σ012 w,12 +σ022 w,22 . Note, that all the function marked with tilde don’t depend on x3 . Using the constructions which are typical for the plates theory [7, 12], we finally obtain 2 − DΔ2 w + h3 (σ0mβ w,αm ),αβ −2hσ0αβ w,αβ + 3 S
2 + h3 ρω2 Δw − 2hρω2 w − q δwdS − 3
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Fig. 13.2 Directions of the tangent normal τ and the outer normal n at an arbitrary point of the closed contour ∂S [7]
? − ∂S
∂ Gt (δw)d(∂S ) + ∂n
? ∂S
∂ 2 ∂w (Ht ) − Nt + h3 ρω2 δwd(∂S ) = 0, ∂τ 3 ∂n
(13.20)
2 2Eh3 where h3 (λ∗ + 2μ) = = D is the cylindrical rigidity of plate; 3 3(1 − ν2 ) 2 G t = h3 T˜ αβ nα nβ — bending moment on ∂S ; 3 2 Ht = h3 (T˜ 2β nβ n1 − T˜ 1β nβ n2 ) — twisting moment on ∂S ; 3 ∂(Δw) 2 3 0 Nt = −D − h (σmβ w,αm ),β nα + 2hσ0αβ nα w,β — shear force on ∂S ; ∂n 3 T˜ αβ = λ∗ δαβ w,kk +2μw,αβ +w,αm σ0mβ , ∂ ∂n
and
∂ ∂τ
α, β, k, m = 1, 2;
— normal and tagentional derivatives to ∂S relatively (Fig. 13.2).
Equating the coefficients by independent variations to zero, we get the boundary problem: 2 2 DΔ2 w + h3 (σ0mβ w,αm ),αβ −2hσ0αβ w,αβ + h3 ρω2 Δw − 2hρω2 w − q = 0, 3 3
∂w
= 0, G t |lσ = 0, w|lu = 0, (13.21) ∂n lu
∂ 2 ∂w
(Ht ) − Nt + h3 ρω2
= 0. ∂τ 3 ∂n l σ
Note, that in case of homogeneous prestress field (σ0αβ = const) and neglect of rotatory inertia 23 h3 ρω2 Δw, we obtain the equation given in [12, 13]. In case of band plate the formulation (13.21) and the formulation which given in [20] are congruent.
13.4 Solution of the Direct Problem The computations of the direct problems solutions were obtained using FEM in the package FreeFem++. The series of computations experiments on the solving of direct problem (13.21) was made for various section shapes (rectangles, ellipses,
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Fig. 13.3 Image of the prestressed cantilever with rectangular section under uniformly distributed periodic load of intensity q. At that l is the cantilever’s length, b and h are the section dimensions. This cantilever may be considered as a band plate of hight h and section with dimensions l and b which is clamped at one side edge
rings, T-shaped regions and others), boundary conditions and types of prestress state (uniaxial and 2-dimensional). All the components of the prestress tensor σ0αβ , α, β = 1, 2 were taken as the functions of coordinates; at that the continuous functions were used (linear, quadratic, exponential, trigonometric) and the piecewise functions with several break points were used too. As a result of problem solving the displacement field w(x1 , x2 , ω) was calculated for the given vibration frequency ω. The accuracy analysis was made for the FreeFem+ solution of the problem (13.21). For that the obtained solution in case of band plate (the section shape is the elongate rectangular) with homogeneous characteristics (the material parameters, the density, the components of the prestress tensor) was compared with the analytical solution of the analogous problem of bending vibrations of the prestressed cantilever with rectangular section under uniformly distributed load (Fig. 13.3) [20]. At that, the case of uniaxial prestress state was considered, i.e. the only nonzero component of the prestress tensor is the component σ011 = const. The prestressed cantilever problem is described by the following boundary problem for the differential equation of the fourth order [20] (J(E + σ011 )w ) − (Fσ011 w ) − ρFω2 w − q = 0, w(0) = 0, w (0) = 0, w (l) = 0, (J(E + σ011 )w (l)) − (Fσ011 w (l)) = 0,
(13.22) (13.23) (13.24)
where l — rod length, J = 2bh3 /3 — moment of inertia, b — section width, 2h — section hight, F = 2bh — section area, q — uniformly distributed load value. Note, that we use slightly modified formulation of the prestressed cantilever problem — instead of girder characteristics Young modulus E we take the value E/(1 − ν2 ), which is included in the cylindrical rigidity formula (ν is the Poisson coefficient).
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Remark. You can derive the formulation of the prestressed cantilever vibration problem (13.22)-(13.24) with homogeneous material parameters (E and ν) from the formulation of the direct vibration problem of thin prestressed plate (13.21), deduced in Sect. 13.3 . For that the special case must be considered: w = w(x1 ), σ011 (x2 ) 0, the rest components of the prestress tensor equal zero. The solution of the boundary problem (13.22)–(13.24) in dimensionless form is w(ξ) = C1 cosh βξ + C2 sinh βξ + C3 cos θξ + C4 sin θξ + where ξ = x/l ∈ [0, 1] and @ A B8 κ14 + 4(1 + τ)κ22 + κ12 β= , 2(1 + τ) σ011
@ A B8 θ=
q0 , κ24
κ14 + 4(1 + τ)κ22 − κ12 2(1 + τ)
(13.25)
,
Fl2 ρFl4 ω2 ql2 τ, κ22 = τ, q0 = , E J EJ EJ and Ci — constants revealed from the boundary conditions (13.23)–(13.24) (since the last one are not manageable, these constants are not presented). The various ratios of rod dimensions were considered for carrying out the accuracy analysis. The result for the rod with following parameters is shown below: l = 0.2 m, b = 0.013 m, h = 0.01 m, E = 1.96 · 1011 Pa, ν = 0.29, ρ = 7.8 · 103 kg/m3 , q = 300 kg/m, σ011 = 98 MPa. The frequency response function for the tag of the rod with foregoing parameters is illustrated in Fig. 13.4. ω The whole frequency range f = 2π ([ f ] = Hz) consists of few region divided by resonance frequencies f1 = 15.1 Hz, f2 = 87.3 Hz, f3 = 241.1 Hz, f4 = 471 Hz. Below the range 0 < f < f1 is termed as the first frequency range, the range f1 < f < f2 — as the second frequency range, the range f2 < f < f3 — as the third frequency range, the range f3 < f < f4 — as the fourth frequency range. τ=
,
κ12 =
Fig. 13.4 Frequency response function of the cantilever’s tag. Frequencies f1 = 15.1, f2 = 87.3, f3 = 241.1 are the bend resonances
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The comparing results of the analytical solution wr (x1 ) for the rod with the numerical solution wp (x1 , b/2) for the plate are illustrated in Fig. 13.5. δ is given as the numerical solution error in percent δ=
|wp (x1 , b/2) − wr(x1 )| 100% max |wr (x1 )|
The comparing analysis of the solutions of two problems revealed that the relative error in the nodes of the numerical solution wp (x1 , x2 ) in comparison with the analytical solution wr (x1 ) for the frequencies from the first and the second frequency ranges is less than 0.07% for the finite elements mesh 20 × 100. This implies very close approximation scheme and allows to use FreeFem++ for solving other problems in future. The isolines graphics of the FEM-solution wp (x1 , x2 ) for the plate with parameters a = 2 m, b = 1 m, h = 0.1 m, E = 1.96 · 1011 Pa, ν = 0.29, ρ = 7.8 · 103 kg/m3, q = 300 kg/m, for different prestress states are shown in Figs 13.6–13.8 below (the axis x1 is directed vertically downward, the axis x2 is directed horizontally to the right). The results for the plate with a clamped boundary (w = ∂w ∂n = 0 at the ∂S ) in case of statics ( f = 0 Hz) are shown in Fig. 13.6. The results for the pre-stretched plate ∂w (σ011 = const) which is clamped at one side edge (w = ∂x = 0 at the boundary x1 = 1 0), for various frequency values, are illustrated in Fig. 13.7. The isolines graphics for the plate with same boundary conditions as in Fig. 13.7 in the case of nonhomogeneous uniaxial prestresses σ011 (x2 ) for the frequency value f = 0.7 Hz are given in Fig. 13.8.
13.5 Analysis of the Prestress Level Influence on the Frequency Response Functions of the Plate Points The investigation of the prestress σ011 = const influence on the frequency response functions of the rectangular plate point {x1 = a; x2 = b/2; x3 = 0} was made for the first and the second frequency ranges, for the various ratios of plate dimensions a : b : h (Fig. 13.9). The graphics of the prestresses influences on the frequency response functions for the ratio of plate dimensions a : b : h = 20 : 10 : 1 are shown in the Fig. 13.10. The investigation of the prestress level influence on the frequency response functions of the rectangular plate points was made for the various ratios of plate dimensions a : b : h. It is significant that I almost does not depend on ratio a : b; on the other hand it depends essentially on ratios h : b and h : a. The plate is thicker (i.e. the coefficient h/a or h/b is more) the I is less (i.e. the prestress level σ011 influences weaker on the frequency response functions) At the same time, with increase of these coefficients all the resonance frequencies increase too; the frequencies change
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Fig. 13.5 The comparing results of the analytical solution wr (x1 ) (shown as firm line) for the rod with the numerical solution wp (x1 , x2 ) (shown as small squares) for the plate. Four different vibration modes are considered: a — from the first frequency range, for f = 7 Hz; b — from the second frequency range, for f = 40 Hz; c — from the third frequency range, for f = 120 Hz; d — from the fourth frequency range, for f = 300 Hz (see also Fig. 13.4)
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is often used in the identification problem in the case of the homogeneous stress fields. Thus, the experimental results demonstrated that the discrepancy of the frequency response functions quite enough for using prestress reconstruction procedure [21].
13.6 Formulation of the Inverse Problem of the Non-homogeneous Prestress Reconstruction in the Solid In [17] the reciprocity relation was built, which allows to construct the iterative process in the prestresses identification problem. Set a problem of function σ0i j (x1 , x2 ) identification using some additional information. The most popular ways of its definition are the following two: 1) the assignment of the displacement field w inside the region for a fixed frequency; 2) the assignment of the displacement field w at
Fig. 13.6 From left to right: σ0αβ = 0; σ011 = const; σ022 = const; σ012 = const; σ0αβ = const
Fig. 13.7 From left to right: f = 0.5 Hz; f = 5 Hz; f = 16 Hz; f = 30 Hz; f = 80 Hz
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π Fig. 13.8 From left to right: σ011 = 0 Pa; σ011 (x2 ) = 109 sin( 100 x2 ) Pa; σ011 (x2 ) = 106 (e0.05x2 − 0.5) 0 6 0.035x 0 6 0.045x 2 Pa; σ (x ) = −10 (e 2 − 100) Pa, x ∈ [0, 100] Pa; σ11 (x2 ) = −5 · 10 sin(0.04x2 )e 2 11 2
the boundary part lσ in the frequency set ωk ∈ [ω− , ω+ ]. In the limits of the first definition way the inverse problem is linear, in the limits of the second definition it’s nonlinear [18]. Consider the second way of the additional information assignment. Let’s apply the fundamental reciprocity relation to the out-of-plane vibration problem of a plate. Using the hypotheses and assumptions for plates, we have * 2 2 δσ011 h3 (w,211 +w,212 ) + 2hw,21 + 2δσ012 h3 w,12 Δw + 2hw,1 w,2 3 3 S
+δσ022
2 3 2 h (w,12 +w,222 ) + 2hw,22 3
+
dS =
q(w − f )dS ,
(13.26)
∂S
where δσ0αβ — corrections to the prestress functions σ0αβ . Consider the case of uniaxial prestretch along the axis x1 — the only one nonzero component of the prestress tensor is σ011 . For satisfaction of the equilibrium equations we regard that the function σ011 depends only on the coordinate x2 . In that case the equation (13.26) takes the form
Fig. 13.9 Image of the prestretched rectangular plate in Cartesian coordinates. At that σ011 is the only nonzero prestress tensor component. The plate is clamped
at one ∂w
edge: w| x1 =0 = ∂x1 =0 x1 =0
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Fig. 13.10 The analysis of the prestress level influence on the frequency response functions of the plate point x1 = a; x2 = b/2; x3 = 0. At these pictures the following plate parameters are taken: a = 2 m, b = 1 m, h = 0.1 m, E = 1.96 · 1011 Pa, ν = 0.29, ρ = 7.8 · 103 kg/m3 , q = 300 kg/m. On the left figure the first frequency range (0, f1 ) is considered, on the right figure — the second one ( f1 , f2 ) ( f1 = 1.6 Hz — the first resonance frequency, f2 = 8.8 Hz — the second resonance frequency). The I value characterizes the prestress influence value in comparison with no-prestress state (σ011 = 0) in
percentage terms. This value is defined by formula I =
w(a,b/2)| 0 −w(a,b/2)|σ0 =0
σ 0
11
11 100%.
w(a,b/2)|σ0 =0
11
The influ-
ence graphics are shown using different line styles corresponding to the values of the dimensionless parameter τ = σ011 /E. Several values of τ lying in the range 10−5 ÷ 10−3 were considered
δσ011 S
2 3 2 h (w,11 +w,212 ) + 2hw,21 dS = 3
q(w − f )dS .
(13.27)
∂S
The presented equation is the Fredholm integral equation of the first kind relatively the correction δσ011 to the unknown uniaxial prestress function. Remark. In the paper [21] the nonlinear inverse problem of in-plane vibration of the plate with non-homogeneous uniaxial prestress field was formulated and solved. At that the various ways of loading at the plate’s lateral surface were considered. The Fredholm integral equation of the first kind was constructed analogously to (13.27). The scheme of the non-homogeneous uniaxial prestress function identification was proposed, which was based on the iterative process construction; on the base of computing experiments for the different laws of variation, the conclusion about the efficiency of the inverse problem solving scheme was drawn. The preferable loads and frequency ranges were given for that the reconstruction of the prestress function was the most efficient.
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13.7 Formulation of the Inverse Problem of the Uniaxial Non-Homogeneous Prestress Reconstruction in the Rectangular Plate Now consider the inverse problem for the rectangular plate with uniaxial nonhomogeneous prestress field existing inside it (Fig. 13.9). The problem takes the following form 2 2 DΔ2 w + h3 (σ0mβ w,αm ),αβ −2hσ0αβ w,αβ + h3 ρω2 Δw − 2hρω2 w − q = 0, 3 3 ∂w w|lu = 0, |l = 0, Gt |lσ = 0, ∂n u
∂ 2 3 2 ∂w
(Ht ) − Nt + h ρω
= 0, ∂τ 3 ∂n l σ
where lu = {x1 = 0; x2 ∈ [0, b]}, lσ = {x1 = a; x2 ∈ [0, b]} ∪ {x1 ∈ [0, a]; x2 = 0} ∪ {x1 ∈ [0, a]; x2 = b}. As given parameters we use the plate’s dimensions, the material constants (the q cylindrical rigidity D and the density ρ), the body forces intensity 2h . Let’s set the 0 identification problem of the prestress function σ11 (x2 ). In the event that the additional information is the assignment of the displacement field w at the whole area S in the frequency set ωk ∈ [ω− , ω+ ], the inverse problem becomes linear and reduces to the solving of the Fredholm integral equation of the first kind (on the basis of (13.27)) b K(x2 , ω)σ011 (x2 )dx2 = F(ω), ω ∈ [ω− , ω+ ], (13.28) 0
where
a K(x2 , ω) =
2 3 2 h (w,11 +w,212 ) + 2hw,21 dx1 , 3
(13.29)
0
b a F(ω) =
q(w − f )dx1dx2 , 0
(13.30)
0
Thereby the operator relation is constructed, which binds together the desired quantity (i.e. the prestress component) and the given displacement field functions in the frequency set. Note, that solving of the presented Fredholm integral equation of the first kind (13.28) is ill-posed problem. For its solving the Tikhonov’s regularization method with the automatic regularization parameter searching was used [11].
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13.8 Numerical Experiments on the Prestress Identification The series of computing experiments on the identification of the uniaxial nonhomogeneous prestress σ011 (x2 ) was conducted. The displacement field w at the area S was given. The various function classes were considered for the reconstruction — linear, polynomial, exponential, trigonometrical and more complicated analytical dependencies [21]. The vibration frequencies f of the applied load were chosen among the first, second and third frequency ranges of the frequency response functions. In all the experiments the prestress functions σ011 (x2 ) were chosen in such way that the fracσ0
tion max E11 (E is the Young modulus) varied in the range 10−5 ÷ 10−3 . In the majority of computations the essential deterioration of the unknown function σ011 (x2 ) identification had a place at the ends of segment x2 ∈ [0, b]. In some cases such error at the ends exceed 30%, whereas in the distance of the ends it varied up to 10%. Such accuracy deterioration of the inverse problem solution relates with the rough calculation of the second order partial derivatives of w function, which appear in the kernel formula (13.29). It is significant that the kernel does not tend to zero near the ends of the segment x2 ∈ [0, b]. In the event that the values σ011 (0) and σ011 (b) are given, the identification quality near these ends is much more high. As it turned out, for the majority of the experiments the consideration of the second frequency range gives a much more accurate identification result, than the consideration of the first or third frequency ranges. The results of uniaxial prestress function identification for the plate with parameters a = 2m, b = 1m, h = 0.1m, E = 1.96 · 1011Pa, ν = 0.29, ρ = 7.8 · 103kg/m3 are shown below. The resonance frequencies are f1 = 1.6Hz, f1 = 8.8Hz, f1 = 23.1Hz, f1 = 43Hz. The identification results in the second frequency range are illustrated in Fig. 13.11–13.13. At that the values σ011 (0) and σ011 (b) are accepted as given. The exact function σ011 (x2 ) is shown at the graphics as a firm line; the identification result — as small squares. The error δ of the inverse problem solution is given in percentage terms by the formula
σ011 (x2 ) − σ0∗
11 (x2 ) δ= % & 100%, 0∗ max |σ11 (x2 )| x2 ∈[0,b]
where σ011 (x2 ) — the identified law of variation, σ0∗ (x ) — the exact law of varia11 2 tion of prestress.
180
Fig. 13.11 Identification results. Exact law σ011 (x2 ) = 2.5 · 106 (e0.035x2 + 4) Pa, x2 ∈ [0, 100]; f− = 1Hz, f+ = 5Hz
Fig. 13.12 Identification results. Exact law σ011 (x2 ) = −106 (e0.045x2 − 100) Pa, x2 ∈ [0, 100]; f− = 3 Hz, f+ = 8 Hz
Fig. 13.13 Identification results. Exact law σ011 (x2 ) = −5 · 106 sin(0.04x2 )e0.035x2 Pa, x2 ∈ [0, 100]; f− = 2.9 Hz, f+ = 5 Hz
R.D. Nedin and A.O. Vatulyan
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13.9 Conclusion The direct problem formulation of the steady-state vibration of a thin elastic isotropic plate with non-homogeneous prestress field is described using the variational principle of Lagrange. The solving calculations of the model direct problems were made using FEM in the package FreeFem++. The investigation of the prestress level influence on the frequency response functions of the plate points was made. It revealed that the frequency ranges near the resonances are the most efficient in the view of the identification. Within the limits of Newton method ideology of solving nonlinear inverse problems, the first kind integral Fregholm’s equation with continuous kernel with respect to correction of the prestress functions is formulated. The series of computation experiments on the identification of the smooth laws (linear, polynomial, exponential, trigonometrical and more complicated analytical dependences) of the uniaxial pretension in the rectangular plate was conducted. On the base of computing experiments, the conclusion about the efficiency of the proposed scheme of solving the important inverse problems class was drawn. The preferable frequency ranges are given for that the reconstruction of the initial stresses is most efficient. Acknowledgements The present work is done with the support of Russian Foundation of Basic Research “Development of new methods for identification of localized and dispersed heterogeneousness in solid bodies” (project code 10-01-00194-a) and Federal Target Program “Scientific and research-and-educational staff of innovation Russia” on 2009 - 2013 years (state contract P596).
References 1. Birger, I.A.: Residual Stresses (in Russian). State scientific-and-technical publishing of machine-building literature, Moscow (1963) 2. Bittencourt, M.L., Pereiray C.E.L.: Procedures for Teaching Variational Formulation and Finite Element Approximation of Mechanical Problems applied to the Kirchhoff Plate Model. Department of Mechanical Design, State University of Campinas, Campinas, Brazil (2004) 3. Deaconu, V.: Finite Element Modelling of Residual Stress – A Powerful Tool in the Aid of Structural Integrity Assessment of Welded Structures. 5th Int. Conference Structural Integrity of Welded Structures (ISCS2007), Timisora, Romania (2007) 4. Feng, Zh.: Processes and Mechanisms of Welding Residual Stress and Distortion. Woodhead Publishing Limited, Cambridge, England (2005) 5. Guz, A. N.: Elastic Waves in Compressible Materials with Initial Stresses and Nondestructive Ultrasonic Method of Two-Ply Residual Stresses Definition (in Russian). Int. Appl. Mechanics, v. 30, 1, 3–17 (1994) 6. Hecht, F., Pironneau, O., Le Hyaric A., Ohtsuka K.: FreeFem++. Version 2.17-1.. (Available via DIALOG, 1999), http://www.freefem.org/ff++ 7. Leybenzon, L. S.: Course of Elasticity Theory (in Russian). State Publishing of Technicaland-Theoretical Literature, Leningrad (1947) 8. Mikhlin, S.G.: Variational Methods in Mathematical Physics (in Russian). Nauka, Moscow (1970) 9. Novatsky, V.: Theory of Elasticity (in Russian). Mir, Moscow (1975)
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10. Robertson, R.L.: Determining Residual Stress from Boundary. Measurements: A Linearized Approach. Journal of Elasticity, vol 52. Kluwer Academic Publishers, Netherlands, 63–73 (1998) 11. Tikhonov, A.N., Arsenin, V.Ya.: Methods of Ill-Posed Problems Solving (in Russian). Nauka, Moscow (1979) 12. Timoshenko, S.P.: Course of Elasticity Theory (in Russian). Naukova Dumka, Kiev (1972) 13. Timoshenko, S., Woinowsky-Krieger, S.: Theory of Plates and Shells (in Russian). Nauka, Moscow (1966) 14. Tovstik, P.E.: Vibration and Stability of Prestressed Plate Lying on Elastic Foundation. Appl. mechanics and mathematics, v.1, 106–120 (2009) 15. Vatulyan, A.O.: Identification Problems of Inhomogeneous Properties of Solids. Bulletin of Samara State University, issue 54, 4, 93–103 (2007) 16. Vatulyan, A.O.: Inverse Problems in Mechanics of Deformable Solids (in Russian). Phizmatlit, Moscow (2007) 17. Vatulyan, A.O.: Iterative Processes in Inverse Coefficient Problems. Abstracts of XIV international conference ”Present-day problems of continuum mechanics”. South Federal University Publishing, Rostov-on-Don, Russia, 81–85 (2010) 18. Vatulyan, A.O.: Regarding Inverse Coefficient Problems in Linear Mechanics of Deformable Solids. Appl. mechanics and mathematics, 6, 911–918 (2010) 19. Vatulyan, A.O.: Regarding Variational Formulation of Inverse Coefficient Problems for Elastic Bodies. Proceedings of RAS, v. 422, 2, 182–184 (2008) 20. Vatulyan, A.O., Dudarev, V.V.: Concerning Some Problems of Non-Homogeneous Prestress State in Elastic Bodies. The publishing of Saratov University. New series. S. Mathematics. Mechanics. Computer Science, v. 9, issue 4, part 2, 25–32 (2009) 21. Vatulyan, A.O., Dudarev, V.V., Nedin, R.D., Saakyan, Ya.G.: Regarding Some Problems of Prestress Identification. Abstracts of XIV international conference ”Present-day problems of continuum mechanics”. South Federal University Publishing, Rostov-on-Don, 86–90 (2010) 22. Zhukov, M.Yu., Shiryaeva, E.V.: Using Finite Element Package FreeFem++ for Problems of Hydrodynamics, Electrophoresis and Biology. South Federal University Publishing, Rostovon-Don (2008)
Chapter 14
Dynamic Response of Pre-Stressed Spatially Curved Thin-Walled Beams of Open Profile Impacted by a Falling Elastic Hemispherical-Nosed Rod Yury A. Rossikhin and Marina V. Shitikova
Abstract The dynamic stability with respect to small perturbations, as well as the local damage of geometrically nonlinear elastic spatially curved open section beams with axial precompression have been analyzed. Transient waves, which are the surfaces of strong discontinuity and wherein the stress and strain fields experience discontinuities, are used as small perturbations, in so doing the discontinuities are considered to be of small values. Such waves are initiated during low-velocity impacts upon thin-walled beams. The theory of discontinuities and the method of ray expansions, which allow one to find the desired fields behind the fronts of the transient waves in terms of discontinuities in time-derivatives of the values to be found, are used as the methods of solution for short-time dynamic processes. The example of using the ray expansions for analyzing the impact response of spatially curved thin-walled beams of open profile is demonstrated by solving the problem about the normal impact of an elastic hemispherical-nosed rod upon an elastic arch representing itself a channel-beam curved along an arc of the circumference. The influence of the initial stresses on the dynamic fields has been investigated. Keywords Spatially curved thin-walled beam of open profile · Surface of strong discontinuity · Ray method · Impact · Shock interaction
14.1 Introduction Thin-walled beams of open section are extensively used as structural components in different structures in civil, mechanical and aeronautical engineering fields. These structures have to resist dynamic loads such as wind, traffic and earthquake Y. A. Rossikhin (B) · M. V. Shitikova Research Center of Wave Dynamics, Voronezh State University of Architecture and Civil Engineering, Voronezh 394006, Russia e-mail:
[email protected],
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 14, © Springer-Verlag Berlin Heidelberg 2011
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loadings, so that the understanding of the dynamic behavior of the structures becomes increasingly important. The classical theory of thin-walled uniform open cross-section straight beams as well as horizontally curved ones was developed by Vlasov [30] in the early 60-s without due account for rotational inertia and transverse shear deformations. The Vlasov theory is the generalization of the Bernoulli-Navier law to the thin-walled open section beams by including the sectorial warping of the section into account by the law of sectorial areas, providing that the first derivative of the torsion angle with respect to the longitudinal axis serves as a measure of the warping of the section. However, as it has been recently shown [27] Vlasov’s equations are inappropriate for use in the problems dealing with the transient wave propagation. Many researchers have tried to modify the Vlasov theory for dynamic analysis of elastic isotropic thin-walled beams with uniform cross-section by including into consideration the rotary inertia and transverse shear deformations ([2], [4], [9], [15], [19], [20], [22]), and/or considering coupled bending–torsional vibration of axially loaded thin-walled beams ([3], [8], [17], [31]). The increasing use of curved thin-walled beams in highway bridges and aircraft has resulted in considerable effort being directed toward developing accurate methods for analyzing the dynamic behaviour of such structures ([5], [10], [13], [14], [16], [21]). Curved members in modern bridges and architectural structures continue to predominate because of emphasis on aesthetics and transportation alignment restrictions in metropolitan areas. It is well known that Timoshenko [29] in order to generalize the Bernoulli-Euler beam model has proposed the transverse shear angle to be the independent variable. This starting point was the basis for the derivation of a set of two hyperbolic differential equations describing the dynamic behaviour of a beam, resulting in the fact that two transient waves propagate in the Timoshenkobeam with finite velocities: the longitudinal wave with the velocity equal to G L = E/$, and the wave of trans verse shear with the velocity equal to GT = Kμ/$, where E and μ are the elastic moduli, $ is the density, and K is the shear coefficient which is weakly dependent on the geometry of the beam. Many of the up-to-date technical articles involve the derivation of the equations which, from the authors viewpoint, should describe the dynamic behaviour of thinwalled beams of the Timoshenko type. The critical overview of the cited above papers has been carried out recently in [27]. It has been emphasized that only inclusion into consideration of three factors at a time, namely: (1) transverse shear deformations, (2) rotary inertia, and (3) warping deformations as the independent field – could lead to the correct system of hyperbolic equations of the Timoshenko type for describing the dynamic behaviour of thin bodies. Ignoring one of the factors or its incomplete consideration immediately results in an incorrect set of governing equations. It has been shown that all papers in the field can be divided into three groups. The papers, wherein the governing set of equations is both hyperbolic and correct from the viewpoint of the physically admissible magnitudes of the velocities of the transient waves resulting from these equations, fall into the first category,
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i.e. the velocity of the longitudinal wave is G L = E/$, while the velocities of the three transverse shear waves, in the general case of arbitrary cross sections of thinwalled beams with open profile, depend essentially of the geometry of the beam ([7], [15], [20], [21]). There are seven independent unknowns in the displacement field in the general case if only primary warping is included into consideration [15], or with additional three generalized displacements describing the variation of the secondary warping due to non-uniform bending and torsion [7], or with additional three variables describing a “complete homogeneous deformation of the microstructure” [20]. As this takes place, different authors obtain different magnitudes for the velocities of transverse shear waves. The second category involves the articles presenting hyperbolic but incorrect equations from the above mentioned viewpoint due to ignoring at least one of the aspects required, i.e. resulting in incorrect magnitudes of the transient waves. This concerns, first of all, the velocity of the longitudinal waves which should not deviate from G L = E/ρ, nevertheless, there are some examples [19] where such a situation takes place. Secondly, in some papers one can find equations looking like hyperbolic ones ( [3], [22], [30]) but from which it is impossible to obtain the velocity, at least, of one transient wave at all. In such papers, usually six generalized displacements are independent while warping is assumed to be dependent on the derivative of the torsional rotation with respect to the beam axial coordinate [22] or is neglected in the analysis [3]. In other words, there is a hybrid of two approaches: Timoshenko’s beam theory and Vlasov’s thin-walled beam theory, some times resulting to a set of equations wherein some of them are hyperbolic, while others are not. Thirdly, not all inertia terms are included into consideration [16]. The papers providing the governing system of non-hyperbolic equations belong to the third group. In such papers, the waves of transverse shear are the diffusion waves possessing infinitely large velocities, and therefore the dynamic equations presented in [2], [4] and [17] cannot be named as the Timoshenko type equations. The simple but effective procedure for checking for the category, within which this or that paper falls in, has been proposed by Rossikhin and Shitikova [27] based on the theory of discontinuities [28] and illustrated by several examples. It has been shown that only the theories of the first group, such as the Korbut-Lazarev theory which provides the physically admissible velocities of propagation of transient waves, could be used for solving the problems dealing with transient wave propagation, while the theories belonging to the second and third group could be adopted for static problems only. However, the presence of three or four transverse shear waves, which propagate with different velocities dependent strongly on geometric characteristics of the thinwalled beam, severely limits the application of such theories in solving engineering problems. As for the experimental verification of the existence of the three shear waves in thin-walled beams of open section, then it appears to be hampered by the fact that the velocities of these waves depend on the choice of the beam’s cross section. This immediately raises the question of whether it is possible to create such a theory of dynamic behaviour of thin-walled beams, within the framework of which,
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instead of three transverse shear waves, only one shear-like wave will propagate with the velocity independent of the characteristics of the beam’s cross section, as it takes place in the case of the longitudinal-flexural-warping wave. The positive answer to this question will be given below.
14.2 Problem Formulation and Governing Equations For investigating the dynamic behaviour of thin-walled beams of open section, we shall proceed from three-dimensional equations of isotropic elasticity written in the Cartesian system of coordinates x1 , x2 , and x3 . Let us consider a certain unperturbed equilibrium of an elastic body characterized by the displacement vector u0i , stress tensor σ0i j , and the vector of volume forces Xi0 (surface forces are absent). The characteristics of the unperturbed equilibrium state satisfy the following geometrically nonlinear equations and boundary conditions on the surface [6]: {σ0jk (δik + u0i,k )}, j + Xi0 = 0 ,
(14.1)
σ0jk (δik + u0i,k )ν j = 0 ,
(14.2)
where δik is the Kronecker’s symbol, ν j are the components of the unit vector normal to the boundary surface, Latin indices take on the magnitudes 1,2,3, a Latin index after comma denotes the partial derivative with respect to the corresponding spatial coordinate x1 , x2 , x3 , and the summation is understood over the repeated indices. Let us perturb the body with some small deviations from the unperturbed equilibrium: ui , σi j , Xi = −$¨ui , where $ is the material density, and an overdot denotes the time-derivative. The characteristic components of the perturbed motion u˜ i , σ ˜ ij , and X˜ i take the form u˜ i = u0i + ui ,
σ ˜ i j = σ0i j + σi j ,
X˜ i = −$˙vi ,
(14.3)
where vi is the displacement velocity. The components of the perturbed state satisfy the following equations and boundary conditions: , σ ˜ ik (δik + u˜ i,k ) , j + X˜ i = 0 , ' ( σ˜ jk δik + u˜ i,k ν j = 0 .
(14.4) (14.5)
Substituting (14.3) in (14.4) and (14.5), carrying out the linearization of the resulting equations, and taking (14.1) and (14.2) into account, we find 3 % & 4 σ jk δik + u0i,k + σ0jk ui,k = $¨ui , (14.6) ,j 3 % & 4 σ jk δik + u0i,k + σ0jk ui,k ν j = 0 . (14.7)
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Suppose that the unperturbed state deviates a little from the initial undeformed state. In the majority of engineering problems the given assumption is found to be valid. In this case, the terms u0i,k can be neglected, and Eqs (14.2), (14.6) and (14.7) at σ0i j = ci j , where ci j (i, j = 1, 2, 3) are certain constant values (moreover, some of them could be vanished according to the conditions of the problem under consideration), take the following form: σi j, j + σ0jl ui,l j = $˙vi , σi j ν j = 0 ,
σ0i j ν j
=0.
(14.8) (14.9)
Equations (14.8)–(14.9) should be considered together with the relationship % & σ ˙ i j = λvl,l δi j + μ vi, j + v j,i , (14.10) where λ and μ are Lam´e constants. Suppose that the wave surface of strong discontinuity exists in a spatially curved thin-walled beam of open section. Then let us differentiate (14.8), (14.10) and (14.9) k times with respect to time, write them on the both sides of the wave surface, and take their difference. As a result we obtain % & . / σi j,(k+1) = λ vl,l(k) δi j + μ vi, j(k) + v j,i(k) , (14.11) . / σi j, j(k) + σ0jl eil, j(k−1) = $ vi,(k+1) , (14.12) σi j,(k) ν j = 0 , (14.13) &+ % &− . / % where ei j = vi, j , and Z,(k) = ∂k Z/∂tk − ∂k Z/∂tk . Equation (14.12) lacks the discontinuities in the external forces, since they are continuous on the wave surface of strong discontinuity. For the ease of further treatment, let us introduce two sets of coordinates: λ , τ, ξ with the unit vectors λ {λi }, τ {τi }, and ξ {ξi }, and λ, x, y with the unit vectors λ, k {ki }, and s {si }. The axes λ, τ, ξ are the natural axes for the curved axis of the beam, in so doing the λ-axis is the tangent to the beam’s axis, the τ-axis is its binormal, the ξ-axis is its main normal, s is the arc length calculated from a certain point with the coordinate s0 along the beam axis (Fig. 14.1), while the x− and y−axes are the main central axes of the beam’s normal section. The angle ϕ(s) is the angle between the x− and τ−axes and the y− and ξ−axes. Following [26], it can be shown that dsi = ki (K + τ) − κλi cos ϕ(s) , ds dki = −si (K + τ) + κλi sin ϕ(s) , ds dλi = −ki κ sin ϕ(s) + si κ cos ϕ(s) , ds
(14.14) (14.15) (14.16)
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Fig. 14.1 Scheme of a spatially curved linear elastic beam of arbitrary open cross-section
where K = dϕ/ds, while κ(s) and τ(s) are the curvature and the torsion of the beam’s axis, respectively. Note that it is precisely these three values, K(s), κ(s) and τ(s), that are of prime engineering interest in studying the spatially curved thin-walled beams of open profile. Moreover, as it will be shown in the further analysis, these values produce new features in the dynamic response of such beams as compared with straight thin-walled beams of open cross-section. Considering the conditions of compatibility ∂ σi j,(k) d[σi j,(k) ] ∂[σi j,(k) ] −1 σi j, j(k) = −G σi j,(k+1) λ j + λj + kj + s j (14.17) ds ∂x ∂y . / . / d[eil,(k) ] ∂ eil,(k) ∂ eil,(k) . / eil, j(k) = −G −1 eil,(k+1) λ j + λj + kj + s j (14.18) ds ∂x ∂y . / . / . / d vi,(k) ∂ vi,(k) ∂ vi,(k) . / . / . / eil,(k) = vi,l(k) = −G−1 vi,(k+1) λl + λl + kl + sl (14.19) ds ∂x ∂y as well as formulas (14.14)–(14.16), Eqs (14.11) and (14.12) can be rewritten as % & [σi j,(k+1) ] = −G−1 λ[vl,(k+1) ]λl δi j − G−1 μ [vi,(k+1) ]λ j + [v j,(k+1) ]λi d[vl,(k) ] ∂[vl,(k) ] ∂[vl,(k) ] +λ λl + kl + sl δ i j ds ∂x ∂y d[v j,(k) ] d[vi,(k) ] ∂[vi,(k) ] +μ λj + λi + kj ds ds ∂x ∂[v j,(k) ] ∂[v j,(k) ] ∂[vi,(k) ] + ki + sj + si , (14.20) ∂x ∂y ∂y
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d[σi j,(k) ] ∂[σi j,(k) ] ∂[σi j,(k) ] λj + kj + sj ds ∂x ∂y d[vi,(k) ] 0 ∂[vi,(k−1) ] 0 + G−2 [vi,(k+1) ]σ0λλ − 2G−1 σλλ − 2G −1 σλx ds ∂x ∂[vi,(k−1) ] 0 d2 [vi,(k−1) ] 0 ∂ d[vi,(k−1) ] 0 − 2G−1 σλy + σ + 2 σλx λλ ∂y ∂x ds ds2 4 ∂ d[vi,(k−1) ] 0 ∂[vi,(k−1) ] 3 +2 σλy + −(K + τ)σ0λy + κσ0λλ sin ϕ ∂y ds ∂x 4 ∂[vi,(k−1) ] 3 + (K + τ)σ0λx − κσ0λλ cos ϕ ∂y & d[vi,(k−1) ] % 0 −1 + G [vi,(k) ] − κ σλx sin ϕ − σ0λy cos ϕ , (14.21) ds
$[vi,(k+1) ] = −G −1 [σi j,(k+1) ]λ j +
where σ0λλ = σ0i j λi λ j , σ0λx = σ0i j λi k j , and σ0λy = σ0i j λi s j . Analysis of the influence of the preliminary shear stresses σ0λx and σ0λy and the axial compression or tension stresses σ0λλ and their different combinations on the dynamic response of thin-walled rods of open profile is a very complicated problem which is of great importance in civil engineering and bridge construction. However, reference to Eq. (14.21) shows that the value σ0λλ is multiplied by the k + 1-order jump and by the s-derivative of the k-order, i.e. it has a strong impact both on the velocities of the surfaces of strong discontinuity and on the discontinuities, in contrast to the values σ0λx and σ0λy which are multiplied only by the k- and k − 1-order jumps, i.e. they exert weak effect only on the jumps. That is why, in the further part of the present paper, we shall neglect σ0λx and σ0λy with respect to the value σ0λλ .
14.2.1 Dynamic Response of Axially Prestressed Spatially Curved Thin-Walled Rods of Open Profile Thus, assuming further that σ0λx = σ0λy = 0 and σ0λλ 0, from (14.21) we have d[σi j,(k) ] ∂[σi j,(k) ] ∂[σi j,(k) ] ρ[vi,(k+1) ] = −G−1 [σi j,(k+1) ]λ j + λj + kj + sj ds ∂x ∂y d[vi,(k) ] d2 [vi,(k−1) ] ∂[vi,(k−1) ] + G−2 [vi,(k+1) ] − 2G −1 + +κ sin ϕ 2 ds ∂x ds ∂[vi,(k−1) ] − κ cos ϕ σ0λλ . (14.22) ∂y To satisfy Eq. (14.13), it is sufficient to put σi j,(k) ki k j = 0 , σi j,(k) si s j = 0 ,
σi j,(k) si k j = 0 .
(14.23)
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Then the boundary surface will be free from the normal and tangential stresses. Now we expand the value [vi,(k) ] entering into (14.20) and (14.22)–(14.23) in terms of three mutually orthogonal vectors λ, k, s. As a result we obtain .
/ vi,(k) = ω(k) λi + θ(k) ki + η(k) si ,
(14.24)
where ω(k) = [vi,(k) ]λi , θ(k) = [vi,(k) ]ki , and η(k) = [vi,(k) ]si . Suppose that for a thin-walled beam of open section the following velocity fields are fulfilled [30], [11]: 1y
ω(k) = ω(k) (s, s1 ) = ω0(k) (s) + ω ˜ 1x ˜ (k) (s)x(s1 ) + ψ˜ (k) ωA (s1 ) , (k) (s)y(s1 ) + ω
(14.25)
0 θ(k) = θ(k) (s, s1 ) = θ(k) (s) − ω1λ (k) (s)(y(s1 ) − ay ) ,
(14.26)
η(k) = η(k) (s, s1 ) = η0(k) (s) + ω1λ (k) (s)(x(s1 ) − a x ) ,
(14.27)
where s1 is the arc length measured along the cross-section profile from the point M0 , which corresponds to the arc length of s01 , to the point M[x(s1 ), y(s1 )], which corresponds to the arc length of s1 (Fig. 14.2).
Fig. 14.2 Scheme of the cross section of a thin-walled beam with a generic open cross-section
Gol’denveizer [11] proposed that the angles of in-plane rotation do not coincide with the first derivatives of the lateral displacement components and, analogously, warping does not coincide with the first derivative of the torsional rotation. It should be emphasized that it was precisely Gol’denveizer [11] who pioneered in combining Timoshenko’s beam theory [29] and Vlasov thin-walled beam theory [30] (note that the first edition of Vlasov’s book was published in Moscow in 1940) and who suggested to characterize the displacements of the thin-walled beam’s cross-section by seven generalized displacements. In order to refine the structure of formula (14.25) in the case of a spatially curved thin-walled beam, let us differentiate (14.24), which involves only the terms
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0 , and η0 , with respect to s with due account for (14.14)–(14.16). As a result ω0(k) , θ(k) (k) we obtain ⎛ 0 ⎞ . / ⎟⎟⎟ d vi(k) 0 ⎜⎜⎜⎜ dω(k) 0 0 = ⎜⎝⎜ + θ(k) κ sin ϕ − η(k) κ cos ϕ⎟⎟⎠⎟ λi ds ds ⎛ 0 ⎞ ⎜⎜⎜ dθ(k) ⎟⎟⎟ 0 0 ⎜ + ⎜⎜⎝ − ω(k) κ sin ϕ + η(k) (K + τ)⎟⎟⎟⎠ ki ds ⎛ 0 ⎞ ⎜⎜⎜ dη(k) ⎟⎟⎟ 0 0 + ⎜⎜⎝⎜ + ω(k) κ cos ϕ − θ(k) (K + τ)⎟⎟⎠⎟ si . (14.28) ds
Moreover, we shall use the relationship for the discontinuity in the k-order derivative with respect to time t of the angular velocity . / 1y Ωi(k) = ω1λ ˜ 1x ˜ (k) si . (k) ki + ω (k) λi + ω
(14.29)
Differentiating (14.29) with respect to s and considering (14.14)–(14.16) yields ⎛ 1λ ⎞ ⎟⎟⎟ d[Ωi(k) ] ⎜⎜⎜⎜ dω(k) 1y 1x = ⎜⎜⎝ +ω ˜ (k) κ sin ϕ − ω ˜ (k) κ cos ϕ⎟⎟⎟⎠ λi ds ds ⎛ 1x ⎞ ⎜⎜⎜ dω ⎟⎟⎟ ˜ (k) 1y + ⎜⎜⎝⎜ − ω1λ ˜ (k) (K + τ)⎟⎟⎠⎟ ki (k) κ sin ϕ + ω ds ⎛ 1y ⎞ ⎜⎜⎜ dω ⎟⎟⎟ ˜ (k) ⎜ 1λ 1x + ⎜⎜⎜ + ω(k) κ cos ϕ − ω ˜ (k) (K + τ)⎟⎟⎟⎟ si . (14.30) ⎝ ds ⎠ If we suppose that transverse shear strains are absent, then we are led to the following relationships, which are in compliance with the Vlasov theory: ⎛ 0 ⎞ ⎜⎜⎜ dη(k) ⎟⎟⎟ 1x 0 0 ⎜ ω ˜ (k) = − ⎜⎜⎝ + ω(k) κ cos ϕ − θ(k) (K + τ)⎟⎟⎟⎠ , (14.31) ds 1y ω ˜ (k)
ψ˜ (k)
⎛ 0 ⎞ ⎜⎜⎜ dθ(k) ⎟⎟⎟ 0 0 = − ⎜⎜⎜⎝ − ω(k) κ sin ϕ + η(k) (K + τ)⎟⎟⎟⎠ , ds
(14.32)
⎛ 1λ ⎞ ⎜⎜⎜ dω(k) ⎟⎟⎟ 1y 1x ⎜ = − ⎜⎜⎝ +ω ˜ (k) κ sin ϕ − ω ˜ (k) κ cos ϕ⎟⎟⎟⎠ ds =−
dω1λ (k)
+
dη0(k)
κ sin ϕ −
0 dθ(k)
κ cos ϕ ds ds% ds & 0 + ω0(k) κ2 sin ϕ − θ(k) sin ϕ + η0(k) cos ϕ κ(K + τ).
(14.33)
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If we now substitute (14.31)–(14.33) in (14.25), then we obtain a formula for defining the k-order time-derivative in the discontinuity in the velocity of translation along the λ-axis without account for transverse shear deformations. Following Gol’denveizer [11], in order to take the transverse shear deforma0 tions into account, it is essential to substitute the derivatives dω1λ (k) /ds, dη(k) /ds, and 1y
0 /ds by the independent functions ψ , ω1x , and ω , respectively, i.e. to rewrite dθ(k) (k) (k) (k) Eqs (14.31)–(14.33) as % & 1x 0 0 ω ˜ 1x (14.34) (k) = − ω(k) + ω(k) κ cos ϕ − θ(k) (K + τ) ,
% 1y & 1y ω ˜ (k) = − ω(k) − ω0(k) κ sin ϕ + η0(k) (K + τ) ,
(14.35)
% & 1y ψ˜ (k) = − ψ(k) + ω ˜ 1x κ sin ϕ − ω ˜ κ cos ϕ (k) (k) 1y
= −ψ(k) + ω1x (k) κ sin ϕ − ω(k) κ cos ϕ % & 0 2 0 + ω(k) κ sin ϕ − θ(k) sin ϕ + η0(k) cosϕ κ(K + τ) .
(14.36)
As a result, instead of formula (14.25), we obtain 0 0 ω(k) (s, s1 ) = ω0(k) (s) − ω1x (k) (s) + ω(k) κ cos ϕ − θ(k) (K + τ) y(s1 ) 1y − ω(k) (s) − ω0(k) κ sin ϕ + η0(k) (K + τ) x(s1 ) 1y 0 2 + −ψ(k) (s) + ω1x (k) (s)κ sin ϕ − ω(k) (s)κ cos ϕ + ω(k) κ sin 2ϕ % & 0 − θ(k) sin ϕ + η0(k) cos ϕ κ(K + τ) ωA (s1 ). (14.37) Formulas (14.26) and (14.27) remain unchanged. If we put K = κ = τ = 0 in (14.37), then we are led to the case of a straight thin-walled rod of open profile. In further treatment we will use formulas (14.25)–(14.27), but in the final relationships we 1y will carry out the substitution of the values ψ˜ (k) , ω ˜ 1x ˜ (k) by their magnitudes (k) , and ω defined by formulas (14.34)–(14.36). At k = 0, the values entering in (14.26), (14.27), and (14.37) have the following 0 physical meaning: ω0(0) (s), θ(0) (s) and η0(0) (s) are the discontinuities in the velocities of translatory motion of the section as a rigid body together with the point C (the center of gravity of the cross-section) along the λ−, x− and y−axes, respectively, 1y 1λ ω1x (0) (s), ω(0) (s) and ω(0) (s) are the discontinuities in the angular velocities of the cross-section’s rotation as a rigid whole around the x−, y− and λ−axes, respectively, ψ(0) (s) is the discontinuity in the velocity of warping of the cross section, ωA (s1 ) is twice the area of the sector AM0 M, the point A is the center of bending (Fig. 14.2) with the coordinates [30] 1 1 ax = ωC (s1 )y(s1 )dF , ay = − ωC (s1 )x(s1 )dF . (14.38) Ix Iy F
F
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Since the section is referred to the main central axes and the coordinates of the point A are defined by formulas (14.38), then I xy = Iωx = Iωy = S y = S x = 0. Note that the sectorial area, generally speaking, depends on two coordinates, namely: the initial point s01 and the terminal point s11 , i.e. ωA (s01 , s11 ). Let us choose s11 in such a way that the relationship 1 ωA (s01 , s11 ) = ωA (s01 , s1 )dF (14.39) F F
will be valid. Then, as it is shown in [30], taking the point with the coordinate s11 as the origin of the arc length measuring, we obtain S ω = ωA (s11 , s1 )dF = 0 . (14.40) F
Let us name this point as the null sectorial point [30]. Since there may exist several such points, the null sectorial point nearest to the point A can be named as the main null sectorial point, and we shall take it as the initial point of reading. Thus, using the above mentioned choice of the coordinates and the initial points of measuring, the functions 1, x(s1 ), y(s1 ), and ωA (s11 , s1 ) occur to be orthogonal. If we substitute (14.20) in the first two equations of (14.23) and consider (14.24), then we are led to the relationships dω(k−1) λ −1 ε x,(k) = εy,(k) = G ω(k) − + κη(k−1) cos ϕ − κθ(k−1) sin ϕ , (14.41) 2(λ + μ) ds where
' ( ' ( ∂ [vi,(k−1) ]ki ∂ [vi,(k−1) ]si ε x,(k) = , εy,(k) = . ∂x ∂y
Equation (14.23), after the substitution of (14.20), (14.24), (14.26), and (14.27) in it, is fulfilled unconditionally. Multiplying (14.20) successively by λi λ j , λi k j and λi s j and considering (14.14)– (14.16), after integration of the final equations over the cross-sectional area of the beam, we could ) obtain the relationships for) discontinuities in the generalized forces Nλ(k+1) = [σi j,(k+1) ]λi λ j dF, Qλx(k+1) = [σi j,(k+1) ]λi k j dF, and Qλy(k+1) = F F ) [σi j,(k+1) ]λi s j dF. In a similar way we could find the relations for discontinuities in F
the ) derivatives of the bending moments ) with respect to the x− and y−axes M x(k+1) = [σi j,(k+1) ]λi λ j ydF and My(k+1) = [σi j,(k+1) ]λi λ j xdF, in the derivatives of the F F ) bimoment B(k+1) = [σi j,(k+1) ]λi λ j ωA dF, and in the derivatives of the bendingF ) ) torsional moment M A(k+1) = [σi j,(k+1) ]λi s j (x − a x )dF − [σi j,(k+1) ]λi k j (y − ay)dF. F
F
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Multiplying (14.22) successively by λi , ki , and si with due account for (14.14)– (14.16), then substituting (14.25)–(14.27) in the obtained equations and integrating them over the cross-sectional area of the beam, after elimination of the generalized forces we are led to the following equations: −2
G ($G −
$G21 − σ0λλ )ω0(k+1)
−1
%
$G 21 + σ0λλ
& dω0(k)
= −2G ds & % & % −1 2 2 0 − G $G 1 + 2$G 2 + 2σλλ κ a x cosϕ + ay sin ϕ ω1λ (k) 0 − η0(k) cos ϕ + θ(k) sin ϕ + F 1(k−1) ,
(14.42)
0 −1 2 0 G −2 ($G2 − $G22 − σ0λλ )(θ(k+1) + ay ω1λ (k+1) ) = −2G ($G 2 + σλλ ) * % + & % & d 0 × θ(k) + ay ω1λ + (K + τ) η0(k) − a x ω1λ (k) (k) ds % & + G−1 $G 21 + $G22 + 2σ0λλ κω0(k) sin ϕ 3 1y 4 + G−1 $G22 ω(k) − ω0(k) κ sin φ + η0(k) (K + τ) + F2(k−1) ,
(14.43)
% & −1 G−2 ($G 2 − $G22 − σ0λλ )(η0(k+1) − a x ω1λ $G22 + σ0λλ (k+1) ) = −2G * % & % &+ d 0 0 1λ × η(k) − a x ω1λ − (K + τ) θ + a ω y (k) (k) (k) ds % & − G−1 $G 21 + $G22 + 2σ0λλ κω0(k) cos ϕ 3 4 0 0 + G−1 $G22 ω1x (k) + ω(k) κ cos φ − θ(k) (K + τ) + F3(k−1) ,
(14.44)
2
0 0 − G −2 ($G2 − $G 21 − σ0λλ ) ω1x (k+1) + ω(k+1) κ cos ϕ − θ(k+1) (K + τ) % & d 0 = 2G −1 ρG 21 + σ0λλ ω1x + ω0(k) κ cos ϕ − θ(k) (K + τ) (k) ds & % + κ $G 21 + 2$G 22 + 2σ0λλ G−1 ω1λ (14.45) (k) sin ϕ + F 4(k−1) , 1y − G−2 ($G 2 − $G 21 − σ0λλ ) ω(k+1) − ω0(k+1) κ sin ϕ + η0(k+1) (K + τ) % & d 1y = 2G −1 ρG 21 + σ0λλ ω(k) − ω0(k) κ sin ϕ + η0(k) (K + τ) ds & % 2 2 + κ $G 1 + 2$G 2 + 2σ0λλ G −1 ω1λ (14.46) (k) cos ϕ + F5(k−1) ,
14 Dynamics of Spatially Curved Thin-Walled Beams
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195
1y $G21 − σ0λλ ) ψ(k+1) − ω1x (k+1) κ sin ϕ + ω(k+1) κ cos ϕ % & 0 ω0(k+1) κ2 sin 2ϕ + θ(k+1) sin ϕ + η0(k+1) cosϕ κ(K + τ) % & d 1y 2G−1 ρG21 + σ0λλ ψ(k) − ω1x (k) κ sin ϕ + ω(k) κ cos ϕ ds % & 0 ω0(k) κ2 sin 2ϕ + θ(k) sin ϕ + η0(k) cos ϕ κ(K + τ) + F 6(k−1) (14.47)
% & 0 0 G−2 ($G 2 − $G22 − σ0λλ ) IPA ω1λ (k+1) + ay Fθ(k+1) − a x Fη(k+1) % &* d % & −1 2 0 0 0 = −2G ρG2 + σλλ IPA ω1λ (k) + ay Fθ(k) − a x Fη(k) ds % &4 0 + F(K + τ) a x θ(k) + ayη0(k) % & + G−1 ρG22 ay F + κIy cos ϕ $G21 + $G 22 + 2σ0λλ 1y × ω(k) − ω0(k) κ sin ϕ + η0(k) (K + τ) % & + G−1 −$G22 a x F + κI x sin ϕ $G21 + $G 22 + 2σ0λλ (14.48) 1x 0 0 × ω(k) + ω(k) κ cos ϕ − θ(k) (K + τ) % &% & + G−1κF $G21 + $G 22 + 2σ0λλ a x cos ϕ + ay sin ϕ ω0(k) + F7(k−1) , where G 21 = E$−1 , G22 = μ$−1 , and F j(k−1) ( j = 1, 2, ..., 7) are the functions of the k − 1-order. The system of seven equations, (14.42)–(14.49), involves seven unknown 1y 0 1λ values: ω0(k) , θ(k) , η0(k) , ω1x (k) , ω(k) , ω(k) , and ψ(k) , which are defined uniquely from this set of equations. To find these values on the quasi-longitudinal shock wave and 8 on the quasi-transverse shock 8 wave, we should put in all equations G = G I = G21 + $−1 σ0λλ and G = G II =
G22 + $−1 σ0λλ , respectively.
14.2.2 Construction of the Desired Wave Fields in Terms of the Ray Series Following the previous two papers by Rossikhin and Shitikova [24] and [26] devoted to the dynamic behaviour of thin elastic bodies, where thin plates and shells have been considered in [24], and spatially curved and twisted slender rod-like solids have been studied in [26], as a method applicable for solving dynamic problems resulting in propagation of wave surfaces of strong and weak discontinuity we will use the method of ray expansions [1]. This method is one of the methods of perturbation technique, where time is used as a small parameter. Thus, knowing the discontinuities of the desired stress and velocity fields determined above within an accuracy of arbitrary constants on the two waves of strong discontinuity, quasi-longitudinal and quasi-transverse, propagating in the
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thin-walled beam of open profile, we could construct the fields of the desired functions also with an accuracy of the arbitrary constants utilizing the ray series [1], which are the power series with variable coefficients and which allow one to construct the solution behind the wave fronts of strong discontinuity [24], [26]: k ∞ 1 α s s Z(t, s) = [Z ](s) t − H t− , k! ,(k) Gα Gα α=I,II
(14.49)
k=0
where Z is the desired value, H(t − s/Gα ) is the unit Heaviside function, and the index α = I, II labels the ordinal number of the wave propagating with the velocity G α . The arbitrary constants entering into the ray expansions are determined from the initial and boundary conditions. The example of using the ray expansions (14.49) for analyzing the impact response of spatially curved thin-walled beams of open cross-section will be demonstrated below by solving the problem about the normal impact of an elastic spherically-headed rod upon an elastic arch, representing itself a channel-beam curved along an arc of the circumference.
14.3 Impact Response of a Thin-Walled Beam of Open Profile During the past two decades foreign object impact damage to structures has received a great deal of attention, since thin-walled structures are known to be susceptible to damage resulting from accidental impact by foreign objects. Impact on aircraft structures or civil engineering structures, for instance, from dropped tools, hail, and debris thrown up from the runway, poses a problem of great concern to designers. Since the impact response is not purely a function of material’s properties and depends also on the dynamic structural behaviour of a target, it is important to have a basic understanding of the structural response and how it is affected by different parameters [25]. From this point of view, analytical models are useful as they allow systematic parametric investigation and provide a foundation for prediction of impact damage. An impact response analysis requires a good estimate of contact force throughout the impact duration. Low velocity impact problems, which also took the local indentation into account, have been solved by many authors. Reference to the stateof-the-art paper [25] shows that in most studies it was assumed that the impacted structure was free of any initial stresses. But this does not adequately reflect the real multidirectional complex loading states that the materials experience during their service life. In practice, the composite facing of a structure may be under a preload, e.g. a sandwich structure with laminate facing under bending loads, jet engine fan blades subjected to centrifugal forces. Even when stationary on the runway a composite airframe is under pre-stress. The dynamic response of straight thin-walled beams of generic open profile impacted by an elastic sphere was investigated in [23] utilyzing the technical
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theory by Korbut and Lazarev [15]. A numerical approach to determining the transient response of straight nonrectangular bars subjected to transverse elastic impact was described in [18] using finite element method. Below in order to analyze the impact response of spatially curved thin-walled beams of open profile we shall implement the theoretical results which have already been described in Sect. 14.2, since the derived hyperbolic system of recurrent equations together with the ray expansions allow one to describe the short-time processes, in particular, the processes of shock interaction.
14.3.1 Impact of an Elastic Hemispherical-Nosed Rod against a Thin-Walled Beam of Open Profile Let us consider the normal impact of an elastic thin spherically-headed rod of circular cross-section with the radius r0 against a lateral surface of the initially stressed thin-walled beam of open section (Fig. 14.3). For simplicity, we shall suppose −1 that the thin-walled beam has the constant curvature κ = R−1 t = R , while the torsion τ = 0. Moreover, the beam possesses one axis of symmetry, along which the impactor moves.
Fig. 14.3 Scheme of shock interaction
At the moment of impact, the velocity of the impacting rod is equal to V0 , and the longitudinal shock wave begins to propagate along the rod with the velocity 8
G 0 = E 0 $−1 0 , where E 0 is its elastic modulus, and $0 is its density. Using the dynamic condition of compatibility on the front of this wave coupling the stress σ− and velocity v− fields [25] ' ( σ− = ρG0 V0 − v− , (14.50)
− the contact stress σ =σ could be defined as cont
y=0
% & σcont = ρG 0 V0 − α˙ − η0 ,
(14.51)
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where α˙ + η0 = v−
y=0 is the normal velocity of the beam’s points within the contact domain, α is the value governing the local bearing of the target’s material during the process of its contact interaction with the impactor, and an overdot denotes the time-derivative. The interacting bodies are considered to be of rather long extent in order to ignore reflected waves, since they arrive at the place of contact after the rebounce of the impactor from the target. Formula (14.51) allows one to find the contact force % & P = πr02 $0 G 0 V0 − α˙ − η0 . (14.52) However, the contact force can be determined not only via Eq. (14.52), but using the Hertz’s law as well [12], [25] P = kα3/2 ,
(14.53)
where k is the contact stiffness coefficient depending on the geometry of colliding bodies, as well as on their elastic constants 4 √ k = E ∗ R , 3
1 1 1 = + , R r 0 Rt
1 − ν20 1 − ν2t 1 = + . E∗ E0 Et
(14.54)
Here, r0 is the radius of the rod’s rounded end (Fig. 14.3), Rt is the radius of curvature of the target in the place of impact, and E0 , ν0 and Et , νt are the elastic modulus and the Poisson’s ratio of the indenter and the target, respectively. Eliminating the force P from (14.52) and (14.53), we are led to the equation for determining the value α(t) α˙ + η0 +
k α1/2 = V0 . πρ0 G0 R
(14.55)
Since the contours of the beam’s cross sections remain rigid during the process of impact, then all sections involving by the contact domain form a layer which moves as rigid whole. Let us name it as a contact layer (Fig. 14.3). If we write the series (14.49) on the unknown moving boundary a(t) of the contact domain, and consider the effect of ‘retardation’ [25] implying in the fact that the transient waves detach from the boundary of the contact domain not immediately at the moment of impact t = 0, but after some time duration t = t∗ = r0 V0 /2G22 elapsed from the moment of impact, then in the series (14.49) we should change the values t and s by t − t∗ and a − a∗ , respectively, where s = a∗ is the location of the contact region boundary at t = t∗ , a = a∗ + a0 (t − t∗ ) + a1 (t − t∗ )2 + ..., (14.56) −1 ∗ −1 ∗ 2 t − t∗ − (a − a∗)G −1 α = (1 − a0G α )(t − t ) − a1Gα (t − t ) + ...,
and a0 , a1 , ... are yet unknown constant values, and
a∗
= r0 V0G−1 II .
(14.57)
14 Dynamics of Spatially Curved Thin-Walled Beams
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With the assumptions made, the motion of the contact domain is described by one equation M η˙ 0 = 2Qλy + P , (14.58) where M = 2$Fa(t − t∗ ) and 2a(t − t∗ ) are the mass and the width of the contact spot, respectively, while the value α is connected with the value a according to the Hertz’s theory by the following relationship: α = R −1 a2 ,
(14.59)
Equation (14.53), the initial condition α˙ = V0 ,
(14.60)
which may be written at t = t∗ due the smallness of the time t∗ , as well as the boundary conditions θ0 = ω1y = ω1λ = ψ = 0 , ω1x = −˙aR−1 , ω0 = a˙
(14.61)
should be added to Eqs (14.55) and (14.58). Representing the value α in terms of a series α = α∗ + α1 (t − t∗ ) + α2 (t − t∗ )2 + α3 (t − t∗ )3 + ...,
(14.62)
where α∗ ≈ 2V0 t∗ = r0 V02 G−2 II , and α1 , α2 , α3 , ... are yet unknown constants, and substituting (14.56) and (14.62) in (14.59), we could find with due account for the initial condition (14.60) all coefficients of the series (14.56), as well as α1 = V0 . Since the solution behind the wave fronts up to the contact domain is constructed in terms of the ray series (14.49), then we should first determine the ray series coefficients for the desired values. This could be carried out by solving Eqs (14.42)– (14.49) at k = 0, 1, 2, ... putting K = τ = 0, κ = R−1 , and ϕ = χ = 0 (Fig. 14.3). Knowing the ray series coefficients, we could write the ray series for all desired values, and then substitute them in Eqs (14.56)–(14.62). Finally equating the coefficients at equal powers of t − t∗ , we could find the coefficients αk , among them, ⎛ ⎞ ⎜⎜⎜ V0 πρ0 R 1/2 α∗3/2 2G2 α∗1/2 ⎟⎟⎟ 1 k α2 = − + + ⎝⎜ ⎠⎟ < 0. 2 πρ0 R α∗1/2 G 0 ρF G 0 R 1/2 Substituting the found arbitrary constants in the ray series, we obtain the final expressions for the desired fields. Thus, for example, knowing the value α2 , it is possible to determine α(t) (14.62) and a(t) (14.56), and therefore to obtain the typical time-dependence of the contact force (14.53) within an accuracy of (t − t∗ )2 , since α2 is a negative value: 3 43/2 P(t − t∗ ) − P∗ = k V0 (t − t∗ ) + α2 (t − t∗ )2 ,
(14.63)
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where P∗ = P|t=t∗ = kα∗3/2 . Equating to zero the expression for the contact force (14.63), we obtain the approximate formula for the duration of contact of the impacting rod with the thin-walled beam of open section. 14.3.1.1 Numerical Example As an example, let us consider the impact of a steel rod with a rounded end upon a steel arch with a constant radius of curvature and zero torsion, the cross section of which represents a channel (Fig. 14.3). The dimensionless time t˜ − t˜∗ = (t − t∗ )( 25 )2/3 V01/3 k2/3 (πr02 $G0 )−2/3 dependence of the dimensionless contact force C− P C∗ = (P − P∗ )(πr2 $G 0 V0 )−1 is presented in Fig. 14.4 for different levels of the P 0 initial axial compression σ Cλλ = σ0λλ ($G 22 )−1 . Reference to Fig. 14.4 shows that the increase in the initial axial compression results in the increase of both the maximal.
Fig. 14.4 Dimensionless time-dependence of the dimensionless contact force
Fig. 14.5 C σλλ -dependence of the dimensionless initial velocity of impact in the case when the contact stress is equal to the yield limit
The curve of the C σλλ -dependence of the dimensionless initial velocity of impact C0 = V0G−1 , resulting in the local damage of the thin-walled open-section beam in V 0 the place of contact is shown in Fig. 14.5 at the given magnitude of the dimensionless yield limit C σy = σy ($0 G20 )−1 . From Fig. 14.5 it is evident that with the increase in the initial axial compression the initial velocity of impact, which may lead to the local damage of the structure, decreases.
14.4 Conclusion Starting from the three-dimensional dynamic theory of linear elasticity and the Vlasov and Gol’denveizer theories, the dynamic theory of thin-walled beams of open section has been proposed with due account for the axial precompression. As this takes place, the derived hyperbolic system of recurrent equations together with
14 Dynamics of Spatially Curved Thin-Walled Beams
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the ray expansions allow one to describe the short-time processes, in particular, the processes of shock interaction. The dynamic stability with respect to small perturbations, as well as the local damage of geometrically nonlinear elastic spatially curved open section beams with axial precompression have been analyzed. Transient waves, which are the surfaces of strong discontinuity and wherein the stress and strain fields experience discontinuities, have been used as small perturbations, in so doing the discontinuities have been considered to be of small values. Such waves could be initiated during lowvelocity impacts upon thin-walled beams. The approach proposed permits to solve analytically for the first time, to the authors’ knowledge, the problem of normal impact of a rod against an elastic thinwalled beam of open section making allowance for the beam’s translatory and rotational motions, warping, rotary inertia, shear deformation, the local bearing due to the Hertz’s theory, and the initial axial compression of the beam. The method of expansion along the rays behind the wave front is valid for short times after the wave front has passed. That is why it is well suited for solving the problems of shock interactions, since the duration of contact is very short, and thus a small number of terms in ray expansions is sufficient to achieve reasonable accuracy of the solution. The theory constructed allows one to investigate the influence of the initial stresses on the dynamic fields and to answer a set of major questions, among them: (1) What magnitude should the initial velocity of impact take on at the given axial precompression of the thin-walled open section beam in order to produce its local damage at the place of shock interaction between the target and the impactor? (2) How does the level of the initial axial compression influence the maximal contact force and the duration of contact? The graphs presented in Figs 14.4 and 14.5 are the answers to the above questions. Acknowledgements The research described in this publication was made possible in part by Grants No. 2.1.2/520 and No. 2.1.2/12275 from the Russian Ministry of High Education.
References 1. Achenbach JD and Reddy DP (1967) Note on wave propagation in linearly viscoelastic media. ZAMP 18:141–144 2. Arpaci A, Bozdag SE and Sunbuloglu E (2003) Triply coupled vibrations of thin-walled open cross-section beams including rotary inertia effects. J Sound Vibr 260:889–900 3. Banerjee JR and Williams FW (1994) Coupled bending-torsional dynamic stiffness matrix of an axially loaded Timoshenko beam element. Int J Solids Struct 31:749–762 4. Bersin AN and Tanaka M (1997) Coupled flexural-torsional vibrations of Timoshenko beams. J Sound Vibr 207:47–59 5. Bhattacharya B (1975) Coupled vibrations of thin-walled open section curved beams. ASCE J Struct Eng 13:29–35 6. Bolotin VV (1961) Nonconservative problems of the theory of elastic stability (in Russian). Fizmatlit, Moscow 7. Capuani D, Savoia M and Laudiero F (1992) A generalization of the Timoshenko beam model for coupled vibration analysis of thin-walled beams. Earthquake Eng Struct Dyn 21:859–879
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8. Cort´inez VH and Rossi RE (1998) Dynamics of shear deformable thin-walled open beams subjected to initial stresses. Rev Internac M´etod Num´er C´alc Dise˜n Ingr 14(3):293–316 9. de Boer R and Sass HH (1975) Der Stoβ auf gerade d¨unnwandige Tr¨ager. Ingenieur-Archiv 44(3):177–188 10. Gendy AS and Saleeb AF (1994) Vibration analysis of coupled extensional/flexural/torsional modes of curved beams with arbitrary thin-walled sections. J Sound Vibr 174:261–274 11. Gol’denveizer AL (1949) To the theory of thin-walled beams (in Russian). Prikl Mat Mekh 13:561–596 12. Goldsmith W (1960) Impact. Arnold, London 13. Kim NI, Seo KJ and Kim MY (2003) Free vibration and spatially stability of non-symmetric thin-walled curved beams with variable curvatures. Int J Solids Struct 40:3107–3128 14. Kim NI and Kim MY (2005) Exact dynamic stiffness matrix of non-symmetric thin-walled curved beams subjected to initial axial force. J Sound Vib 284:851–878 15. Korbut BA and Lazarev VI (1974) Equations of flexural-torsional waves in thin-walled bars of open cross section. Int Appl Mech 10:640–644 16. Korbut BA and Lazareva GV (1982) Dynamic theory of thin curvilinear beams. Int Appl Mech 18:476–482 17. Li J, Shen R, Hua H and Jin X (2004) Coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams Int J Mech Sciences 46:299–320 18. Lin Y, Lai WK and Lin KL (1998) A numerical approach to determining the transient response of nonrectangular bars subjected to transverse elastic impact. J Acoust Soc Am 103:1468– 1474 19. Mescherjakov VB (1968) Free vibrations of thin-walled open section beams with account for shear deformations (in Russian). Proc Moscow Inst Railway Transport Eng 260:94–102 20. Muller P (1983) Torsional-flexural waves in thin-walled open beams. J Sound Vibr 87:115– 141 21. Piovan MT, Cort´inez VH and Rossi RE (2000) Out-of-plane vibrations of shear deformable continuous horizontally curved thin-walled beams. J Sound Vibr 237:101–118 22. Proki´c A (2006) On fivefold coupled vibrations of Timoshenko thin-walled beams. Eng Struct 28:54–62 23. Rossikhin YuA, Shitikova MV (1999) The impact of a sphere on a Timoshenko thin-walled beam of open section with due account for middle surface extension. ASME J Pressure Vessel Tech 121:375–383 24. Rossikhin YuA, Shitikova MV (2007) The method of ray expansions for investigating transient wave processes in thin elastic plates and shells. Acta Mech 189:87–121 25. Rossikhin YuA, Shitikova MV (2007) Transient response of thin bodies subjected to impact: Wave approach. Shock Vibr Digest 39:273–309 26. Rossikhin YuA, Shitikova MV (2008) The method of ray expansions for solving boundaryvalue dynamic problems for spatially curved rods of arbitrary cross-section. Acta Mech 200:213–238 27. Rossikhin YuA, Shitikova MV (2010) The analysis of the transient dynamic response of elastic thin-walled beams of open section via the ray method. Int J Mech 4:9–21 28. Thomas TY (1961) Plastic flow and fracture in solids. Academic Press, New York 29. Timoshenko SP (1928) Vibration problems in engineering. Van Nostrand, New York 30. Vlasov VZ (1961) Thin-walled elastic beams. Published for the National Science Foundation, Washington, D.C. by the Israel Program for Scientific Translations, Jerusalem (Eng. trnsl. from the 2d Russian Ed. published in 1959 in Moscow). 31. V¨or¨os GM (2009) On coupled bending–torsional vibrations of beams with initial loads. Mech Res Com 36:603–611
Chapter 15
On Stability of Elastic Rectangular Sandwich Plate Subject to Biaxial Compression Denis Sheydakov
Abstract In the present paper, in the framework of a general stability theory for three-dimensional bodies the buckling analysis has been carried out for a rectangular sandwich plate subject to biaxial compression. The sandwich plate consists of a porous core, covered by a hard and stiff shell. The behavior of a coating is investigated in the framework of a classic (non-polar) continuum model, while to describe the properties of a core the Cosserat continuum model is used. Using the linearization method in the vicinity of a basic state, the neutral equilibrium equations have been derived, which describe the perturbed state of a sandwich plate. The linearized boundary-value problems have been formulated both for a case of a general sandwich plate and for a sandwich plate with identical top and bottom coatings. By solving these problems numerically for some specific materials, the critical curves and corresponding buckling modes can be found, and the stability regions can be constructed in the plane of loading parameters (relative axial compressions). Keywords Buckling analysis · Sandwich plate · Biaxial compression
15.1 Introduction The problem of equilibrium stability for deformable bodies is of major importance both from theoretical and practical point of view, because the exhaustion of loadcarrying capability and collapse of buildings and engineering structures quite often occurs due to buckling under external loads. In the case of elastic medium, the stability theory is extensively developed for classic non-polar materials. There is large number of papers on stability both for thin and thin-walled bodies in the form of D. Sheydakov (B) South Scientific Center of Russian Academy of Sciences, Chekhova Ave. 41, 344006 Rostov-on-Don, Russia e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 15, © Springer-Verlag Berlin Heidelberg 2011
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rods, plates and shells, and for massive (three-dimensional) bodies. However, due to the increasing number of new constructional materials, the problem of stability for bodies with a complex structure becomes relevant. The present research is dedicated to the buckling analysis of nonlinearly elastic plates of highly porous materials, such as metallic and polymeric foams. As a rule, these plates have a sandwich structure (porous core covered by hard and stiff shells), which is necessary for corrosion protection and optimization of mechanical properties during loading. In majority of papers the behavior of foams is studied in the framework of a classic continuum model. But it seems more reasonable to use a model of micropolar medium or Cosserat continuum model [1, 3, 4, 7, 12] for analysis of highly porous materials, because it allows describing their distinctive properties more adequately. Any particle of the Cosserat continuum has an additional inner degree of freedom a microrotation. As a result of this, a deformation in the couple-stress elasticity theory is described not by one, but by two strain tensors-the stretch tensor and wryness tensor-which greatly complicates the study of micropolar bodies.
15.2 Equilibrium of the Rectangular Sandwich Plate Subject to Biaxial Compression We consider the rectangular sandwich plate of length b1 , width b2 and thickness H. The behavior of the porous core of thickness 2a is described by the model of micropolar elastic body. The top coating of thickness h+ and bottom coating of thickness h− are made of classic non-polar materials. In the case of biaxial compression of the plate, the position of a particle in the strained state is given by the radius vectors R, R+ and R− (here and below, by subscripts ‘+’ and ‘−’ we denote the quantities related to the top and bottom coatings, respectively, without subscripts ‘+’ or ‘−’ – related to the core of the sandwich plate) [6, 13]: X1 = α1 x1 ,
0 ≤ x1 ≤ b1
X2 = α2 x2 , 0 ≤ x2 ≤ b2 ⎧ ⎪ ⎪ f+ (x3 ), a ≤ x3 ≤ a + h+ ⎪ ⎪ ⎪ ⎨ X3 = ⎪ f (x3 ), |x3 | ≤ a ⎪ ⎪ ⎪ ⎪ ⎩ f (x ), −(a + h ) ≤ x ≤ −a − 3 − 3
(15.1)
R+ = α1 x1 e1 + α2 x2 e2 + f+ (x3 )e3 ,
a ≤ x3 ≤ a + h+
R = α1 x1 e1 + α2 x2 e2 + f (x3 )e3 ,
|x3 | ≤ a
R− = α1 x1 e1 + α2 x2 e2 + f− (x3 )e3 ,
−(a + h−) ≤ x3 ≤ −a
(15.2)
15 On Stability of Elastic Rectangular Sandwich Plate Subject to Biaxial Compression
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Here x1 , x2 , x3 are Cartesian coordinates in the reference configuration (Lagrangian coordinates), X1 , X2 , X3 are Eulerian coordinates, {e1 , e2 , e3 } are orthonormal vector basis of Cartesian coordinates, α1 and α2 are compression ratios along the axes x1 and x2 , respectively, f (x3 ) and f± (x3 ) are unknown functions, which characterize the strain in thickness direction of the sandwich plate. In addition, for |x3 | ≤ a a proper orthogonal tensor of microrotation H is given, which represents the rotation of the particle for micropolar medium and at the considered strain is equal to the unit tensor E: H = E = e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3
(15.3)
According to expressions (15.1), (15.2), the deformation gradients C and C± are (hereinafter denotes the derivative with respect to x3 ): C+ = grad R+ = α1 e1 ⊗ e1 + α2 e2 ⊗ e2 + f+ e3 ⊗ e3 ,
a ≤ x3 ≤ a + h +
C = grad R = α1 e1 ⊗ e1 + α2 e2 ⊗ e2 + f e3 ⊗ e3 ,
|x3 | ≤ a
C− = grad R− = α1 e1 ⊗ e1 + α2 e2 ⊗ e2 + f− e3 ⊗ e3 ,
−(a + h−) ≤ x3 ≤ −a
(15.4)
where grad is gradient in the Lagrangian coordinates. It follows from relations (15.3), (15.4) that for the porous core (|x3 | ≤ a) of the plate the wryness tensor L is equal to zero [8, 9] '
(
L × E = − grad H · HT = 0
and stretch tensor Y is expressed as follows Y = C · HT = α1 e1 ⊗ e1 + α2 e2 ⊗ e2 + f e3 ⊗ e3
(15.5)
According to (15.4), for the top (a ≤ x3 ≤ a + h+) and bottom (−a − h− ≤ x3 ≤ −a) coatings the expressions for stretch tensors U± and macrorotation tensors A± have the form [6]: %
U± = C± · CT±
&1 2
= α1 e1 ⊗ e1 + α2 e2 ⊗ e2 + f± e3 ⊗ e3 A± = U−1 ± · C± = E
(15.6)
We assume that the elastic properties of the sandwich plate are described by the physically linear material, whose specific strain energy in the case of micropolar body is a quadratic form of the tensors Y − E and L [2, 5]:
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% & 1 1 W (Y, L) = λtr2 (Y − E) + (μ + κ) tr (Y − E) · (Y − E)T + 2 2 % & 1 1 1 1 + μtr (Y − E)2 + γ1 tr2 L + γ2 tr L · LT + γ3 tr L2 2 2 2 2 μ + κ > 0,
λ + 2μ + κ > 0,
γ2 ≥ 0,
(15.7)
γ1 + γ2 + γ3 > 0
and in the case of classic non-polar medium – quadratic form of the tensors U± − E [6]: 1 W± (U± ) = λ± tr2 (U± − E) + μ± tr (U± − E)2 , μ± > 0, λ± + 2μ± > 0 (15.8) 2 Here λ, μ and λ± , μ± are Lame coefficients for the plate core and coatings, respectively, κ, γ1 , γ2 , γ3 are micropolar elastic coefficients. It follows from expressions (15.3), (15.5), and (15.7) that for the porous core the Piola-type couple stress tensor is equal to zero in the case of biaxial compression of the sandwich plate G=
% & ∂W · H = γ1 (tr L) E + γ2 L + γ3 LT · H = 0 ∂L
and Piola-type stress tensor D is D=
% % & & ∂W · H = λtr (Y − E) E + μ YT − E + (μ + κ) (Y − E) · H = ∂Y
= (λs + χ (α1 − 1)) e1 ⊗ e1 + (λs + χ (α2 − 1)) e2 ⊗ e2 + ' ' (( + λs + χ f − 1 e3 ⊗ e3 ; s = α1 + α2 + f − 3, χ = 2μ + κ
(15.9)
According to (15.6), (15.8), the expressions of Piola stress tensors D± for coatings have the form: D± =
∂W± · A± = (λ± tr (U± − E) E + 2μ± (U± − E)) · A± = ∂U±
= (λ± s± + 2μ± (α1 − 1)) e1 ⊗ e1 + (λ± s± + 2μ± (α2 − 1)) e2 ⊗ e2 + ' ' (( + λ± s± + 2μ± f± − 1 e3 ⊗ e3 ; s± = α1 + α2 + f± − 3
(15.10)
The equilibrium equations of nonlinear micropolar elasticity in the absence of mass forces and moments are written as follows [2, 13] % & divD = 0, divG + CT · D = 0 (15.11) ×
where div is the divergence in the Lagrangian coordinates. The symbol × represents the vector invariant of a second-order tensor: K× = (Kmn em ⊗ en )× = Kmn em × en
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The equilibrium equations for classic non-polar continuum in the absence of mass forces have the form [6]: divD± = 0 (15.12) Boundary conditions e3 · D+ | x3 =a+h+ = 0, e3 · D± | x3 =±a = e3 · D| x3 =±a ,
e3 · D− | x3 =−(a+h− ) = 0
f± (±a) = f (±a) ,
f (0) = 0
(15.13)
express the absence of external loads on the faces of the sandwich plate (x3 = a + h+ , x3 = −a − h−), rigid coupling of top and bottom coatings with the core (x3 = ±a) and the absence of vertical displacement on the plane x3 = 0. By solving the boundary problem (15.11) – (15.13) while taking into account the relations (15.9), (15.10), we found the unknown functions f (x3 ) and f± (x3 ) f (x3 ) = α3 x3 ,
f+ (x3 ) = α+3 (x3 − a) + aα3, α3 = 1 +
λ(2 − α1 − α2 ) , λ+χ
f− (x3 ) = α−3 (x3 + a) − aα3 α±3 = 1 +
λ± (2 − α1 − α2 ) λ± + 2μ±
15.3 The Perturbed State Suppose that in addition to the above-described state of equilibrium for the sandwich plate, there is an infinitely close equilibrium state under the same external loads, which is determined by: 1) for the micropolar core–the radius vector R + ηv, and microrotation tensor H − ηH × ω, 2) for the coatings–the radius vectors R± + ηv± . Here η is a small parameter, v and v± are vectors of additional displacements, ω is a linear incremental rotation vector, which characterizes the small rotation of the particles for micropolar medium, measured from the initial strain state. The perturbed state of equilibrium for the micropolar medium is described by the equations [2]: divD• = 0, divG• + gradvT · D + CT · D• = 0 (15.14) ×
d
D (R + ηv, H − ηH × ω)
dη η=0
d
• G = G (R + ηv, H − ηH × ω)
dη η=0 D• =
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where D• and G• are the linearized Piola-type stress tensor and couple stress tensor. In the case of physically linear micropolar material (15.7) for these tensors the following relations are valid: • & ∂W ∂W • % ' • ( D• = ·H+ · H = λ tr Y E + (μ + κ) Y• + μY•T · H− ∂Y ∂Y (15.15) % % & & T − λtr (Y − E) E + μ Y − E + (μ + κ) (Y − E) · H × ω
∂W G = ∂L •
•
·H+
& ∂W • % ' • ( · H = γ1 tr L E + γ2 L• + γ3 L•T · H− ∂L % & − γ1 (tr L) E + γ2 L + γ3 LT · H × ω
'
(
Y• = gradv + C × ω · HT ,
(15.16)
L• = grad ω · HT
Here Y• is the linearized stretch tensor, L• is the linearized wryness tensor. The equations of neutral equilibrium in the framework of non-polar nonlinear theory of elasticity have the form [6]:
d
div D•± = 0, D•± = D± (R± + ηv± )
(15.17) dη η=0 Representations of the linearized Piola stress tensors D•± for physically linear material (15.8) are obtained by linearization of constitutive relations (15.10) with regard to (15.6): • ' ' ( ( ∂W± ∂W± D•± = · A± + · A•± = λ± trU•± E + 2μ± U•± · A± + ∂U± ∂U± (15.18) ( −1 ' • + (λ± tr (U± − E) E + 2μ± (U± − E)) · U± · gradv± − U± · A± Here U•± are the linearized stretch tensors, which can be expressed in terms of the linearized Cauchy-Green deformation tensors: % &• (U± · U± )• = C± · CT± ⇓
'
U•± · U± + U± · U•± = gradv± · CT± + C± · gradv±
(T
Linearized boundary conditions on the faces of the sandwich plate (x3 = a + h+ , x3 = −a − h−) and on the interfaces (x3 = ±a) are written as follows:
e3 · D•+
x =a+h = 0, e3 · D•−
x =−(a+h ) = 0 + − 3 3 (15.19)
•
•
•
e3 · D± x =±a = e3 · D x =±a , v± | x3 =±a = v| x3 =±a , e3 · G x =±a = 0 3
3
3
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We assume that at the ends of the plate (x1 = 0, b1 ; x2 = 0, b2 ) there is no friction and constant normal displacements are given. This leads to the following linearized boundary conditions: 1) for the porous core of the plate (|x3 | ≤ a):
e1 · D• · e2
x
1 =0, b1
e1 · G• · e1
x
= e1 · D• · e3
x
1 =0, b1
1 =0, b1
e2 · D · e1
x •
2 =0, b2
e2 · G• · e2
x
= e2 · ω| x1 =0, b1 = e3 · ω| x1 =0, b1 = 0 (15.20)
= e2 · D · e3
x •
2 =0, b2
= e1 · v| x1 =0, b1 = 0
2 =0, b2
= e2 · v| x2 =0, b2 = 0
= e1 · ω| x2 =0, b2 = e3 · ω| x2 =0, b2 = 0
2) for the top and bottom coatings (a ≤ x3 ≤ a + h+, −a − h− ≤ x3 ≤ −a):
e1 · D•± · e2
x =0, b = e1 · D•± · e3
x =0, b = e1 · v± |x1 =0, b1 = 0 1 1 1 1
• • e2 · D± · e1 x =0, b = e2 · D± · e3 x =0, b = e2 · v± |x2 =0, b2 = 0 2
2
2
(15.21)
2
We write the vectors of additional displacements v and v± , and incremental rotation ω in the basis of Cartesian coordinates: v = v1 e1 + v2 e2 + v3 e3 v± = v±1 e1 + v±2 e2 + v±3 e3 ω = ω 1 e1 + ω 2 e2 + ω 3 e3
(15.22)
With respect to representation (15.22), the expressions for the linearized stretch tensors Y• and U•± , and wryness tensor L• have the form: ∂v1 ∂v2 ∂v3 Y• = e1 ⊗ e1 + − α1 ω 3 e1 ⊗ e2 + + α1 ω 2 e1 ⊗ e3 + ∂x1 ∂x1 ∂x1 ∂v1 ∂v2 ∂v3 + + α2 ω 3 e2 ⊗ e1 + e2 ⊗ e2 + − α2 ω 1 e2 ⊗ e3 + (15.23) ∂x2 ∂x2 ∂x2 ∂v1 ∂v2 ∂v3 + − α3 ω 2 e3 ⊗ e1 + + α3 ω 1 e3 ⊗ e2 + e3 ⊗ e3 ∂x3 ∂x3 ∂x3 ∂v±2 ∂v±3 1 ± (e2 ⊗ e3 + e3 ⊗ e2 ) + α + α 2 3 ∂x1 α2 + α±3 ∂x3 ∂x2 ∂v±3 ∂v±2 ∂v±1 1 ± (e1 ⊗ e3 + e3 ⊗ e1 ) + + e2 ⊗ e2 + α1 + α3 ∂x2 α1 + α±3 ∂x3 ∂x1 ∂v±3 ∂v±1 ∂v±2 1 (e1 ⊗ e2 + e2 ⊗ e1 ) + e3 ⊗ e3 + α1 + α2 ∂x3 α1 + α2 ∂x2 ∂x1
U•± =
∂v±1
e1 ⊗ e1 +
(15.24)
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∂ω 1 ∂ω 2 ∂ω 3 ∂ω 1 e1 ⊗ e1 + e1 ⊗ e2 + e1 ⊗ e3 + e2 ⊗ e1 + ∂x1 ∂x1 ∂x1 ∂x2 ∂ω 2 ∂ω 3 ∂ω 1 ∂ω 2 ∂ω 3 + e2 ⊗ e2 + e2 ⊗ e3 + e3 ⊗ e1 + e3 ⊗ e2 + e3 ⊗ e3 ∂x2 ∂x2 ∂x3 ∂x3 ∂x3 L• =
(15.25)
According to relations (15.3) – (15.6), (15.15), (15.16), (15.18), (15.22) – (15.24), the components of the linearized Piola-type stress tensor D• and couple stress tensor G• , and Piola stress tensors D•± are written as follows: e1 · D• · e1 = (λ + χ)
∂v1 ∂v2 ∂v3 +λ +λ ∂x1 ∂x2 ∂x3
∂v2 ∂v1 +μ +(λs + μ(α1 + α2 )− χ)ω3 ∂x1 ∂x2 ∂v3 ∂v1 e1 · D• · e3 = (μ + κ) +μ − (λs + μ (α1 + α3 ) − χ) ω2 ∂x1 ∂x3 ∂v1 ∂v2 e2 · D• · e1 = (μ + κ) +μ −(λs + μ(α1 + α2 ) − χ) ω3 ∂x2 ∂x1 ∂v1 ∂v2 ∂v3 e2 · D• · e2 = λ + (λ + χ) +λ ∂x1 ∂x2 ∂x3 ∂v3 ∂v2 e2 · D• · e3 = (μ + κ) +μ + (λs + μ (α2 + α3 ) − χ) ω1 ∂x2 ∂x3 ∂v1 ∂v3 e3 · D• · e1 = (μ + κ) +μ + (λs + μ (α1 + α3 ) − χ) ω2 ∂x3 ∂x1 ∂v2 ∂v3 e3 · D• · e2 = (μ + κ) +μ − (λs + μ (α2 + α3 ) − χ) ω1 ∂x3 ∂x2 ∂v1 ∂v2 ∂v3 e3 · D• · e3 = λ +λ + (λ + χ) ∂x1 ∂x2 ∂x3 e1 · D• · e2 = (μ + κ)
∂ω1 ∂ω2 ∂ω3 + γ1 + γ1 ∂x1 ∂x2 ∂x3 ∂ω ∂ω ∂ω ∂ω 2 1 1 2 e1 · G• · e2 = γ2 + γ3 , e2 · G• · e1 = γ2 + γ3 ∂x1 ∂x2 ∂x2 ∂x1 ∂ω3 ∂ω1 ∂ω1 ∂ω3 e1 · G• · e3 = γ2 + γ3 , e3 · G• · e1 = γ2 + γ3 ∂x1 ∂x3 ∂x3 ∂x1 ∂ω ∂ω ∂ω 1 2 3 e2 · G• · e2 = γ1 + (γ1 + γ2 + γ3 ) + γ1 ∂x1 ∂x2 ∂x3 ∂ω3 ∂ω2 ∂ω2 ∂ω3 e2 · G• · e3 = γ2 + γ3 , e3 · G• · e2 = γ2 + γ3 ∂x2 ∂x3 ∂x3 ∂x2 ∂ω ∂ω ∂ω 1 2 3 e3 · G• · e3 = γ1 + γ1 + (γ1 + γ2 + γ3 ) ∂x1 ∂x2 ∂x3
(15.26)
e1 · G• · e1 = (γ1 + γ2 + γ3 )
(15.27)
15 On Stability of Elastic Rectangular Sandwich Plate Subject to Biaxial Compression
∂v ∂v1 ∂v + λ± 2 + λ± 3 ∂x1 ∂x2 ∂x3 ± ± ± ∂v ∂v ∂v λ s − 2μ ± ± ± 2 e1 · D•± · e2 = 2μ± 2 + − 1 ∂x1 α1 + α2 ∂x1 ∂x2 ± ± ∂v λ± s± − 2μ± ∂v3 ∂v±1 e1 · D•± · e3 = 2μ± 3 + − ∂x1 α1 + α±3 ∂x1 ∂x3 ∂v±1 λ± s± − 2μ± ∂v±1 ∂v±2 • e2 · D± · e1 = 2μ± + − ∂x2 α1 + α2 ∂x2 ∂x1 ∂v± ∂v± ∂v± e2 · D•± · e2 = λ± 1 + (λ± + 2μ± ) 2 + λ± 3 ∂x1 ∂x2 ∂x3 ± ± ± ∂v ∂v ∂v λ s − 2μ ± ± ± 3 3 • e2 · D± · e3 = 2μ± + − 2 ∂x2 α2 + α±3 ∂x2 ∂x3 ∂v± λ± s± − 2μ± ∂v±1 ∂v±3 e3 · D•± · e1 = 2μ± 1 + − ∂x3 α1 + α±3 ∂x3 ∂x1 ∂v± λ± s± − 2μ± ∂v±2 ∂v±3 e3 · D•± · e2 = 2μ± 2 + − ∂x3 α2 + α±3 ∂x3 ∂x2 ∂v± ∂v± ∂v± e3 · D•± · e3 = λ± 1 + λ± 2 + (λ± + 2μ± ) 3 ∂x1 ∂x2 ∂x3 e1 · D•± · e1 = (λ± + 2μ± )
±
±
211
±
(15.28)
Expressions (15.14), (15.17), describing the perturbed state of equilibrium for a porous sandwich plate, constitute a system of twelve partial differential equations in the twelve unknown functions v1 , v2 , v3 , v±1 , v±2 , v±3 , ω1 , ω2 , ω3 . Substitutions v1 = V1 (r) sin β1 x1 cos β2 x2 , v2 = V2 (r) cos β1 x1 sin β2 x2 , v3 = V3 (r) cosβ1 x1 cos β2 x2 ,
v±1 = V1± (r) sin β1 x1 cos β2 x2 v±2 = V2± (r) cos β1 x1 sin β2 x2 v±3 = V3± (r) cos β1 x1 cos β2 x2
ω1 = Ω1 (r) cos β1 x1 sin β2 x2 ω2 = Ω2 (r) sin β1 x1 cos β2 x2 ω3 = Ω3 (r) sin β1 x1 sin β2 x2 β1 = πm1 /b1 ,
β2 = πm2 /b2 ,
(15.29)
(15.30)
m1,2 = 1, 2, ...
lead to the separation of variables x1 , x2 in these equations and allows to satisfy the linearized boundary conditions (15.20), (15.21) at the ends of the plate. By taking into account the relations (15.4), (15.9), (15.22), (15.26)–(15.29), equations of neutral equilibrium (15.14), (15.17) are written as follows:
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% % && (μ + κ) V1 − (λ + μ) β21 + (μ + κ) β21 + β22 V1 − − (λ + μ)β1 β2 V2 − (λ + μ) β1 V3 + B2Ω2 − B3 β2 Ω3 = 0 % % && (μ + κ) V2 − (λ + μ) β22 + (μ + κ) β21 + β22 V2 − − (λ + μ)β1 β2 V1 − (λ + μ) β2 V3 − B1Ω1 + B3 β1 Ω3 = 0 % & (λ + χ) V3 − (μ + κ) β21 + β22 V3 + (λ + μ) β1 V1 + + (λ + μ)β2 V2 + B1β2 Ω1 − B2β1 Ω2 = 0
(15.31)
% % & & 2 2 2 γ2 Ω 1 − (γ1 + γ3 ) β1 + γ2 β1 + β2 − B1 (α2 + α3 ) Ω1 − − (γ1 + γ3 )β1 β2 Ω2 + (γ1 + γ3 ) β1 Ω3 + B1V2 + B1β2 V3 = 0 % % & & 2 2 2 γ2 Ω 2 − (γ1 + γ3 ) β2 + γ2 β1 + β2 − B2 (α1 + α3 ) Ω2 − − (γ1 + γ3 )β1 β2 Ω1 + (γ1 + γ3 ) β2 Ω3 − B2V1 − B2β1 V3 = 0 % % & & 2 2 (γ1 + γ2 + γ3 ) Ω 3 − γ2 β1 + β2 − B3 (α1 + α2 ) Ω3 − − (γ1 + γ3 )β1 Ω1 − (γ1 + γ3 ) β2 Ω2 − B3 β2 V1 + B3β1 V2 = 0 %
& % % & & 2μ+ + B+2 (V1+ ) − (λ+ + 2μ+ ) β21 + 2μ+ + B+3 β22 V1+ − % & % & − λ+ − B+3 β1 β2 V2+ − λ+ − B+2 β1 (V3+ ) = 0
%
& % % & & 2μ+ + B+1 (V2+ ) − (λ+ + 2μ+ ) β22 + 2μ+ + B+3 β21 V2+ − % & % & − λ+ − B+3 β1 β2 V1+ − λ+ − B+1 β2 (V3+ ) = 0
(15.32)
%% & % & & (λ+ + 2μ+ ) (V3+ ) − 2μ+ + B+2 β21 + 2μ+ + B+1 β22 V3+ + % & % & + λ+ − B+2 β1 (V1+ ) + λ+ − B+1 β2 (V2+ ) = 0 %
& % % & & 2μ− + B−2 (V1− ) − (λ− + 2μ− ) β21 + 2μ− + B−3 β22 V1− − % & % & − λ− − B−3 β1 β2 V2− − λ− − B−2 β1 (V3− ) = 0
%
& % % & & 2μ− + B−1 (V2− ) − (λ− + 2μ− ) β22 + 2μ− + B−3 β21 V2− − % & % & − λ− − B−3 β1 β2 V1− − λ− − B−1 β2 (V3− ) = 0
(15.33)
15 On Stability of Elastic Rectangular Sandwich Plate Subject to Biaxial Compression
213
%% & % & & (λ− + 2μ− ) (V3− ) − 2μ− + B−2 β21 + 2μ− + B−1 β22 V3− + % & % & + λ− − B−2 β1 (V1− ) + λ− − B−1 β2 (V2− ) = 0 Here we use the following notations B1 = λ (α1 − 3) + (λ + μ) (α2 + α3 ) − χ B2 = λ (α2 − 3) + (λ + μ) (α1 + α3 ) − χ B3 = λ (α3 − 3) + (λ + μ) (α1 + α2 ) − χ B±1 =
λ± s± − 2μ± , α2 + α±3
B±2 =
λ± s± − 2μ± , α1 + α±3
B±3 =
λ± s± − 2μ± α1 + α2
The expressions for the linearized boundary conditions (15.19) take the form: 1) for x3 = a + h+: % & % & % & % & 2μ+ + B+2 V1+ + β1 B+2 V3+ = 0, 2μ+ + B+1 V2+ + β2 B+1 V3+ = 0 (15.34) % & λ+ β1 V1+ + λ+ β2 V2+ + (λ+ + 2μ+ ) V3+ = 0 2) for x3 = a: % & % & 2μ+ + B+2 V1+ + β1 B+2 V3+ − (μ + κ) V1 + β1 μV3 − B2Ω2 = 0 % & % & 2μ+ + B+1 V2+ + β2 B+1 V3+ − (μ + κ) V2 + β2 μV3 + B1Ω1 = 0 % & λ+ β1 V1+ + λ+ β2 V2+ + (λ+ + 2μ+ ) V3+ − λβ1 V1 − −λβ2 V2 − (λ + χ) V3 = 0
γ2 Ω1 + γ3 β1 Ω3 = 0,
(15.35)
γ2 Ω2 + γ3 β2 Ω3 = 0 γ1 β1 Ω1 + γ1 β2 Ω2 − (γ1 + γ2 + γ3 ) Ω3 = 0 V1 − V1+ = 0, V2 − V2+ = 0, V3 − V3+ = 0
3) for x3 = −a: % & % & 2μ− + B−2 V1− + β1 B−2 V3− − (μ + κ) V1 + β1 μV3 − B2Ω2 = 0 % & % & 2μ− + B−1 V2− + β2 B−1 V3− − (μ + κ) V2 + β2 μV3 + B1Ω1 = 0 % & λ− β1 V1− + λ− β2 V2− + (λ− + 2μ− ) V3− − λβ1 V1 − −λβ2 V2 − (λ + χ) V3 = 0
γ2 Ω1 + γ3 β1 Ω3 = 0,
γ2 Ω2 + γ3 β2 Ω3 = 0 γ1 β1 Ω1 + γ1 β2 Ω2 − (γ1 + γ2 + γ3 ) Ω3 = 0 V1 − V1− = 0, V2 − V2− = 0, V3 − V3− = 0
(15.36)
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4) for x3 = −a − h−: % & % & 2μ− + B−2 V1− + β1 B−2 V3− = 0,
%
& % & 2μ− + B−1 V2− + β2 B−1 V3− = 0 % & λ− β1 V1− + λ− β2 V2− + (λ− + 2μ− ) V3− = 0
(15.37)
Thus, the stability analysis of the sandwich plate is reduced to solving a linear homogeneous boundary problem (15.31) – (15.37) for a system of twelve ordinary differential equations. It is easy to show that in the case of identical top and bottom coatings (h+ = h− , λ+ = λ− , μ+ = μ− ) the boundary problem (15.31) – (15.37) has two independent sets of solutions [10]. The First set is formed by solutions for which the deflection of a plate is an odd function of x3 :
0 ≤ x3 ≤ a
a ≤ x3 ≤
H 2
⎧ ⎪ ⎪ V1 (x3 ) = V1 (−x3 ), ⎪ ⎪ ⎪ ⎨ V2 (x3 ) = V2 (−x3 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ V (x ) = −V (−x ), 3 3 3 3
Ω1 (x3 ) = −Ω1 (−x3 ) Ω2 (x3 ) = −Ω2 (−x3 ) Ω3 (x3 ) = Ω3 (−x3 ) (15.38)
⎧ + ⎪ ⎪ V1 (x3 ) = V1− (−x3 ) ⎪ ⎪ ⎪ ⎨ + V2 (x3 ) = V2− (−x3 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ V + (x ) = −V − (−x ) 3 3 3 3
For the Second set of solutions, on the contrary, the deflection is an even function:
0 ≤ x3 ≤ a
⎧ ⎪ V1 (x3 ) = −V1 (−x3 ), ⎪ ⎪ ⎪ ⎪ ⎨ V2 (x3 ) = −V2 (−x3 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ V (x ) = V (−x ), 3
a ≤ x3 ≤
H 2
3
3
3
Ω1 (x3 ) = Ω1 (−x3 ) Ω2 (x3 ) = Ω2 (−x3 ) Ω3 (x3 ) = −Ω3 (−x3 )
⎧ + ⎪ V1 (x3 ) = −V1− (−x3 ) ⎪ ⎪ ⎪ ⎪ ⎨ + V2 (x3 ) = −V2− (−x3 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ V + (x ) = V − (−x ) 3 3 3 3
(15.39)
Due to this property of boundary problem (15.31) – (15.37), for the study of stability it is sufficient to consider only the upper half of the sandwich plate (0 ≤ x3 ≤ H/2). The boundary conditions at x3 = 0 follows from the evenness and oddness of the unknown functions V1 , V2 , V3 , Ω1 , Ω2 , Ω3 : a) for the First set of solutions: V1 (0) = V2 (0) = V3 (0) = 0,
Ω1 (0) = Ω2 (0) = Ω3 (0) = 0
(15.40)
15 On Stability of Elastic Rectangular Sandwich Plate Subject to Biaxial Compression
215
b) for the Second set of solutions: V1 (0) = V2 (0) = V3 (0) = 0,
Ω1 (0) = Ω2 (0) = Ω3 (0) = 0
(15.41)
Thus, in the case of identical top and bottom coatings, the stability analysis of the sandwich plate is reduced to solving two linear homogeneous boundary-value problems – (15.31), (15.32), (15.34), (15.35), (15.40) and (15.31), (15.32), (15.34), (15.35), (15.41) – for a system of nine ordinary differential equations.
15.4 Conclusion In the framework of bifurcation approach, we studied the stability of an elastic rectangular sandwich plate with a porous core, which is subject to biaxial compression. For the physically linear material, a system of linearized equilibrium equations was derived, which describes the behavior of the sandwich plate in a perturbed state. Using special substitutions, the linearized boundary-value problems were formulated both for a case of a general sandwich plate and for a sandwich plate with identical top and bottom coatings. In the general case, the stability analysis was reduced to solving a linear homogeneous boundary problem (15.31) – (15.37) for a system of twelve ordinary differential equations. In the case of identical top and bottom coatings, the stability analysis of the sandwich plate was reduced to solving two linear homogeneous boundary-value problems – (15.31), (15.32), (15.34), (15.35), (15.40) and (15.31), (15.32), (15.34), (15.35), (15.41) – for a system of nine ordinary differential equations. For specific porous materials and coatings these boundary-value problems can be solved numerically, using methods described in [11]. Acknowledgements This work was supported by the President of Russian Federation (grant MK6315.2010.1) and by the Russian Foundation for Basic Research (grant 11-08-01152-a).
References 1. Cosserat, E., Cosserat, F.: Theorie des Corps Deformables. Hermann et Fils, Paris (1909) 2. Eremeyev, V.A., Zubov, L.M.: On stability of elastic bodies with couple-stresses. Mekhanika Tverdovo Tela (3), 181–190 (1994). In Russian 3. Eringen, A.C.: Microcontinuum Field Theory. I. Foundations and Solids. Springer, New York (1999) 4. Kafadar, C.B., Eringen, A.C.: Micropolar media - I. The classical theory. International Journal of Engineering Science 9, 271–305 (1971) 5. Lakes, R.: Experimental methods for study of Cosserat elastic solids and other generalized elastic continua. In: Muhlhaus, H., Wiley, J. (ed.) Continuum models for materials with microstructure, pp. 1–22. New York (1995) 6. Lurie, A.I.: Non-linear Theory of Elasticity. North-Holland, Amsterdam (1990) 7. Maugin, G.A.: On the structure of the theory of polar elasticity. Philosophical Transactions of Royal Society London A 356, 1367–1395 (1998)
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8. Nikitin, E., Zubov, L.M.: Conservation laws and conjugate solutions in the elasticity of simple materials and materials with couple stress. Journal of Elasticity 51, 1–22 (1998) 9. Pietraszkiewicz, W., Eremeyev, V.A.: On natural strain measures of the non-linear micropolar continuum. International Journal of Solids and Structures 46, 774–787 (2009) 10. Sheydakov, D.N.: Stability of a rectangular plate under biaxial tension. Journal of Applied Mechanics and Technical Physics 48(4), 547–555 (2007) 11. Sheydakov, D.N.: Buckling of elastic composite rod of micropolar material subject to combined loads. In: H. Altenbach, V.I. Erofeev, G.A. Maugin (ed.) Mechanics of Generalized Continua – From Micromechanical Basics to Engineering Applications, pp. 255–271. SpringerVerlag, Berlin et al. (2011) 12. Toupin, R.A.: Theories of elasticity with couple-stress. Archives for Rational Mechanics and Analysis 17, 85–112 (1964) 13. Zubov, L.M.: Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies. Springer, Berlin (1997)
Part III
Nonlinear Models and Coupled Fields
Chapter 16
On the Nonlinear Theory of Two-Phase Shells Victor A. Eremeyev and Wojciech Pietraszkiewicz
Abstract We discuss the nonlinear theory of shells made of material undergoing phase transitions (PT). The interest to such thin-walled structures is motivated by applications of thin films made of martensitic materials and needs of modeling biological membranes. Here we present the resultant, two-dimensional thermodynamics of non-linear theory of shells undergoing PT. The global and local formulations of the balances of momentum, moment of momentum, energy and entropy are given. Two temperature fields on the shell base surface are introduced: the referential mean temperature and its deviation, as well as two corresponding dual fields: the referential entropy and its deviation. Additional surface heat flux and the extra heat flux vector fields appear as a result of through-the-thickness integration procedure. Within the framework of the resultant shell thermodynamics we derive the continuity conditions along the curvilinear phase interface which separates two material phases. These conditions allow us to formulate the kinetic equation describing the quasistatic motion of the interface relative to the shell base surface. The kinetic equation is expressed by the jump of the Eshelby tensor across the phase interface. In the case of thermodynamic equilibrium the variational statement of the static problem of two-phase shell is presented. Keywords Non-linear shell · Shell thermodynamics · Phase transition · Cosserat shell · Micropolar shell · Kinetic equation · Singular curve
V. A. Eremeyev (B) Martin-Luther-Universit¨at Halle-Wittenberg, Kurt-Mothes-Str., 1, 06099 Halle (Saale), Germany and South Scientific Center of RASci & South Federal University, Milchakova St. 8a, 344090 Rostov on Don, Russia e-mail:
[email protected],
[email protected] W. Pietraszkiewicz Institute of Fluid-Flow Machinery of the Polish Academy of Sciences, ul. Gen. J. Fiszera 14, 80-952 Gda´nsk, Poland e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 16, © Springer-Verlag Berlin Heidelberg 2011
219
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16.1 Introduction The interest in thin-walled structures undergoing PT grows recently from their perspective applications in engineering. As examples of such structures martensitic films and biological membranes can be considered. The stress- and temperatureinduced PT are widely observed in thin structures made of superelastic shape memory alloys (SMA) and shape memory polymers, such as NiTi, NiMnGa, AgCd, AuCd, CuAlNi, polyurethane, etc, which are used in various microelectromechanical systems (MEMS). Thin plates, strips, and tubes made of SMA are used as working elements of such MEMS as micropumps, sensors, actuators, microengines etc., see e.g. [8, 48, 53] and books [5, 32, 38]. Experiments on shape memory alloys and other materials undergoing PT are often performed with thin-walled samples such as thin strips, rectangular plates or thin tubes, see [25–28, 33, 41, 52, 56] among others. Other examples of PT in thinwalled structures are tents and tunnels appearing in martensitic thin films during PT, which were discovered and investigated in [6, 8, 24, 29, 47, 48]. The major known theories of PT in deformable solids are related to three-dimensional (3D) thermoelasticity, see the books [1, 4, 5, 22] and references given therein. The first two-dimensional (2D) mechanical model of PT in thin films was proposed in [7, 30, 47], see also [5, 38]. The model consists of the Cosserat membrane with one director, but without taking into account the bending stiffness of the membrane. Alternative membrane models of PT with applications to biomembrane modelling were proposed in [2, 10, 12]. The non-linear resultant equilibrium conditions of elastic shells undergoing PT of martensitic type were formulated by Eremeyev and Pietraszkiewicz [13] within the resultant dynamically exact and kinematically unique theory of shells presented in [11, 18, 34]. These conditions were extended in [44] taking into account the line tension energy of the interface. By analogy to the 3D case, the two-phase shell was regarded in [13,44] as the Cosserat surface consisting of two material phases divided by a sufficiently smooth surface singular curve (phase interface). Existence of such a curve was confirmed by several experiments on thin-walled plates, strips, and tubes, see e.g. [25–28,33,41,52,56]. These experiments demonstrate how the macroscopic domain of the new phase forms, show its further evolution during loading and annihilation after unloading. In the case of plates and strips the new phase forms often as a few bands across the strip. In the case of tubes the new phase may appear as helical or cylindrical bands which width and shape depend on acting loads. The phase boundary between “old” and “new” phases in many cases can be interpreted as a curvilinear, coherent, sharp phase interface. The quasistatic behaviour of twophase shells has recently been analyzed in [15–17]. In this paper we discuss the resultant, two-dimensional thermomechanics of shells undergoing diffusionless, displacive phase transitions of martensitic type of the shell material. In particular, we formulate the corresponding boundary-value problem (BVP). The main attention is paid to formulation of the compatibility conditions across the phase interface and to derivation of the kinetic equation describing propagation of the phase interface during loading and unloading.
16 On the Nonlinear Theory of Two-Phase Shells
221
16.2 Kinematics of 6-Parametric Theory of Shells The kinematics of general resultant theory of shells coincides with the kinematics of 2D Cosserat continuum, see [11, 18, 34] for details. In the undeformed placement the shell is represented by the base surface M with the position vector x(θα ) and the unit normal vector n(θα ), where {θα }, α = 1, 2, are the surface curvilinear coordinates. In the deformed placement the shell is represented by the surface N = χ(M) with the position vector y = χ(x) and with the attached three directors (dα , d). The deformation of the shell is described by the relations y(x, t) = χ(x) = x + u(x, t),
dα (x, t) = Q(x, t)x,α ,
d(x, t) = Q(x, t)n(x),
(16.1)
where t is a time-like scalar parameter, χ the deformation function, u ∈ E the translation vector of M, and Q ∈ S O(3) the proper orthogonal tensor representing the workaveraged gross rotation of the shell cross sections from their undeformed shapes described by (x,α , n), where (. . .),α denotes partial differentiation with respect to θα . ˙ T ) the angular velocity vecThen υ ≡ u˙ is the translation velocity and ω ≡ ax(QQ tors, where ax(. . .) is the axial vector associated with the skew tensor (. . .), and (. ˙. .) denotes the derivative with respect to t. Within the framework of 6-parametric theory of shells considered here, the following two strain measures corresponding to the deformations (16.1) are introduced, see [11, 13, 14, 43]: E = ε α ⊗ aα ,
K = κ α ⊗ aα ,
εα = y,α − dα ,
1 κα = di × Q,α QT d i , 2
(16.2)
where (aα , n) and (di ), i = 1, 2, 3, are bases reciprocal to the base (x,α , n) and the base (dα , d), respectively. We assume that in the deformed placement the shell consists of different material phases occupying different complementary subregions separated by the curvilinear phase interface D ∈ N. For a piecewise differentiable mapping χ we can introduce on M a singular image curve C = χ−1 (D) with the position vector xC . We call a priori unknown curves D and C the phase interfaces in the deformed and reference placements, respectively. Let us note that xC and yD are kinematically independent on u and Q. This means that D and C are non-material curves, in general. For the description of motion of the surface curve C on M we introduce the phase interface velocity V ≡ x˙ C · ν , where ν ∈ T x M is the unit external normal vector to C, and ν · n = 0. Hence, y (or u), Q, and xC constitute the basic kinematic unknown variables in the theory of shells undergoing PT.
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16.3 Integral Balance Equations The resultant 2D equations of the general non-linear theory of shells can be derived exactly by direct through-the-thickness integration of 3D balance laws of linear and angular momentum as well as of the energy balance and the entropy inequality of continuum mechanics, see [11, 13, 17, 34]. In quasi-static problems discussed here the global equilibrium conditions require the total force and total torque of all loads acting upon any part P ⊂ M\C to vanish F = 0,
where F≡ M≡
f da +
P
∂P\∂M f
(16.3)
n∗ ds,
nν ds +
∂P\∂M f
(c + y × f ) da + P
M = 0,
∂P∩∂M f
(mν + y × nν ) ds +
' ∗ ( m + y × n∗ ds.
∂P∩∂M f
Here f and c are the resultant surface force and couple vector fields acting on N\D, but measured per unit area of M\C. Similarly, nν and mν are the internal contact stress and couple resultant vectors defined at an arbitrary edge ∂R of R = χ(P), while n∗ and m∗ are the external boundary resultant force and couple vectors applied along the part ∂N f of N = χ(M), respectively. The latter four vectors are measured per unit length of the corresponding undeformed edges ∂P and ∂M f , respectively. According to the Cauchy postulate, the contact vectors nν and mν can be represented through the respective internal surface stress and couple resultant tensors N and M by nν = Nνν, mν = Mνν. (16.4) The tensors N, M ∈ E ⊗ T x M defined on M\C are the resultant surface stress measures of the 1st Piola–Kirchhoff type, respectively. In the literature various descriptions of shell thermodynamics are known, see e.g. [15, 18, 20, 21, 35, 39, 40, 45, 46, 49–51, 57], where various sets of surface fields responsible for temperature were used and several formulations of the first and second laws of thermodynamics for shells were discussed. The resultant local thermomechanic energy balance and the entropy inequality for the shell can also be derived by direct through-the-thickness integration of the global 3D thermomechanic balance of energy and the entropy inequality, see [35, 49–51]. However, in the construction of 2D thermodynamic relations for shells one cannot simply transfer the notions of temperature, entropy and energy from the 3D case to their averages defined on M. For example, in the 3D shell-like body it is quite natural to associate different temperatures with its lower, upper and lateral boundary surfaces. In the resultant shell model this leads to appearance of several 2D temperature fields defined in the same point of the base surface. Each such field may require
16 On the Nonlinear Theory of Two-Phase Shells
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its own heat conduction law. As a result, the 2D entropy inequality for shells should also be appropriately modified. Similarly, the 3D stress power density of the shell-like body takes into account also a part of stress power not expressible in terms of 2D stress resultants and stress couples, see [43], Sect. 7. In the 2D energy balance equation this may require to introduce additional surface sources of energy with their own constitutive relations, see discussion in [35, 42]. The referential form of energy balance (The 1st Law of thermomechanics) of an arbitrary part P of the shell base surface M\C can be described in analogy to the 3D energy balance, see [54, 55], by the resultant quantities [35] as ˙ + E˙ = A + Q, K
(16.5)
where K is the resultant kinetic energy, E the resultant internal energy, A the resultant mechanical power, and Q is the resultant heating. For the quasistatic process discussed here K = 0, while E, A, and Q can be represented by E≡ ρ# da, A ≡ (f · υ + c · ω ) da + (nν · υ + mν · ω ) ds P
P
∂P\∂M f
(n∗ · υ + m∗ · ω ) ds,
+ Q≡
∂P∩∂M f
ρr da + P
qν ds +
∂P\∂Mh
q∗ ds,
∂P∩∂Mh
where ρ is the resultant surface mass density in undeformed placement, ε the internal resultant surface strain energy density per unit undeformed surface mass, and r the internal surface heat supply minus heat fluxes through the upper and lower shell faces, all per unit mass of M, qν and q∗ are the surface heat fluxes through ∂P and ∂Mh , respectively. The contact heat flux qν can be represented through the surface heat flux vector q by the formula qν = q · ν . The referential form of entropy inequality (The 2nd Law of thermomechanics) of an arbitrary part P of the shell base surface M\C follows from the Clausius-Duhem inequality [54, 55], ˙ ≥ J, H (16.6) where in our case H is the resultant shell entropy and J the resultant entropy supply. For any part P ⊂ M\C these fields are defined as follows: H≡ ρη da, J ≡ ρ j da + jν ds + j∗ ds, P
P
∂P\∂Mh
∂P∩∂Mh
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where η is the resultant internal entropy density, j the resultant entropy supply minus entropy fluxes through the upper and lower shell faces, both per unit undeformed surface mass, and jν and j∗ are the resultant entropy fluxes through the internal ∂P and external ∂Mh boundary contours, respectively. The field jν can be expressed through the referential entropy flux vector j ∈ T x M according to jν = j · ν. The relations between the resultant quantities and their 3D counterparts can be derived by use of the through-the-thickness integration procedure applied to the 3D balance equations [17]. After [39] we introduce the mean referential temperature θ(x, t) > 0 and the temperature deviation ϕ(x, t) by 1 1 1 1 1 1 1 = + , ϕ= − , (16.7) θ 2 Θ+ Θ− h Θ− Θ+ where Θ± > 0 are temperatures of the upper and lower shell faces M± taken to be equal to those prevailing in the adjoining external media, and h is the shell thickness. Unlike in the 3D entropy balance [54, 55], the resultant entropy supply j and the resultant entropy flux j take now the extended form [17], 1 j = r − ϕs, θ
1 j = q − ϕs, θ
(16.8)
where s is the resultant extra heat supply and s is the resultant extra heat flux vector.
16.4 Local Shell Equations and Constitutive Relations From the integral equilibrium equations (16.3), the energy balance equation (16.5) and the entropy inequality (16.6), after appropriate transformations follow the local Lagrangian equilibrium equations and the static boundary conditions % & Div N + f = 0, DivM + ax NFT − FNT + c = 0 in M\C, (16.9) Nνν − n∗ = 0,
Mνν − m∗ = 0 along ∂M f ,
the local resultant thermomechanic balances of energy in the referential description ρε˙ = ρr − Div q + N • E◦ + M • K◦ q · ν − q∗ = 0
in M\C,
along ∂Mh ,
(16.10)
16 On the Nonlinear Theory of Two-Phase Shells
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and the local resultant entropy inequalities r 1 1 ρ˙η − ρ − ϕs + Div q − ϕDiv s + h · s − 2 q · g ≥ 0 in M\C, θ θ θ q q∗ ν ∗ ∗ − ϕ s − − ϕs ≥ 0 along ∂Mh , ν θ∗ θ g = Grad θ,
h = Grad ϕ,
(16.11)
g, h ∈ T x M,
of the non-linear theory of shells. Here F ≡ Grad y = y,α ⊗ aα is the surface deformation gradient, F ∈ E ⊗ T x M, Div N ≡ N ,α · aα means the surface divergence of N, (·)◦ ≡ Q dtd [QT (·)] is the co-rotational time derivative, and the scalar product of two tensors A, B ∈ E ⊗ T x M is defined by A • B ≡ tr (AT B). The fields u, Q, θ, ϕ constitute the basic thermo-kinematic independent variables of the shell boundary value problem in M\C, while the fields N, M, ε, η, χ, q, and s have to be specified by the constitutive equations. Here, as an example we discuss the constitutive equations for thermoelastic shells which take the form [17], ψ ≡ ε − θη − ϕχ = ψ(E, K, θ, ϕ), N = ρψ,E ,
M = ρψ,K ,
q = q(E, K, θ, g, ϕ, h),
η = −ψ,θ ,
χ = −ψ,ϕ ,
(16.12)
s = s(E, K, θ, g, ϕ, h),
where we have introduced the surface free energy density ψ. For thermoelastic shells the local energy balance equation (16.10) reduces to the form ρ(θη˙ + ϕχ) ˙ = ρr − Div q, (16.13) while the local entropy inequality (16.11) results in the equation − ρχ˙ + ρθs − θDiv s = cϕ,
c ≥ 0,
(16.14)
where the new constitutive function c is introduced, and the reduced dissipation inequality becomes 1 − g · q − θh · s ≥ 0. (16.15) θ Both relations (16.13) and (16.14) play the role of thermoconductivity equations in the theory of thermoelastic shells. The two equations are necessary to determine two fields: the surface mean temperature θ and the surface temperature deviation ϕ. When s = 0, the equation (16.14) contains as the special case the equation for temperature deviation established for thermoelastic beams in [50]. The simplest cases of the constitutive equations for q and s satisfying (16.15) may be taken similar to the referential Fourier law in 3D continuum mechanics: q = −c g,
s = −c⊥ h,
(16.16)
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where c is the positive heat conductivity of the shell in tangential direction and c⊥ is the positive heat conductivity of the shell in the transverse normal direction.
16.5 Continuity Conditions Along Phase Interface and Kinetic Equation Since all fields defined on M can be discontinuous across C, the phase interface C can be considered as a surface non-material singular curve. In particular, the curvilinear phase interfaces in shells can be either coherent or incoherent in rotations, see [13]. For the coherent interface both fields y (or u) and Q are supposed to be continuous at C and the kinematic compatibility conditions along C become [[υυ]] + V[[Fνν]] = 0, ω]] + V[[ Kνν ]] = 0, [[ω
(16.17) (16.18)
where the expression [[. . .]] = (. . .)B − (. . .)A means the jump at C. The phase interface is called incoherent in rotations if only y (or u) is continuous at C but Q may be discontinous. In this case the condition (16.17) is still satisfied, but (16.18) may be violated, see [13]. Assuming [[y]] = 0 along C, from (16.3) we obtain the local Lagrangian dynamic compatibility conditions [11], [[Nνν]] = 0,
[[Mνν]] = 0,
(16.19)
which are just the local balances of forces and couples at C in the case of quasistatic deformations. Additionally, we assume that the surface temperature field θ and its deviation ϕ are continuous on the whole M, that is [[θ]] = 0,
[[ϕ]] = 0
along C.
(16.20)
The local energy balance and the entropy inequality along C corresponding to (16.5) and (16.6) are [17], V[[ρε]] + [[Nνν · υ ]] + [[Mνν · ω ]] − [[q · ν ]] = 0, .. // 1 V[[ρη]] − q · ν + ϕs · ν ≡ δ2C ≥ 0 , θ
(16.21) (16.22)
where δ2C ≥ 0 denotes the surface entropy production along C. The second thermoconductivity equation (16.14) leads to the relation along C, 1 V [[ρχ]] − [[s· ν ]] = 0 . θ
(16.23)
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From (16.17)–(16.23) we obtain the compatibility condition in the form 3 4 θδ2C = −V [[ρψ]] − ν · NT [[Fνν]] − ν · MT [[Kνν]] along C (16.24) for the coherent phase interface, and 3 4 θδ2C = −V [[ρψ]] − ν · NT [[Fνν]]
along C
(16.25)
for the phase interface incoherent in rotations. The entropy production δ2C remains always non-negative for all thermomechanical processes. This allows us to postulate the kinetic equation, describing motion of the phase interface for all quasistatic processes, in the form V = −F (νν · [[C]]νν) ,
(16.26)
where F is the non-negative definite kinetic function depending on the jump of C at C, i.e. F(ς) ≥ 0 for ς > 0, where C = Cc ≡ ρψA − N T F − MT K for the coherent interface and C = Ci ≡ ρψA − NT F for the one incoherent in rotations, A = 1 − n ⊗ n, and 1 is the 3D unit tensor. In the nonlinear shell theory the tensors Cc and Ci play the role of the Eshelby tensors or the energy-momentum tensors. In 3D continuum mechanics Eshelby-type tensors have various applications in the configurational mechanics [4, 23, 31, 36]. In particular, in the linear theory of plates and shells the Eshelby tensor was used to formulate the 2D conservation laws and the path-independent integrals, see [31]. Following [1, 4, 16] we take the kinetic function F(ς) in the form ⎧ k(ς − ς ) 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 + ξ(ς − ς0 ) ⎪ ⎪ ⎨ F(ς) = ⎪ 0 ⎪ ⎪ ⎪ ⎪ k(ς + ς0 ) ⎪ ⎪ ⎪ ⎩ 1 − ξ(ς + ς0 )
ς ≥ ς0 , −ς0 < ς < ς0 ,
(16.27)
ς ≤ −ς0 .
Here ς0 describes the effects associated with nucleation of the new phase and action of the surface tension, see [1], ξ is a parameter describing limit value of the phase transition velocity [4], and k is a positive kinetic factor. Equation (16.26) with (16.27) can be considered as the special constitutive equation describing the motion of phase interfaces in shells.
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16.6 Variational Statement of Thermodynamic Equilibrium of Two-Phase Shell Since the contributions of Gibbs [19], the variational formulation is widely used for description of phase transitions in solids, see e.g. [3, 5, 9, 22, 37]. The weak statement of quasistatic problems of non-linear shells undergoing PT was considered by Eremeyev and Pietraskiewicz [13], where the thermodynamic continuity condition was derived and relations for the Eshelby tensor C in shells were obtained. Let us note that the variational approach requires the thermodynamic equilibrium of twophase shell to be the minimizer or the stationary point of the functional of total energy. It does not describe the evolution of C as the external loads or the temperature are changing. The evolution of phase interface can be analyzed using the kinetic equation (16.26) and the BVP presented in previous Sections. Let us consider the isothermal process, i.e. we assume that θ is constant and ϕ = 0 during loading. In this case the thermodynamic equilibrium corresponds to the local or global extremum of the functional of free energy [19, 22]. Hence, the free energy ψ plays the role of the strain energy used in [13], and we have the variational principle in the form δU = A,
U=
ρψ da,
(16.28)
M
where we use the same relation for A as in Sect. 16.3, but here υ = δu is the virtual translation vector and ω = ax (δQQT ) the virtual rotation vector. Using (16.17) and (16.18), from (16.28) follow the equilibrium equations (16.9) and the static compatibility conditions (16.19). Additionally, we obtain the thermodynamic compatibility condition [[νν · Cνν]] = 0,
(16.29)
where C = Cc for the coherent interface and C = Ci for the interface incoherent in rotations. Equation (16.29) can be used to find position of the phase interface C in the thermodynamic equilibrium state. It corresponds to the stationary solution of the kinetic equation (16.26). Assuming adiabatic behaviour, i.e. when η is constant and χ = 0, we use the variational principle (16.28) with the functional of internal energy [19, 22], U= ρε da. M
The continuity condition (16.29) was extended in [44] taking into account the phase interface energy and other line fields defined along C.
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16.7 Conclusions We have presented the resultant, two-dimensional thermomechanics of shells undergoing diffusionless, displacive phase transitions of martensitic type of the shell material. We have used the extended thermodynamics of shells with two temperature fields, i.e. the referential surface mean temperature and the temperature deviation, as well as two dual surface entropy related fields. We have discussed the thermodynamic continuity conditions along the curvilinear phase interface for quasistatic motion and for thermodynamic equilibrium. Summarising, in the case of finite deformations the thermomechanic BVP for thermoelastic shells undergoing phase transition consists of: • the equilibrium equations (16.9)1,2 supplemented by appropriate static and kinematic boundary conditions for N, M, u, and Q, • the thermoconductivity equations (16.13) and (16.14) with appropriate boundary conditions for θ and ϕ, • the compatibility conditions (16.19), (16.20), and (16.23) along the interface C, • the kinetic equation (16.26) or the thermodynamic equilibrium condition (16.29) along C, all supplemented with the proper constitutive equations for N, M, ε, η, χ, q, and s, see [17]. The kinetic equation (16.26) is used to find position of the curvilinear interface C in its quasistatic motion, while (16.29) is used to find the equilibrium position of C. The BVP summarized above was illustrated in [17] by the 1D analytically solved example of stretching and bending of the two-phase circular plate. The somewhat simpler version of the thermomechanic shell model undergoing PT with only one surface mean temperature field was earlier developed in [15] and illustrated by the 1D analytic solution of tension and bending of two-phase tube in membrane [15] and bending [16] approximations. However, realistic 2D experimental observations on thin-walled samples presented in papers cited in Introduction can only be verified numerically by two-dimensional solutions of the BVP developed here. For this purpose one still needs to develop 2D computer codes based on the extended finite element method (XFEM) for shells with moving singular curves. Acknowledgements The first author was supported by the DFG grant No. AL 341/33-1, by the RFBR with the grant No. 09-01-00459, and by the Federal target program “Scientific and pedagogical cadre of the innovated Russia during 2009–2013” (state contract P–361), while the second author was supported by the Polish Ministry of Science and Education under grant No. N506 254237.
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Chapter 17
A Gradient-Enhanced Damage Model for Viscoplastic Thin-Shell Structures An Danh Nguyen, Marcus Stoffel and Dieter Weichert
Abstract A finite element model of non-local damage viscoplasticity for dynamic analysis of thin-walled shell structures is presented. To take void nucleation and growth into account, a non-local implicit gradient formulation is employed. The free energy function includes both a non-local damage variable on the mid-surface of shell structures and a local one in shell space. Local constitutive laws considering viscoplastic behavior, isotropic hardening and isotropic ductile damage leading to softening are used. The performance of the proposed approach is demonstrated through the numerical simulation of shock-wave loaded structures. Keywords Non-local damage · Viscoplasticity · Non-linear shells · Finite element
17.1 Introduction Simulation of metallic thin-shell structures subjected to impulsive loadings plays a very important role in engineering design of many fields, i.e. mechanical and civil engineering, aerospace, crash tests, explosions. In such conditions, materials of the structures may exhibit inelastic behavior, where the impact, internal damage of microstructure can develop overtime and lead to collapse of the structure instantly. It is well-known that softening and damage effects, which are always accompanied by localization of deformation, lead to the ill-posedness of the boundary value problem and, consequently, mesh-dependence, incorrect size effect, and excessive damage localization. A possibility to overcome such the mathematical difficulty is to introduce the gradient of plastic strain or the damage parameter fields in order A. D. Nguyen (B) · M. Stoffel · D. Weichert Institute of General Mechanics, Templergraben 64, D-52062 Aachen, Germany e-mail:
[email protected],
[email protected] e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 17, © Springer-Verlag Berlin Heidelberg 2011
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to penalize possible sharp localization. The use of gradients in the localization of deformation and fracture allows to obtain the thickness of shear band and to simulate the effect of surface tension forces (c.f. [15–17]. Among many strain-gradient and non-local damage theories, the approach proposed by Dimitrijevic and Hackl [4] are particularly interesting for our study to investigate vibration of thin-shell structures. The approach introduces a new nonlocal damage variable to transfer the values of the inelastic variables across the element boundaries, while preserving the C0 interpolation order of displacements. Such advantages help to preserve the shell theory and to reuse existing local constitutive laws of material behavior. In this paper, based on the above approach we develop a gradient-enhanced damage model for dynamic finite element computation of viscoplastic thin-shell structures. To take void nucleation and growth into account, an enhanced free energy function, which includes both a non-local damage variable on the mid-surface of shell structures and a local one in shell space, is introduced to penalize the difference between the non-local and local damage free energy. Consequently, only the potential energy is enhanced while the standard kinetic energy is preserved under assumption that the dynamic effect due to appearance of non-local variable is ignored. The dynamic thin-shell elastic theory proposed in [7] is used to integrate the presented gradient-enhanced model into the finite element program FEAP [1] to capture finite deformation. Local constitutive laws considering viscoplastic behavior, isotropic hardening and isotropic ductile damage leading to softening in [9] are employed. The performance of the proposed formulation is illustrated through numerical simulation. The damage evolution as well as property of mesh-independence are investigated in the first example of a square plate subjected to membrane loading. Then the approach is applied to simulate a copper circular plate subjected to impulse loading. The numerical results of the latter example is validated by comparison with the experimental results obtained in [2].
17.2 Dynamic Thin Shell Model 17.2.1 Basic Kinematics of Shells The non-linear shell model proposed by [6], in which the shell is regarded as an inextensible one-director Cosserat surface, is used. An overview of the shell model is given below. Further information can be found in [5, 8].
17 A Gradient-Enhanced Damage Model for Viscoplastic Thin-Shell Structures
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17.2.1.1 Shell Configurations Reference configuration of a shell is described as D T (ξ1 , ξ2 ) B := X¯ ∈ R3 | X¯ (ξ1 , ξ2 , ξ) := X (ξ1 , ξ2 ) + ξT E X , T ) ∈ C and ξ ∈ [h− , h+ ] with (X
(17.1)
Any configuration x¯ of the shell S ⊂ R3 is assumed to be defined as D S := x¯ ∈ R3 | x¯ (ξ1 , ξ2 , ξ) := x (ξ1 , ξ2 ) + ξtt(ξ1 , ξ2 ) E with (xx, t ) ∈ C and ξ ∈ [h− , h+ ]
(17.2)
where X denotes the position vector of the reference mid-surface, x = X + u denotes the position vector of the current mid-surface, u represents displacement vector of the related mid-surface point, t denotes a unit vector field at each point of the surface, referred to as the director field, h− = −h/2; h+ = h/2 where h denotes the thickness of the shell. A mid-surface A is defined by vector x and points belongs to A are defined by ξ |ξ3 =0 = ξ1 E 1 + ξ2 E 2 . The translational velocity vector of the mid-surface point at time t > 0 is denoted as x˙ and the angular velocity of the shell-director t is denoted as ω . Since the shelldirector is inextensible, the following relation is obtained (c.f. [5]) ω = t × ˙t .
(17.3)
17.2.1.2 Motion of Shell Director Motion of the extensible-shell director may be expressed by using an orthogonal transformation t = RT .
(17.4)
ˆ through an The rotation tensor Θ is associated with a skew-symmetric tensor Θ exponential mapping Θ ˆ 1 − cos Θ Θ ˆ 2 ˆ = I + sin Θ R = exp Θ Θ+ Θ . 2 Θ Θ Θ Θ
(17.5)
E i }i=1,3 be differLet {eei }i=1,3 bestandard orthonormal vector basis in R3 and {E ent orthonormal basis for the mid-surface. The motion of the director then may be expressed in different way t = R T = R Λ 0 T¯ = Λ T¯
(17.6)
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E 1 E 2 T ] represents a transformation matrix to transform between midwhere Λ 0 = [E E i } and standard basis {eei }; Λ represents rotational matrix related to surface basis {E the current state, T¯ = T¯ i E i with T¯ i = Λ0i j .
17.2.2 Strain Measures The covariant base vectors are obtained by partial derivatives of the position vectors with respect to the convective coordinates ξi , (i = 1, 2, 3) (note that ξ3 ≡ ξ) Gi =
X ∂X ,g = ∂ξi i
∂xx , ∂ξ i
i = 1, 2, 3,
(17.7) j
whereas the contravariant base vectors are defined in a standard way by G i · G j = δi j j and gi · g j = δi , here δi denotes Kronecker symbol. Vectors A α = G α |ξ=0 , a α = g α |ξ=0 span the tangent space to the mid-surface at a point X , and at a point x , respectively. For further use, the following notations are introduced 8 G = det Gi j , A = det Aαβ = A = det Ai j , μ0 = GA , dB = μ0 dξ dM0 (17.8) Lagrangian strain tensor at a point in shell space is defined by Ei j =
& 1% gi · g j − G i · G j G i ⊗ G j 2
(17.9)
where G i and g i represent covariant basis vectors at reference and current state respectively; G i and g i represent contravariant basis vectors at reference and current state respectively. Lagrangian strain tensor at a point in shell space can be expressed through the strain measures at the mid-surface by Eαβ = #αβ + ξραβ + ξ2 tαβ 1 E 3α = Eα3 = (δα + ξρ3α ) 2 E 33 = χ
(17.10)
where #αβ , δα , χ, ραβ , ρ3α , tαβ denote the membrane strain, transverse shear, thickness-stretch, bending, couple and shear and director strain measure on the midsurface, respectively. In case of inextensible thickness, the strain measures in (17.10) can be rewritten by E (1) + ξ2 E (2) E = E (0) + ξE
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ tαβ 0 ⎥⎥⎥ ⎢⎢⎢ #αβ 1 δα ⎥⎥⎥ ⎢⎢⎢ ραβ 1 ρα3 ⎥⎥⎥ ⎢ ⎥⎥⎥⎥ 2 ⎥⎥⎥ + ξ2 ⎢⎢⎢⎢⎢ ⎥⎥⎥ = ⎢⎢⎢⎣ 1 2 ⎥⎥⎥⎦ + ξ ⎢⎢⎢⎣ 1 ⎦ ⎢ ⎢⎢⎣ ⎥⎥⎦ 2 δα 0 2 ρα3 0 0 0
(17.11)
17 A Gradient-Enhanced Damage Model for Viscoplastic Thin-Shell Structures
In what follows, we set (0) (0) e := E 11 E 22 . / := # γ κ t
(0) 2E 12
(0) 2E 13
(0) 2E23
(1) E11
(1) E22
(1) 2E 12
237
t (17.12)
where (0) (0) (0) # := E 11 E22 2E 12 (0) (0) γ := 2E13 2E 23 (1) (1) (1) κ := E 11 E22 2E 12
(17.13)
17.2.3 Effective Stress Resultants For simplicity of inelastic computation, the second Piola-Kirchhoff stress resultant tensors are used. h+ n = n i j G i ⊗ G j , ni j = S i j μ¯ dξ (17.14) h− h+
m = mi j G i ⊗ G j ,
ni j =
h−
S i j ξ μ¯ dξ
(17.15)
where S i j denotes components of the second Piola-Kirchhoff stress tensor and 2 A μ¯ = ;G = det Gαβ G Spatial tensors of effective stress resultants on S0 , which are defined on the basis A α , A 3 } ≡ {A Aα , T }, are defined by {A ˜ = J¯q˜ α A α , M ˜ = J¯m N˜ = J¯n˜ αβ A α ⊗ A β , Q ˜ αβ A α ⊗ A β
(17.16)
¯
where J¯ = GG¯ . We will also use the following notation for the symmetric stress 0 resultant tensors n = nαβ A α ⊗ A β ,
q = nα3 A α ⊗ T + n3α T ⊗ A α ,
In what follows, we set r t := n11 n22 n12 . / := n q r t
n13
n23
m11
m = mαβ A α ⊗ A β (17.17)
m22
m12
t
(17.18) (17.19)
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where n := n11 n22 n12 q := n13 n23 m := m11 m22 m12
(17.20) (17.21) (17.22)
17.3 Gradient-Enhanced Damage of Shell Structures 17.3.1 Kinetic Energy The kinetic energy is defined by % & 1 Aρ0 x˙ · x˙ + Iρ0 ˙t · ˙t dA0 2 A0 1 1 1 = p·p+ π · π dA0 2 A0 Aρ0 Iρ0
Π) = K (Π
where Π := (pp , π ), the mid-surface linear momentum p is defined by p = Aρ0 x˙
Aρ0 =
h+
h−
ρ0 μ0 dξ = ρ0 h
(17.23)
and the mid-surface director momentum π is defined by π = Iρ0 ˙t
Iρ0 =
h+
h−
ρ0 ξ2 μ0 dξ = ρ0
h3 12
(17.24)
ρ0 denotes the mass density at reference configuration.
17.3.2 Potential Energy Dimitrijevic and Hackl [4] proposed a gradient-enhanced damaged model by introducing an additional non-local variable to transfer the values of the damage parameter across the element boundaries. This makes the model non-local in nature and avoids the very strong requirement of the C 1 continuous displacement field. Adapting the previous approach, we introduce here an enhanced free energy for the thin-shell structures as 2 1 2 βd ˜ Φ r e Ψ (Φ ) = : + cd ∇ϕ + ϕ − γd d dξ (17.25) 2 2 where Φ := (xx, t , ϕ). The non-local variable ϕ is defined by
17 A Gradient-Enhanced Damage Model for Viscoplastic Thin-Shell Structures
ϕ=
239
h+ h−
d˜ dξ
(17.26)
where d˜ is a non-local damage parameter in the shell space. It should be noted that the proposed free energy here is different from that in [4]. First, the proposed non-local variable is introduced as a mid-surface quantity. Consequently, a local constitutive law for the non-local variable in shell space is not necessary. Second, the elastic strain energy takes the standard form. A difference between non-local and local energy is adjusted by the parameter βd representing a penalty of energy difference between non-local and local field. The degree of gradient regularization and the internal length scale is represented by the parameter cd . The final parameter γd is used as a switch between the local and non-local model for the reason of numerical computation. With such the definition of enhanced free ˜ the potential energy is written as energy Ψ, Φ Φ) dA0 − Pext (Φ Φ) P (Φ ) = Ψ˜ (Φ (17.27) A0
Φ) is the potential energy of the external loads. where Pext (Φ
17.3.3 Weak Formulation of Motion Equations By the above definitions of kinetic and potential energy, the Lagrangian differential equations of motion can be transformed to the Hamiltonian canonical equations subjected to initial conditions of configuration, momenta and non-local variable. The weak form of these equation are written as Z) ∂H(Z Φ dA0 + Φ=0 Π˙ · δΦ · δΦ (17.28) Φ ∂Φ A0 Z) ∂H(Z ˙ · δΠ Π dA0 − Π=0 Φ · δΠ (17.29) Π ∂Π A0 Φ, Π ) and H(Z Z ) := K(Π Π ) + P(Φ Φ) where Z := (Φ From (17.28) the weak form of momentum equation may be expressed as % & % & ¨ , Φ , δΦ Φ = Gine Π˙ , δΦ Φ + G stat (Φ Φ, δΦ Φ) − Gext (Φ Φ) = 0 G dyn Φ (17.30)
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where
% & % & ϕ dA0 G ine Π˙ = p˙ · δuu + π˙ · δtt + p˙ ϕ · δϕ A 0 Φ, δΦ Φ) = (nn : δ## + q : δγγ + m : δκκ) dA0 + G stat (Φ +
A0
A0
⎛ ⎜⎜ βd δϕ ⎜⎜⎝ϕ − γd
h+ h−
⎞2 ⎟⎟ d dξ⎟⎟⎠ dA0
A0
(17.31) cd ∇δϕ ∇ϕ dA0 (17.32)
Following the approach in [4], we can derive an evolution equation of nonlocal variable ϕ from (17.32). βd ϕ − γd d¯ − cd ∇2 ϕ = 0 (17.33) where d¯ =
h+
h−
d dξ
(17.34)
Equation (17.33) expresses an implicit gradient scheme for thin-shell structures.
17.4 Local Constitutive Laws To complete the model, local constitutive laws considering viscoplasticity, isotropic hardening and isotropic ductile damage leading to softening are needed. We limit ourself to investigate cases of small strains. St. Venant-Kirchhoff law of non-linear hyperelasticity and the coupled viscoplastic and damage model proposed in [9] are used. A summary of formulations of these models are given in the sections below. In more detail, readers can refer to [8, 9].
17.4.1 Hyperelasticity Law on the Mid-surface The second Piola-Kirchhoff stress tensor can be rewritten by E ] 1 + 2μE E = CE S = λtr [E % (0) & kl i jkl 2 (2) S =C E i j + ξE i(1) j + ξ Ei j , where λ =
Eν (1+ν)(1−2ν)
and μ =
E 2(1+ν)
(17.35) (17.36)
denote Lam´e’s constants, and
C i jkl = H i jkl − H i j33
H 33kl H 3333
(Case of S 33 = 0).
(17.37)
17 A Gradient-Enhanced Damage Model for Viscoplastic Thin-Shell Structures
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Stress resultant components in a general case are defined by
h+
h− h+
n = ij
mi j = where
Cj =
h+ h−
h−
% & (1) 2 (2) C i jkl Ei(0) μ¯ dξ, j + ξE i j + ξ E i j
(17.38)
% (0) & (1) (2) C i jkl Ei j + ξE i j + ξ2 E i j ξ μ¯ dξ,
(17.39)
C i jkl ξ j μdξ, ¯
j = 0, 1, 2, 3, 4,
i j 33, kl 33
(17.40)
For thin-shell structures the following conditions are valid, G i = A i and G i = A i , i.e. 8
G i j = Ai j , G i j = Ai j , and μ¯ = GA = 1 and the components of the constitutive tensor become independent on ξ. Then the stress resultant components become ni j = m
αβ
h+
h−
=
h+
h−
(0) C i jkl E kl dξ,
i j 33, kl 33,
(1) 2 C αβγδ E γδ ξ dξ,
i j 33, kl 33
(17.41) (17.42)
The constitutive law of hyperelasticity on mid-surface may be expressed by r = C¯ e where
(17.43)
⎡ ⎤ ⎢⎢⎢ C 0(5×5) ⎥⎥⎥ 0 ⎥⎥⎥ C¯ (5×5) = ⎢⎢⎢⎣ 0 C 2(3×3) ⎦ Cj =
h+
h−
C i jkl ξ j μdξ, ¯
i, j, k, l = 0, 1, 2, 3, 4,
i j 33, kl 33
(17.44)
(17.45)
17.4.2 Viscoplastic Damage To take material behavior of local viscoplastic damage the constitutive laws proposed in [9] is used. A brief summary of these laws are given in Eqs (17.46)–(17.55). The overstress S ex which depends on the deviatoric stress S d and the hydrostatic in stress 13 trS is used to control the evolution of the inelastic train rate E˙ [10, 12, 13]. The evolution of the local damage variable d˙ based on the work by [12–14] takes in into account the inelastic volumetric strain rate tr E˙ as well as transient and saturation behavior. Isotropic hardening and inelastic deformation in case of undamage materials are considered by von Mises yield criterion. More details can be found in [9]
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E = E e + E in F Gn ∂S ex ˙ S ex in E˙ = E0 ∂SS Sp
H I in in d˙ = (c1 + c2 e−c3 Eν )E˙ νin + c5 (c4 − d) trE˙ F Gn 1 S ex E˙ νin = 1−d S p S ex = S eq − S F − K 1 S eq = f 1−d K˙ = b(Q − K)˙#νin S = w(d) C E 1 S d = S − m trSS 3 trSS = m S
(17.46) (17.47) (17.48) (17.49) (17.50) (17.51) (17.52) (17.53) (17.54)
(17.55)
in (d) := (1 − d)md denotes a softening function, E , E e , E in , E˙ , S ex , d, E˙ νin , where w S eq , K, f , S , S d , m denote total strain tensor, elastic strain tensor, inelastic strain tensor, over stress, damage parameter, equivalent inelastic strain, isotropic hardening stress, effective von Mises yield function, second Piola-Kirchhoff stress, deviatoric stress, unity matrix, respectively. The symbols E˙ 0 , S p , n, c1 , c2 , c3 , c4 , S F , a1 , a2 , b, Q, md denote material parameters. For finite element computation, the system of differential equations (17.47), (17.48) and (17.50) in terms of the second Piola Kirchhoff stress, isotropic hardening stress, and damage variables is solved by using the implicit backward Euler integration scheme at each Gauss point across the thickness of the shell. Then, the second Piola Kirchhoff stress as well as the local consistent tangent operator are employed to update the new stress resultants and the consistent tangent operator in the mid-surface.
17.5 Examples The presented formulation of the gradient-enhanced damage model for viscoplastic thin-shell structures is implemented in the user element subroutine compiled with FEAP [1]. The implementation is valid for thin-shell structures in general, but here two examples of plates subjected to dynamical loading are chosen to show the efficiency of the suggested approach without loosing the generality of the presented shell model. We note that all herein presented results are of preliminary nature and will have to be validated by future studies.
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1 0.8 0.6
Displacement (mm)
0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0
0.5
1
1.5
2
2.5
3
3.5
4
Time (s)
Fig. 17.1 Example 1 - History of prescribed displacement 0.2 0.18
Damage parameters
0.16 0.14 0.12 0.1 0.08 0.06 d
0.04
in ν
ε
0.02 0 0
0.5
1
1.5
2
2.5
3
3.5
4
Time (s)
Fig. 17.2 Case of 144 elements - Evolution of damage parameter d and equivalent inelastic strain #vin at bottom fiber of critical section
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17.5.1 Example 1 A square plate of 100mm× 100mm× 5mm, which is subjected to the prescribed horizontal displacements (Figure 17.1) at the right edge of the plate, is investigated. The material parameters given in Table 17.1 are artificial only for academic study. Two regular-structured meshing of 144 and 576 elements are considered. The evolution of the damage parameter, equivalent inelastic strain, membrane strain measure and membrane stress resultants are depicted in Figs 17.2-17.6. The response of reaction and displacement at the right edge of the plate are depicted in Figs 17.6-17.7 is quite coincident with mesh refinement. Table 17.1 Example 1 - Material parameters Parameters
Values
Units
Parameters
Values
Units
h
5
mm
κ
0.8
1
E
1.98 × 105
N mm2
ν
0.3
1
ρ
106
kg m3
Ip
0.8
1
SF
167.88
N mm2
E˙ 0
1.0
1
Sp
163.12
N mm2
np
1.0
1
Q
75
N mm2
b
175
1
a1
1.3
1
a2
1.0
1
c1
1.2
1
c2
-0.02
1
c3
60.
1
c4
40.0
1
c5
0.1
1
md
2.0
1
cd
1.0
1
β
1.0
N mm2
· mm
d0
0.01
1
γd
1.0
N mm2
· mm
I p : Loading intensity factor
17.5.2 Example 2 A copper circular plate of 138 mm diameter which is clamped on the outer-ring area of 15 mm radius length is investigated as in [2]. Due to the clamped area, the loading area is of 108 mm diameter. Alternate shock pressures are shown in Fig. 17.9 and
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0.025 0.02
Membrane strain measure
0.015 0.01 0.005 0 −0.005 −0.01 −0.015 −0.02 0
ε11 22 ε 12 ε 0.5
1
1.5
2
2.5
3
3.5
4
2.5
3
3.5
4
Time (s)
Fig. 17.3 Case of 144 elements - Membrane strain measures 1500
Membrane stress resultant
1000
500
0
−500
−1000
−1500 0
11
n 22 n 12 n 0.5
1
1.5
2 Time (s)
Fig. 17.4 Case of 144 elements - Membrane stress resultants
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Stress resultant n
1000
500
0
−500
−1000
11
11
ε −n ε22−n22 ε12−n12
−1500 −0.02
−0.015
−0.01
−0.005
0
0.005
Strain measure ε
0.01
0.015
0.02
0.025
0.6
0.8
1
Fig. 17.5 Case of 144 elements - (## − n ) curve 5
1.5
x 10
1
Reaction (N)
0.5
0
−0.5
−1 144 elem. 576 elem. −1.5 −1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
Displacement (s)
Fig. 17.6 Displacement-reaction curve at the right edge of the plate
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5
1.5
x 10
144 elem. 576 elem. 1
Reaction (N)
0.5
0
−0.5
−1
−1.5 0
0.5
1
1.5
2
Time (s)
Fig. 17.7 History of reaction at the right edge of the plate
Fig. 17.8 Example 2 - Sketch of a quarter of copper plate
2.5
3
3.5
4
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x 10
6
2 1.5
2
Pressure (N/mm )
1 0.5 0 −0.5 −1 −1.5 −2 −2.5 0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
Time (s)
Fig. 17.9 Example 2 - History of prescribed pressure −3
8
x 10
Experiment Viscoplasticity Viscoplastic damage Viscoplastic nonlocal damage
6 4
Displacement (m)
2 0 −2 −4 −6 −8 −10 −12 0
0.005
0.01
0.015
Time (s)
Fig. 17.10 Displacements at center of the copper plate
0.02
0.025
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Table 17.2 Example 2 - Material parameters Parameters
Values
Units
Parameters
Values
Units
h
2
mm
κ
0.833
1
E
113 × 109
N mm2
ν
0.33
1
ρ
8858
kg m3
Iρ
1
1
SF
180
N mm2
E˙ 0
1.0
1
Sp
40.5
N mm2
np
8.15
1
Q
64.52
N mm2
b
153
1
a1
1.3
1
a2
1.0
1
c1
1.2
1
c2
-0.02
1
c3
60.
1
c4
40.0
1
c5
0.1
1
md
2.0
1
cd
1.0
1
β
1.0
N mm2
· mm
d0
0.01
1
γd
1.0
N mm2
· mm
I p : Loading intensity factor
material parameters of copper is given in Table 17.2. Due to axisymmetry, only a quarter of the plate as in Fig. 17.8 is considered in numerical computation. The obtained results, which are compared with other two models of viscoplasticity and damage coupled viscoplasticity, are depicted in Fig. 17.10. Vibrations are observed at the beginning of loading and loading, and dissipate quickly. This is reasonable since the energy and momentum conserving algorithm proposed in [7] is used with the values of numerical dissipation β = 0.65, γ = 0.95 and α = 0.55.
17.6 Conclusions This work proposes a gradient-enhanced damage model for viscoplastic thin-shell structures subjected to dynamically loading. The previous approach gives a possibility to remove pathological mesh dependence and with numerical difficulties that appeared in existing simulations of softening and damage phenomena. However, it should be noted that the presented numerical results are preliminary and need future systematical studies.
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References 1. FEAP-8.2.k, Finite Element Analysis Program (2008). University of California. 2. Stoffel M. (2007). An experimental method for validating mechanical models. Habilitation thesis, RWTH-Aachen. 3. Stoffel M. (2004). Evolution of plastic zones in dynamically loaded plates using different elastic-viscoplastic laws. Int. J. Sol. Struc., 41 (24-25), 6813-6830. 4. Dimitrijevic B.J., Hackl K. (2008). A method for gradient enhancement of continuum damage models. Tech. Mech., 28(1), 43-52. 5. Simo J.C., Fox D.D.(1989). On a stress resultant geometrically exact shell model. Part I. Formulation and optimal parametrization. Comput. Meth. Appl. Mech. Engng., 72, 267-304. 6. Simo J.C., Rifai M.S., Fox D.D. (1992). On stress resultant geometrically exact shell model. Part VI. Conserving algorithms for non-linear dynamics. Int. J. Numer. Meth. Engng., 34, 117-164. 7. Simo J.C., Tarnow N. (1994). A new energy and momentum conserving algorithm for the non-linear dynamics of shells. Int. J. Numer. Meth. Engng., 37, 2527-2549. 8. Brank B., Briseghella L., Tonello N., Damjanic F.B. (1998). On non-linear dynamics of shells: Implementation of energy-momentum conserving algorithm for a finite rotation shell model. Int. J. Numer. Meth. Engng, 42, 409-442. 9. Velde J., Kowalsky U., Z¨umendorf T., Dinkler D. (2009). 3D-FE-analysis of CT-speciemens including viscoplastic material behaviour and nonlocal damage. Comp. Mat. Sci., 46, 532-357. 10. Chaboche J.L., Rousselier G. (1983). On the Plastic and Viscoplastic Constitutive Equations– Part I: Rules Developed With Internal Variable Concept. J. Pres. Vess. Tech., 105, 153-158. 11. Chaboche J.L., Rousselier G. (1983). On the Plastic and Viscoplastic Constitutive Equations– Part II: Application of Internal Variable Concepts to the 316 Stainless Steel. J. Pres. Vess. Tech., 105, 159-164. 12. Gurson A.L. (1977). Continuum theory of ductile rupture by void nucleation and growth: Part I–Yield criteria and flow rules for porous ductile media. J. Eng. Mat. Tech., 99, 2-15. 13. Tvergaard V., Needleman A. (1984). Analysis of cup-cone fracture in a round tensile bar. Act. Metal., 32, 157-169. 14. Lemaitre J., Desmorat R., Sauzay M. (2000). Anisotropic damage law of evolution. Eur. J. Mech. A/Solids, 19(2), 187-208. 15. Nguyen A.D., Stoffel M., Weichert D. (2010). A one-dimensional dynamic analysis of straingradient viscoplasticity. Eur. J. Mech. A/Solids, 29, 1042-1050. 16. Aifantis E.C. (1992). On the role of gradients in localization of deformation and fracture. Int. J. Eng. 30, No. 30, 1279-1299. 17. Nguyen Q.S., Andrieux S. (2005). The non-local generalzed standard approach: a consistent gradient theory. Compt. Rend. Meca. 333, 139-145.
Chapter 18
On Constitutive Restrictions in the Resultant Thermomechanics of Shells with Interstitial Working Wojciech Pietraszkiewicz
Abstract We analyse the restrictions imposed by the recently derived refined, resultant 2D entropy inequality on the forms of 2D constitutive equations of viscous shells with heat conduction and of thermoelastic shells. Due to the presence of additional non-classical surface fields, the entropy inequality does allow the constitutive equations to capture some kinematic and thermal longer-range spatial and temporal interactions. We also propose several forms of the 2D kinetic constitutive equations. Keywords Shell thermomechanics · Interstitial working restrictions · Kinetic constitutive equations
·
Constitutive
18.1 Introduction The resultant two-dimensional (2D) balance laws of mass, linear momentum, angular momentum, and energy as well as entropy inequality of the non-linear theory of shells were first formulated by Simmonds [1] by direct through-thethickness integration of the corresponding 3D laws of continuum thermomechanics. Pietraszkiewicz et al. [2] proved that the 3D stress power of the shell can in general be given through the 2D effective stress power expressed only by the 2D shell quantities as in [1] plus an additional stress power not expressible through 2D resultant fields. Hence, in order to balance the shell energy an additional 2D stress power, called an interstitial working after Dunn and Serrin [5] and Dunn [6], was added in [3, 4] to the 2D resultant balance of energy equivalent to one of [1]. Then appropriately refined 2D laws of shell thermomechanics of [3] can be regarded as W. Pietraszkiewicz (B) Institute of Fluid-Flow Machinery of the Polish Academy of Sciences, ul. Gen. J. Fiszera 14, 80-952 Gda´nsk, Poland e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 18, © Springer-Verlag Berlin Heidelberg 2011
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exact implications of the corresponding 3D laws of rational thermomechanics discussed by Truesdell and Toupin [7] and Truesdell and Noll [8]. In this paper we analyse the restrictions imposed by the refined, resultant 2D entropy inequality on the forms of 2D constitutive equations of viscous shells with heat conduction and of thermoelastic shells. For this purpose the procedure proposed by Coleman and Noll [9] is used. It is found, in particular, that due to the presence of additional 2D fields the refined 2D entropy inequality does allow the constitutive equations to depend also on the first surface gradients of the shell strain measures as well as on the second surface gradient of the mean referential surface temperature.
18.2 Local Laws of Refined Resultant Thermomechanics of Shells Let M be a regular surface modelling the base surface of the shell in the reference (undeformed) placement. We also assume that all surface fields discussed here are smooth on any part Π ⊂ M, and that mass is not created during the thermodynamic process. Then the refined, local 2D balance laws of linear momentum, angular momentum, and energy as well as the entropy inequality of the shell derived in [3] for any Π ⊂ M read: N F T − F N T ) + ρcc = k˙ + y˙ × l , Div N + ρ f = ˙l , Div M + ax (N o o N • E + M • K + Div w ) − (ρr − Divqq) = 0 , ρε˙ − (N ˙ + N • E o + M • K o + Div w − ρθs − 1 q · g + θ Div s ≥ 0 . ˙ −ρψ − ρθη θ
(18.1)
In the mechanical balance laws (18.1)1,2 , ρ(x, t) > 0 is the resultant referential surface mass (density field), where x is a point of M and t ∈ T is time, l (x, t) and k (x, t) are the resultant surface linear momentum and angular momentum vectors per unit area of M, f (x, t) and c (x, t) are the resultant surface force and couple vectors per unit mass of M, respectively. Additionally, N (x, t) ∈ V ⊗ T x M and M (x, t) ∈ V ⊗ T x M are the respective stress resultant and stress couple tensors of the first Piola-Kirchhoff type, where V is the 3D vector space and T x M is the 2D tangent space at x ∈ M, F = Grad y ∈ V ⊗ T x M is the surface deformation gradient, where y (x, t) is the position vector of the actual (deformed) base surface, while Grad and Div are the surface gradient and divergence operators with respect to x ∈ M as defined by Gurtin and Murdoch [19], and ax (T) means the axial vector of the skew tensor T ∈ V ⊗ V, TT = −T. In the thermodynamic 2D laws (18.1)3,4 , ε(x, t), η(x, t), ψ(x, t), and r(x, t) are the resultant surface internal energy, entropy, free energy, and heat supply per unit mass of M, θ(x, t) > 0 is the surface mean referential temperature with the surface gradient g = Grad θ ∈ T x M, E (x, t) ∈ V ⊗T x M and K (x, t) ∈ V ⊗T x M are the natural stretch and bending tensors of the general six-field shell model, q (x, t) is the resultant referential heat flux vector, w (x, t) is the interstitial working flux vector, s(x, t) and s (x, t) are the respective extra surface heat supply and entropy supply vectors, ˙l is the material
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time derivative of l (x, t), (.)o means the co-rotational time derivative of (.), and for A T D ). For the description of all resultant any A , D ∈ V ⊗ T x M we denote A • D = tr (A 2D fields given above in terms of the corresponding 3D fields, and for the additional relations not presented here, we refer to [3, 4, 10]. The starting point of our resultant shell thermomechanics formulated in [3] had been the classical 3D Cauchy continuum and the 3D Clausius–Duhem inequality. But our refined, resultant 2D thermomechanic shell relations (18.1) are far from being the classical ones. The kinematic structure of the resultant shell theory is that of the Cosserat [12] surface, with y (x, t) and Q (x, t) as independent kinematic field variables of the shell motion, where Q (x, t) is the proper orthogonal tensor field. The thermodynamic structure of our refined, resultant 2D laws (18.1) also does not remind that of 3D rational thermomechanics developed in [7, 8]. In particular, our 2D laws (18.1)3,4 contain the extra surface scalar field s(x, t) and divergences of the additional surface vector fields w (x, t) and s (x, t), which are not present in corresponding 3D laws of rational thermomechanics but are somewhat similar to those present in 3D extended thermodynamics, see for example [13].
18.3 Restrictions on the Form of Constitutive Equations Our exact, local, resultant 2D balance laws (18.1) are expressed through 16 surface fields, which together form the shell thermomechanic process over the domain M × T . Different groups of fields play different roles in the process. The fields y , Q , θ constitute the basic thermo-kinematic independent field variables of the initial-boundary value problem of shell thermomechanics. That only seven scalar fields can be taken as independent field variables here follows from the fact that there are only seven scalar resultant field equations (18.1)1−3 to determine them. The fields N , M , q , ε, η, w , s, s have to be specified by appropriate (material) constitutive equations and the fields l , k by appropriate kinetic constitutive equations. When all the fields above are settled, the fields f , c , r are supposed to be adjusted so as to satisfy the refined 2D balance equations (18.1)1−3 . Any such process is called an admissible thermomechanic process; it is completely determined by the evolution of deformation and temperature of the shell base surface. General requirements that the constitutive equations of our resultant shell thermomechanics must obey are analogous to those of 3D rational thermomechanics formulated by Truesdell and Toupin [7], Sect. 293. These are briefly: 1) consistency, 2) coordinate invariance, 3) well posedness, 4) material frame-indifference, 5) material symmetry, and 6) equipresence. Additional quite general requirements were proposed by Truesdell and Noll [8]: 7) determinism, 8) local action, and 9) fading memory. But in the literature one can find many other specific requirements which define particular classes of material behaviour, such as for example hyperand hypoelasticity, viscosity, incompressibility, inextensibility in some direction, slow motions, small deformations, etc. Either of these requirements can be used to define particular classes of shell constitutive behaviour as well.
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Let us introduce the referential shell stress measures N(x, t) and M(x, t) with corresponding referential shell strain measures E(x, t) and K(x, t) defined by N = QT N ,
M = QT M ,
E = QT E ,
K = QT K ,
(18.2)
so that the local 2D inequality (18.1)4 becomes ˙ + Div w − ρθs − 1 q · g + θ Div s ≥ 0 ˙ + N • E˙ + M • K − ρψ˙ − ρθη θ
in Π ⊂ M . (18.3)
If y , Q , θ are taken as the independent field variables, then from the requirement of determinism the 2D thermomechanic response described by the fields N , M , q , ε, η, w, s, s, l, k ) at the shell particle, whose position in the reference Σ = (N placement is x ∈ M, can mathematically be defined by the response functionals ∞ Σ(x, t) = Σˆ τ=0 y t (z, τ), Q t (z, τ), θt (z, τ); z, B (z) . (18.4) History φt (x, τ) ≡ φ(x, t − τ) up to the present time t of any field φ(x, t) in (18.4) is defined for all τ ∈ [0, +∞), and z ∈ M is any other place than x on M. The explicit dependence of the right-hand side of (18.4) on z and on the referential structure tensor B (z) defined in [2] indicates that thermomechanic properties at x ∈ M may depend also on properties of other shell particles in Π ⊂ M, thus allowing for material inhomogeneneity and non-uniformity in shell behaviour, in general. In the special case of (18.4), which we wish to investigate in more detail as an example, we allow the fields Σ to depend on y t , Q t , and θt only locally through the first time derivatives at τ = 0 and locally through the first surface gradients at x. Then the corresponding constitutive assumption would be ˙ , K, ˙ θ, g ; x, B (x) . Σ(x, t) = Σˆ κ E, K, E (18.5) Any function in Σˆ κ is called the response or the constitutive function and is assumed to be differentiable with respect to the fields enclosed in brackets as many times as required. The lower index κ in Σˆ κ indicates that the form of each response function depends on the choice of the reference placement.
18.3.1 Viscous Shells with Heat Conduction The shell-like body described by the 2D response functions (18.5) is usually called a viscous shell with heat conduction. The structure of its constitutive equations based on the approximate resultant 2D entropy inequality analogous to that of [1] has recently been discussed by Eremeyev and Pietraszkiewicz [10]. But in our refined 2D entropy inequality (18.3) there is the interstitial working flux vector field w (x, t) which was not present in [10]. Within 3D continuum mechanics a similar interstitial working vector field allows for modelling the material behaviour also with
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mechanical longer-range spatial interactions at x ∈ M, see [5, 6]. Additionally, the presence of θ Div s in (18.3) suggests that also the thermal longer-range spatial interactions at x ⊂ M may be accounted for in the refined resultant shell thermomechanics. Hence, the more appropriate non-classical constitutive assumption for the viscous shell with heat conduction compatible with (18.3) is Σ(x, t) = Σˆ κ [Λ; x, B (x)] ,
˙ θ, g , G ) , Λ = (E, K, Grad E, Grad K, E˙ , K,
(18.6)
where G = Grad g = G T ∈ T x M ⊗ T x M. The thermomechanic process, in which the fields E, K, θ are constant in space and time, is usually called the equilibrium process. The equilibrium response functions of the shell stress measures N E , ME are defined by E
E
ˆ κ [Λ0 ; x, B (x)] , ME = M ˆ κ [Λ0 ; x, B (x)] , Λ0 = (E, K, 0 , 0 , 0 , 0 , θ, 0 , 0 ). NE = N (18.7) Then the dynamic, dissipative parts of the response functions of the shell stress measures are such that ˆ κD [Λ; x, B (x)] , MD = M ˆ κD [Λ; x, B (x)] , ND = N N = NE +N D , M = ME + MD .
(18.8)
According to Coleman and Noll [9], in 3D rational thermodynamics the Clausius– Duhem inequality plays the role of a restriction placed on allowable forms of the response functions. The functions must be so chosen as to satisfy the inequality in every smooth thermomechanic process compatible with the 3D balance equations. To reveal the restrictions placed on the 2D shell constitutive equations (18.6) by our refined resultant 2D entropy inequality (18.3), let us just introduce (18.6) into (18.3) to obtain ˙ + (M − ρψ,K ) • K ˙ − (ρη + ρψ,θ )θ˙ − ρψ,g ·g˙ (N − ρψ,E ) • E ˙ − ρψ,E˙ •E¨ − ρψ,K˙ •K ¨ −ρψ,Grad E •Grad E˙ − ρψ,Grad K •Grad K 1 ˙ −ρψ,G •G − ρθs − q · g + Div w + θ Div s ≥ 0 . θ
(18.9)
Any deformation–temperature path can be realised in a thermomechanic process, so that values of Λ and Λ˙ can be chosen arbitrarily. Since the left-hand side of (18.9) depends on Λ˙ linearly, in order to satisfy (18.9) the coefficients in front of time rates of variables must vanish, η = − ψ,θ , ψ,Grad E = 0 , ψ,Grad K = 0 , ψ,E˙ = 0 , ψ,K˙ = 0 , ψ,g = 0 , ψ,G = 0 .
(18.10)
These relations imply that ψ = ψˆ κ [E, K, θ; x, B (x)], and by (18.10)1 and definition ψ = ε − θη the same structure should have the response functions εˆ κ and ηˆ κ . With (18.10) the inequality (18.9) reduces to
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˙ + (M − ρψ,K ) • K ˙ − ρθηs − 1 q · g + Div w + θ Div s ≥ 0 . (N − ρψ,E ) • E θ
(18.11)
Please note that in (18.11), N and M still depend only on Λ, while ψ,E and ψ,K depend on E, K, θ. If the decompositions (18.7) and (18.8) are used in (18.11), the following constitutive equations for the equilibrium part of the shell stress measures are obtained: NE = ψˆ κ ,E [E, K, θ; x, B (x)] ,
ME = ψˆ κ ,K [E, K, θ; x, B (x)] .
(18.12)
Let us now apply the chain rule to the surface vector fields w and s , which allows one to expand (18.11) with (18.12) into 1 1 1 D ˙ D ˙ N • E + M • K − ρθs − q − w ,θ − s ,θ · g + w ,g + s ,g • G θ θ θ 1 G + tr w ,G + s ,G ◦ GradG θ 1 1 + tr w ,E + s ,E ◦ Grad E + tr w ,K + s ,K ◦ Grad K (18.13) θ θ 1 + tr w ,Grad E + s,Grad E Grad2 E θ 1 + tr w ,Grad K + s ,Grad K Grad2 K ≥ 0 , θ where for two 3rd-order surface tensors Γ , Δ and for two 4th-order surface tensors Φ , Ψ the 2nd-order surface tensors Γ ◦ Δ and Φ Ψ are defined such that in Cartesian components associated with T x M we have Γ ◦ Δ )αβ = Γαiλ Δiλβ , (Γ
Φ Ψ )αβ = Φαiλμ Ψiλμβ , (Φ
α, β = 1, 2.
(18.14)
Let us remind that according to [3, 5, 6] the interstitial working vector w is of entirely mechanical origins. Hence, we may additionally assume that w = wˆ κ [E, K, Grad E, Grad K; x, B(x)] , so that w ,θ = w ,g = 0 . Similarly, according to [3] the extra surface vector field s is of entirely thermal origins and we may additionally assume that . / s = sˆ κ θ, g , G ; x, B (x) , so that s,E = s,K = s,Grad E = s ,Grad K = 0. With these additional assumptions the inequality (18.13) reduces to 1 ˙ − ρθs − 1 (qq − s ,θ ) · g + 1 s ,g •G G + tr G ND • E˙ + MD • K s ,G ◦GradG θ θ θ (18.15) w w (w + tr (w , ◦Grad E) + tr , ◦Grad K) E K % & % & 2 2 + tr w ,Grad E Grad E + tr w ,Grad K Grad K ≥ 0 .
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18.3.2 Thermoelastic Shells An important special case of viscous shells with heat conduction are thermoelastic ˙ K. ˙ In this case shells, which constitutive functions in (18.6)1 do not depend on E, the refined 2D entropy inequality takes the form (18.9), only now without terms containing ψ,E˙ and ψ,K˙ . To satisfy it we should have ψ,Grad E = 0 ,
ψ,Grad K = 0 ,
ψ,g = 0 ,
ψ,G = 0 ,
(18.16)
so that also in this case ψ = ψˆ κ [E, K, θ; x, B (x)], with the same structure of εˆ κ and ηˆ κ . As a result, the constitutive equations for N, M, and η of the thermoelastic shells are N = ρψ,E , M = ρψ,K , η = − ψ,θ . (18.17) With the constitutive equations (18.17) the reduced form of our 2D entropy inequality (18.9) becomes 1 1 1 −ρθs − q − w ,θ − s ,θ · g + w ,g + s ,g • G θ θ θ 1 G + tr w ,G + s ,G ◦ GradG θ 1 1 + tr w ,E + s,E ◦ Grad E + tr w ,K + s ,K ◦ Grad K (18.18) θ θ 1 + tr w ,Grad E + s ,Grad E Grad2 E θ 1 + tr w ,Grad K + s ,Grad K Grad2 K ≥ 0 . θ If again wˆ κ in (18.18) is assumed not to depend on θ, g and sˆ κ on E, K, Grad E, Grad K then the inequality (18.18) reduces further to 1 1 1 G + tr G −ρθs − (qq − s ,θ ) · g + s ,g •G s ,G ◦GradG θ θ θ (18.19) w, ◦Grad E) + tr (w w, ◦Grad K) + tr (w % E & K% & + tr w ,Grad E Grad2 E + tr w,Grad K Grad2 K ≥ 0 . In both cases of viscous shells with heat conduction and of thermoelastic shells the reduced 2D entropy inequalities (18.15) and (18.19) still put some constraints on allowable forms of wˆ κ and sˆ κ , detailed discussion of which we leave for future work. But the main message following from (18.15) and (18.19) is that our refined, resultant 2D entropy inequality (18.3) does allow the constitutive equations of both types of shells to depend on Grad E, Grad K, and G indeed. However, before fully revealing such an explicit dependence and its thermodynamic significance much research has still to be done.
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18.4 Kinetic Constitutive Equations In the resultant shell thermomechanics it is not apparent how the resultant linear momentum l (x, t) and angular momentum k (x, t) vector fields should be related to the ˙ Q T ) velocities of the shell as well as to 2D translational υ = y˙ and angular ω = ax (Q other kinematic and thermal surface fields. Since the fields l , k do not enter into our refined entropy inequality (18.3), it does not place any thermodynamic restrictions on their functional forms. In [3, 11] some heuristic arguments were proposed on how the 2D kinetic constitutive equations can be constructed within the classes of shell models discussed in Subsects 5.1 and 5.2. Let us remind here some of those arguments and propose some forms of the 2D kinetic constitutive equations for the general six-field theory of shells. Since l and k are of entirely mechanical origins, the 2D constitutive equations for them may be postulated in the reduced form ˙ x, B (x) , l = ˆl κ ρ, υ , ω , E, K, Grad E, Grad K, E˙ , K; (18.20) ˙ , K; ˙ x, B (x) . k = kˆ κ ρ, υ , ω , E, K, Grad E, Grad K, E In analogy to the 3D case, it is reasonable to assume that the constitutive functions ˆlκ and kˆ κ are linear functions of υ, ω, l = J 1υ + J 2ω ,
k = J 3υ + J 4ω .
(18.21)
˙ Here J A ∈ V ⊗ V, A = 1, 2, 3, 4, are tensor functions of ρ, E, K, Grad E, Grad K, E, ˙ in general. K, If the tensors J A satisfy the symmetry conditions J T1 = J 1 ,
J T2 = J 3 ,
J T4 = J 4 ,
(18.22)
we may expect existence of the 2D kinetic energy density κ(x, t) per unit area of M, which is the quadratic positive-definite function of υ , ω , κ=
1 υ · J 1υ + υ · J 2ω + ω · J 3υ + ω · J 4ω) . (υ 2
(18.23)
The function κ may serve as potential for l , k , so that (18.21) can be found from κ according to l = κ,υ , k = κ,ω . (18.24) The tensors J A may be taken to be quite complex functions of all their arguments, in general. But because of mechanical origins of l and k we may additionally assume that J A = J A (ρ, E, K). With such J A the kinetic constitutive equations (18.21) was proposed in [11]. Even simpler forms of (18.21) follow when, in analogy to the rigid-body motion, the J A are assumed in the forms
18 Constitutive Restrictions in the Resultant Thermomechanics of Shells
J 1 = ρ11 ,
QI 3 QT , J T2 = J 3 = ρQ
QI 4 QT , J 4 = ρQ
259
(18.25)
where 1 ∈ V ⊗V is the 3D identity tensor, while I 3 and I 4 are constant inertia tensors of the shell cross section in the reference (undeformed) placement. In this case, according to (18.24), we obtain Q I T3 Q T ω , l = ρυυ + ρQ
Q I 3 Q T υ + ρQ QI 4 QT ω . k = ρQ
(18.26)
Explicit formulas for I 3 and I 4 depend on the specific internal structure of the shell across the reference thickness. The kinetic constitutive equations (18.26) were proposed by Zhilin [14] and applied, for example, in Altenbach and Zhilin [15] and Eremeyev and Zubov [16]. In many papers on shell dynamics the inertia tensors are assumed to be I 3 = 0 , I 4 = I11, where I is the transverse geometric moment of inertia of the shell cross section. Then (18.26) reduce to the ultimate simple forms l = ρυυ ,
ω. k = ρIω
(18.27)
The kinetic constitutive equations equivalent to (18.27) were used, for example, by Zhilin [17], Simmonds [1], and Chr´os´cielewski et al. [18].
18.5 Conclusions The refined, resultant 2D balance of energy and entropy inequality derived in [3, 4] for the general six-field theory of shells contain three non-classical surface fields w , s, s , which play the role of somewhat similar fields appearing in 3D extended thermodynamics. We have briefly analysed restrictions imposed by our refined 2D entropy inequality on the 2D constitutive equations of viscous shells with heat conduction and of thermoelastic shells. Applying the procedure of Coleman and Noll [9], it has been shown that in the both cases our refined 2D entropy inequality allows the constitutive equations to depend also on the first surface gradients of shell strain measures E, K and on the second surface gradient of the surface mean referential temperature θ. We have also provided several forms of the 2D kinetic constitutive equations obtained with the help of heuristic arguments. More definite properties of the non-classical fields w , s, s as well as reduced explicit forms of the constitutive equations, following from requirements of material frame-indifference and material symmetry for these and other types of shells, should be discussed separately. Due to the non-classical forms of the reduced inequalities (18.15) and (18.19) as well as a number of independent fields in the constitutive equations (18.6), it is natural to expect that the corresponding analyses be quite involved.
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Acknowledgements This research was supported by the Polish Ministry of Science and Education under grant No. N506 254237.
References 1. Simmonds, J.G.: The nonlinear thermodynamical theory of shells: Descent from 3-dimensions without thickness expansions. In: Axelrad, E.L., Emmerling, F.A. (eds.), Flexible Shells, Theory and Applications, pp. 1–11. Springer-Verlag, Berlin (1984). 2. Pietraszkiewicz, W., Chr´os´cielewski, J., Makowski J.: On dynamically and kinematically exact theory of shells. In: Pietraszkiewicz, W., Szymczak, C. (eds.), Shell Structures: Theory and Applications, pp. 163–167. Taylor & Francis, London (2005). 3. Pietraszkiewicz, W.: Refined resultant thermomechanics of shells. Int. J. Engng Sci. (2011) (submitted). 4. Pietraszkiewicz, W.: On non-linear shell thermodynamics with interstitial working. In: Wilma´nski, K., Je¸drysiak, J., Michalak, B. (eds.), Mathematical Methods in Continuum Mechanics. Politechnika Ł´odzka, Ł´od´z (2011) (in print). 5. Dunn, J.E., Serrin J.: On the thermodynamics of interstitial working. Arch. Rational Mech. Anal. 85 (1985) 95–133. 6. Dunn, J.E.: Interstitial working and a non-classical continuum. In: Serrin, E. (ed.), New Perspectives in Thermodynamics, pp. 187-222. Springer-Verlag, Berlin et al. (1986) 7. Truesdell, C., Toupin R.: The Classical Field Theories. In: Fl¨ugge, S. (ed.), Handbuch der Physik, Band III/1, Springer-Verlag, Berlin (1960). 8. Truesdell, C., Noll W.: The Non-Linear Field Theories of Mechanics. In: Fl¨ugge, S. (ed.), Handbuch der Physik, Band III/3, Springer-Verlag, Berlin-Heidelberg-NewYork (1965). 9. Coleman, B.D., Noll W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rational Mech. Anal. 13 (1963), 1, 167–178. 10. Eremeyev, V.A., Pietraszkiewicz, W.: Phase transitions in thermoelastic and thermoviscoelastic shells. Arch. Mech. 61 (2009), 1, 125–152. 11. Chr´os´cielewski, J., Makowski, J., Pietraszkiewicz, W.: Statics and Dynamics of Multi-Shells: Non-Linear Theory and the Finite Element Method. Institute of Fundamental Technological Research Press (in Polish). Warsaw (2004). 12. Cosserat, E., Cosserat, F.: Th´eorie des Corps Deformables. Hermann et Fils, Paris (1909). 13. M¨uller, I., Ruggeri, T.: Rational Extended Thermodynamics (2nd Edition). Springer, New York (1998). 14. Zhilin, P.A.: Basic equations of non-classical theory of elastic shells (in Russian). Dinamika i Prochnost’ Mashin, Trudy Leningr. Polit. In-ta 386 (1982) 29–46. 15. Altenbach, H., Zhilin, P.A.: Theory of elastic simple shells (in Russian). Advances in Mechanics 11 (1988), 107–148. 16. Eremeyev, V.A., Zubov, L.M.: Mechanics of Elastic Shells (in Russian). Nauka, Moskva (2008). 17. Zhilin, P.A.: Mechanics of deformable directed surfaces. Int. J. Solids Struct. 12 (1976), 9-10, 635–648. 18. Chr´os´cielewski, J., Makowski, J., Pietraszkiewicz, W.: Non-linear dynamics of flexible shell structures. Comp. Assisted Mech. Engng. Sci. 9 (2002) 341-357. 19. Gurtin, M. E., Murdoch, A. I.: A continuum theory of elastic material surfaces. Arch. Rational Mech. Anal. 57 (1975), 291–323.
Chapter 19
Free Finite Rotations in Deformation of Thin Bodies Leonid I. Shkutin
Abstract The term ”thin bodies” includes shells, plates and rods. Such bodies are divided in two groups: shell-like and rod-like bodies. The first group includes shells, plates and thin-walled rods, and the second group includes beams and rods with rigid cross-sections. Two approaches to model thin bodies deformation are developed in the scientific literature: axiomatic and approximate. The axiomatic approach was developed by Bernoulli, Euler, and Cosserat brothers. The paper by Ericksen and Truesdell [1] stimulated a general interest to axiomatic models of the deformation in mechanics. The review of the relevant publications is given in the references [2–5]. This lecture is devoted to construction and application of the approximate deformation models for rod-like and shell-like bodies. Keywords Finite rotations · Rod-like structures · Shell-like structures
19.1 Separation of Local Rotations in Cauchy Continuum Let a material continuum (or a solid body) in its reference state occupies a region (volume) B with a boundary (surface) Aν . The region is parameterized by the triplets of Cartesian coordinates xK and Lagrangian local coordinates t J . Let iK is the orthonormal basis of Cartesian coordinates. The position of a point in the reference state is given by the positional vector r(t J ) measured from the origin of Cartesian coordinates. The equality g J = r, J defines the holonomic local basis g J (r) of the local coordinate system in the reference state. The upper-case Latin indices take values 1, 2, 3; the lower-case indices after comma denote derivation with respect to Lagrangian coordinates; the traditional rules of tensor analysis are used in the following. L. I. Shkutin (B) Institute of Computational Modeling, Akademgorodok, 660036 Krasnoyarsk, Russia e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 19, © Springer-Verlag Berlin Heidelberg 2011
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Displacement of an arbitrary point r of the continuum in the deformed state is measured by a vector w(r), and deformation of a point vicinity is given by the transformation r → y , g J → yJ , y = r + w , yJ = y, J . (19.1) Here y(r), yJ (r) are positional and gradient vectors of a material point in the actual state (dependence on time is not shown). For separation of local rotations accompanying the deformation, in a point of the continuum we input an inholonomic basis a J (r) which rotates like a solid body. In the reference state this basis is given by the local vectors a◦J (r), in the actual state it is given by the local orthogonal transformation a J = a◦J · Θ = Θ · a◦J ,
Θ ≡ a◦J a J ,
Θ ·Θ ≡ 1,
(19.2)
where Θ (r) and Θ (r) are the conjugate tensors-rotators. The initial local basis a◦J can differ from the holonomic basis g J . In particular, as ◦ a J an orthogonal basis can be assumed if g J is not such. It is convenient to distinguish theses bases in deformation models of thin bodies. If we take the local orthonormal bases e◦J and e J corresponding to the bases a◦J and a J , then the orthogonal transformations e◦J = i J · O◦ ,
e J = e◦J · Θ = i J · O ,
O = O◦ · Θ
(19.3)
will relate these local bases with each other and with the Cartesian basis i J . Here O◦ (a) is the given orientation tensor of the initial local basis e◦J relative to the Cartesian one. This tensor has the same components in both these bases. The components of the tensor-rotator Θ (a) coincide in the bases e◦J and e J . It is convenient to realize a rotation from the Cartesian basis i J to the local basis e J with the multiplicative tensor O = O1 · O2 · O3 , where O1 , O2 , O3 are the base rotators with orthogonal matrices O1 , O2 , O3 : ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢1 0 0 ⎥⎥⎥ ⎢⎢⎢ cos θ2 0 sin θ2 ⎥⎥⎥ ⎢⎢⎢cos θ3 − sin θ3 0⎥⎥⎥ ⎢⎢⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ ⎢⎢⎢0 cos θ1 − sin θ1 ⎥⎥⎥⎥⎥, ⎢⎢⎢⎢⎢ 0 1 0 ⎥⎥⎥⎥⎥, ⎢⎢⎢⎢⎢ sin θ3 cos θ3 0⎥⎥⎥⎥⎥. ⎢⎢⎣ ⎥⎥⎦ ⎢⎢⎣ ⎥⎥⎦ ⎢⎢⎣ ⎥⎥⎦ 0 sin θ1 cos θ1 − sin θ2 0 cos θ2 0 0 1 Here θ1 , θ2 , θ3 are angles of the right rotations with respect to corresponding base vectors. The unknown orientation tensor O(a) has the same components OKL in the Cartesian and rotated bases : O11 = cos θ2 cos θ3 , O12 = − cos θ2 sin θ3 , O13 = sin θ2 , O21 = cos θ1 sin θ3 + sin θ1 sin θ2 cos θ3 , O22 = cos θ1 cos θ3 − sin θ1 sin θ2 sin θ3 , O23 = − sin θ1 cos θ2 , O31 = sin θ1 sin θ3 − cosθ1 sin θ2 cos θ3 , O32 = sin θ1 cos θ3 + cosθ1 sin θ2 sin θ3 , O33 = cos θ1 cos θ2 θN are unknown angles of rotation from the Cartesian basis.
(19.4)
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Derivation of the rotated basis eK is realized with the formula eK, J = d J × eK = d JL eL × eK = d JL eLKM e M ,
(19.5)
where d J are Darboux vectors, d JL are their components in the chosen basis, and eLKM are components of the Levi-Civita tensor. From (19.5) follow Poisson’s type formulae: d J1 = e3 · e2, J = −e2 · e3, J , d J2 = e1 · e3, J = −e3 · e1, J ,
(19.6)
d J3 = e2 · e1, J = −e1 · e2, J . Using formulae (19.4), (19.6) and the expansion eK, J = OKM, J i M = OKM, J OLM eL ,
(19.7)
we obtain the relations d J1 = −θ1, J − θ3, J sin θ2 , d J2 = −θ2, J cos θ1 + θ3, J sin θ1 cos θ2 , d J3 = −θ2, J sin θ1 − θ3, J cos θ1 cos θ2 ,
(19.8)
which express the components of Darboux vectors via the rotation angles θN . In the general case these angles are represented by sums θN = θN◦ + ϑN , where θN◦ are orientation angles of the initial local basis, and ϑN are their increments during deformation. Darboux vectors and their components form the local curvature tensor of the postrotational space. The gradient vector transformation yI = a◦I · G = GIJ a J ,
G ≡ a◦I yI = GIJ a◦I a J ,
GIJ = y, I · a J
(19.9)
introduce an asymmetric tensor-gradient G(r) with two-basis expansion: in initial and rotated bases. In the reference state this tensor takes the value G◦ = a◦I gI . An objective (asymmetric) tensor of strains W(r) is given by the formulae W ≡ G − G◦ · Θ = WIJ a◦I aJ = a◦I wI .
(19.10)
Here wI (r), WIJ (r) are vectorial and scalar components of this tensor: wI = y, I − gI · Θ = a◦I · W = WIJ a J ,
WIJ = wI · a J = GIJ − GIJ◦ .
(19.11)
Formulae (19.10), (19.11) show dependence of the strain tensor on two independent kinematic fields: displacements and rotations. The second field is given by an independent orthogonal tensor, which has three independent degrees of freedom. These rotational degrees of freedom in Cauchy continuum can be eliminated with
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the help of three scalar relations on the strain tensor. In particular, the classic conditions of its symmetry lead to the relations: WJI = WIJ .
(19.12)
The strain tensor subjected to these relations will have six independent scalar components. The variant (19.12) realizes the polar decomposition of asymmetric strain tensor (19.10). Instead of (19.12), the alternative relations can be used W12 = W13 = W23 = 0 ,
(19.13)
W21 = W31 = W32 = 0 ,
(19.14)
which eliminate a triple of tensorial components. The variants (1.12)–(1.14) give the strain tensors by symmetric and triangular matrices of components: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢W11 W12 W13 ⎥⎥⎥ ⎢⎢⎢W11 0 0 ⎥⎥⎥ ⎢⎢⎢W11 W12 W13 ⎥⎥⎥ ⎢⎢⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ ⎢⎢⎢W12 W22 W23 ⎥⎥⎥⎥⎥ ∼ ⎢⎢⎢⎢⎢W21 W22 0 ⎥⎥⎥⎥⎥ ∼ ⎢⎢⎢⎢⎢ 0 W22 W23 ⎥⎥⎥⎥⎥ . ⎢⎢⎣ ⎥⎥⎦ ⎢⎢⎣ ⎥⎥⎦ ⎢⎢⎣ ⎥⎥⎦ W13 W23 W33 W31 W32 W33 0 0 W33 Application of the relations (19.13) and (19.14) can be justified in describing deformation of thin bodies. The strength of Cauchy’s continuum in a vicinity of point r is measured with the stress vectors σI (r) on unit coordinate areas tI = const . The equality √ sI = σI / g II (19.15) gives the Lagrangian contravariant stress vectors s I (r). Balance of the external and internal forces in an actual state of Cauchy’s continuum can be expressed with the virtual equation (f · δy − δw) dB + f ν · δydAν = 0 , (19.16) B
Aν
δw = s · δ wI = S δWIJ , S IJ = s I · a J . I
IJ
Here: δw is the virtual strain energy density per unit initial volume; δ is the corotational variation operator; S IJ are two-base components of the stress tensor; f, f ν are densities of the volume and surface forces.
19.2 A Rod Deformation Model with One-Dimensional Field of Finite Rotations A volume B occupied by a rod is usually bounded by a tubular surface A3 and two end surfaces An . A spatial local system of coordinates t J is connected with the base curve C3 of the rod so that t3 is the longitudinal parameter of the line, t1 and t2 are the transverse orthogonal coordinates. Let A is an arbitrary cross section of a rod.
19 Free Finite Rotations in Deformation of Thin Bodies
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The rod-like solid body can be described by the equation r = a + ti e◦i .
(19.17)
It expresses a 3D position vector r(t J ) via a positional vector a(t3 ) of a point of the base line and two orientation unit vectors e◦i (a), which are orthogonal to the base line and its tangential unit vector e◦3 = a, 3 . The relations gi = e◦i , g3 = e◦3 + ti e◦i, 3 , e◦3 = a, 3 (19.18) express a 3D local basis g I (r) via the orthonormal basis e◦I (a) of the base line. The rod deformation in the Cauchy model is defined with the vectors (19.1), y = r+w ,
yI = y, I = gI + w, I ,
(19.19)
where w(r) is the displacement vector field. The basis of the base line is deformed together with the rod. The local orthogonal transformation eI = e◦I · Θ ,
Θ ≡ e◦J eJ ,
Θ, i ≡ 0 ,
Θ · Θ ≡ 1,
(19.20)
gives a tensor-rotator Θ (a) and a rotating basis eI (a) with the reference value e◦I (a). Assuming that a rod remains a thin body, its actual configuration is approximated with a vector function similar to (19.17), y ≈ x + ti ei = x + ti e◦i · Θ ,
(19.21)
where x(a) is an actual position vector of a point of the base line. Approximation (19.21) gives the gradient vectors yI (19.19) of the actual state yi ≈ ei ,
y3 ≈ x, 3 + ti ei, 3 .
(19.22)
Formula (19.11) gives a nonzero strain vector corresponding to approximations (19.18) and (19.22), w3 = y, 3 − g3 · Θ ≈ u3 + ti vi3 . (19.23) The vectors of metric and torsion-bending strains at the base line are given as follows: u3 ≡ x, 3 − e◦3 · Θ = x, 3 − e3 , (19.24) vi3 ≡ ei, 3 − e◦i, 3 · Θ = ei · Ω 3 = ω 3 × ei ,
Ω 3 ≡ Θ · Θ, 3 .
(19.25)
The metric strain vector u3 (a) depends on the position vector and the tensor-rotator, while the vectors of torsion-bending strains vi3 (a) depend on the tensor-rotator, or on the axial vector ω 3 (a) generated by the skew-symmetric tensor Ω 3 . The approximation (19.21) eliminates the vectors wi which measure the strains of transversal fibres of the rod. The vector w3 measures the strains of longitudinal
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fibres. For accounting the Poisson effect, the generalized approximation of strain field is wI ≈ uI + ti viI . (19.26) Here the vectors u3 , vi3 are given by functions (19.24), (19.25), and the vectors ui , vi j are arbitrary and play the role of Lagrange multipliers in the virtual work equation. The expansions of vector-functions (19.26) are fulfilled in the rotating basis uI = u IJ e J ,
viI = viIJ e J ,
wI = wIJ e J
| ∀ JI
(19.27)
with the triangular matrices of components ⎡ ⎤ ⎢⎢⎢u11 0 0 ⎥⎥⎥ ⎢⎢⎢ ⎥ ⎢⎢⎢u u 0 ⎥⎥⎥⎥⎥ , ⎢⎢⎢ 21 22 ⎥⎥⎥ ⎣ ⎦ u31 u32 u33
⎡ ⎤ ⎢⎢⎢vi11 0 0 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢v v ⎥ ⎢⎢⎢ i21 i22 0 ⎥⎥⎥⎥⎥ , ⎣ ⎦ vi31 vi32 vi33
⎡ ⎤ ⎢⎢⎢w11 0 0 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢w w ⎥ ⎢⎢⎢ 21 22 0 ⎥⎥⎥⎥⎥ . ⎣ ⎦ w31 w32 w33
As a result, the approximations (19.21)–(19.27) permit to realize the variant (19.13) for representation of the rod strain tensor. Derivation of rotating basis eK is fulfilled with the formula (19.5): eK, 3 = d3 × eK = d3L eLKM e M ,
(19.28)
where d3 is a Darboux vector, while d3L are its components in the chosen basis. The formulae (19.8) express these components via the rotation angles θN (from Cartesian basis to rotated one): d31 = −θ1,3 − θ3,3 sin θ2 , d32 = −θ2,3 cos θ1 + θ3,3 sin θ1 cos θ2 ,
(19.29)
d33 = −θ2,3 sin θ1 − θ3,3 cos θ1 cos θ2 . ◦ of components d When θN =θN◦ these formulae give the reference values d3K 3K . The physical components of metric (19.24) and torsion-bending (19.25) strains in the rotating basis are defined by the formulae ◦ u3J = O JK xK, 3 − e3J , vi3J = eiJK ω3K , ω3K = d3K − d3K .
(19.30)
Here xK , O JK are unknown components of the position vector and orientation tensor (19.4) in the Cartesian basis. The generalized dynamic equations of the rod are constructed by the standard method from the virtual equation (19.16) and approximations (19.21), (19.23), (19.26). As a result, we obtain the one-dimensional equations of force and moment balance t3, 3 + p = 0 , m3, 3 + x, 3 × t3 + q = 0 , (19.31)
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with the resultant vectors of internal forces and couples t3 = σ 3 dA , m3 = ai × σ 3 ti dA , A
A
(19.32)
and with the resultant vectors p, q of external forces and couples along the center line. Representing in (19.31) t3 = T 3J e J , m3 = M3J e J , x, 3 × t3 = (e3K + u3K ) e JKL T 3L e J , we obtain the force scalar equilibrium equations T 3J, 3 + e JKL d3K T 3L + p · eJ = 0
(19.33)
and the couple scalar equilibrium equations M3J, 3 + e JKL d3K M3L + (e3K + u3K ) e JKL T 3L + q · eJ = 0 .
(19.34)
In these equations T 3J , M3J are physical components of internal forces and couples. The last terms (in form of scalar product) show that expansion of the vectorial equations is fulfilled in the rotating basis. The constitutive relations for the elastic rod are formulated by the generalized equations T 3J =
A
CJK w3K dA , M3L = eiJL
A
CJK w3K ti dA ,
(19.35)
in which the coefficients CJK can depend on strains w3L = u3L + ti vi3L (in a nonlinear manner).
19.3 A Shell Deformation Model with Two-Dimensional Field of Finite Rotations A spatial local coordinate system t J is connected with a base surface A of a shell, so that t1 and t2 are internal parameters of the surface, t3 is the transverse orthogonal coordinate, and dt3 is an element of shell transverse fibre. A volume B occupied by a shell is usually bounded by two face surfaces A1 , A2 and a boundary surface A3 , which includes boundary contour C of the base surface. Surfaces A and Am are given by the equations t3 = 0 t3 = hm , so that h1 t3 h2 (h1 and h2 are known functions of the base surface points or are constants). The volume and surface elements of the shell are given by dB = η dt3 dA ,
dA = a dt1 dt2 ,
dAm = αm dA , dA3 = η dt3 dC , η = g/a , (19.36) where g = det[g IJ ] , a = det[ai j ] are volume and surface Jacobians of the local coordinate system, αm is the metric parameter of the surface Am , dC is
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the boundary contour element of the base surface. Equations (19.36) can be used for computation of parameters αm . A shell-like solid body can be given by the equation r = a + t3 e◦3 .
(19.37)
It expresses spatial position vector r(t J ) via the positional vector a(ti ) of the surface point and the unit vector e◦3 (a), which is orthogonal to the surface and its tangential unit vectors e◦i = a, i . The equations g3 = e◦3 ,
gi = a◦i + t3 b◦i ,
a◦i = a, i ,
b◦i = e◦3, i
(19.38)
express spatial local basis g I (r) via the orthonormal surface basis e◦I (a). The shell deformation in the Cauchy model is defined by the vectors (19.1): y = r+ w,
yI ≡ y, I = gI + w, I ,
(19.39)
where w(r) is the displacement vector field. The surface basis is deforming together with the shell. The local orthogonal transformation aI = a◦I · Θ ,
Θ, 3 ≡ 0 ,
Θ·Θ ≡ 1
(19.40)
relates the surface tensor-rotator Θ (a) and the rotating basis eI (a) with the reference value e◦I (a). Assuming that the shell remains a thin body, its actual configuration is approximated by the vector function similar to (19.37), y ≈ x + t3 e 3 ,
(19.41)
where x(a) is the actual position vector of the surface point, e3 is the unit vector. The linear (relative to transverse coordinate) displacement field corresponds to the function (19.41), w = y − g ≈ u + t3 v ,
u = x−a ,
v = e3 − e◦3 .
(19.42)
Approximation (19.41) gives the gradient vectors yI (19.39) of the actual state y3 = e3 ,
yi = x, i + t3 bi ,
bi ≡ e3, i .
(19.43)
Formula (19.11) gives two nonzero strain vectors related by approximations (19.38) and (19.43), wi = y, i − gi · Θ ≈ ui + t3 vi , w3 ≈ 0 . (19.44)
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The surface vectors of metric and torsion-bending strains are given here by ui = x, i − a◦i · Θ = x, i − ai ,
x = a+u ,
vi = a3, i − b◦i · Θ = a3 · Ω i = ω i × a3 ,
Ω i ≡ Θ · Θ, i .
(19.45) (19.46)
The vectors of metric strains ui (a) depend on the position vector and tensorrotator, and the vectors of torsion-bend strains vi (a) depend on tensor-rotator, or on two axial vectors ω i (a) generated by two skew-symmetric tensors Ω i ≡ Θ · Θ, i . The approximation (19.41) eliminates the vector w3 (19.44) which measures the strains of transversal fibres of the shell. The vectors wi measure the strains of longitudinal fibres. For accounting the Poisson effect, the generalized approximation of the strain field is assumed w I ≈ u I + t3 v I . (19.47) Here the vectors ui ,vi are given by functions (19.45), (19.46), and the vectors u3 ,v3 are arbitrary and play the role of Lagrange multipliers in the virtual work equation. The expansion of vector-functions (19.47) is fulfilled in the rotated basis uI = u IJ e J , vI = vIJ e J , wI = wIJ e J with the matrix of components ⎡ ⎤ ⎢⎢⎢u11 u12 u13 ⎥⎥⎥ ⎢⎢⎢⎢ ⎥⎥ ⎢⎢⎢u21 u22 u23 ⎥⎥⎥⎥⎥ , ⎢⎢⎣ ⎥⎥⎦ 0 0 u33
⎡ ⎤ ⎢⎢⎢v11 v12 0 ⎥⎥⎥ ⎢⎢⎢⎢ ⎥⎥ ⎢⎢⎢v21 v22 0 ⎥⎥⎥⎥⎥ , ⎢⎢⎣ ⎥⎥⎦ 0 0 v33
| ∀ JI
(19.48)
⎡ ⎤ ⎢⎢⎢w11 w12 w13 ⎥⎥⎥ ⎢⎢⎢⎢ ⎥⎥ ⎢⎢⎢w21 w22 w23 ⎥⎥⎥⎥⎥ . ⎢⎢⎣ ⎥⎥⎦ 0 0 w33
The approximation (19.44), in general, does not allow to fulfill condition w21 = 0 (19.9) because rotator Θ (a) is independent of coordinate t3 . But this condition is the only possible variant of elimination of one degree of rotation freedom. It is possible, for example, to require realization of condition u21 = 0 instead of w21 = 0 or condition u21 = u12 instead of w21 = w12 . The most simple variant is the elimination of one degree of freedom in the rotator. In particular, one can eliminate a basis rotation with respect to e3 (a drilling rotation). In this case the rotation field will have two degree of freedom and can be represented by two independent scalar components. Derivation of the rotated basis eK is fulfilled with the formula (19.5), eK, i = di × eK = diJ e JKL eL ,
(19.49)
where di are Darboux vectors, diJ are their components in the chosen basis. The formulae (19.8) express these components via the rotation angles θN (from Cartesian basis to rotated one): di1 = −θ1, i − θ3, i sin θ2 , di2 = −θ2, i cos θ1 + θ3, i sin θ1 cos θ2 , di3 = −θ2, i sin θ1 − θ3, i cos θ1 cos θ2 .
(19.50)
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◦ of components d . When θN =θN◦ these formulae give the reference values diJ iJ The physical components of metric (19.45) and torsion-bending (19.46) strains in the rotated basis are defined by the formulae −1 ◦ uiJ = A−1 i O JK x K, i − eiJ , viJ = Ai e3JK ωiK , ωiK = diK − diK .
(19.51)
Here xK , O JK are unknown components of position vector and orientation tensor ◦ are known Darboux parameters of the basis e◦ . (19.4) in the Cartesian basis, diK K The generalized dynamic equations of the shell are constructed by the standard method as a consequence of the virtual equation (19.16) and approximations (19.44)–(19.47). As a result, we obtain the two-dimensional equations of force and moment balance C i + x, i × ti +C ∇i ti + p = 0 , ∇i m q=0, (19.52) with the resultant surface vectors of internal forces and couples defined by tI =
h2
sI ηdt3 ,
mI =
h1
h2
sI ηt3 dt3 ,
C i = e3 × mi , m
(19.53)
h1
and with the surface vectors p, C q of external forces and couples (η is known geometric parameter of the shell). In components Eq. (19.52) read ∇i ti = a−1 (ati ), i , ti = A−1 i T iJ e J , a = A1 A2 , i i −1 −1 C CiJ = e3KJ MiK , C = A MiJ e J , M m = A MiK eK , m i
i
(19.54)
and we obtain the scalar formulation of shell dynamic equations: −1 (A1 A2 A−1 i T iJ ), i + e JKL diK A1 A2 Ai T iL + A1 A2 p · e J = 0 , CiJ ), i + e JKL diK A1 A2 A−1 M CiL + (A1 A2 A−1 M i
i
(19.55)
+A1 A2 e JKL (eiK + uiK )T iL + A1 A2 C q · eJ = 0 . CiJ are physical components of internal forces and couples, In these equations T iJ M the last terms (in the form of scalar product) show that expansion of the vectorial equations (19.55) is fulfilled in the rotated basis. The resulting set of differential equations (19.50), (19.51), (19.55) in the general formulation contains first order partial derivatives of five kinematic and five dynamic functions. Conditions along the shell boundary contour are formulated with the virtual equality Ci j − Q Cj ) δω j = 0 . (n◦i T iJ − PJ ) δu J + (n◦i M
(19.56)
C j are given vectors of forces and couples, δu J , δω j are virtual displaceHere PJ , Q ments and rotations in the contour points, n◦i are components of the external unit normal to the contour. The equality (19.56) allows one to correctly formulate five boundary conditions of dynamic, kinematic and mixed types.
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Constitutive relations for the elastic shell are formulated by the generalized equations T iJ =
h2
σiJ ηi dt3 ,
h1
MiJ =
h2
σiJ ηi t3 dt3 ,
CiK = e3JK MiJ , M
(19.57)
h1
in which the stresses σiJ have to be given by linear or nonlinear functions of strains (19.48), ηi are the known functions weakly depending on the coordinate t3 .
19.4 Axisymmetric Bending Modes of Plates and Shells We consider a dome-like shell with axisymmetric base surface A. The local coordinates t J with orthonormal basis e J (t1 , t2 ) are given on the surface, so that t1 ∈ [0 , l] , t2 ∈ [0 , 2π], t3 ∈ [−h , h] are meridional, circumferential, and normal coordinates (2h is a dome thickness, l is the length of meridian). In addition, the cylindrical coordinates (x1 , t2 , x3 ) are introduced with orthonormal basis i J (t2 ) . The surface basis differs from the cylindrical one by an angle θ2 (t1 ) relative to the vector i2 . The corresponding transformation of the surface basis is realized by the rotator O2 . The dome shape in the reference state is given by parametric equations x1 = r(t1 ) , x3 = z(t1 ) , θ2 = θ(t1 ) ,
∀ t1 ∈ [0 , l] ,
(19.58)
dr/dt1 = cos θ , dz/dt1 = − sin θ , where r, z are the point coordinates in cylindrical system. We consider the axisymmetric deformation of a dome when the base surface remains the surface of revolution, so that x1 = x1 (t1 ) , x3 = x3 (t1 ) , θ2 = θ2 (t1 ) ,
∀ t1 ∈ [0 , l] ,
(19.59)
where x1 , x3 are the sought coordinates of a point in cylindrical system, θ2 is a sought rotation angle. The dome material is isotropic and linearly elastic. The full system of equations includes the kinematic equations following from (19.50) and (19.51), x1,1 = (1 + u11) cos θ2 + u13 sin θ2 , x3,1 = −(1 + u11) sin θ2 + u13 cos θ2 , v11 = (θ2 − θ),1 , v22 = r−1 (sin θ2 − sin θ) , u22 = r−1 (x2 − r) , (19.60) the dynamic equations following from (19.55), (rX1 ),1 − T 22 + r p · i1 = 0 , (rX3 ),1 + r p · i3 = 0 , (rM11 ),1 − M22 cos θ2 − r T 13 + r q · i2 = 0 , T 11 = X1 cos θ2 − X3 sin θ2 , T 13 = X1 sin θ2 + X3 cos θ2 ,
(19.61)
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and the generalized constitutive relations following from (19.57), u11 = (1 − ν2 )F −1 T 11 − νu22 , v11 = (1 − ν2)H −1 M11 − νv22 , u13 = γF −1 T 13 , T 22 = νT 11 + Fu22 , M22 = νM11 + Hv22 .
(19.62)
The following notations are used in this system: γ = E/G, F = 2hE, 3H = 2h3 E ; 2h is the shell thickness; E, G are modules of extensional and transverse shear; ν is Poisson’s ratio; uiJ (t1 ), vii (t1 ) are components of strains in the rotating basis; X1 (t1 ), X3 (t1 ) are components of internal forces in the cylindrical basis; T iJ (t1 ), Mii (t1 ) are components of internal forces and couples in the rotating basis; derivation is fulfilled with respect to the coordinate t1 . The resulting set of differential equations (19.60), (19.61) contains first order derivatives of three kinematic functions x1 (t1 ), x3 (t1 ), θ2 (t1 ), and three dynamic functions X1 (t1 ), X3 (t1 ), M11 (t1 ). Therefore, the values of three unknown functions have to be given on shell boundary contour.
19.4.1 Circular Plate Under Radial Compression In the cylindrical coordinates (x1 , t2 , x3 ), a circular plate is described by the parametric equalities x1 = R t1 , x3 = 0 , ∀ t1 ∈ [0 , 1] , ∀ t2 ∈ [0 , 2π] ,
(19.63)
where R , t1 are radius and radial coordinate. For the system (19.60)–(19.62), we formulate two boundary-value problems on axisymmetric deformation of the circular plate under the uniform radial compressive load of intensity P. Since the surface load is absent, one should set p = q = 0 in the system. At the pole t1 = 0, the following symmetry and regularity conditions should hold: T 11 (0) = T 22 (0) , M11 (0) = M22 (0) , X3 (0) = 0 . (19.64) At the boundary contour (t1 = l), the two variants of conditions are specified: (a) x3 (1) = 0 , X1 (1) = −P , M11 (1) = 0 , (b) x3 (1) = 0 , X1 (1) = −P , θ1 (1) = 0 . The nonlinear boundary-value problems with boundary conditions (a) and (b) were solved by the shooting method with the Mathcad software package. The branching solutions of these boundary-value problems were obtained by a variation of the load parameter P. The main results of numerical solution are presented in Figs 19.1 and 19.2 for the plate with parameters E/G = γ = 2.5 , h2 /(3R2 ) = ε = 0.025 , ν = 0.25 . The following three bifurcation values of the numerical load parameter p = P/C (C = 2Ehε) were obtained:
19 Free Finite Rotations in Deformation of Thin Bodies
273
0.2
1a
w
1b
2a
2b
3a
3b
0.1
10
0
20
30
40
λ
Fig. 19.1 Bifurcation maps of plate deformation in terms of parameters λ = P/P1a and w = −u3 (0)/R, 4
3a
2a
1a
u,% 0
3 2 1
0
0.5
1.0
1.5
2.0
p
2.5
Fig. 19.2 Bifurcation maps in terms of parameters p = P/C and u = −u2 (1)/R
Problem (a): p1a 0.1085, p 2a 0.772, p 3a 1.958. Problem (b): p1b 0.3915, p 2b 1.313, p 3b 2.763. The bifurcation maps of plate deformation are shown in Fig. 19.1 in terms of parameters λ = P/P1a , w = −u3 (0)/R, and in Fig. 19.2 in terms of parameters p = P/C, u = −u2 (1)/R. Here P1a = C p1a is the classical critical load for the simply supported plate, so that λ1a = 1 for the classic value of the parameter λ. The line 0 in Fig. 19.2 is the trajectory of base (plane) strains, and lines 1a, 2a, 3a are trajectories of buckling modes. These modes are shown in [6].
19.4.2 Conical Shell Under Radial Compression In the cylindrical coordinates, a meridian of the conical shell (dome) is given by the parametric equations (19.58) of the form
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x1 = r = t1 cos α , x3 = z = (l − t1 ) sin α , θ2 = α , ∀ t1 ∈ [0 , l] ,
(19.65)
where l is length of the meridian, α is angle in the base of the dome. We study an axisymmetric deformation (19.60)– (19.62) of the dome under a compressive radial load P uniformly distributed along the boundary contour r = b = l cos α. The surface load is absent, so that p = q = 0 in the system (19.60)–(19.62). At the pole t1 = 0, the following symmetry and regularity conditions should hold: T 11 (0) = T 22 (0) , M11 (0) = M22 (0) , X3 (0) = 0 .
(19.66)
At the boundary point t1 = l, the conditions of simple supported movable contour are: x3 (l) = 0 , X1 (l) = −P , M11 (l) = 0 . (19.67) 3a
2a
1a
4
u,% 0
3 2 1
0
0.5
1.0
1.5
2.0
p
2.5
Fig. 19.3 Bifurcation map for the dome with angle α = π/180.
4
u,%
1
2
2
3 2 1 0
3 0.5
1.0
-1
1.5
p
2.0
Fig. 19.4 Bifurcation map for the dome with angle α = π/12.
The nonlinear boundary-value problems (19.66), (19.67) were solved by the method of shooting from the point t1 = l to the point t1 = δl, where δl is a small varying value. Some results for isotropic conical domes are shown in Figs 19.3 and 19.4 h2 /(3l2 ) = ε = 0.025 , E/G = γ = 2.5 , ν = 0.25. for the domes with parameters
19 Free Finite Rotations in Deformation of Thin Bodies
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The results are represented by diagrams in the plane (p , u) where p = P/C and u is the percentage value of boundary strain |u22(l)| (19.60) . Figure 19.3 compares the bifurcation map for the dome with angle α = π/180, with that for the plate. Solid curves 1, 2 and 3 are the first three branches (trajectories) of bending modes of the dome. They are separated by the trajectories of the plate buckling modes (dashed curves of Fig. 19.2). Figure 19.4 shows these solutions for a dome with angle α = π/12. In this case, the branches of higher modes are below the line that represents the plane modes of the plate, and even below the abscissa axis. It means that the dome can have the buckling modes with the stretched boundary contour. Bending modes corresponding to all these branches are shown in [7].
19.4.3 A Dome Axisymmetric Deformation in Thermal Cycle of Phase Transformations As the constitutive relations for a shape-memory alloy in the phase transformation cycle, we use the micromechanical relations, which are formulated in [9]: w11 = φ11 + (σ11 − νσ22 )/E , 1 2 , w13 = φ13 + σ13 /G , dφii /dq = (1 − q)λ(κ φii + sii /σ) , dφ13 /dq = (1 − q)λ(κ φ13 + σ13 /σ) , π T+ − T q = sin , T− T T+ . 2 T+ − T−
(19.68)
Here wiJ and φiJ are total and phase strains, respectively, σiJ and sii are the stresstensor and stress-deviator components, E, G and ν are elasticity moduli, κ, λ and σ are experimental parameters of thermocycle, T + and T − are the initial and final temperatures of the direct transformation, T is the current temperature, 0 q 1 is an internal state parameter defined as the volume fraction of the martensite phase. According to (19.68), growing of the phase strains is completed at q = 1. The phase transformation is treated as a quasi-static process with temperature uniformly distributed over the sample, so that the parameter q does not depend on coordinates. In the phase transformation interval, the elastic moduli in (19.68) are not constant but change from their austenite to martensite values. For the phase transformations considered, these moduli can be represented as the averaged relations E = q E− + (1− q ) E+ , ν = q ν− + (1− q ) ν+ , G = E/(2 + 2ν) , where the subscripts minus and plus refer to martensite and austenite phases, respectively. Assuming that the stresses depend on q much more weakly as compared to the phase strains, we can find an approximate solution of the differential equations (19.68):
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2σ11 − σ22 2σ22 − σ11 σ13 , φ22 η , φ13 η , (19.69) 3 σκ 3 σκ σκ D κ E η(q) ≡ exp 1 − (1 − q)1+λ − 1 . 1+λ This solution satisfies the following physical conditions: the phase strains are absent in the austenite phase and reach maximal values in the martensite phase. Substitution of the functions (19.69) into the first three equations (19.68) gives three-dimensional constitutive relations φ11 η
E 0 w11 η1 σ11 − η2 σ22 ,
E0 w22 η1 σ22 − η2 σ11 ,
E 0 w13 η3 σ13 ,
(19.70)
E0 2E 0 E0 E0 E0 E0 +η , η2 (q) ≡ ν +η , η3 (q) ≡ +η , E 3 σκ E 3 σκ G σκ where E 0 is a constant that has dimension of stress and which is conveniently identified as E − or E + . Equations (19.70) describe one-directional phase transformation as thermoelastic deformation with an implicit temperature dependence (via the parameter q). Integration of these equations through the shell thickness leads to the generalized constitutive relations η1 (q) ≡
C0 u11 η1 T 11 − η2 T 22 ,
C0 u22 η1 T 22 − η2 T 11 ,
l−1 H0 v11 η1 M11 − η2 M22 ,
C0 u13 η3 T 13 ,
l−1 H0 v22 η1 M22 − η2 M11 ,
(19.71)
where C0 = 2hE0 and H0 = 0 /3 are the stiffness parameters. We consider the simple supported spherical dome loaded in the austenite phase by the uniform normal pressure of intensity P. The reference shape (19.58) of the dome meridian is given by the parameters 2h3 E
r = lα−1 sin θ, z = lα−1 (cosθ − cos α) , θ = αt1 , ∀ t1 ∈ [0 , 1] , where l is length of the meridian, α is angle in the base of the dome. In the bending state the load vector p is directed along internal normal to the base surface so that p · i1 = −P sin θ2 , p · i3 = −P cos θ2 , q · i2 = 0 .
(19.72)
At the boundary point t1 = 1, the conditions of simple supported contour are specified as x1 (1) = b , x3 (1) = 0 , M11 (1) = 0 . (19.73) At the pole t1 = 0, the conditions (19.66) should hold. The boundary problem (19.66), (19.73) for the set of equations (19.60), (19.61), (19.71) and (19.72) is solved by the shooting method, for discrete values of the state parameter q ∈ [0 , 1]. The complete thermocycles, obtained by solving the directand inverse-transformation problems, are shown in Fig. 19.5a for the shell without transient buckling (α = π/36, ε = 0.025, λ = 0), and in Fig. 19.5b for the shell with
19 Free Finite Rotations in Deformation of Thin Bodies
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2 W
1
2
1
0 40
20
60
T,°C
a W 2
1
2
1
0 20
40
60
T,°C
b Fig. 19.5 Phase diagrams: a) for a shell without transient buckling (α = π/36, ε = 0.025, λ = 0), b) for a shell with transient buckling (α = π/36, ε = 0.01, λ = 1)
transient buckling (α = π/36, ε = 0.01, λ = 1) as trajectories of the maximal axial displacement (in relation to the dome height) depending on temperature. The direct transition (curves 1) was computed for the load parameter p = 0.002, much smaller than upper critical loads of the dome in the austenite and martensite phases. The inverse transition (curves 2) was computed with zero load parameter. The solid curve 1 in Fig. 19.5a shows the trajectory of direct transition with a dynamic jump at the upper critical point (T 33.3◦C, q 0.69). The solid curve 2 shows the trajectory of inverse transition with a dynamic jump at the lower critical
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L.I. Shkutin
point (T 68.5◦C, q 0.04). These curves form the dynamic hysteresis loop with the instantaneous jumps of the dome from one equilibrium shape to another at fixed temperature. The dashed curves 1 and 2 in Fig. 19.5b form the static hysteresis loop provided that the dome deformation is directed along this loop.
Concluding Remark The papers [6–9] allow to acquaint in more detail with earlier author’s results. Acknowledgements This research was supported by the RFBR under grant No. 11-01-00053.
References 1. Ericksen, J. L., Truesdell, C.: Exact theory of stress and strain in rods and shells. Arch. Rat. Mech. Anal. 1 (1), 295–323 (1958) 2. Shkutin, L. I.: Deformation Mechanics of Flexible Bodies. Nauka, Novosibirsk (1988) 3. Shkutin, L. I.: Generalized models of the Cosserat type for finite deformation analysis of thin bodies. J. Appl. Mech. Tech. Phys., 37 (4), 400–410 (1996) 4. Eremeyev, V. A., Zubov, L. M.: Mechanics of Elastic Shells. Nauka, Moscow (2008) 5. Pietraszkievicz, W., Eremeyev, V. A.: On natural strain measures of the non-linear micropolar continuum. Int. J. Solids Struct., 46, 774–787 (2009) 6. Shkutin, L. I.: Numerical analysis of axisymmetric buckling of plates under radial compression. J. Appl. Mech. Tech. Phys., 45 (1), 89–95 (2004) 7. Shkutin, L. I.: Numerical analysis of axisymmetric buckling of a conical shell under radial compression. J. Appl. Mech. Tech. Phys., 45 (5), 741–746 (2004) 8. Shkutin, L. I.: Analysis of axisymmetric phase strains in plates and shells. J. Appl. Mech. Tech. Phys., 48 (2), 285–291 (2007) 9. Shkutin, L. I.: Axisymmetric deformation of plates and shells with phase transformations under thermal cycling. J. Appl. Mech. Tech. Phys., 49 (2), 330–335 (2008)
Chapter 20
On Universal Deformations of Nonlinear Isotropic Elastic Shells Leonid M. Zubov
Abstract A two-dimensional nonlinear model of the elastic shell is considered. It is assumed that the shell is the deformable surface with kinematically independent fields of translations and rotations. Within this model, several sets of finite nonuniform deformations are found, which for any isotropic shell satisfy equilibrium equations without surface loads. Universal solutions are obtained for six families of deformations, which are characterized by certain fields of surface translations. Each of these families consists of several subfamilies, which differ by the rotation field. It is found out that the closed isotropic spherical shell without external loads has four different equilibrium states, and the shell remains spherical in each of these states. The general expression for the rotation field of the isotropic micropolar plate is found, using the equilibrium equations and the distorsion tensor for uniform deformations. Equilibrium of the Cosserat membrane, which has the form of minimal surface, is studied. Keywords Universal solution · Nonlinear elasticity · Cosserat shell · Micropolar shell
20.1 Basic Statements Let σ be the reference (undeformed) base surface of the shell. We refer the particle of σ to Gaussian coordinates qα , α = 1, 2, with the position vector r(q1 , q2 ) = x1 i1 + x2 i2 + x3 i3 . Here xk are the Cartesian coordinates of the point of σ, while ik is the orthonormal Cartesian frame (k = 1, 2, 3). The natural and reciprocal bases on σ are given by the following relations: L. M. Zubov (B) South Federal University, Milchakova str., 8a, 344090, Rostov on Don, Russia e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 20, © Springer-Verlag Berlin Heidelberg 2011
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rα =
∂r , ∂qα
β
rβ · rα = δα ,
rβ · n = 0 ,
(20.1)
β
where δα is the Kronecker symbol and n is the unit normal to σ. The base surface Σ of the shell after deformation is referred to the same convected coordinates qα as well, and the position of the point of Σ is given by the position vector R(q1 , q2 ) = X1 i1 + X2 i2 + X3 i3 , where Xk are the Cartesian coordinates. The boundary value problem of the nonlinear micropolar shell consists of the equilibrium equations, see [1–6], div D + f = 0 ,
divG + (FT · D)× + l = 0 ,
(20.2)
the constitutive equations D = P·H , G = K·H , ∂W(E, L) ∂W(E, L) P= , K= , ∂E ∂L
(20.3)
and the kinematic relations
1 ∂H T L = rα ⊗ · H , 2 ∂qα × ∂Φ ∂Φ Δ Δ gradΦ = rα ⊗ α , div Φ = rα · α . ∂q ∂q E = F · HT ,
(20.4)
Here W is the 2D elastic strain energy density of the shell, grad and div are the surface gradient and divergence operators with regard to points on σ, respectively, D and G are the surface stress resultant and stress couple tensors, F = gradR, and H is the microrotation tensor characterizing the rotational degrees of freedom of the shell particles. E and L are the surface stretch and wryness tensors, while f and l are the external surface force and moment vectors, respectively. A× denotes the vectorial invariant of the second-order tensor A defined by the relation A× = (Amn im in )× ≡ Amn im × in . In what follows only isotropic elastic shells are considered. This means that functions W(E, L), P(E, L), K(E, L) have the form W(QT · E · Q, εQT · L · Q) = W(E, L) ,
ε = det Q ,
P(QT · E · Q, εQT · L · Q) = QT · P(E, L) · Q ,
(20.5)
K(QT · E · Q, εQT · L · Q) = εQT · K(E, L) · Q . In (20.5), Q is any orthogonal tensor satisfying the restriction n·Q = n .
(20.6)
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Equations (20.5) indicate that E and P are true second-order tensors, while L and K are second-order pseudotensors. We study finite deformations of isotropic shells without surface loads, i.e. we find solutions of (20.2) with f = l = 0. The solutions discussed below for isotropic micropolar shells form six families of non-uniform deformations. Each family is characterized by the mapping R = R(q1 , q2 ) of the surface σ into the surface Σ, which sets the translation field of the shell. Since the field of microrotations in the micropolar shell is kinematically independent upon the translation field, the proper orthogonal tensor field H(q1 , q2 ) needs to be specified for determining the shell deformed state. Therefore, each of the mentioned six families of non-uniform deformations consists of several subfamilies, which differ from each other by microrotation fields. For all subfamilies the expressions for tensors E and L are presented, which are determined with the help of (20.4). The distorsion tensor F is identical for all solutions of the family.
20.2 Families of Non-Uniform Deformations 20.2.1 Cylindrical Bending of a Rectangular Plate Let the surface σ be plain x1 = const such that n = i1 . Let also q1 = x2 , q2 = x3 . We denote by R, Φ, Z the circular cylindrical coordinates of a point on the deformed surface Σ. The cylindrical bending of the plate is specified by the equations R = R0 ,
Φ = κx2 ,
Z = αx3 ,
(20.7)
where R0 , Φ, α are constants. The tensor F corresponding to (20.7) is F = κR0 i2 ⊗ eΦ + αi3 ⊗ i3 ,
Δ
eΦ = −i1 sin Φ + i2 cos Φ .
(20.8)
Since H is the kinematically independent field, one needs to specify also the microtation field which is consistent with the mapping (20.7). Here we consider the following subfamilies for H: Subfamily 1a: H = i1 ⊗ eR + i2 ⊗ eΦ + αi3 ⊗ i3 ,
Δ
eR = i1 cosΦ + i2 sin Φ ,
E = κR0 i2 ⊗ i2 + αi3 ⊗ i3 ,
L = κi2 ⊗ i3 .
(20.9)
Subfamily 1b: H = i1 ⊗ eR − i2 ⊗ eΦ − i3 ⊗ i3 , E = − (κR0 i2 ⊗ i2 + αi3 ⊗ i3 ) ,
L = −κi2 ⊗ i3 .
(20.10)
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Subfamily 1c: H = −i1 ⊗ eR + i2 ⊗ eΦ − i3 ⊗ i3 , E = κR0 i2 ⊗ i2 − αi3 ⊗ i3 ,
L = −κi2 ⊗ i3 .
(20.11)
Subfamily 1d: H = −i1 ⊗ eR − i2 ⊗ eΦ + i3 ⊗ i3 , E = −κR0 i2 ⊗ i2 + αi3 ⊗ i3 ,
L = κi2 ⊗ i3 .
(20.12)
Microrotations in subfamilies 1a–1d differ from each other by rotations about various axes. Let us introduce components P sm = i s · P · im , Ksm = i s · K · im , (s, m = 1, 2, 3). Then Eqs (20.3), (20.4) yield P1m = K1m = 0. In Eq. (20.5) we consider Q = Q1 = 2i1 ⊗ i1 − I ,
det Q1 = 1 .
Here I is the identity tensor of the three-dimensional Euclidean space. From Eqs (20.9)–(20.12) we obtain QT1 · E · Q1 = E ,
QT1 · L · Q1 = L .
(20.13)
Equations (20.5) and (20.13) yield the relations P · Q1 = Q1 · P ,
K · Q1 = Q1 · K ,
which are equivalent to Pt1 = Kt1 = 0 ,
t = 1, 2, 3 .
(20.14)
Further, on the basis of Eqs (20.9)–(20.12) we have QT2 · E · Q2 = E ,
QT2 · L · Q2 = −L ,
Q2 = I − 2i2 ⊗ i2 ,
det Q2 = −1 , (20.15)
QT3 · E · Q3 = E ,
QT3 · L · Q3 = −L ,
Q3 = I − 2i3 ⊗ i3 ,
det Q3 = −1 .
Assuming Q = Q2 and Q = Q3 in Eq. (20.5), with the help of (20.15) in addition to (20.14) we obtain the relations P23 = P32 = 0 ,
K22 = K33 = 0 .
(20.16)
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283
From Eqs (20.3), (20.8)–(20.12), (20.14), (20.15) it follows that for the uniform plate the following formulas are valid: D = D22 i2 ⊗ eΦ + D33 i3 ⊗ i3 ,
G = G23 i2 ⊗ i3 + G 32 i3 ⊗ eΦ ,
(20.17)
and the tensor components in (20.17) are constant. Considering that the tensor FT · D is symmetric in this case, we see that the moment equilibrium equation (20.2)2 with l = 0 is identically satisfied for any values of R0 , κ, α. Since Eq. (20.7) leads to ∂eΦ = −κeR , ∂x2 the first equilibrium equation of (20.2) with f = 0 results in D22 = 0 and is satisfied by choosing the constant R0 . Hence, it is proved that deformations characterized by the relations (20.7), (20.9)–(20.12) satisfy the equilibrium equations of any homogeneous isotropic plate without surface loads.
20.2.2 Straightening of a Circular Cylindrical Shell Sector We introduce the circular cylindrical coordinates r, ϕ, z of a point on the surface σ by the relations x1 = r cos ϕ , x2 = r sin ϕ , x3 = z . Let us study the circular cylindrical shell of radius r0 , and let ϕ, z be Gaussian coordinates q1 , q2 . Straightening (unbending) and stretching of the shell is set by the following transformation: X1 = A ,
X2 = ηϕ ,
X3 = αz ,
q1 = ϕ ,
q2 = z ,
(20.18)
where A, η, α are constants. Introducing the unit vectors er = i1 cos ϕ + i2 sin ϕ ,
eϕ = −i1 sin ϕ + i2 cos ϕ ,
with the help of Eqs (20.18) we obtain F=
η eϕ ⊗ i2 + αi3 ⊗ i3 . r0
(20.19)
In addition to (20.19) we introduce the following subfamilies for the microrotations: Subfamily 2a: H = er ⊗ i1 + eϕ ⊗ i2 + i3 ⊗ i3 , η 1 E = eϕ ⊗ eϕ + αi3 ⊗ i3 , L = − eϕ ⊗ i3 . r0 r0
(20.20)
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Subfamily 2b: H = −er ⊗ i1 + eϕ ⊗ i2 − i3 ⊗ i3 , η 1 E = eϕ ⊗ eϕ − αi3 ⊗ i3 , L = − eϕ ⊗ i3 . r0 r0
(20.21)
H = er ⊗ i1 − eϕ ⊗ i2 − i3 ⊗ i3 , η 1 E = − eϕ ⊗ eϕ − αi3 ⊗ i3 , L = − eϕ ⊗ i3 . r0 r0
(20.22)
H = −er ⊗ i1 − eϕ ⊗ i2 + i3 ⊗ i3 , η 1 E = − eϕ ⊗ eϕ + αi3 ⊗ i3 , L = − eϕ ⊗ i3 . r0 r0
(20.23)
Subfamily 2c:
Subfamily 2d:
As in Subsect. 20.2.1, one proves that for the uniform isotropic shell under deformations (20.18), (20.20)–(20.23) the surface stress resultant and stress couple tensors are D = D22 eϕ ⊗ i2 + D33 i3 ⊗ i3 ,
G = G23 eϕ ⊗ i3 + G32 i3 ⊗ i2 ,
(20.24)
where D22 , D33 ,G 23 ,G 32 are constants. With Eqs (20.24) and f = l = 0 both equilibrium equations (20.2) are satisfied identically for any A, η, α.
20.2.3 Bending or Eversion of a Cylindrical Shell Let us study the following family of deformations of a circular cylindrical shell with initial radius r0 : R = R0 ,
Φ = κϕ ,
Z = αz ;
q1 = ϕ ,
q2 = z ,
(20.25)
where R0 , κ, α are constants. Under such deformation a sector of the undeformed circular cylindrical shell becomes a sector of the cylindrical shell of radius R0 . Besides, a stretching or a compression along the cylinder axis occurs. The cases κ > 0, α < 0 or κ < 0, α > 0 correspond to eversion of the cylindrical shell sector (or the closed cylinder). The distorsion tensor takes the form F=
κR0 eϕ ⊗ eΦ + αi3 ⊗ i3 . r0
(20.26)
We introduce the following subfamilies of H which differ from each other again by rotations about various axes.
20 On Universal Deformations of Nonlinear Isotropic Elastic Shells
285
Subfamily 3a: H = er ⊗ eR + eϕ ⊗ eΦ + i3 ⊗ i3 , κR0 κ−1 E= eϕ ⊗ eϕ + αi3 ⊗ i3 , L = eϕ ⊗ i3 . r0 r0
(20.27)
H = −er ⊗ eR + eϕ ⊗ eΦ − i3 ⊗ i3 , κR0 1+κ E= eϕ ⊗ eϕ − αi3 ⊗ i3 , L = − eϕ ⊗ i3 . r0 r0
(20.28)
H = er ⊗ eR − eϕ ⊗ eΦ − i3 ⊗ i3 , κR0 1+κ E=− eϕ ⊗ eϕ − αi3 ⊗ i3 , L = − eϕ ⊗ i3 . r0 r0
(20.29)
Subfamily 3b:
Subfamily 3c:
Subfamily 3d: H = −er ⊗ eR − eϕ ⊗ eΦ + i3 ⊗ i3 , κR0 κ−1 E=− eϕ ⊗ eϕ + αi3 ⊗ i3 , L = eϕ ⊗ i3 . r0 r0
(20.30)
Applying Eq. (20.5), one proves that in the case of homogeneous isotropic cylindrical shell under deformations (20.25), (20.27)–(20.30) the surface stress resultant and stress couple tensors have the following expressions with constant coefficients: D = D22 eϕ ⊗ eΦ + D33 i3 ⊗ i3 ,
G = G23 eϕ ⊗ i3 + G32 i3 ⊗ eΦ ,
(20.31)
According to (20.25), (20.26) and (20.31) we have div D = −
κ D22 eR , r0
divG = 0 ,
(FT · D)× = 0.
(20.32)
From Eqs (20.32) it follows that the moment equilibrium equation (20.2)2 with l = 0 is satisfied for any values of constants R0 , κ, α, and the force equilibrium equation with f = 0 leads to D22 = 0 and can be used to determine the constant R0 .
20.2.4 Straightening and Bending of a Cylindrical Shell Let us discuss a circular cylindrical shell which deformation is given by the relations R = R0 ,
Φ = sz ,
Z = tϕ ,
(20.33)
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where R0 , s, t are constants. Under this deformation a cylindrical shell sector is straightened to a rectangular plate, which is bent afterwards into another cylinder sector with the axis orthogonal to that of the initial cylinder. The distorsion tensor corresponding to (20.33) takes the form F=
t eϕ ⊗ i3 + sR0 i3 ⊗ eΦ . r0
(20.34)
Consistent with (20.34) the fields H consist of the following subfamilies: Subfamily 4a: H = er ⊗ eR + eϕ ⊗ i3 − i3 ⊗ eΦ , t 1 E = eϕ ⊗ eϕ − sR0 i3 ⊗ i3 , L = si3 ⊗ eϕ − eϕ ⊗ i3 . r0 r0
(20.35)
Subfamily 4b: H = er ⊗ eR − eϕ ⊗ i3 + i3 ⊗ eΦ , t 1 E = − eϕ ⊗ eϕ + sR0 i3 ⊗ i3 , L = −si3 ⊗ eϕ − eϕ ⊗ i3 . r0 r0
(20.36)
Subfamily 4c: H = −(er ⊗ eR + eϕ ⊗ i3 + i3 ⊗ eΦ ) , t 1 E = − eϕ ⊗ eϕ − sR0 i3 ⊗ i3 , L = −si3 ⊗ eϕ − eϕ ⊗ i3 . r0 r0
(20.37)
Subfamily 4d: H = −er ⊗ eR + eϕ ⊗ i3 + i3 ⊗ eΦ , t 1 E = eϕ ⊗ eϕ + sR0 i3 ⊗ i3 , L = si3 ⊗ eϕ − eϕ ⊗ i3 . r0 r0
(20.38)
For deformations described by (20.33) and (20.35)–(20.38), the surface stress resultant and stress couple tensors of the uniform isotropic cylindrical shell have the form (20.31), which garantees that the equilibrium equations become satisfied.
20.2.5 Radially Symmetric Deformation of a Spherical Shell Let the initial state of the shell be a spherical segment of radius r0 which, in particular, may be a closed sphere. We describe position on the surface σ by the spherical coordinates r, ϕ, θ, which are related with the Cartesian coordinates by
20 On Universal Deformations of Nonlinear Isotropic Elastic Shells
x1 = r cos ϕ cos θ ,
x2 = r sin ϕ cos θ ,
287
x3 = r sin θ .
Let us consider longitude q1 = ϕ (0 ϕ 2π) and the geographic % the geographic & π π 2 latitude q = θ − 2 θ 2 to be the Gaussian coordinates on σ. At poles of the sphere we have θ = ± π2 . We denote spherical coordinates of the surface point in the deformed state as R, Φ, Θ. Further we use two orthonormal frames er , eϕ , eθ and eR , eΦ , eΘ , which are related with the Cartesian base by er = (i1 cos ϕ + i2 sin ϕ) cos θ + i3 sin θ , eϕ = −i1 sin ϕ + i2 cos ϕ , eθ = (−i1 cos ϕ + i2 sin ϕ) sin θ + i3 cos θ , (20.39) eR = (i1 cos Φ + i2 sin Φ) cos Θ + i3 sin Θ , eΦ = −i1 sin Φ + i2 cos Φ , eΘ = (−i1 cos Φ + i2 sin Φ) sin Θ + i3 cos Θ . We set the deformation of the sphere using the relations R = R0 = const ,
Φ=ϕ,
Θ=θ.
(20.40)
On the basis of Eqs (20.39), (20.40) we have F=
R0 (eϕ ⊗ eϕ + eθ ⊗ eθ ) . r0
The mapping (20.40) is supplemented by two subfamilies of H. Subfamily 5a: H = er ⊗ er + eϕ ⊗ eϕ + eθ ⊗ eθ = I , R0 E = (eϕ ⊗ eϕ + eθ ⊗ eθ ) , L = 0 . r0
(20.41)
(20.42)
Subfamily 5b: H = er ⊗ er − eϕ ⊗ eϕ − eθ ⊗ eθ , R0 2 E = − (eϕ ⊗ eϕ + eθ ⊗ eθ ) , L = (eθ ⊗ eϕ − eϕ ⊗ eθ ) . r0 r0
(20.43)
20.2.6 Eversion of a Spherical Segment In this case the coordinate conversion and the tensor of distortion are the following: R = R0 ,
Φ=ϕ,
Θ = −θ ,
(20.44)
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F=
R0 (eϕ ⊗ eΦ − eθ ⊗ eΘ ) . r0
(20.45)
Two subfamilies of microrotations are given by the following relations: Subfamily 6a: H = −er ⊗ eR + eϕ ⊗ eΦ − eθ ⊗ eΘ , R0 2 E = (eϕ ⊗ eϕ + eθ ⊗ eθ ) , L = (eθ ⊗ eϕ − eϕ ⊗ eθ ) . r0 r0
(20.46)
Subfamily 6b: H = −er ⊗ eR − eϕ ⊗ eΦ + eθ ⊗ eΘ , R0 E = − (eϕ ⊗ eϕ + eθ ⊗ eθ ) , L = 0 . r0
(20.47)
Let us assume that Q = Q4 = 2er ⊗ er − I, det Q4 = 1 in Eqs (20.5). For deformations (20.40), (20.42), (20.43) and (20.44), (20.46), (20.47) the following equalities hold true: QT4 · E · Q4 = E , QT4 · L · Q4 = L . (20.48) From Eqs (20.5) and (20.48) it follows that Q4 · P = P · Q4 ,
Q4 · K = K · Q4 ,
from which we obtain eϕ · P · er = eθ · P · er = eϕ · K · er = eθ · K · er = 0 .
(20.49)
Supposing Q = Q5 = I − 2eϕ ⊗ eϕ , Q = Q6 = I − 2eθ ⊗ eθ , det Q5 = −1, detQ6 = −1 in Eqs (20.5), on the basis of (20.42), (20.43), (20.46), and (20.47) we get P = Pϕϕ eϕ ⊗ eϕ + Pθθ eθ ⊗ eθ ,
K = Kϕθ eϕ ⊗ eθ + Kθϕ eθ ⊗ eϕ .
(20.50)
Let us consider the orthogonal tensor Q7 = er ⊗ er + eϕ ⊗ eθ − eθ ⊗ eϕ ,
detQ7 = 1 .
According to Eqs (20.42), (20.43), (20.46), and (20.47) the following equalities hold true: Q7 · P = P · Q7 , Q7 · K = K · Q7 . (20.51) Using Eqs (20.50), Eqs (20.51) yield Pϕϕ = Pθθ ,
Kθϕ = −Kϕθ .
Hence, it is proved in the case of homogeneous isotropic spherical shell undergoing deformations (20.40), (20.42), (20.43), (20.44), (20.46), and (20.47), the following relations hold true:
20 On Universal Deformations of Nonlinear Isotropic Elastic Shells
P = a(eϕ ⊗ eϕ + eθ ⊗ eθ ) ,
K = b(eϕ ⊗ eθ − eθ ⊗ eϕ ) ,
289
(20.52)
where a and b are constants. With the help of Eqs (20.41), (20.45), and (20.52) one can check that the moment equilibrium equation (20.2) is identically satisfied with l = 0, and the force equilibrium equation (20.2) is reduced to a = 0 for f = 0 and can be used to determine the radius R0 . One can implement the eversion not only to a segment of the spherical shell but to the closed sphere as well. For example, one can perform the eversion of the whole sphere by cutting it into two halfs, everting each of them and then sticking them together along the edges. Therefore, the obtained solutions (20.40), (20.42), (20.43), and (20.44), (20.46), and (20.47) mean, in particular, that there are four different states of the closed isotropic micropolar spherical shell in the absence of external force and moment loads. The shell maintains its spherical form in each of these states.
20.3 Uniform Deformations of an Isotropic Plate Let us assume that the surface σ is a part of a plane, i.e. we study a flat shell, which is located in the plane x1 x2 and has the normal vector n = i3 . The uniform deformation of the plate is described by the relations X s = aαs xα + b s , α = 1, 2 ,
aαs = const ,
s = 1, 2, 3 ,
b s = const ,
H = const .
(20.53)
Obviously, the transformation (20.53) leaves the plate flat. Since under the uniform deformation due to Eqs (20.4), (20.53) we have L = 0 and E = const, the surface stress resultant tensor D and the stress couple tensor G are also constant in the homogeneous plate. So, the first vector equilibrium equation (20.2)1 with f = 0 is identically satisfied. Assuming Q = −I in the condition of isotropy (20.5), we get K(E, 0) = 0. Hence, G = 0 under the uniform deformation. The moment equilibrium equation (20.2)2 with l = 0 is transformed into (FT · D)× = 0 .
(20.54)
Unlike the force equilibrium equation, the moment one (20.54) is not identically satisfied under the uniform deformation but is used as the equation determining the constant field of proper orthogonal microrotation tensor H for the given distorsion tensor F. One can find the solution of nonlinear equation (20.54) if and only if a certain constitutive equation of the elastic shell is known, i.e. if the relationship P(E, 0) is specified, in general. Nevertheless, as is shown below, there is a set of solutions in
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the class of isotropic plates, which do not depend on this constitutive equation. Such solutions one can call universal for the isotropic plates. Theorem 20.1. In the case of isotropic plate, Eq. (20.54) is satisfied under such uniform deformatons, for which the surface stretch tensor E is symmetric. Proof. With the help of Eqs (20.3) and (20.4), Eq. (20.54) is transformed into (HT · ET · P · H)× = 0 . The equation is equivalent to requiring the tensor ET · P to be symmetric. Under hypothesis of the theorem, the tensor E is symmetric and the relationship P(E) satisfies Eq. (20.5) with L = 0. Since, according to definition (20.4), the tensor E satisfies the condition i3 · E = 0, its symmetry property yields E · i3 = 0. Therefore, E is the plane symmetric tensor having the spectral decomposition E = E 1 d1 ⊗ d1 + E2 d2 ⊗ d2 ,
(20.55)
where d1 , d1 are orthonormal vectors in the surface σ. In Eq. (20.5) with L = 0, assuming consequently Q = S1 , Q = S2 , Q = S3 , where S1 = I − 2d1 ⊗ d1 , S2 = I − 2d2 ⊗ d2 , S3 = 2i3 ⊗ i3 − E, and taking into account Eq. (20.55), we have the equalities P · St = St · P , t = 1, 2, 3 . (20.56) From Eq. (20.56) follow the relations dα · P · i3 = 0 ,
dα · P · dβ = 0 ,
α, β = 1, 2,
αβ,
which mean that the tensor P has the structure P = P1 d1 ⊗ d1 + P2 d2 ⊗ d2 .
(20.57)
According to Eq. (20.57), the tensor P is symmetric and coaxial with the tensor E. Hence, the tensor ET · P is symmetric as well, Q. E. D. Taking into consideration definition (20.4) of the surface stretch tensor E, the theorem just proved means that the moment equilibrium condition with l = 0 will be satisfied, if the microrotation tensor H satisfies the equation F · HT = H · FT .
(20.58)
Since we are interested only in proper orthogonal solutions of Eq. (20.58), one should add to it the following relations: H · HT = I ,
det H = 1 .
(20.59)
The shell distorsion tensor F, by definition, has the properties n·F = 0 ,
a·F 0
∀a : a· n = 0 .
(20.60)
20 On Universal Deformations of Nonlinear Isotropic Elastic Shells
291
It results from Eq. (20.60) that the symmetric tensor F · FT has the following spectral decomposition: F · FT = U 12 e1 ⊗ e1 + U22 e2 ⊗ e2 ,
U1 > 0, U2 > 0 ,
(20.61)
where e1 and e2 are the orthogonal unit eigenvectors located within the plane σ. The third eigenvector, relevant to zero eigenvalue of the tensor F · FT , is the normal vector n = i3 . The system of Eqs (20.58), (20.59) with the conditions (20.61) was studied earlier [7] under completely different circumstances. The solution found in [7] is nonunique and for U1 U 2 contains the following four branches: H = ±(U1−1 e1 ⊗ e1 + U2−1e2 ⊗ e2 ) · F + e3 ⊗ e3 , H = ±(U1−1 e1 ⊗ e1 − U2−1e2 ⊗ e2 ) · F − e3 ⊗ e3 , e3 = e1 × e2 ,
(20.62)
e3 = (U 1−1 e1 · F) × (U 2−1e2 · F) .
One can transform the expressions (20.62) to the explicit form % & H = J −1 F∗T ± (2J + I1 )−1/2 (I1 + J)F − F · FT · F ,
(20.63)
1 I2 = tr 2 (F · FT ) − tr (F · FT )2 , 2 & 1 % 2 ∗ 2 F = F − Ftr F + I tr F − tr F2 . 2 ∗ Here F is the tensor associated with F, [8]. Let us study the nonsingular tensor C = F + n ⊗ N in the three-dimensional Euclidian space, where N is the unit vector normal to the surface Σ of the deformed shell, and write down the polar decomposition of the tensor C, C = (F · FT )1/2 + n ⊗ n · A . (20.64) J = ± I2 ,
I1 = tr (F · FT ) ,
The proper orthogonal tensor A in Eq. (20.64) is defined by the distorsion tensor and is called the macrorotation tensor of the shell. The explicit expression is known [8] for this tensor,
% &−1/2 I2 A = 2 I2 + I1 (I1 + I2 )F − F · FT · F + F∗T .
(20.65)
Comparing Eq. (20.63) with (20.65) we see that one of solutions of the system (20.58), (20.59) coincides with the macrorotation tensor A. In the degenerate case U 1 = U 2 the solution set of the system (20.58), (20.59) becomes infinite. It is described by the formulas (20.62), where e1 , e2 are any two orthogonal unit vectors in the plane σ. If σ remains undeformed, i.e. translations Δ
are equal zero, X s = x s , we have F = g, where g = I − n ⊗ n is the first fundamental
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tensor of σ. In this case the set of rotations described by formulas (20.62) consists of the identical transformation, rotations about the normal n and about any axis orthogonal to n. These microrotations, according to (20.4), cause nontrivial values of deformation tensor E − g of the plate and, consequently, invoke stresses. Thus, the model of elastic plate of Cosserat type allows the existence of a nontrivial stress field without the metric deformation.
20.4 Equilibrium of an Isotropic Cosserat Membrane If the strain energy density W does not depend on the wryness tensor L, a micropolar shell is called a Cosserat membrane [3]. In this case, according to (20.3), K=G=0,
P = P(E) =
dW(E) . dE
(20.66)
The equilibrium equations (20.2) in the case of membrane without surface loads transform to div D = 0 ,
(20.67)
(FT · D)× = 0 .
(20.68)
We call the membrane the isotropic Cosserat membrane, if for any orthogonal tensor Q satisfying (20.6), the following condition holds true: P(QT · E · Q) = QT · P(E) · Q .
(20.69)
Bellow we study one simple solution of the equilibrium equations (20.67), (20.68) for the isotropic elastic membrane. Let us assume that particles of the membrane do not undergo translations, R = r, and the field of microrotations is set by the relation H = 2n ⊗ n − I ,
(20.70)
which describes the rotation about normal to the surface σ. The distorsion tensor and the surface stretch tensor in this case are F=g,
E = −g .
(20.71)
Assuming Q = 2n ⊗ n − I in Eq. (20.69), we get P · n = 0. Further, in Eq. (20.69) we first substitute Q = j1 ⊗ j2 − j2 ⊗ j1 + n ⊗ n , and then Q = j1 ⊗ j2 + j2 ⊗ j1 + n ⊗ n , where j1 , j2 is any orthonormal frame in the tangent plane to the surface σ. It leads to the relations
20 On Universal Deformations of Nonlinear Isotropic Elastic Shells
j1 · P · j2 = j2 · P · j1 = 0 ,
293
j1 · P · j1 = j2 · P · j2 .
Hence it follows that P = αg with α = const due to homogeneity of the membrane. As far as according to Eqs (20.3) and (20.71), D = −αg, FT · D = −αg, the moment equilibrium equation (20.68) is fulfilled and the force equilibrium equation (20.67) transforms to divg = 0 or (tr b)n = 0 .
(20.72)
Δ
Here b = −gradn is the second fundamental tensor [8] of the surface σ. Evidently, Eq. (20.72) is fulfilled if and only if the surface σ has the mean curvature equal zero. Such surfaces are called minimal surfaces and can be realized experimentally with the help of the soap film pulled on a wire frame. So, for the isotropic Cosserat membrane in the form of minimal surface, a nontrivial equilibrium stressed state is found, which exists without any metric deformation.
20.5 Conclusion The explicit solutions of nonlinear equilibrium equations of isotropic micropolar shells found in this work represent families of large nonuniform deformations, which are defined with the help of some parameters. These solutions are analogous to universal deformations of isotropic incompressible three-dimensional micropolar bodies [9]. The solutions obtained here differ from universal deformations of elastic shells described in [3], particularly in the fact that they are realized without action of surface moments for any values of parameters mentioned above. This property of the constructed universal solutions makes them easy-to-use for experimental finding of constitutive equations of the micropolar shells. Let us note that determination of uniform deformations of the Cosserat-type shells studied in this paper is not a trivial problem, because this leads to determination of uniform microrotation field from the nonlinear moment equilibrium equations. The equilibrium state of Cosserat membrane having the minimal surface shape just illustrates some peculiarities of the micropolar shell mathematical model. Acknowledgements This work was supported by the Russian Foundation for Basic Research (grant No 09-01-00459) and the Federal target programme “Research and Pedagogical Cadre for Innovative Russia” for 2009–2013 years (state contract No P-596).
References 1. Altenbach, H., Zhilin, P.A.: General theory of elastic simple shells (in Russian). Uspekhi Mekhaniki (Advances of Mechanics) 11 (4), 107–148 (1988) 2. Chr´os´cielewski, J., Makowski, J., Pietraszkiewicz, W.: Statyka i dynamika powłok wielopłatowych: Nieliniowa teoria i metoda element´ow sko´nczonych. Wydawnictwo IPPT PAN, Warszawa (2004)
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3. Eremeyev, V.A., Zubov, L.M.: Mechanics of Elastic Shells (in Russian). Nauka, Moscow (2008) 4. Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells (2nd ed.). Cambridge University Press, Cambridge (1998) 5. Zhilin, P.A.: Mechanics of deformable directed surfaces. Int. J. Solid. Structures 12(9–10), 635–648 (1976) 6. Zubov, L.M.: Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies. Springer, Berlin (1997) 7. Zubov, L.M.: The representation of the displacement gradient of isotropic elastic body in terms of the Piola stress tensor. J. Appl. Math. Mech. 40(6), 1012–1019 (1976) 8. Zubov, L.M.: Methods of Nonlinear Elasticity in Shell Theory (in Russian). Izdatelstvo Rostovskogo Universiteta, Rostov-on-Don (1982) 9. Zubov, L.M.: Universal solutions for isotropic incompressible micropolar solids. Doklady Physics 55 (11), 551–555 (2010)
Part IV
Numerical Analysis
Chapter 21
Application of Genetic Algorithms to the Shape Optimization of the Nonlinearly Elastic Corrugated Membranes Mikhail Karyakin and Taisiya Sigaeva
Abstract Corrugated membranes are extremely important structural parts of a great number of devices, highly sensitive pressure sensors in particular. In engineering of corrugated shapes different factors could be taken into account as crucial. Among them – membrane’s work without buckling, buckling for the prescribed load, flatness of the membrane characteristic – dependence of the applied pressure on the liquid or gas volume, a lack of the plastic deformation and so on. In this paper an approach to different problems of optimization and design using genetic algorithm is proposed. To model elastic behavior of the membrane the nonlinear equations based on the Kirchhoff-Love hypothesis and not imposed constrains on the shallowness of the membrane’s shape are used. Several multi-parameter families of the axially symmetric ”design”, i.e. functions of the shape that describe the dependence of the middle surface prominence on the membrane radius, are considered. These designs were obtained due to special modifications of spherical dome, conical membrane and their integration. The value of the linear section length of the membrane characteristic served as an optimality criterion. The paper presents modified genetic algorithm, describes refinement mechanism of its settings and observes examples of the designs, which are optimal in the specified shape classes. Keywords Genetic algorithm theory · Shape optimization
·
Corrugated membrane
·
Nonlinear shell
M. Karyakin (B) Southern Federal University, Milchakova str., 8a, 344090, Rostov-on-Don, Russia & Southern Mathematical Institute, Vladikavkaz, Russia e-mail:
[email protected] T. Sigaeva Southern Federal University, Milchakova str., 8a, 344090, Rostov-on-Don, Russia e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 21, © Springer-Verlag Berlin Heidelberg 2011
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21.1 Introduction Membrane structures are among the most common and the most efficient structural elements in nature and technology due to such properties as high resistance, minimum material, large spans, shelter function, maximum load carrying capacity, specific frequency response. The amount of deflection for a given pressure difference depends on the membrane shape (e.g. diameter and thickness), the nature of the support (e.g. strongly fixed, weakly pinned), and the membrane material properties (e.g. elastic modulus, Poisson ratio, density). One of the key problems of the membrane study is closely connected with shape optimization and deserves great attention, as it plays with the interaction between shape and structural response in order to achieve certain objectives fulfilling some constrains. It also has a wide range of application in automotive and aircraft industry, civil engineering and etc. Shape optimization problems are usually solved numerically, by using iterative methods [4]. That is, one starts with an initial guess for a shape, and then gradually evolves it, until it morphs into the optimal shape. A number of prominent specialists have been applying this technique to the shells and membranes. In [8] the optimal damping set of a two-dimensional membrane is estimated using an iterative shape optimization scheme based on level set methods and the gradient descent algorithm. Another strategy in the optimization process, based on discretization and linearization techniques, is introduced by reliable finite element formulation and used in [3] and [6] for the membrane form-finding. A more recent alternative in dealing with the shape optimization problems is the genetic algorithm (GA), developed by Holland in 1975 [7], a programming technique that mimics biological evolution as a problem-solving strategy. In [11], for example, author uses it for optimization of membrane separation modules, while in [2] it is applicated to problems of designing axisymmetric shells of minimal weight. This paper discusses implementation GA for the shape optimization of the nonlinearly elastic corrugated membranes and includes four sections. In Sect. 21.2 the theoretical approach to the membrane modeling is described. Next, modified GA can be seen in Sect. 21.3 and results of testing its effectiveness are given in Sect. 21.4. Finally, in Sect. 21.5 some conclusion are proposed.
21.2 Membrane Modeling Let the surface profile of a membrane with the thickness h and radius ρ h be the function given by z = f (r) in the cylindrical coordinate system or by points ri , zi (i = 1..N). These points can be determined, for instance, from the optical or mechanical profilometer experiments. If the profile is denoted by the points, then we can use spline approximation to retrieve it. Principal surface curvatures along the meridian and parallels respectively are defined by
21 Application of GA to the Optimization of the Corrugated Membranes
k1 = −
f (1 + f 2)
3 2
,
k2 = −
f r(1 + f 2 )
1 2
,
f =
299
∂f . ∂r
(21.1)
Nonlinear behavior of these membranes can adequately be represented by twodimensional nonlinear equations based on the Kirchhoff hypotheses [10]: 1. Each line which is initially perpendicular to the middle surface remains straight and normal to the deformed middle surface. 2. The stresses σα3 are negligibly small and are taken to be zero in the generalized Hooke‘s law, which in turn describes the relationship between the stresses T αβ and moments Mαβ with the infinitesimal strain tensor components #αβ and principal curvature tensor component καβ : T 11 = B(#11 + ν#22 ),
T 22 = B(#22 + ν#11 ),
T 12 = B(1 − ν)#12,
M11 = D(κ11 + νκ22 ),
M22 = B(κ22 + νκ11 ), M12 = B(1 − ν)κ12, 1 2 1 1 #11 = e11 + θ1 , #22 = e22 + θ22 , #12 = e12 + θ1 θ2 , 2 2 2 ∂u1 1 ∂u2 u1 e11 = C + k1 u3 , e22 = + C + k2 u3 , ∂r r ∂φ r 1 ∂u1 ∂ u2 ∂θ1 ∂ θ2 1 ∂θ1 2e12 = + rC ( ), κ11 = C , 2κ12 = rC ( ) + , r ∂φ ∂r r ∂r ∂r r r ∂φ 1 ∂θ2 C ∂u3 1 ∂u3 κ22 = + θ1 , θ1 = −C + k1 u1 , θ2 = − + k 2 u2 , r ∂φ r ∂r r ∂φ 1 Eh Eh3 C = C(r) , B , D= , (21.2) 1 2 1−ν 12(1 − ν2) (1 + f 2 ) 2 where u1 , u2 , u3 are the displacements of the membrane middle surface point along the meridian, parallel and normal respectively, while θ1 , θ2 represent angles of the normal rotation, E is the Young modulus and ν denotes the Poisson ratio; cylindrical c.s. is used. In this special case the nonlinear equilibrium equations are expressed as [5] ∂(rT 11 ) ∂T 12 S1 sin(θ1 ) − + rk1 Q1 − − rp = 0, ∂r ∂φ C C ∂(rT 12 ) ∂T 22 S2 sin(θ2 ) + + rk2 Q2 − − rp = 0, ∂r ∂φ C C ∂(rCQ1 − rS 1 ) ∂Q2 r 1 ∂S 2 cos(θ1 ) + − (k1 T 11 + k2 T 22 − − rp = 0. (21.3) ∂r ∂φ C C ∂φ C Here Q1 =
1 ∂(rM11 ) M22 1 ∂(M12 ) 1 ∂(rM12 ) M12 1 ∂(M22 ) − + , Q2 = + + , r ∂r r rC ∂φ r ∂r r rC ∂φ S 1 = T 11 θ1 + T 12 θ2 , S 2 = T 12 θ1 + T 22 θ2 ,
where Q1 , Q2 are transverse loads.
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Boundary conditions on the membrane contour at x = 0 for the case of rigid clamping are u1 = 0, u2 = u3 = 0, θ1 = 0. (21.4) Note that equations were obtained without taking into account the shell flatness hypothesis (21.2,21.3). In addition, according author’s experience it can be claimed that for the membranes with the thickness of h/ρ < 1/50 equations based on the higher order models give almost the same result but are more complicated for a solving. To reduce this problem to the differential equations system of the first order let consider an augmented eight-dimensional vector Y = (Y1 , Y2 , ..., Y8 ) with the dimensionless components u1 u2 u3 rT 11 , Y2 = , Y3 = , Y4 = θ1 , Y5 = , h h h Bh rT 12 rM11 r(CQ1 − T 11 θ1 − T 12θ2 ) Y6 = , Y7 = , Y8 = , 2 Bh Bh Bh Y1 =
(21.5)
which in turn are proportional to the membrane middle surface displacements along the meridian, parallel and normal, angle of the normal rotation, axial and shear forces, bending moment and transverse forces. These components are continuous functions in the points, where the curvature undergoes a jump, what has an essential role in the numerical integration. In the case of axisymmetrical deformation (u2 = 0, ∂φ Yi = 0) obtained equations can be simplified and turned into the system of ordinary differential equations of first order: dY1 ν K1 + νK2 1 1 2 = − Y1 − Y3 + Y5 − Y , dx x C Cx 2C 4 Y2 ≡ 0, dY3 K1 1 = − Y1 − Y4 , dx C C dY4 ν 12 = − Y4 + Y8 , dx x Cx dY5 (1 − ν2 )C px ν K1 = Y1 − (1 − ν2)K2 Y3 + Y4 + Y5 − Y7 , dx x C x C Y6 ≡ 0, (1 − ν2 )K22 x K1 + ν2 K22 dY7 px 2 px = (1 − ν2 )K2 Y1 + Y3 + Y5 − Y + , dx C C 2C 4 C dY8 (1 − ν2 )C 1 ν 1 = Y4 + Y7 − − Y8 + Y4 Y5 . dx 12x C x C
(21.6)
Here Ki = ki h are dimensionless curvatures. Boundary conditions at point x1 = ρ/h for rigid fixing boundary are Y1 (x1 ) = 0,
Y3 (x1 ) = 0,
Y4 (x1 ) = 0.
(21.7)
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For the numerical integration of this system let take into account the commonly used method, in which boundedness of solution at x = 0 is replaced by boundary conditions at x = x0 (x0 ≤ 1): Y5 (x0 ) = 0,
Y7 (x0 ) = 0,
Y8 (x0 ) = 0,
(21.8)
what is quite natural, since these continuous in the neighborhood of zero components are vanished at x = 0. Numerical calculating for this nonlinear two-point boundary problem (21.6, 21.7, 21.8) can effectively be done with the shooting method, in which loading parameter is a value of vertical deflection Y3 at the point x = x0 , while the set of other parameters consists of Vi , (i = 1, 2, 3): Y1 (x0 ) = V1 ,
p = V2 ,
Y4 (x0 ) = V3 ,
(21.9)
that can be determined after solving Cauchy problem using boundary conditions. This problem has the stable and durable solution, which can be found in an acceptable time even for thin shells (h/ρ < 1/300).
21.3 Genetic Algorithm The genetic algorithm [1] has been inspired by evolutionary biology and incorporates techniques such as inheritance, mutation, inverse, selection and crossover to find a better solution. In this part the scheme of the genetic algorithm processing for the case of three parameters optimization is presented. A potential solution is introduced by a “chromosome” including three “genes” or coded parameters (Fig. 21.1a). Eight bits are used to represent the type value of each gene using the reflected binary coding, also known as Gray code, a binary numerical system where two successive values differ in only one bit [9]. GA follows the basic workflow: • Initialization. By selecting N random points from the search space and converting them to the chromosomes, we get initial population of N individuals. • Crossover. At first, this operator selects two parent’s individuals and swaps their bits randomly with the small probability pc1 (Fig 21.1b). Then with the probability pc2 it is determined whether or not they are crossed using the three-stepwise crossover (the single-point crossover for each of the genes) or just copied without any modifications to the next generation. Three-stepwise crossover means the exchanging genetic material between individuals by copying parts of the genes from one parent to another in order to produce children (the single-point crossover for each of the genes) (Fig. 21.1c). After this, the population of parents is extended by children individuals.
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• Mutation. This operator serves to introduce diversity into the children population and to look for a better solution in yet unexplored areas. An occasional bit in each gene of the chromosome has the probability pm to get inverted (Fig. 21.1d). • Inverse. The next step for the modification of the new generation is inverse, what means the exchanging parts of a chromosome before and after randomized bit with the probability pi (Fig. 21.1e). As well as mutation, this operator extends the search on the new areas of the search place. • Selection. In this part it is necessary to guarantee that the best individuals survive from generation to generation. This technique is called elitism. The parent’s and children’s generations are sorted by the fitness function. Then by the extending the selection zone from the beginning of sorted population we choose occasional individuals and if they have no analogues – we add them to the new population of N individuals. Therefore the fitter individuals have greater chance to proceed and cross, while the fact that population does not become occupied by one individual ensure the maintaining of the diversity. Finally we replace the combined population with the obtained one and repeat the process until the termination condition (maximum number of generations) is met. This algorithm can be called modified as it compares different techniques, such as three-stepwise single-point crossover, elitism strategy and truncation method in order to achieve certain results in an acceptable time.
Fig. 21.1 Genetic algorithm: (a) chromosome, (b)–(c) crossover, (d) binary mutation, (e) inverse
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21.4 Numerical Results To demonstrate using of the proposed approach let consider two membrane form finding problems described by three parameters. For the first issue let examine membrane form of the ”double dome” type, which is sketched at the Fig. 21.2a (where for seeing imaginary the radius and height scales are different). Some of parameters describing profile and material properties for this membrane are fixed, while other ones are varying values we have to determine. This data can be clearly seen from the Fig. 21.2b. Three real-valued parameters (a, b, c) form the chromosome of 24 genes that gives appropriate accuracy. Each of the estimated parameters has its own upper and lower limits, and these intervals present the search area in the GA. This information is presented in Table 21.1. Number of chromosomes in population was variable, but we noticed that population size of 100 was good choice in all experiments. Varying of the probabilities we have obtained the optimal result using next values: pc1 = 0.3, pc2 = 0.9, pm = 0.3, pi = 0.25. Termination condition was maximum generation number, and this was set to 30. Optimization problem in this case was finding the membrane with characteristic (i.e. the dependence between the applied pressure and the change of the volume under the membrane surface) containing the highest possible flatness extensive zone. To evaluate the fitness function of each chromosome the special program modeling the compression of the membrane using information about the profile and material (a)
(b) a
c
b r
h
Fig. 21.2 Double dome: (a) 3-d membrane model, (b) membrane profile. Fixed values: E, ν, h, ρ. Parameters: a, b, c Table 21.1 Doubled dome optimization input values and results E, Pa
ν
h, mm
ρ, mm
[a1 , a2 ]
[b1 , b2 ]
[c1 , c2 ]
1.73 · 105
0.3
0.1
25
[0.04, 0.4]
[1.1, 19.1]
[0.2, 0.5]
Curve
a, mm
b, mm
c, mm
Fitness function value
1
0.1339440918
9.612588334
0.47890625
initial
2
0.0890390625
1.131196269
0.43500000
average
3
0.9143066406
5.554626929
0.41796875
best
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parameters was developed. On the basis of the numerical solving of the boundary problem (21.6) - (21.8) this program builds the dependence of the applied pressure on the volume below the membrane and then searches the nearly linear zone with the least pitch on this ready-built curve. Nearness to the linearity means that the mean-square deflection of the function from the straight line passing through its initial and end points is not exceeding some small given value. Minimizing by GA fitness function magnitude was determined by this zone pitch-length ratio. At the Fig. 21.3a certain membranes load distribution lines (characteristics) observed in numerical experiments are shown (values of estimated parameters can be found in Table 21.1). Membrane 1 is a representative of the initial population, the second one presents typical chromosome after 10 iterations and the last one fits with the best individual from the population occurred just before meeting termination condition. Load distribution lines of the optimal membrane, plate of the same diameter and spherical dome of the same camber are compared at the Fig. 21.3b. Obtained results clearly show efficiency of the GA and possibility of the application it to the form optimization problem under given conditions. However, form implementation of the found membrane for the real devices becomes complicated due to the necessity of the preloading, since the found zone satisfying required conditions is lying in the large-scale deflections region far from the no-load state. For the next example of the using GA let consider shape optimization of the corrugated membrane already in existence (Fig. 21.4a) by fixing mechanical properties and geometric subset and varying three real-valued parameters (a, b, c) as well (Fig. 21.4b). Initial design fits with parameters values presented in the row 1 of the Table 21.2, while its characteristic is illustrated by curve 1 at the Fig. 21.5. During optimization characteristic was improved according next conditions: 1. The lead of the diagram should be nearly linear curve with the length overcoming that of the existing curve by at the least 20%. 2. Linearity of this zone must not be worse compared with existing membrane.
(a)
2
,10 Pa
(b)
2
,10 Pa
1
2 3 , mm
3
Fig. 21.3 P-V curves: (a) double dome (curves 1, 2, 3 are for an initial chromosome, for the individuals after a few iteration and for the best set of parameters respectively); (b) comparison of the spherical dome (1), plate (2) and optimal membrane’s (3) characteristics.
21 Application of GA to the Optimization of the Corrugated Membranes (a)
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(b) a
c
b r
s h
Fig. 21.4 Truncated sinusoidal dome: (a) 3-d membrane model, (b) membrane profile. Fixed values: E, ν, h, ρ, s. Parameters: a, b, c
3. Declivity angle of this linear zone should be smaller by at the least 20% than the same angle of the membrane template. Results of the GA implementation for this problem are proposed at the Fig. 21.5 and in Table 21.2. As well as in the previous issue, curve 2 represents individual from the population of the iteration 10, while curve 3 fits with the best individual in the end of the algorithm. Optimization of the linear zone length was of about 30% (600 mm3 against 450 mm3 ); angle optimization was of about 30% (0.108 against 0.153). 2
,10 Pa 2
1 3
Fig. 21.5 P-V curves of the truncated sinusoidal domes. Curves 1, 2, 3 are for the membrane template characteristic, for the individuals after a few iteration and for the best set of parameters respectively
, mm
3
Table 21.2 Truncated sinusoidal dome optimization input values and results E, Pa
ν
h, mm
ρ, mm
s, mm
[a1 , a2 ]
[b1 , b2 ]
1.73 · 105
0.3
0.1
25
1
[2.25, 22.45] [0.2, 0.8]
Curve
a, mm
b, mm
c, mm
1
2.676803243
0.7203125000
0.09972656250
initial
2
9.745247582
0.5445312500
0.03307812500
average
3
12.45468988
0.5937500000
0.06882812250
best
[c1 , c2 ] [0.03, 0.1]
Fitness function value
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It also must be mentioned that this research is out of the scope of the stability analysis, although this approach can be extended and for this case by means of numerical scheme proposed in [5], if the fitness function expression will be modified according some conditions, that will definitely slow the optimization program process. Stability test can be realized regardless of the GA program by analyzing only the optimal forms. Such analysis had been particularly shown that load distribution line 3 at the Fig. 21.5 has no bifurcation points. Therefore working process over the indicated load range is stable.
21.5 Conclusion Presented genetic algorithm was developed to estimate the corrugated membrane profile characteristics responding desired conditions based on the membrane compression experiment modeling results. The methodology was demonstrated with a simple model to illustrate the details of procedures involved. For the cases studied, results from simulation shows that it is possible to determine these characteristics with good accuracy. Acknowledgements This research was supported by the Federal target program “Scientific and pedagogical cadre of the innovated Russia during 2009–2013” (state contract P–361).
References 1. Back, T.: Evolutionary algorithms in heory and practice: evolution strategies, evolutionary programming, genetic algorithms. Oxford Univ. Press, NY (1996) 2. Banichuk, N. V., Ivanova, S. Yu., Makeev, E. V.: Some problems of optimizing shell shape and thickness distribution on the basis of a genetic algorithm. Mech. Solids. 42(6), 956–964 (2007) 3. Bletzinger, K.-U.: Form Finding and Optimization of Membranes and Minimal Surfaces. Tech. Univ. of Denmark, Lyngby, (1998) 4. Delfour, M.C., Zolesio, J.P.: Shapes and Geometries: Analysis, Differential Calculus and Optimization. SIAM, Philadelphia (2001) 5. Getman, I. P., Karyakin, M. I., Ustinov, U. A.: Nonlinear behavior analysis of circular membranes with arbitrary radius profile. J. Appl. Math. Mech. 74(6), 917–927 (2010) 6. Gosling, P. D., Lewis, W. J.: Form-finding of prestressed membranes using a curved quadrilateral finite element for surface definition. Comput.Structs. 61(5), 871–883 (1996) 7. Holland, J. H.: Adaptation in natural and artificial systems. Univ. of Michigan Press, Ann Arbor MI (1975) 8. Lassila T.: Optimal damping of a membrane and topological shape optimization. Struct. Multidisc. Optim. 38(1), 3438–3452 (2009) 9. Rowe, J., Whitley, D., Barbulescu, L., Watson, J. P.: Properties of Gray and binary representations. Evol. Comput. 12, 47–76 (2004) 10. Vorovich, I. I.: Nonlinear Theory of Shallow Shells. Springer, NY (1999) 11. Yuen, C.C., Aatmeeyata, Gupta, S.K. and Ray, A.K.: Multi-objective optimization of membrane separation modules using genetic algorithm. J. of Membrane Sci. 176, 177–196 (2000)
Chapter 22
Advances in Quadrilateral Shell Elements with Drilling Degrees of Freedom Stephan Kugler, Peter A. Fotiu and Justin Murin
Abstract A unique derivation of quadrilateral shell elements with six degrees of freedom at each node is presented. The theoretical and numerical formulation is based on the combination of a membrane element with drilling degrees of freedom and a shear deformable plate element. The predictive quality and the computational efficiency is improved by applying multifield variational principles in connection with suitable assumed strain fields. The resulting element formulation does not require any Gaussian quadrature since all parts of the stiffness matrix can be integrated analytically. Furthermore, the derivation is generalized to geometrical and physical nonlinearities according to a corotational updated Lagrangian description. Keywords Quadrilateral shell element · Multifield variational principles
22.1 Introduction The numerical analysis of shell structures requires stable and robust finite element formulations. The development of suitable procedures represents a challenging task in finite element research. The field of applications is widely spread and reaches from linear static analysis in structural engineering to nonlinear applications in the simulation of sheet metal manufacturing processes where geometrical and physical nonlinearities are coupled with instabilities and contact phenomena. S. Kugler (B) · P. A. Fotiu Department of Applied and Numerical Mechanics, University of Applied Sciences Wiener Neustadt, Wiener Neustadt, Austria e-mail:
[email protected],
[email protected] J. Murin Department of Mechanics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology in Bratislava, Slovak Republic e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 22, © Springer-Verlag Berlin Heidelberg 2011
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a) 4
ˆ1 Φ zˆ
Φ3zˆ
yˆ
b)
F
F
zˆ
F
ˆ2 Φ zˆ
F
Fig. 22.1 Drilling degrees of freedom: a) In-plane rotational degrees of freedom, b) Beam to shell connection
A general purpose shell element requires six nodal degrees of freedom (three translational and three rotational components) in order to model beam to shell connections correctly: Classical shell elements can be derived combining a membrane element and a plate element with altogether three displacements and only two rotational degrees of freedom with respect to axes in plane. Consequently, the rotational degree of freedom with respect to the shell’s normal, frequently referred to as drilling degree of freedom (Fig. 22.1a), is missing and the stiffness matrix of a structure as depicted in Fig. 22.1b is singular. Many different procedures have been proposed to introduce this missing drilling degree of freedom (e.g. [1, 2]). However, the most rigorous attempt is based on a functional that includes drilling degrees of freedom directly. Such a strategy has been done by Hughes and Brezzi [3], who used Reissner’s functional [4] including rotations as separate degrees of freedom. They showed mathematically that Reissner’s proposal can not be used for low order finite element interpolations since the LBB conditions as well as the constraint count condition for the mixed patch test are failed. Hence, they enriched Reissner’s functional by an additional term to ensure stability of low order finite element interpolations. It has to be noted that the motivation for this additive functional term arises from purely mathematical considerations and no physical interpretation is given. A different derivation of such functional suitable for the derivation of low order finite elements is given in [5,6]. Here, in contrast to [3], Cosserat’s micropolar theory of elasticity [7, 8] is used as a starting point where rotations of a point are included naturally from the outset. By neglecting couple stresses μi j within a weak form of this boundary value problem leads to the internal energy, 1 int s s Π = ci jkl εkl εi j dΩ + α εaij εaij dΩ, (22.1) 2 Ω
Ω
where Ω is the elemental volume, ci jkl represents an elasticity tensor and the constant α is an added material dependent parameter that relates to the asymmetric strains εaij . The symmetric and antimetric strain tensors εisj and εaij obey the following geometrical conditions ε sji =
& & 1% 1% ui, j + u j,i and εaji = ui, j − u j,i + ei jk ϕk . 2 2
(22.2)
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In (22.2) the variable ui represents displacements in the i-th Cartesian direction, ϕi is the ith- component of the rotation and ei jk denotes the alternator tensor. As indicated, a large number of element formulations are proposed in literature with respect to the potential (22.1), e.g. [9]. Most of these procedures are optimized with respect to high coarse mesh accuracy as e.g. in Ibrahimbegovic et al. [10–12]. However, there are not many efficient element technologies which are well suited for explicit time integration (e.g. [13]) and advances in that field of application will be presented in this paper. Within our approach a new general purpose quadrilateral shell element is derived based on flat element geometries where membrane and plate bending properties are decoupled. If the element configuration is initially warped a rigid body projection [14, 15] is used to ensure the correct number of rigid body modes and to couple membrane and bending properties. Drilling degrees of freedom are introduced within the membrane part of the stiffness matrix in relation to functional (22.1). The plate bending element is formulated within the Mindlin-Reissner theory and, therefore, is valid for thin to moderately thick shells. It has to be pointed out that the herein proposed element formulation avoids any Gaussian quadrature since all parts of the stiffness matrix can be integrated analytically.
22.1.1 Element Configuration, Shape Functions and Rigid Body Projection Within an initially warped element configuration (dashed quadrilateral with gray nodes in Fig. 22.2) a local Cartesian coordinate system is embedded by simple vector calculus. Thereby, the local base vectors eˆ x and eˆ y enclose equal angles δ with the vectors g1 and g2 that connect the midpoints of opposing elemental sides.The nodal coordinates r¯ i with respect to the elemental Cartesian coordinate system are calculated with the aid of an auxiliary matrix T T T r¯ i = X¯ i Y¯ i Z¯i = T (ri − rm ) with T = eˆ x eˆ y eˆ z ,
projected plane element
g2
zˆ 4
yˆ
δ zg
r1
1
r1
xˆ
rˆ2 rm yg
xg
g1 warped configuration
2
Fig. 22.2 Warped and projected, plane element configuration
(22.3)
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and the projected nodal coordinates1 rˆ i (black in Fig. 22.2) are defined by simply omitting the z-component of r¯ i . Therefore, the nodal coordinates of the projected plane element configuration are given with respect to the elemental Cartesian coor T dinate system by rˆ i = X¯ i Y¯ i 0 . In the following sections we discuss on the basis of these nodal coordinates rˆ i (plane element configuration) the independent derivation of a three parameter membrane element and a plate element, respectively. However, it can be shown that in arbitrarily warped element configurations some parasitic strain appears under pure rigid body motion and the corresponding element tends to be too stiff [6, 14]. Rankin and Nour-Omid [15] constructed a projection matrix that extracts the pure deformation from a given displacement field, i.e. the displacement components that are free from any rigid body motion. The projection matrix transforms an element that has non zero rigid body modes to one with invariance ˆ pro j is accordproperties under rigid body motion. The projected stiffness matrix K ingly given by ˆ pro j = Pˆ T K ˆˆ K (22.4) R PR , ˆ denotes the unprojected stiffness matrix derived from the plane element where K configuration and Pˆ R is the projection matrix reading −1 ˆPR = I(24×24) − R ˆ R ˆTR ˆ ˆT . R (22.5) ˆ is evaluated by In (22.5) the rigid body matrix R ⎡ ⎢⎢⎢ 1 0 ⎢⎢⎢ ⎡ ⎤ ⎢⎢⎢ 0 1 ˆ 1 ⎥⎥⎥ ⎢⎢⎢ R ⎢⎢⎢ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎢⎢⎢ R ˆ 2 ⎥⎥⎥ ˆ = ⎢⎢⎢⎢ ⎥⎥⎥⎥ with R ˆ i = ⎢⎢⎢⎢ 0 0 R ⎢⎢⎢ 0 0 ⎢⎢⎢ R ˆ ⎥⎥ ⎢⎢⎢ ⎢⎢⎢⎣ 3 ⎥⎥⎥⎥⎦ ⎢⎢⎢ ˆ R4 ⎢⎢⎢⎢ 0 0 ⎣ 00
0 0 0 −riz 1 riy 0 1 0 0 0 0
⎤ riz −riy ⎥⎥⎥ ⎥⎥⎥ 0 rix ⎥⎥⎥⎥ ⎥⎥⎥ −rix 0 ⎥⎥⎥⎥ ⎥⎥ . 0 0 ⎥⎥⎥⎥⎥ ⎥⎥ 1 0 ⎥⎥⎥⎥⎥ ⎥⎦ 0 1
(22.6)
The assembly of the total stiffness matrix has to be carried out with respect to the global Cartesian base eg . Therefore, the projected stiffness matrix has to be transformed according to (22.3) which reads ⎡ ⎤ ⎢⎢⎢ ⎥⎥⎥ CT K C with T C(24,24) = diag ⎢⎢⎢⎢T, . . . , T⎥⎥⎥⎥ . ˆ pro j T K=T (22.7) ⎢⎣;<=>⎥⎦ 8×
The derivation of isoparametric quadrilateral six parameter shell elements requires a suitable plane four-node element kinematic (see Fig. 22.3). The shape functions read 1
Note that throughout the following chapters a superposed hat indicates that the quantity refers to the elemental Cartesian coordinate system.
22 Advances in Quadrilateral Shell Elements
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N = N1 N2 N3 N4 = Δ + bx xˆ + by yˆ + γ ξη, with
1 1 yˆ 24 yˆ 31 yˆ 42 yˆ 13 , by = xˆ42 xˆ13 xˆ24 xˆ31 , 2Ae 2Ae ⎡ ⎤T ⎢⎢⎢ Xˆ 2 yˆ 34 + Xˆ 3 yˆ 42 + Xˆ 4 yˆ 23 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎡ ⎤ ⎢⎢⎢ γ 0 ⎥⎥⎥ 1 ⎢⎢⎢⎢⎢ Xˆ 1 yˆ 43 + Xˆ 3 yˆ 14 + Xˆ 4 yˆ 31 ⎥⎥⎥⎥⎥ ⎥⎥⎥ γ= ⎢ ⎥ , Υ = ⎢⎢⎢⎣ 4Ae ⎢⎢⎢⎢⎢ Xˆ 1 yˆ 24 + Xˆ 2 yˆ 41 + Xˆ 4 yˆ 12 ⎥⎥⎥⎥⎥ 0 γ⎦ ⎢⎢⎣ ⎥⎥⎦ Xˆ 1 yˆ 32 + Xˆ 2 yˆ 13 + Xˆ 3 yˆ 21 ⎡ ⎛ 4 ⎞ ⎛ 4 ⎞ ⎤ ⎜⎜⎜ ⎟⎟⎟ ⎥⎥⎥ 1 ⎢⎢⎢⎢ ⎜⎜⎜⎜ ˆ ⎟⎟⎟⎟ Δ = ⎢⎢⎣t − ⎜⎜⎝ Xi ⎟⎟⎠ b x − ⎜⎜⎜⎝ Yˆ i ⎟⎟⎟⎠ by ⎥⎥⎥⎦ , with t= 1 1 1 1 , 4 bx =
i=1
(22.8) (22.9)
(22.10)
(22.11)
i=1
1 ( xˆ24 yˆ 31 + xˆ31 yˆ 42 ) . (22.12) 2 Note that all upper case letters refer to nodal variables i.e. Xˆ J and Yˆ J depict the nodal xˆ and yˆ coordinates for J = 1, 2, 3, 4. According to Fig. 22.3 the element kinematics of an isoparametric element is given by ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢⎢⎢ xˆ ⎥⎥⎥ ⎢⎢⎢ N 0 ⎥⎥⎥ ⎢⎢⎢ X ˆ ⎥⎥ T T ˆ = Xˆ 1 Xˆ 2 Xˆ 3 Xˆ 4 , Y ˆ = Yˆ 1 Yˆ 2 Yˆ 3 Yˆ 4 . ⎥⎥⎥⎦ ⎢⎢⎢⎣ ⎥⎥⎥⎥⎦ , X xˆ (ξξ ) = ⎢⎢⎢⎣ ⎥⎥⎥⎦ = ⎢⎢⎢⎣ ˆ yˆ 0 N Y (22.13) The Jacobian J of the elements kinematics reads ⎡ ⎤ ⎢⎢⎢ ∂ xˆ ∂ˆy ⎥⎥⎥ ∂ξ ∂ξ J = ⎢⎢⎣⎢ ∂ xˆ ∂ˆy ⎥⎥⎦⎥ . (22.14) xˆ IJ = Xˆ I − Xˆ J , yˆ IJ = Yˆ I − Yˆ J , Ae =
∂η ∂η
xˆ (ξ ) = NXˆ e
η
3
yˆ
4
3
η ξ 4
ξ xˆ
1
2
1
Fig. 22.3 Plane quadrilateral with skew angled coordinates ξ and η and node numbering
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22.2 Linear Element Formulation In this section a geometrically linear shell element with a uniform thickness h is proposed which is based on the flat element geometry discussed in Sect. 22.1.1. First we derive a three-parameter membrane element including drilling degrees of freedom (Sect. 22.2.1) and independently a plate bending element (Sect. 22.2.2). A coupling of these element properties may occur by the rigid body projection (22.4) if the element is in an arbitrarily warped configuration.
22.2.1 Membrane Element In order to introduce drilling degrees of freedom within a membrane element we refer to the lately proposed functional (22.1) (see [5, 6]) with the corresponding strain definitions (22.2). This framework can even be generalized by using Hu-Washizu multi-field variational principles, where all equations of the boundary value problem are only fulfilled in a weak sense (see e.g. [16]). By requiring certain orthogonalities between the strain displacement relations and the assumed stress fields [5, 6] an enlarged potential can be given which reads 1 s s ci jkl ε¯ kl ε¯ i j dΩ + α ε¯ aij ε¯ aij dΩ − fi ui dΩ + ti ui dΓ. (22.15) 2 Ω
Ω
Ω
Γt
In (22.15) an overbar indicates that the corresponding strain field can be assumed independently. In order to avoid any condensation procedure on element level we relate the interpolations of strain fields to nodal degrees of freedom and avoid the application of internal non-conforming degrees of freedom. Equation (22.15) has to be reduced to two-dimensional problems and the variational statement for equilibrium reads in Voigt notation, T Tˆ ˆ ˆ ˆ ˆ ˆ δε s Cε s dΩ + 2α δεa εa dΩ = δuˆ fdΩ + δuˆ T ˆtdΓ, (22.16) Ω
Ω
Ω
Γt
ˆ represents the where Γt denotes the boundary where tractions are applied and C plane stress elasticity matrix. The two dimensional strain displacement relations in the strong form are given by T & 1% εˆ s = u xˆ,xˆ uyˆ ,ˆy u x,ˆ , εˆ a = u xˆ,ˆy − uyˆ,xˆ + ϕ. ˆ ˆ y + uy, ˆ xˆ 2
(22.17)
The keystep within this derivation is to find the assumed strain interpolations εˆ s and εˆ a in two steps. Firstly, we propose them by simply using the classical bilinear shape functions for the displacements and in-plane rotations in relation to (22.17). Within the second step these corresponding results are modified for sake of accuracy and
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efficiency. The bilinear shape functions enable the strain interpolations to be divided into a constant and a non-constant (hourglass) part denoted by 0 and H in the index and we state ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢⎢⎢ U xˆ ⎥⎥⎥ ˆ ⎢⎢⎢ γ 0 ⎥⎥⎥ ⎢⎢⎢ U xˆ ⎥⎥⎥ ˆ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ , εˆ s = B su0 ⎢⎢⎢⎣ ⎥⎥⎥⎦ + B suH ⎢⎢⎢⎣ (22.18) Uyˆ 0 γ ⎦ ⎣ Uyˆ ⎦ ⎛ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎞ ⎜ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎟ ˆεa = 1 ⎜⎜⎜⎜⎜Bˆ au0 ⎢⎢⎢⎢⎢ U xˆ ⎥⎥⎥⎥⎥ + Bˆ auH ⎢⎢⎢⎢⎢ γ 0 ⎥⎥⎥⎥⎥ ⎢⎢⎢⎢⎢ U xˆ ⎥⎥⎥⎥⎥⎟⎟⎟⎟⎟ + Nϕ Φ ˆ, (22.19) ⎝ ⎣ ⎦ ⎣ 2 Uyˆ 0 γ ⎦ ⎣ Uyˆ ⎦⎠ T T ˆ = φ1 φ2 φ3 φ4 . The correspondwith Ui = Ui1 Ui2 U i3 U i4 for i = xˆ, yˆ and Φ ing B-matrices and the shape function for the drilling degree freedom are given by ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ b x 0 ⎥⎥⎥ ⎢⎢⎢ (ξη), xˆ −e (ξη),ˆy ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ Bˆ su0 = j0 / j ⎢⎢⎢⎢ 0 by ⎥⎥⎥⎥ , Bˆ suH = j/ j0 ⎢⎢⎢⎢ −e (ξη), xˆ (ξη),ˆy ⎥⎥⎥⎥ , ⎢⎢⎣ ⎥⎥⎦ ⎢⎢⎣ ⎥⎥⎦ by b x 0 0 ˆ Bˆ au0 = j0 / j by −b x , B auH = κ j/ j0 (ξη),ˆy − (ξη), xˆ , Nϕ = j0 / j N1 N2 N3 N4 .
(22.20)
(22.21) (22.22)
√ √ Introducing the prefactors j/ j0 and j0/ j ( j = det J and j0 = det J(ξ = η = 0), see (22.14)) renders all integrands of the stiffness matrix to polynomials which can be evaluated analytically. Since numerical integration requires at least four Gauß points, this procedure is roughly four times faster compared to classical proposals. Further, two additional scalar constants e and κ are introduced within the nonconstant B-matrices to improve the accuracy of the proposed element. Following closely the derivations of [5, 6] those constants can be identified according to the analytical representation of a higher order patch test and read e = ν , κ = 2.
(22.23)
Remark 22.1. This procedure is strongly related to selective integration where shear terms are underintegrated while the remaining terms are fully integrated. However, those procedures have at least to the author’s knowledge never been discussed for elements with drilling degrees of freedom. Consequently, the stiffness matrix of a three parameter membrane element reads ⎡ ˆ ⎢⎢⎢ K ˆ K = ⎢⎢⎢⎢⎣ 11 ˆ K 21 with
⎤ ˆ T ⎥⎥⎥ K 21 ⎥⎥ , ˆ ⎥⎥⎦ K
(22.24)
22
ˆ ˆ ˆ ˆ ˆ K11 = K suu0 + KsuuH + Kauu0 + KauuH ,
(22.25)
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ˆ =K ˆ ˆ ˆ ˆ K 21 aϕu0 + KaϕuH and K22 = Kaϕϕ ,
(22.26)
where s and a indicate that the correspondence with the symmetric and the antimetric strain fields. The analytically integrated parts of the stiffness matrix can be given explicitly, ⎡ ⎤T ⎡ ⎤ ⎢⎢⎢⎢ b x 0 ⎥⎥⎥⎥ ⎢⎢⎢⎢ b x 0 ⎥⎥⎥⎥ ⎢⎢ ⎥⎥ ⎢⎢⎢ ⎥ ˆ ˆ ⎢⎢ 0 by ⎥⎥⎥⎥ , K suu0 = 4h j0 ⎢⎢⎢⎢ 0 by ⎥⎥⎥⎥ C (22.27) ⎢ ⎥⎥⎥ ⎢⎢⎣ ⎥⎥⎦ ⎢⎢⎣ ⎦ by b x by b x ⎡ ⎤ % & ⎢ yˆ 213 + yˆ 224 ν ( xˆ13 yˆ 13 + xˆ24 yˆ 24 ) ⎥⎥⎥⎥ ˆ T Eh ⎢⎢⎢⎢ % & ⎥⎦⎥ Υ , KsuuH = Υ (22.28) ⎢ 6 j0 ⎣ ν ( xˆ13 yˆ 13 + xˆ24 yˆ 24 ) xˆ213 + xˆ224 T ˆ K (22.29) by −bx , auu0 = 2αh j0 by −b x ⎡ % ⎤ & ( xˆ13 yˆ 13 + xˆ24 yˆ 24 ) ⎥⎥⎥⎥ xˆ213 + xˆ224 αh ⎢⎢⎢⎢ ˆ % & ⎥⎥⎦ Υ , ⎢⎢⎣ KauuH = Υ T (22.30) 3 j0 ( xˆ13 yˆ 13 + xˆ24yˆ 24 ) yˆ 213 + yˆ 224 ⎡ ⎤ ⎢⎢⎢ xˆ42 xˆ13 xˆ24 xˆ31 yˆ 42 yˆ 13 yˆ 24 yˆ 31 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ αh ⎢⎢⎢⎢⎢ xˆ42 xˆ13 xˆ24 xˆ31 yˆ 42 yˆ 13 yˆ 24 yˆ 31 ⎥⎥⎥⎥⎥ ˆ Kaϕu0 = (22.31) ⎢ ⎥, 8 ⎢⎢⎢⎢⎢ xˆ42 xˆ13 xˆ24 xˆ31 yˆ 42 yˆ 13 yˆ 24 yˆ 31 ⎥⎥⎥⎥⎥ ⎢⎢⎣ ⎥⎥⎦ xˆ42 xˆ13 xˆ24 xˆ31 yˆ 42 yˆ 13 yˆ 24 yˆ 31 ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ xˆ42 yˆ 42 ⎥⎥⎥ ⎢⎢⎢ 4 2 1 2 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎥ ⎢⎢⎢ xˆ yˆ ⎥⎥⎥ ⎢⎢⎢ 2 4 2 1 ⎥⎥⎥⎥⎥ αh 2αh j 31 31 0 ˆ ˆ ⎢⎢⎢ ⎥⎥⎥ Υ , K ⎢⎢⎢ ⎥⎥⎥ . KaϕuH = (22.32) aϕϕ = 3 ⎢⎢⎢⎢ xˆ24 yˆ 24 ⎥⎥⎥⎥ 9 ⎢⎢⎢⎢ 1 2 4 2 ⎥⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎣ ⎦ ⎣ ⎦ xˆ13 yˆ 13 2124
22.2.2 Plate Bending Element In this section we derive a robust and highly efficient plate bending element based on the Mindlin-Reissner theory which is valid for thin to moderately thick shells. In this derivation the bending and transverse shear energies are treated independently: While the bending performance of the element is improved according to the B-bar method [17], the transverse shear forces are interpolated separately and linked by discrete collocation constraints. The resulting formulation is similar to that of Bathe and Dvorkin [18, 19] and relies on mixed interpolations of tensorial components. The element should satisfy the following requirements: • No internal degrees of freedom should be included to achieve a computational efficient formulation. • The shear energy has to vanish in the thin plate limit.
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The kinematic assumptions read u xˆ = zˆϕyˆ , uyˆ = −ˆzϕ xˆ , uzˆ = uzˆ ,
(22.33)
where the variable uˆi for ˆi = x, ˆ yˆ , zˆ depicts the deformation in i-th direction while ϕˆi for ˆi = xˆ, yˆ represents the corresponding bending angles2 . The strain definitions are given by % & # xˆ xˆ = zˆϕy,ˆ xˆ , #yˆ yˆ = −ˆzϕ xˆ,yˆ , γ xˆyˆ = zˆ ϕyˆ ,ˆy − ϕ xˆ, xˆ , γ xˆzˆ = uzˆ,xˆ + ϕyˆ , γyˆ zˆ = uzˆ,ˆy − ϕ xˆ , (22.34) % & where #ˆi ˆj are the components of the symmetric strain tensor #ˆi ˆj = 12 uˆi, ˆj + u ˆj,ˆi and γ xˆzˆ and γyˆ zˆ are the mean shear strains according to the Mindlin theory. ˆ b denotes the plane stress elasticity tensor) We assume an isotropic material (C with the constitutive equation in bending ˆ b #ˆ b , σˆ b = C
(22.35)
T with σˆ b = σ xˆ xˆ σyˆ yˆ σ xˆyˆ ]T and #ˆ b = [ # xˆ xˆ #yˆ ˆ y # xˆ yˆ ] . The constitutive equation in shear reads 5 sˆ = Ghγˆ s , (22.36) 6 T with sˆ = [ s xˆ syˆ ]T denoting the vector of distributed shear forces (γˆ s = [ γ xˆzˆ γyˆ ˆz ] ) E and G represents the shear modulus, i.e. G = 2(1+ν) . Equilibrium requires the potential Π to be a minimum, i.e.
Πb =
1 2
Ω
Π = Πb + Π s − Πext → min, 1 σˆ Tb #ˆ b dΩ , Π s = sˆT γˆ s dA , Πext = puzˆ dA. 2 Ae
(22.37) (22.38)
Ae
Here, Ae is the area of the reference plane (mid surface) and the variable p represents the lateral load pressure acting on the plate. The functional (22.38) violates the thin plate limit (i.e. the Kirchhoff patch test) in a classical displacement based plate element. This deficiency can be resolved according to a Hu-Washizu multi field variational principle. 22.2.2.1 The Bending Part of the Functional The bending part Πb of the functional (22.37) can be improved by introducing the orthogonality % &T ˆ b dΩ = 0, #ˆ b − #ˆ b σ (22.39) Ω 2
The hat on top of the variables indicates that we refer to an elemental Cartesian base of Sect. 22.1.1 (see Fig. 22.2)
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where an overbar indicates the independent assumption of the corresponding field. The resulting potential reads T 1 Πb = #ˆ b Cb #ˆ b dΩ (22.40) 2 Ω
and gives the variational basis to assume strain fields improving the element’s performance and computational efficiency. We interpolate ⎡ ⎤ ⎡ ⎤⎡ ⎤ T ˆ ⎢⎢⎢ Φ xˆ ⎥⎥⎥ ˆ ⎢⎢⎢ γ 0 ⎥⎥⎥ ⎢⎢⎢ Φ xˆ ⎥⎥⎥ ⎥⎥⎥ + BbH ⎢⎢⎢⎣ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ , #ˆ b = # xˆ xˆ #yˆ yˆ γ xˆyˆ = Bb0 ⎢⎢⎢⎣ (22.41) Φ yˆ ⎦ 0 γ ⎦ ⎣ Φ yˆ ⎦ with 9 Bˆ b0 = zˆ
⎡ ⎤ ⎡ ⎤ 9 ⎢⎢ ν (ξη) ⎢⎢⎢ 0 bTx ⎥⎥⎥ ⎥ ,ˆy (ξη), xˆ ⎥⎥⎥ ⎢ ⎢ ⎥ ⎢ ⎥⎥⎥ ˆ ⎥ j0 ⎢⎢⎢ T j ⎢⎢⎢ ⎢⎢ −b 0 ⎥⎥⎥ , BbH = zˆ ⎢⎢ − (ξη),ˆy −ν (ξη), xˆ ⎥⎥⎥⎥⎥ , ⎥⎥⎦ ⎥⎥⎦ j ⎢⎢⎢⎣ y j0 ⎢⎢⎢⎣ −bTx bTy 0 0
(22.42)
ˆ where the modifications√of Bˆ b0 and √BbH are motivated by the considerations of Sect. 22.2.1. The prefactors j0/ j and j/ j0 again ensure that the stiffness matrix can be integrated analytically.
22.2.2.2 The Shear Part of the Functional Considering the shear part Π s of the potential (22.37), we use a modified HuWashizu formulation, where the equilibrium equations and the strain displacement relations are fulfilled in a weak sense, while the constitutive relation is satisfied in strong (differential) form. Accordingly, we obtain ˆ − 1 sˆT 6 sdA, ˆ Π s = γˆ Ts sdA (22.43) 2 5Gh A
A
where the overbar indicates an independent interpolation of the corresponding field [6]. According to the well known count condition [16] it can be shown that we have ˆ i.e. two to apply four degrees of freedom for the assumed distributed shear forces s, interpolation points for each s xˆ and syˆ , which have to be located on the elements boundary. The satisfaction of the Kirchhoff patch test can be ensured by a covariant representation of transverse shear strain, γξ3 = uzˆ,ξ + det|J| ϕη , γη3 = uzˆ,η − det|J| ϕξ ,
(22.44)
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where the contravariant components of the rotations (ϕξ and ϕη ) are given by ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢⎢⎢ ϕξ ⎥⎥⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢⎢⎢ ⎥⎥⎥ = J−T ⎢⎢⎢⎢⎢ ϕ xˆ ⎥⎥⎥⎥⎥ = 1 ⎢⎢⎢⎢⎢ yˆ ,η − xˆ,η ⎥⎥⎥⎥⎥ ⎢⎢⎢⎢⎢ ϕ xˆ ⎥⎥⎥⎥⎥ . (22.45) ⎣ ϕη ⎦ ⎣ ϕ ⎦ det |J| ⎣ −ˆy xˆ ⎦ ⎣ ϕ ⎦ yˆ ,ξ ,ξ yˆ In (22.44) γξ3 vanishes at ξ = 0 and η = ±1 while γη3 vanishes for ξ = ±1 and η = 0 if pure bending deformations are applied3. Hence, linear shape functions may be used for the interpolation, T covariant γˆ = γ γ = Ns Γ , (22.46) ξ3
s
⎡ ⎢⎢⎢ NξI 0 NξIII N s = ⎢⎢⎢⎣ 0 NηII 0 1 1 NξI = (1 − η) , NξIII = (1 + η), 2 2
η3
⎡ ⎤ ⎢⎢⎢ ΓξI ⎥⎥⎥ ⎢ ⎥⎥ ⎤ ⎢⎢⎢ ⎢⎢⎢ ΓηII ⎥⎥⎥⎥⎥ 0 ⎥⎥⎥⎥ ⎥ ⎥⎥ , Γ = ⎢⎢⎢ ⎢⎢⎢ ΓξIII ⎥⎥⎥⎥⎥ , NηIV ⎦ ⎢⎢⎢⎣ ⎥⎥⎥⎦ ΓηIV
(22.47)
1 1 NηII = (1 + ξ) , NηIV = (1 − ξ). (22.48) 2 2
covariant
Now, γˆ s is related to the nodal transverse displacements Uzˆ and the two nodal bending angles Φ xˆ and Φ yˆ by inserting the corresponding parametric location into (22.44), Γ = Auz Uzˆ + Aϕx Φ xˆ + Aϕy Φ yˆ , (22.49) where the coefficients Auz , Aϕx and Aϕy depend on the element geometry only, ⎡ ⎢⎢⎢ −1 1 0 ⎢⎢⎢ 1 ⎢⎢⎢⎢⎢ −1 0 0 Auz = ⎢⎢⎢ 2 ⎢⎢⎢ 0 0 1 ⎢⎢⎣ 0 −1 1 ⎡ ⎢⎢⎢ yˆ 12 ⎢⎢⎢ 1 ⎢⎢⎢⎢⎢ yˆ 14 Aϕx = ⎢⎢⎢ 4 ⎢⎢⎢ 0 ⎢⎢⎣ 0
⎤ 0 ⎥⎥⎥⎥ ⎥⎥ 1 ⎥⎥⎥⎥ ⎥⎥⎥ , −1 ⎥⎥⎥⎥ ⎥⎥⎦ 0
⎤ ⎡ ⎢⎢⎢ xˆ21 xˆ21 0 ⎥⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢ xˆ 0 0 0 yˆ 14 ⎥⎥⎥ ⎥⎥⎥ , Aϕy = 1 ⎢⎢⎢⎢⎢ 41 4 ⎢⎢⎢⎢ 0 0 0 yˆ 43 yˆ 43 ⎥⎥⎥⎥ ⎥⎥⎦ ⎢⎢⎣ yˆ 23 yˆ 23 0 0 xˆ32 yˆ 12 0
⎤ 0 ⎥⎥⎥⎥ ⎥⎥ 0 xˆ41 ⎥⎥⎥⎥ ⎥⎥⎥ . xˆ34 xˆ34 ⎥⎥⎥⎥ ⎥⎥⎦ xˆ32 0 0
(22.50)
According to the transformation rules between covariant and Cartesian components of a vector we finally evaluate the transverse shear strain field with respect to the elemental system ⎡ ⎤ ⎢⎢⎢ yˆ ,η −ˆy,ξ ⎥⎥⎥ covariant 1 −1 ⎢⎢⎢ ⎥⎥⎥ N s Γ . γˆ s = J γˆ s = (22.51) det|J| ⎣ − xˆ,η xˆ,ξ ⎦ 3
Here, the transverse displacements uzˆ and the bending angles ϕ xˆ and ϕyˆ are interpolated in the classical bilinear form.
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For sake of efficiency and the strong request of avoiding numerical quadrature by applying an analytical integration of transverse shear part we further modify the transformation (22.51) according to covariant ˆ covariant . γˆ s = J−1 γˆ s ≈ J−1 0 γs
(22.52)
It has to be noted that this approximation causes a violation of the constant transverse shear patch test, i.e. γ xˆzˆ = γyˆ zˆ = 1 for uzˆ = xˆ + yˆ and vanishing bending angles, in arbitrarily distorted element configurations. However, we may view the transverse shear strains as penalty functions ensuring the thin plate limit and the modifications of (22.52) will not introduce significant detoriations of performance.
22.2.2.3 The Stiffness Matrix The stiffness matrix of the quadrilateral plate element is now found from the potential T 5Gh 1 ˆT ˆ ˆ 1 Π = Πb + Πs = # b Cb # b dΩ + γˆ s γˆ dA (22.53) 2 2 6 s Ω
and its variation
δΠ b + δΠ s = Ω
T ˆ b #ˆ b dΩ + δ#ˆ b C
A
T
δγˆ s
5Gh ˆ γ dA. 6 s
(22.54)
A
Here, #ˆ b represents the assumed bending strain depending on nodal variables Φ xˆ and Φ yˆ (see (22.41) and (22.42)) while γˆ s denotes the assumed transverse shear according to (22.52). The variation of internal transverse shear energy (22.54) reads ˆ ΓT K δΠ s = δΓ sΓ ,
(22.55)
with ⎡ ⎢⎢⎢ K11 K12 K13 ⎢⎢⎢ ⎢⎢⎢ K22 K12 5Gh 5Eh ˆ = ⎢⎢⎢ −1 K NTs J−T J N dA = s s ⎢ 0 0 ⎢ 6 576 (1 + ν) j0 ⎢⎢ K11 ⎢⎢⎢ A ⎣ sym
⎤ K12 ⎥⎥⎥⎥ ⎥⎥ K24 ⎥⎥⎥⎥ ⎥⎥⎥ K12 ⎥⎥⎥⎥ ⎥⎥⎦ K22
K11 = 4 (x13 + x24 )2 + (y13 + y24)2 , K12 = 3 x224 − x213 − y213 + y224 , K13 = 2 (x13 + x24 )2 + (y13 + y24)2 , K22 = 4 (x13 − x24 )2 + (y13 − y24 )2 , K24 = 2 (x13 − x24 )2 + (y13 − y24 )2 . (22.56)
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The variation of internal bending energy δΠ b (22.54) leads to the constant part of the bending stiffness matrix
ˆ Kb0 =
h/2
−h/2 A
⎡ ⎤T ⎡ ⎤ ⎢⎢⎢ 0 bTx ⎥⎥⎥ ⎢⎢⎢ 0 bTx ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢⎢ T ⎥⎥ h3 ⎢⎢⎢ ˆT ˆ ˆ ˆ ⎢⎢ −b 0 ⎥⎥⎥⎥ , Bb0 C Bb0 dAdˆz, = j0 ⎢⎢⎢⎢ −bTy 0 ⎥⎥⎥⎥ C y ⎢ ⎥⎥⎥ ⎥⎥⎦ ⎢⎢⎣ 3 ⎢⎢⎣ ⎦ −bTx bTy −bTx bTy
(22.57)
while the modifications of the hourglass part (22.42) allow for analytical integration since the integrand of h/2 ˆ ˆB ˆ bH dAdˆz, KbH = Bˆ TbH C (22.58) −h/2 A
is not fractional anymore, ⎡ ⎤ 2 + xˆ 2 3 ⎢⎢ ⎥⎥⎥ ( ) x ˆ −ν x ˆ y ˆ + x ˆ y ˆ Et 24 24 13 13 ⎢ ˆ 13 24 ⎥⎥⎥ Υ , ⎢⎢⎢⎣ KbH = Υ T ⎦ 72 j0 −ν ( xˆ24 yˆ 24 + xˆ13yˆ 13 ) yˆ 213 + yˆ 224
(22.59)
Hence, the total stiffness matrix of the plate element reads ⎡ ˆ ˆ ⎢⎢⎢ Kuu K uϕx ⎢⎢⎢ ˆ ˆ K = ⎢⎢⎢⎢⎢ Kϕxϕx ⎢⎢⎣ symm
⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥ ϕxϕy ⎥⎥⎥ , ⎥⎥⎦ ˆ K ϕyϕy
ˆ Kuϕy ˆ K
(22.60)
with ˆ ˆ Kuu = ATuz K s Auz , and
ˆ ˆ Kuϕx = ATuz K s Aϕx ,
⎡ b ⎢ˆ ˆ = ⎢⎢⎢⎢⎢ Kϕxϕx K b ⎢⎢⎣ ˆ b Kϕyϕx ˆ ˆb ˆ Kϕxϕx = Kϕxϕx + ATϕx K s Aϕx , b ˆ ˆ ˆ A . K =K + AT K ϕyϕy
ϕyϕy
ϕy
s
ˆ ˆ Kuϕy = ATuz K s Aϕy ,
⎤ ˆ b ⎥⎥⎥ K ˆ +K ˆ ϕxϕy ⎥⎥ ⎥=K b0 bH ˆ b ⎥⎥⎦ Kϕyϕy ˆ ˆb ˆ Kϕxϕy = Kϕxϕy + ATϕx K s Aϕy ,
(22.61)
(22.62)
(22.63)
ϕy
22.3 Generalization to Geometrical Nonlinearities In this chapter we generalize the previous formulation to large deformations and large strains. Within an updated Lagrangian scheme this generalization simply amounts to an interpolation of velocities and angular velocities instead of displacements and rotations. The analytical integration of membrane and bending stiffness
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matrices requires the spin and the material tangent stiffness to be constant within an element [20]. However, this is not the case for arbitrary deformation fields. Nevertheless, these two hypotheses greatly reduce the computational effort for the calculation of the internal force vector fiint , which is related to the virtual internal power δΠ int by δΠ int = δvi fiint . A key step is the use of an elastic-plastic update only within the constant part of the symmetric membrane and bending proportions to evaluate the constant internal force vector fˆint 0 discussed in Sect. 22.3.1. All remaining parts, i.e. the symmetric hourglass membrane part, the complete unsymmetric drilling rotation part, the total transverse shear part and the hourglass bending part, are assigned to the hourglass internal force vector fˆint H given in Sect. 22.3.2.
22.3.1 The Constant Part of the Internal Force Vector Following the procedures of [6] the virtual internal power at the current time step tn reads4 % & n ˆ n T σˆ n dΩ, δΠint 0 = δD (22.64) 0 Ωn
ˆ n denotes the constant part of the corotational rate of deformation tenwhere D 0 sor. The corotational Cauchy stress tensor at the current time step σˆ n is evaluated according to a hypo-elastic-plastic central difference stress update. There, the constant part of the rate of deformation tensor at the time tn−1/2 = tn − 1/2Δt is calculated by ⎡ n−1/2 ⎤ ⎡ n−1/2 ⎤ ⎢⎢ ˆ ⎥⎥⎥ ⎢ˆ ⎥⎥⎥ 1 1 ⎥ n− /2 n− /2 ⎢⎢⎢ V x n−1/2 ⎢⎢⎢ ω x ⎥ ˆ ˆ ˆ D0 = B su0 ⎢⎢⎢ n−1/2 ⎥⎥⎥ + zˆBb0 ⎢⎢⎣ n−1/2 ⎥⎥⎥⎦ , (22.65) ⎣ˆ ⎦ ωˆ y Vy with
⎡ ⎤ ⎢⎢⎢ b x 0 ⎥⎥⎥
⎥⎥⎥ ⎢⎢⎢⎢ 1/2 1/2 ⎥⎥
⎢ Bˆ n− and Bˆ n− = su0 = ⎢⎢⎢ 0 by ⎥⎥⎥ b0 ⎢⎣ ⎥⎦
by bx n−1/2 t
⎡ ⎤ ⎢⎢⎢ 0 bTx ⎥⎥⎥
⎢⎢⎢ ⎥ ⎢⎢⎢ −bT 0 ⎥⎥⎥⎥⎥
. ⎢⎢⎢ y ⎥ ⎣ T T ⎥⎥⎦
−b x by n−1/2
(22.66)
t
n−1/2
n−1/2 ˆ The variables Vi and ωˆ i for i = x, y refer to the projected nodal velocities and nodal angular velocities (22.5) at tn−1/2 , respectively, while b x and by are given in (22.9). Further, we write in relation to (22.64) ⎧ h ⎫ ⎡ ⎤T ⎪ ⎪ /2 ⎪ ⎪ ⎪ ⎢⎢⎢ δV ˆ nx ⎥⎥⎥ % &T ⎪ ⎪ ⎪ ⎨ ⎬ n n n ⎢ ⎥ ˆ ˆ δΠint 0 = ⎢⎢⎣ n ⎥⎥⎦ B σ dˆ z dA+ ⎪ ⎪ su0 ⎪ ⎪ ⎪ ⎪ ˆy ⎪ ⎪ δV ⎩ ⎭ Ae
−h/2
;<=> f
4
In this section superscripts refer to the corresponding time step.
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⎧ h ⎫ ⎡ ⎤T ⎪ ⎪ /2 ⎪ ⎪ n ⎥⎥ % ⎪ ⎪ ⎢⎢⎢ δω & ⎪ ⎪ ˆ x ⎥⎥ n T⎨ n ⎬ ⎢ ˆ ⎢ ⎥ + ⎢⎣ n ⎥⎦ Bb0 ⎪ zˆσˆ dˆz⎪ dA, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˆy δω ⎩−h/2 ⎭ Ae ;<=>
(22.67)
m
since the current element area is usually assumed to be constant in thickness direction. Here, the B-matrices Bˆ nsu0 and Bˆ nb0 have to be related to the current time tn . Accordingly, the constant internal force vector consisting of nodal forces (f n ) and nodal moments (mn ) is given by ⎡ ⎤ ⎢⎢⎢ f n ⎥⎥⎥ ˆn f int 0 = ⎢⎢⎢⎣ n ⎥⎥⎥⎦ , (22.68) m with
% & ˆn T f f n = Ae B su0
% &T mn = Ae Bˆ nb0 m.
and
(22.69)
The resultant forces f and the resultant moments m are evaluated using Simpson’s rule. Finally, a rigid body projection (22.4) and the coordinate transformation (22.7) has to be performed, i.e. ˆn fˆn = C TPˆ T f int 0 . (22.70) int 0
R
22.3.2 The Internal Stabilization Force Vector n The internal stabilization force vector fint H is determined by a central difference update using the analytically integrated stiffness matrices of Sect. 22.2, n−1/2 n−1/2 f˙int , H = KV
(22.71)
with a local to global coordinate transformation (22.7) in connection with a rigid body projection (22.4) ˆ K=C TT Pˆ TR KPˆ R C T. (22.72) ˆ we state For the in-plane part of K ⎡ ˆ ⎢⎢⎢ K ˆ˙f ⎢⎢⎢⎢ 11 = int H in ⎣ˆ K21 n−1/2
⎤⎡ ⎤ ˆ T ⎥⎥⎥ ⎢⎢ V ˆ n−1/2 ⎥⎥⎥ K 21 ⎥⎥ ⎢⎢⎢ ⎥⎥ 1 ⎥, ˆ ⎥⎥⎦ ⎢⎣ ω ˆ n− /2 ⎦ K 22
(22.73)
z
1/2 1/2 1/2 T ˆ n−1/2 are the nodal in-plane velocities, i.e, V ˆ n− ˆ n− where V , while ωˆ n− is V z x y 1/2 n− the drilling rotational velocity at t . The corresponding stiffness matrices read ˆ ˆ ˆ ˆ K11 = K suuH + Kauu0 + KauuH ,
(22.74)
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ˆ =K ˆ ˆ ˆ ˆ K 21 aϕu0 + KaϕuH , K22 = Kaϕϕ ,
(22.75)
which have to be found in terms of the projected nodal coordinates at tn−1/2 . The related submatrices are given in (22.28)–(22.32). ˆ For the out of plane part of K at tn−1/2 consisting of a shear and a bending part we write ⎡ ˆ ⎤⎡ ⎤ 1/2 ˆ ˆ ⎢⎢⎢ Kuu K ⎥⎥⎥ ⎢⎢ V ˆ n− ⎥⎥ K uϕx uϕy z ⎢ ⎢ ⎥⎥⎥⎥ ⎢⎢⎢ n−1/2 ⎥⎥⎥⎥ ˆ˙f n−1/2 = ⎢⎢⎢⎢⎢ ˆ ˆ (22.76) ˆ x ⎥⎥⎥⎥ , Kϕxϕx Kϕxϕy ⎥⎥⎥⎥ ⎢⎢⎢⎢ ω ⎢⎢⎢ int H Out ⎥⎦ ⎥⎥⎦ ⎢⎢⎣ n− ⎢⎣ 1/2 ⎥ ˆ symm Kϕyϕy ωˆ y where the submatrices given in (22.61)–(22.63) are evaluated at tn−1/2 . Here the ˆ ˆ bending part Kb in (22.62) only consists of the non-constant hourglass part KbH given in (22.59) because the corresponding constant part is already introduced in (22.69).
22.4 The Mass Matrix Within an efficient shell formulation the mass matrix has to be optimized for an explicit time integration. A common procedure in explicit transient dynamics is described in Hughes, Liu and Levit [21]. Their way of scaling the rotatory inertia terms will be inherited here. The scaling ensures that the critical time step is not bounded by rotary inertia terms. The consistent mass matrix is given by Mdisp = h ρNT NdA, (22.77) Ae
where N is the matrix of shape functions of the bilinear quadrilateral, ρ denotes the density of the material and h and Ae are the initial shell thickness and the initial element area, respectively. The translational masses are evaluated according to the row summing technique. The nodal rotatory inertia is computed by scaling the translational mass at the node by the factor β (see [6, 21]), i.e., * 2 + h Aelem Mrot = βMdisp , where β = max , . (22.78) 4 8
22.5 Numerical Examples In order to investigate the quality of the derivations the proposed element is tested within this section. Challenging linear and nonlinear test examples are discussed and the results in terms of the developed element technology are compared to
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literature. More test cases can be found in Kugler [6]. Note that we usually omit units in the description of the benchmark problems, however, unit-consistency will be understood. An important issue of the proposed shell element is the proposition of suitable values for the added material constant α, that relates the skew symmetric strains to the corresponding antimetric stresses (22.1). Hughes and Brezzi [3] advise to relate this constant to the shear modulus of the material, i.e. α = μ. However, it is shown in Kugler et al. [5] that the proposed membrane element is rather insensitive to α for in-plane loadings. If a curved surface (see Fig. 22.4) is discretized with the proposed element technology, we observe that the membrane and bending stiffnesses are coupled at the inter-element boundaries. A nodal rotation that is associated to the bending strains of element two (ϕb (E2)) has a component about the axis normal to element one (E1). Therefore, the drilling rotation stiffness of E1 is multiplied by ϕ = ϕm (E1) + ϕb(E2) sin β,
(22.79)
where ϕm (E1) represents the original drilling rotation of element one (the index m indicates membrane) and β is the angle between the element normals. This observation was made by Cook [22] and was detailed in [23]. Since the bending stiffness decreases of order O(h3 ) while the membrane stiffness varies only with O(h) the large membrane stiffness of element one tends to lock the bending rotations of element two in the case of very thin shells. With increasing mesh refinement and β approaching zero this phenomenon disappears. However, when very thin smooth surfaces are discretized with coarse meshes this undesirable membrane-bending interaction influences the overall stiffness substantially. Within the present theory the coupling between drilling rotations and membrane stiffnesses can be directly manipulated by the material constant α. Consequently, we propose the usage of high values of α if the drilling stiffness is important (e.g. α = 250μ) and low values in all other cases (e.g. α = 0.005μ)5. Remark 22.2. This rarely tackled mechanism emanates directly from the six parameter formulation since only the presence of drilling stiffnesses causes that problem. Therefore, the term “membrane locking” ( [22,23]) leads to misinterpretations and,
ϕ m (E1)
ϕb (E2)sin β
ϕb (E2)
β E1
smooth curved surface
E2
Fig. 22.4 A smooth curved surface modeled with two flat shell elements 5
The drill stiffness is important under three circumstances: at beam to shell connections, if shell surfaces which has to be discretized exhibits a “true” kink or if the load is applied by concentrated couples and drill rotations are postprocessed.
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consequently, the author proposes “drill rotation locking” as a suitable term describing that mechanism.
22.5.1 Slab Supported by a Central Column As indicated, it is a main issue of this paper to derive shell elements consisting of six degrees of freedom at each node to resolve problems with beam to shell connections. To check the performance within those kind of problems a square plate 10 × 10 with a unit thickness is supported by single column of length l = 10 and a circular cross section with radius of r = 0.25. The slab is rotated by forces (load case 1) and by concentrated couples (load case two) at the vertices of the plate (see Fig. 22.5), while the beam end is fully clamped. According to St. Venant’s theory of torsion both load cases yield a tip rotation of 0.42373, where we used E = 1 · 106 for Young’s modulus and ν = 0.3 for Poisson’s ratio. Gray shaded elements denote the application of α = 250μ while for all other elements α = 0.005μ is used. The Table 22.1 Simulation results: Slab supported by a central column a) Tip rotation of the column b) Vertex rotation Vertex rotation Frey [24] Ibrahimbegovic [12] Present 0.4238
0.4237
0.4237
Forces
0.4238
Moments 0.4238
0.4237
0.4237
Moments
0.4240
Forces
results for the beam tip rotation are given in Tab. 22.1 and are compared to results from literature. Since the square plate rotates almost rigidly, we expect an almost constant drilling rotation distribution, i.e. the vertex rotation of the plate is identical compared to the beam tip rotation. All results indicate the capability of the proposed shell formulation to reproduce the exact solution in a beam to shell connection.
Fig. 22.5 Slab supported by a central column
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22.5.2 Hemispherical Shell Within this benchmark problem a smooth hemispherical shell is loaded by opposing radial point loads alternating at 90°. Since the resulting deformation mode is mainly inextensional bending, we use this problem to evaluate the ability of the shell to handle double curved shell geometries under inextensional bending deformations correctly. Two different geometries with a thickness of h = 0.04 are given in Fig. 22.6. The first one has a hole at top and bottom (Fig. 22.6a) resulting in a mesh consisting of non warped elements. The second geometry is a full hemisphere (Fig. 22.6b) with initially warped elements where the rigid body projection (22.4) influences the results substantially. The elasticity properties are taken as E = 6.825 · 107 and ν = 0.3. The normalized radial displacements ur/u∗r (u∗r = 0.0924 at point load [25]) are given in Tab. 22.2 where α is set to α = 0.005μ. Table 22.2 Simulation results: Hemispherical shell a) With hole on top No. of elements Simo [26] SQ2 [12]
Present
8×8
1.00
1.00
1.00
16 × 16
1.01
1.01
1.01
b) Without hole on top YASE [28] MITC4 [18]
No. of elements
SRI [27]
48
0.93
0.34
192
0.98
0.84
QPH [25]
Present
0.91
0.86
0.96
0.99
0.99
0.99
The geometrically nonlinear transient problem is simulated with a density of ρ = 2.5 · 10−4 (at the initial configuration). The load is assumed to be constant with time. In order to examine the performance of the present element formulation
a)
Fig. 22.6 Hemispherical shell: a) With hole; b) Without hole
b)
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QPH
Belytschko-Tsay
0
displacement
-0.05
-0.1
-0.15
-0.2
-0.25
0
0.01
0.02
0.03
0.04
0.05
time
Fig. 22.7 Dynamic response of the hemispherical shell
according to warped element geometries the hemisphere without a hole on the top is meshed with 192 elements as shown in Fig. 22.6b. The results are compared to the Belytschko et al. [29] formulation as well as to the Belytschko and Leviathan proposals [25] (benchmark results are calculated with Ls-Dyna). The Belytschko– Lin–Tsay formulation indicates a severe amplitude decay which is atypical for explicit transient solution procedures with an elastic constitutive relation. The present formulation, however, shows an excellent over all performance.
22.5.3 Roll-Up of a Clamped Beam An initially flat shell with length L = 12, width w = 1 and height h = 1 is clamped at one end and a bending moment is applied at the other end. According to the Euler beam theory, the analytical solution for a beam rolled up to a circular arc of 0 radius r is given by 1r = M EI , where M0 is the end moment and I is the moment of inertia. Within the proposed shell theory the load can either be applied by bending moments causing an out of plane bending deformation Fig. 22.8a (load case 1), or by concentrated in-plane drilling moments causing an in-plane membrane deformation Fig. 22.8b (load case 2). We are using a regular 12x1 mesh to obtain the corresponding solutions where a Young’s modulus of E = 30 · 106 , a Poisson’s ratio ν = 0 and a density ρ = 1.2 is used. The load is applied smoothly by corresponding concentrated couples at the two end nodes M = M0/2 with M0 = 1308996.939. The deformation characteristics of both load cases are given in Fig. 22.8 for 25%, 50%, 75% and 100% of M0 , respectively. We see that both deformation fields are in excellent agreement with the analytical findings even for such a coarse meshes.
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a)
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b)
Fig. 22.8 Roll-up of a clamped beam: a) Shell bending (load case 1); b) Membrane drilling (load case 2)
22.6 Summary Within this paper a quadrilateral shell element with six degrees of freedom per node is derived. The problem is tackled by independent descriptions of the membrane and the bending properties while a coupling is only introduced for warped element geometries based on a rigid body projection scheme. The membrane part consists of drilling degrees of freedom, i.e. membrane inplane rotational degrees of freedom which are derived using a unique functional that introduces rotational degrees of freedom naturally. The bending properties are derived using the Mindlin-Reissner assumption. All parts of the shell element’s linear stiffness matrix can be evaluated analytically and no Gaussian quadrature has to be carried out at all. The proposed element fulfills all patch test conditions and gives excellent results for typical benchmark problems. Acknowledgements This work has been supported by Grant Agency VEGA No. 1/0093/10.
References 1. D. Allman, A compatible triangular element including vertex rotations for plane elasticity analysis, Computers & Structures 19 (1984) 1–8. 2. A. Cazzani, S. Atluri, Four-noded mixed finite elements, using unsymmetric stresses for linear and nonlinear analysis of membranes, Computational Mechanics 11 (1993) 229–251. 3. T. J. Hughes, F. Brezzi, On drilling degrees of freedom, Computer Methods in Applied Mechanics and Engineering 72 (1989) 105–121. 4. E. Reissner, A note on variational priciples in elasticity, International Journal of Solids and Structures 1 (1965) 93–95. 5. S. Kugler, P. Fotiu, J. Murin, A highly efficient membrane finite element with drilling degrees of freedom, Acta Mechancia 213 (2010) 323–348. 6. S. Kugler, Development of a laterally pressed quadrilateral shell element, Ph.D. thesis, STU Bratislava (2010). 7. E. Cosserat, F. Cosserat, Th´eorie des corps d´eformables, A. Hermann, Paris, 1909. 8. A. Eringen, Nonlocal continuum field theories, Springer, 2002.
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9. T. Hughes, A. Masud, I. Harari, Numerical assessment of some membrane elements with drilling degrees of freedom, Computers & Structures 55 (1995) 297–314. 10. A. Ibrahimbegovic, A novel membrane finite element with an enhanced displacement interpolation, Finite Elements in Analysis and Design 7 (1990) 167–179. 11. A. Ibrahimbegovic, R. Taylor, E. Wilson, A robust quadrilateral membrane finite element with drilling degrees of freedom, International Journal for Numerical Methods in Engineering 30 (1990) 445–457. 12. A. Ibrahimbegovic, F. Frey, Membrane quadrilateral finite elments with rotational degrees of freedom, Engineering Fracture Mechanics 43 (1992) 12–24. 13. Y. Zhu, T. Zacharia, A new one-point quadrature, quadrilateral shell element with drilling degrees of freedom, Computer Methods in Applied Mechanics and Engineering 136 (1996) 165–203. 14. T. Belytschko, I. Leviathan, Projection schemes for one-point quadrature shell elements, Computer Methods in Applied Mechanics and Engineering 115 (1994) 277–286. 15. C. Rankin, B. Nour-Omid, The use of projectors to improve finite element performance, Computers & Structures 30 (1988) 257–267. 16. O. C. Zienkiewicz, R. L. Taylor, Finite Element Method: Volume 1, The Basis (Finite Element Method ), Butterworth-Heinemann, 2000. 17. J. Simo, T. Hughes, On the variational foundations of assumed strain methods, Computer Methods in Applied Mechanics and Engineering 53 (1986) 1685–1695. 18. E. Dvorkin, K. Bathe, A continuum mechanics based four-node shell element for general nonlinear analysis, Eng. Comput. 1 (1984) 77–88. 19. K. Bathe, E. Dvorkin, A formulation of general shell elements - the use of mixed interpolation of tensorial components, International Journal for Numerical Methods in Engineering 22 (1986) 697–722. 20. Q. Zeng, A. Combescure, A new one-point quadrature, general non-linear quadrilateral shell element with physical stabilization, International Journal for Numerical Methods in Engineering 42 (1998) 1307–1338. 21. T. Hughes, W. Liu, I. Levit, ”Nonlinear dynamics finite element analysis of shells”. Nonlinear finite element analysis in Struct. Mech., Springer-Verlag Berlin, 1981. 22. R. Cook, Four-node ’flat’ shell element: Drilling degrees of freedom, membrane-bending coupling, warped geometry, and behavior, Computers & Structures 50 (1994) 549–555. 23. C. Choi, T. Lee, Efficient remedy for membrane locking of 4-node flat shell elements by nonconforming modes, Computer Methods in Applied Mechanics and Engineering 192 (2003) 1961–1971. 24. F. Frey, Shell finite elements with six degrees of freedom per node, Analytical and Computational Models for Shells (Edited by A.K. Noor, T. Belytschko, J.C. Simo), ASME (1989) 291–317. 25. T. Belytschko, I. Leviathan, Physical stabilization of the 4-node shell element with one point quadrature, Computer Methods in Applied Mechanics and Engineering 113 (1994) 321–350. 26. J. Simo, D. Fox, M. Rifai, On a stress resultant geometrically exact shell model. part ii: The linear theory; computational aspects, Computer Methods in Applied Mechanics and Engineering 73 (1989) 53–92. 27. T. Hughes, W. Liu, Nonlinear finite element analysis of shells: Part i. three dimensional shells, Computer Methods in Applied Mechanics and Engineering 26 (1981) 331–362. 28. B. Engelmann, R. Whirley, A new elasto-plastic shell formulation for dyna 3d,, Lawrence Livermore National Laboratory Report UCRL-JC-104826. 29. T. Belytschko, J. Lin, C. Tsay, Explicit algorithms for the nonlinear dynamics of shells, Computer Methods in Applied Mechanics and Engineering 42 (1984) 225–251.
Chapter 23
Invariant-Based Geometrically Nonlinear Formulation of a Triangular Finite Element of Laminated Shells Stanislav V. Levyakov
Abstract A non-standard approach is proposed to develop a simple and computationally effective triangular finite element applicable to geometrically nonlinear analysis of composite shells. The approach is based on the natural components of the stress and strain tensors and their invariants which allow one to express the strain energy of the shell in a compact form without coordinate transformations. The natural components of the tensors are referred to fibers oriented along the triangle edges. An advantage of using the natural strains is that it suffices to approximate one-dimensional functions rather than two-dimensional strain fields over the elemental area. To this end, analytical solutions of auxiliary beam bending problems are used. The validity and accuracy of the approach proposed are verified using benchmark solutions and numerical data available in the literature. Keywords Invariants · Triangular finite element · Natural strain · Composite shell · Geometrical nonlinearity
23.1 Introduction Investigation of nonlinear deformation of shells poses challenging problems to analysts and structural engineers. Except for simple geometry, loading, and boundary conditions, solution of shell problems arising in practice can be obtained only by numerical methods. Since the advent of computers, the finite element method has become a dominant numerical technique for analysis of thin-walled structures, and much effort has been spent to develop computationally effective shell finite elements. Contribution of mechanical engineers and mathematicians in this field is S. V. Levyakov (B) Department of Engineering Mathematics, Novosibirsk State Technical University, 630092 Novosibirsk, Russia e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 23, © Springer-Verlag Berlin Heidelberg 2011
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immense. Review of various approaches and techniques for constructing shell elements can be found in excellent papers by Bucalem and Bathe [6], Yang et al. [18], and Gal and Levy [8]. Modern day computers allow one to handle systems of equations with millions of unknowns and implement highly sophisticated finite-element models with large number of degrees of freedom. In the context of engineering analysis, however, simple and reliable shell finite elements are still desirable. For practical applications, there should be a sensible balance between accuracy of the numerical solution and computational cost. Much research has been focused on a triangular finite element owing to its ability to model general shell geometry and irregular boundaries. A promising approach to construct an effective finite element is to consider the element as a mechanical object and not just a domain for local approximation of unknown functions. This idea was implemented in the natural mode method proposed by J. H. Argyris [2]. The method is based on (1) decomposition of the deformation of a body into purely deformation modes and rigid-body motion and (2) introduction of the coordinates defined by the triangle edges. It was shown that the deformation modes and the triangular element stiffness are convenient to determine using three membrane strains, three bending strains, and three transverse shear strains in the direction of the triangle edges. The normal components of the tensors referred to the triangle edges are called the natural strains. It is worth noting that the idea of using natural strains corresponds to the well-known fact that the plane stress (strain) states are fully determined by three direct stresses (strains) measured in three nonparallel directions. This theoretical statement has long been applied to investigate experimentally stresses in structures using strain gauges since the direct strains are much easier to measure than shear strains. The natural-mode approach was successfully implemented in a series of facet triangular elements for solving linear and nonlinear problems of laminated shells [1–4]. However, derivation of the elements involves lengthy matrix formulas relating the stress and strain components determined in different coordinate systems. The present study is in line with the natural mode approach. It shares the idea of using natural components of strain and separating rigid-body motion from pure deformation displacements of the finite element. Distinctive features of the finiteelement formulation given below are that (a) invariants of the natural strains are used; (b) finite displacements and rotations are allowed within the element; (c) geometry of initially curved element is taken into account using nodal directors; (d) deformation of the element in the presence of arbitrarily large rotations is described using the concept of kinematic group [10]. A computational benefit of the approach is that no operations on the coordinate transformation are required since no local coordinates are introduced for the elements. The paper is a continuation of the authors previous work [11–13] and organized as follows. In Sect. 23.2, assumptions upon which the finite element is constructed are summarized. In Sect. 23.3, relations between Cartesian and natural components of the strain tensor are given in an explicit form. In Sect. 23.4, strain invariants are obtained in terms of three natural components. In Sect. 23.5, template formulas are given for calculating invariants of two-dimensional symmetric tensors. In Sects 23.6
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and 23.7, the strain energy densities are expressed in terms of invariants for isotropic and composite materials. In Sect. 23.8, the total strain energy of a laminated shell subject to the Reissner–Mindlin assumptions is obtained in terms of natural strains. In Sect. 23.9, approximations of the natural strains within the element are considered. Sect. 23.10 contains formulas for computing the stiffness matrix, gradient, and Hessian of the element. Sect. 23.11 describes a technique for evaluating variations of the strain energy which are necessary to determine equilibrium states of shells. Finally, Sect. 23.12 demonstrates capabilities and accuracy of the finite element using linear and nonlinear problems of plates and shells.
23.2 Basic Requirements and Assumptions In the derivation of the shell finite element, we adopt the following requirements: • the element should be based on mechanistically clear assumptions; • the finite element should be applicable to linear and geometrically nonlinear analysis of composite and isotropic shells; • the finite element should be simple and inexpensive with a minimum number of degrees of freedom; • the finite element should not lock in the case of very thin structures described by the Kirchhoff–Love theory; • the finite element should be insensitive to mesh distortion. We determine configuration of a shell by geometry of its middle surface. Let a triangular element be cut from the shell by three planes normal to the middle surface. In general, the material of the shell is anisotropic and linearly elastic. In the derivation of the finite element, the following assumptions are employed: (a) strains are small compared to unity, but no restriction is imposed on the magnitude of displacements and rotations; (b) the area of the curved triangular element is equal to the area of a planar triangle whose side lengths are equal to those of the curved element; (c) each side of the curved triangular element is a planar nearly circular curve which remains planar and nearly circular after deformation; (d) the normal strains in the direction of the element edges are constant; (e) transverse shear strains are taken into account using the first order shear deformation theory which states that straight lines normal to the middle surface at the initial state remain straight but not necessarily normal to the middle surface after deformation, and these lines do not change in length during deformation. We note that assumption (b) is adopted to take into account the shell metrics. It introduces a correction to the faceted element approach where the side lengths and area of the element are smaller compared to those of the real element. Assumption (c) is used to describe finite curvature changes within the element.
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23.3 Natural Components of Strain We consider a small planar triangular element whose vertices are denoted by i, j, and k, and whose lengths of the edges opposite to the vertices are denoted by li , l j , and lk , respectively (Fig. 23.1). Position of any point within the triangle is convenient to determine by the area coordinates Li , L j , and Lk which satisfy the condition Li + L j + Lk = 1 [21]. To describe inhomogeneous properties of the material within the triangle, we introduce material Cartesian coordinates ξ1 and ξ2 . In the particular case of isotropic material, there is no need to use material coordinates. The aim of this section is to obtain explicit relations between the Cartesian components of the strain tensor S mn (m, n = 1, 2) and the natural strains S i (i = 1, 2, 3) of fibers parallel to the triangle edges. These relations are written in the form of a system of linear algebraic equations [15]: S i = αmni S mn αmni = λmi λni ,
(i = 1, 2, 3; λ1i =
ξ1k − ξ1 j , li
m, n = 1, 2), λ2i =
ξ2k − ξ2 j , li
(23.1) (23.2)
where λmi are the direction cosines of the ith edge, ξmk is the mth material coordinate of the kth vertex of the triangle, and summation is performed over m, n = 1, 2. In (23.1), (23.2), and below, subscripts i, j, and k obey the rule of cyclic permutation. Relations (23.1) can be inverted to give (summation over i = 1, 2, 3) S mn = βmni S i ,
Fig. 23.1 a) Shell finite element; b) single layer element; c) isometric triangle
(23.3)
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333
l1 l2 l3 li [λmk λn j + λnk λm j − 2(λ1 j λ1k + λ2 j λ2k )δmn ] (m, n = 1, 2), (23.4) 8F 2 where F is the triangle area and δmn is Kronecker’s delta. For convenience of further derivation of the finite element, we write expressions (23.1) and (23.3) in the matrix notation βmni =
S = AS C ,
⎡ ⎢⎢⎢ l1 l2 l3 ⎢⎢⎢⎢ ⎢⎢ B= 4F 2 ⎢⎢⎢⎣ 1
S C = BS ,
(23.5)
, , S C = S 11 S 22 S 12 T , S = S1 S2 S3 T, ⎡ ⎤ ⎢⎢⎢ λ2 λ2 2λ11 λ21 ⎥⎥⎥ ⎢⎢⎢ 11 21 ⎥⎥⎥ A = ⎢⎢⎢⎢ λ212 λ222 2λ12 λ22 ⎥⎥⎥⎥ , ⎢⎢⎣ ⎥⎥⎦ λ213 λ223 2λ13 λ23 −λ22 λ23 l1
−λ23 λ21 l2
−λ21 λ22 l3
−λ12 λ13 l1
−λ13 λ11 l2
−λ11 λ12 l3
1 2 (λ13 λ22 + λ23 λ12 )l1 2
(λ11 λ23 + λ21 λ13 )l2 12
(23.6)
(23.7) ⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ , ⎥⎥⎥ ⎦
(λ12 λ21 + λ22 λ11 )l3 (23.8) where the subscript C stands for Cartesian. The transformation matrices are written in terms of αmni and βmni as ⎡ ⎤ ⎢⎢⎢ α111 α221 2α121 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ A = ⎢⎢⎢⎢ α112 α222 2α122 ⎥⎥⎥⎥ , ⎢⎢⎣ ⎥⎥⎦ α113 α223 2α123
⎡ ⎤ ⎢⎢⎢ β111 β112 β113 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ B = ⎢⎢⎢⎢ β221 β222 β223 ⎥⎥⎥⎥ . ⎢⎢⎣ ⎥⎥⎦ β121 β122 β123
(23.9)
Thus, we have obtained explicit relations (23.5) which allow one to readily express Cartesian components of the strain tensor in terms of natural strains and vice versa.
23.4 Invariants of Strain Tensor We focus our attention on invariants expressed in terms of quantities independent of material coordinates. The invariants written in curvilinear convective coordinates were discussed by Kuznetsov and Levyakov [13]. The classical invariants of the elasticity theory [15] are written in Cartesian coordinates as 2 IS = S 11 + S 22 , IS S = S 11 S 22 − S 12 . (23.10) Here IS and IS S are the first and second invariants of the strain tensor, respectively. Relations (23.10) can be combined with (23.3) to give invariants in the natural form. This process, however, involves cumbersome expressions. We consider a
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more straightforward technique for obtaining invariants of two-dimensional tensors in terms of natural components. According to [9], we consider the ratio of the squared area of the deformed triangle F ∗ to the squared area of the triangle in the initial undeformed state F. Given the triangle side lengths, the squared area of the triangle is determined by Heron’s formula which can be written as follows (summation over p = 1, 2, 3): F2 =
Δ , 16
Δ = (l p l p )2 − 2l2p l2p ,
(23.11)
where Δ is the discriminant of the triangle. We assume that the squared area of the deformed triangle can be determined in a similar manner(summation over p = 1, 2, 3): Δ∗ ∗2 F ∗2 = , Δ∗ = (l∗p l∗p )2 − 2l∗2 (23.12) p lp , 16 where the lengths of the triangle sides after deformation are given by 2 l∗2 p = l p (1 + 2S p)
(23.13)
In view of (23.11), (23.12), and (23.13), the ratio of the squared areas can be decomposed as follows (summation over p = 1, 2, 3): F ∗2 = 1 + 2IS + 4IS S , F2 IS = 2(aa p S p − 2a2pS p ), a p = l2p Δ−1/2 ,
IS S = (a p S p )2 − 2a2p S 2p . a = l p l p Δ−1/2 ,
(23.14) (23.15) (23.16)
where a and a p are the metric coefficients. Since the metric coefficients (23.16) are functions of side lengths of the triangle only, we call the strain invariants (23.15) the coordinate-free invariants to emphasize the fact that they do not depend on the material system of coordinates.
23.5 Templates for Invariants of Tensors Relations given for the invariants of strains given in Sect. 23.4 are valid for any symmetric two-dimensional tensor umn (m, n = 1, 2) which can be represented by the following vectors comprising Cartesian and natural components: , U C = U11 U22 U 12 T ,
, U = U1 U 2 U3 T .
(23.17)
Below, we consider formulas which we call templates since they can be applied to any symmetric two-dimensional tensor of rank two.
23 Triangular Finite Element of Laminated Shells
335
Relations between Cartesian and natural components of the tensor are given by U i = αmni Umn ,
U mn = βmni U i .
(23.18)
U C = BU .
(23.19)
In matrix notation, relations (23.18) read U = AU C ,
The first and second invariants of the tensor are written in terms of Cartesian and natural components as IU = U11 + U22,
2 IUU = U 11 U22 − U12 .
(23.20)
In the natural components, expressions (23.20) become (summation over p = 1, 2, 3) IU = 2(aa p U p − 2a2p U p ),
IUU = (a p U p )2 − 2a2pU 2p .
(23.21)
Let the tensor U C be a sum of two tensors V C and W C , i.e. U C = V C + W C , where the subscript C denotes Cartesian components. In this case, relation (23.20)2 becomes IUU = IVV + IWW + 2IVW , (23.22) where invariants IVV and IWW are determined by template (23.21) in which U is replaced with V and W, respectively, and IVW is the combined invariant of two tensors. In Cartesian components, the combined invariant is given by IVW =
1 (V11 W22 + V22 W11 − 2V12W12 ). 2
(23.23)
In contrast to the definition of the combined invariant given by Kuznetsov and Levyakov [13], a factor of 1/2 is introduced in (23.23) in order that the second invariant falls out as a particular case of the combined invariant. For example, setting V C = U C and W C = U C in (23.23), we obtain the second invariant IUU given by Eq. (23.20)2 . It is worth noting that the first invariant IU can be obtained from (23.23). To this end, we set W C = U C and V C = {2 2 0}T , the latter vector representing a spherical tensor. Substituting these vectors into (23.23), we arrive at (23.20)1 . Let us express the combined invariant in terms of natural components. Substituting the expression U = V + W into (23.21)2 , we arrive at expression (23.22) in which the combined invariant is given by (summation over p, q = 1, 2, 3) IVW = (a p V p )(aq Wq ) − 2a2p(VW) p .
(23.24)
Here (VW) p = V p W p is simply a product of the components having the same subscript, i.e. no summation is performed over p. In particular case where V = {2 2 2}T
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and W = U , template (23.24) yields the expression for the first invariant IU . Setting V = U and W = U , from (23.24) one obtains the second invariant IUU . In matrix notation, templates (23.21) and (23.24) are written as IU = τ T U , where
μμ T − ρ )U U, IUU = U T (μ
μμ T − ρ )W W, IVW = V T (μ
τ = 2{a1 (a − 2a1) a2 (a − 2a2) a3 (a − 2a3)}T , ⎡ ⎤ ⎢⎢⎢ a2 0 0 ⎥⎥⎥ ⎢⎢⎢ 1 ⎥⎥⎥ μ = {a1 a2 a3 }T , ρ = 2 ⎢⎢⎢⎢ 0 a22 0 ⎥⎥⎥⎥ . ⎢⎢⎣ ⎥⎥⎦ 0 0 a23
(23.25) (23.26)
(23.27)
Consider, for example, the stress tensor σmn . Relations between the Cartesian and natural components of the stress tensor are obtained by replacing U with σ in templates (23.18). The first and second invariants can be obtained using templates (23.20) for Cartesian components and templates (23.21) for natural components of the stress tensor.
23.6 Strain Energy Density for Plane Stress We consider a linear elastic anisotropic body under plane-stress conditions. In this case, the strain energy density is written in Cartesian coordinates as follows (summation over m, n = 1, 2): 1 ΠV = σmn S mn . (23.28) 2 We write this expression in terms of invariants as ΠV =
1 (Iσ IS − 2IσS ) . 2
(23.29)
Here the combined invariant IσS is obtained using template (23.23) in which V and W are replaced with σ and S , respectively. The validity of (23.29) can be directly proved using templates (23.20) and (23.23). Thus, the energy ΠV known to be an invariant quantity is expressed in terms of the two first invariants and one combined invariant whichever material is considered. This can be regarded as a simple proof of the energy invariance with respect to the choice of coordinates. If the natural strains and stresses are given, the strain energy density (23.29) can be determined using templates (23.21) and (23.24) in which V and W are replaced with σ and S , respectively.
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23.6.1 Isotropic Material Isotropic material is characterized by two elastic constants, Young’s modulus E and Poisson’s ratio ν. Expanding (23.29) using templates (23.20) and Hooke’s law relations, we express the strain energy in terms of the strain invariants: ΠV =
1 E (I 2 − 2(1 − ν)IS S ). 2 1 − ν2 S
(23.30)
23.6.2 Six-Constant Anisotropic Material Elastic properties of an anisotropic material are usually determined using coordinates referred to Cartesian anisotropy axes. For the strain and stress tensors represented by the vectors (23.17)1 , we obtain ⎡ ⎤ ⎢⎢⎢ d11 d12 d13 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ σC = DC S C , D C = ⎢⎢⎢⎢ (23.31) d22 d23 ⎥⎥⎥⎥ , ⎢⎢⎣ ⎥⎥⎦ sym. d33 where D C is a matrix of elastic coefficients. For orthotropic materials, the elastic coefficients are given by d11 =
E11 , 1 − ν12ν21
d22 =
E22 , 1 − ν12ν21
d13 = d23 = 0,
d12 =
E 11 ν21 E 22 ν12 = , 1 − ν12ν21 1 − ν12 ν21
d33 = 4G 12 .
(23.32)
We write the stress-strain relations in terms of natural components. Using (23.19), from (23.31) we obtain σ = DS , D = A DC B . (23.33) Substituting the strain and stress invariants into (23.29) and using templates (23.25), we obtain the strain energy density as a bilinear form of the natural stresses and strains: 1 μμ T − ρ ), ΠV = S T H Δ σ , H Δ = τ τ T − 2(μ (23.34) 2 where H Δ is the 3 × 3 invariant symmetric matrix. It follows from (23.26), (23.27), and (23.16) that the components of the matrix H Δ depend only on the lengths of the triangle sides. Inserting (23.33) into (23.34), we express the strain energy density as a quadratic form of natural strains ΠV =
1 T S DNS , 2
D N = H Δ D,
where D N is the 3 × 3 natural matrix of elastic constants.
(23.35)
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S.V. Levyakov
23.7 Transverse Shear Strain Energy Density We introduce symmetric shear strain tensor Γ whose Cartesian and natural components are respectively given by [13] Γmn = γm3 γn3
(m, n = 1, 2),
Γi = γi2 .
(23.36)
Relations between Cartesian and natural components are given by template (23.18) or (23.19).The first invariant of the tensor can be obtained using templates (23.20)1 and (23.21)1 .
23.7.1 Isotropic Material The transverse shear strain energy density of an isotropic shell is written as ΠΓV =
1 2 2 G(γ13 + γ23 ), 2
(23.37)
where G = E/[2(1 + ν)] is the shear modulus. Noting that the sum in parentheses is the first invariant of the tensor Γ , we immediately obtain ΠΓV =
1 GIΓ . 2
(23.38)
23.7.2 Transversely Isotropic Material Transversely isotropic material is characterized by three engineering constants E, ν, and G = G13 = G23 , where G is the transverse shear modulus. Similar to (23.37) and (23.38), we find that 1 ΠΓV = G IΓ . (23.39) 2
23.7.3 Anisotropic Material We write the density of the transverse shear strain energy of an anisotropic material as (summation over m, n = 1, 2) ΠΓV =
1 gmn Γmn , 2
(23.40)
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339
where gmn is the tensor of elastic constants. Note that templates listed in Sect. 23.5 can be applied to the tensors gmn and Γmn . Comparison of (23.40) with (23.28) suggests that expression (23.40) can be written in the form of (23.29): ΠΓV =
1 (Ig IΓ − 2IgΓ ), 2
(23.41)
where the first invariants Ig and IΓ and combined invariant IgΓ can be expressed in terms of Cartesian or natural components using templates (23.20), (23.21), (23.23), and (23.24). In the particular case of orthotropic material, we obtain gC = {G13 G 23 0}T ,
Γ C = {Γ11 Γ22 Γ12 }T ,
1 (G23 Γ11 + G 13 Γ22 ). 2 Substitution of (23.43) into (23.41) yields the well-known expression Ig = G13 + G 23 ,
IΓ = Γ11 + Γ22,
ΠΓV =
IgΓ =
1 (G 13 Γ11 + G23 Γ22 ). 2
(23.42) (23.43)
(23.44)
Using templates (23.25), we write the invariants in (23.41) in matrix form as Ig = τ T g ,
IΓ = τ T Γ ,
μμ T − ρ )Γ Γ. IgΓ = g T (μ
(23.45)
Substituting (23.45) into (23.41), we obtain the density of the transverse shear energy in matrix form 1 ΠΓV = g T H Δ Γ , (23.46) 2 where the matrix H Δ is given by (23.34)2 . The strain density (23.46) can be expressed in terms of Cartesian or natural components of the tensors gmn and Γmn .
23.8 Total Strain Energy of Laminated Shell We consider a laminated shell composed of N orthotropic layers. For each layer, we introduce material coordinates ξ1 and ξ2 measured along the principal axes of the orthotropic material. The layers are assumed to be perfectly bonded such that no slipping occurs at the interlayer boundaries. We use the Reissner–Mindlin assumptions which imply that strains are linear functions of the coordinate z in the direction normal to the middle surface of the shell, whereas the transverse shear strains are constant across the shell thickness. In this case, the natural strains are written as S = ε + zκκ ,
ε = {ε1 ε2 ε3 }T ,
κ = {κ1 κ2 κ3 }T ,
h h − ≤z≤ , 2 2
(23.47)
where εi and κi are the natural components of the tensors of membrane strains and curvature changes of the middle surface, respectively, and h is the shell thickness.
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Combining (23.35)1 , (23.46), and (23.47)1 , we obtain the strain energy of the shell in terms of the natural strain components 1 Π= (εεT D N ε + 2zκκT D N ε + z2 κ T D N κ + g T H Δ Γ ) dV, (23.48) 2 V
where V is the volume occupied by the shell.
23.9 Approximation of Natural Strains We consider a shell finite element cut by three planes normal to the shell middle surface. Three nodes i, j, and k at the element vertices lying on the shell middle surface are determined by position vectors r i , r j , and r k , respectively. Moreover, we introduce three unit vectors n i , n j , and n k normal to the middle surface of the undeformed shell (Fig. 1). The resulting set of the nodal vectors is called the kinematic group and used to describe kinematics of the element [10]. In a deformed state of the shell, these nodal vectors are denoted by r ∗m and n ∗m (m = i, j, k). Note that, according to assumption (e) in Sect. 23.2, the vectors n ∗m are unit but not necessarily normal to the deformed middle surface of the shell. Our aim is to express the natural strains in terms of the scalar products of the nodal vectors which play the role of nodal parameters.
23.9.1 Normal Strains Approximation for the normal strains εi obtained under the assumptions (a), (c), and (d) is given by [12]: e∗ sin−1 (Δχ∗i ) εi = i − 1, (23.49) li Δχ∗i where Δχ∗i
=
χ∗2k − χ∗1 j 2
,
χ∗2k
=
ψ∗2k e∗i
,
χ∗1 j
=
ψ∗1 j e∗i
,
χ2k − χ1 j ψ1 j ψ2k sin−1 (Δχi ) , χ2k = , χ1 j = , li = ei , 2 ei ei Δχi 8 ei = (rr k − r j )2 , ψ1 j = n j (rr k − r j ), ψ2k = n k (rr k − r j ), 8 e∗i = (rr ∗k − r ∗j )2 , ψ∗1 j = n ∗j (rr ∗k − r ∗j ), ψ∗2k = n ∗k (rr ∗k − r ∗j ).
Δχi =
(23.50) (23.51) (23.52) (23.53)
Natural strains calculated by formula (23.49) determine constant membrane strain state within the triangle.
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341
It should be noted that relations (23.49) and (23.51)4 contain uncertainty of the form (0/0) for flat or nearly flat elements. In this case, calculations are performed using Taylor series expansions [12]
23.9.2 Curvature Changes and Transverse Shear Strains To obtain approximations of the curvature changes and transverse shear strains, we consider the bending problem of a simply supported composite beam subjected to end rotations (Fig. 23.2). The differential equations and boundary conditions governing bending of the shear deformable beam oriented along the ith side of the triangle are written as Di
% d 2 ϑi dwi & − C i ϑi + = 0, 2 dsi dsi ϑi = ϑ1 j , ϑi = ϑ2k ,
wi = 0 wi = 0
d % dwi & ϑi + = 0, dsi dsi for si = 0, for si = li .
(23.54)
(23.55)
Here Ci is the transverse shear rigidity, wi is the deflection of the beam, ϑi is the rotation, si is the coordinate measured along the ith side, and Di is the flexural rigidity of the beam. The end rotations ϑ1 j and ϑ2k are determined by the formulas ∗ ϑmn = θmn − θmn ,
θmn = sin−1 (χmn ),
∗ θmn = sin−1 (χ∗mn ).
(23.56)
The end rotations are functions of the nodal parameters of the element, which can be seen from Eqs (23.51)–(23.53) and (23.56). To determine the flexural rigidity Di , we use the coordinate transformation of the elastic tensor upon rotation of the material axes:
Fig. 23.2 Side of the shell element as a planar curve
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S.V. Levyakov
h/2 Di =
E i (z)z2 dz,
E i (z) = d11 λ41i + d22 λ42i + (2d12 + d33 )λ21i λ22i ,
(23.57)
−h/2
where the elastic constants dmn which enter the matrix DC (see Eq. (23.31)) and λmi are the direction cosines of the ith side of the planar triangle which are calculated using the nodal material coordinates for each layer using formulas (23.2). The transverse shear rigidity which enters Eq. (23.54)1 is given by h/2 Ci = ki
gi dz,
(23.58)
−h/2
where gi is the natural component of the tensor gmn which can be calculated by template (23.18)1 and ki is the shear correction factor which takes into account nonuniform distribution of the transverse shear stress across the thickness of the beam. The shear correction factor is calculated by the following formulas [4, 17]: ⎡⎛ h/2 ⎞ h/2 ⎤ ⎟⎟⎟ ⎥⎥⎥ : ⎢⎢⎢⎢⎜⎜⎜⎜ ⎟ ⎥⎥⎥ ki = R2i ⎢⎢⎢⎢⎢⎜⎜⎜⎜⎜ Gi dz⎟⎟⎟⎟⎟ ti2 (z)G−1 dz (23.59) ⎥⎥⎥ , i ⎣⎝ ⎠ ⎦ −h/2
−h/2
h/2 Ri =
h/2 Ei (z − zNi)2 dz,
zNi =
−h/2
−h/2
⎛ h/2 ⎞−1 ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ E i z dz ⎜⎜⎜ Ei z dz⎟⎟⎟⎟⎟ , ⎝ ⎠ −h/2
z ti (z) = −
E i (ζ − zNi )2 dζ.
(23.60)
−h/2
The solution of the boundary-value problem (23.54), (23.55) yields dϑi 1 ηi1 si κi = = (ϑ2k − ϑ1 j ) + (ϑ2k + ϑ1 j ) 2 − 1 , dsi li li li γi = ϑi + where ηi1 =
3 , 1 + ηi3
dwi = ηi2 (ϑ2k + ϑ1 j ), dsi
ηi2 =
1 ηi1 ηi3 , 6
ηi3 = 12
(23.61) (23.62)
Di . Ci l2i
(23.63)
The first, constant term in (23.61) describes pure bending of the beam for arbitrarily large rotations ϑ2k and ϑ1 j . The second term in (23.61) takes into account transverse shear forces which occur due to “unbalanced” end rotations ϑ2k and ϑ1 j . This antisymmetric mode of bending is taken into account in a linear approximation in view of assumption (c) in Sect. 23.2. Relations (23.62) and (23.63) imply that
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343
γi → 0 as Ci → ∞. Hence, the convergence to the Kirchhoff–Love shell is achieved and shear locking is automatically eliminated. To interpolate the curvature change (23.61) into the triangular element, we recall the following relations which hold at the edge opposite to node i: Li = const,
Lj = 1 −
si , li
Lk =
si . li
(23.64)
It follows that
1 li (1 + Lk − L j ). 2 Substitution of (23.65) into (23.61) yields si =
κi =
1 ηi1 (ϑ2k − ϑ1 j ) + (ϑ2k + ϑ1 j )(Lk − L j ). li li
(23.65)
(23.66)
This expression shows that κi is constant in the direction of the median drawn from the ith vertex to the opposite side of the triangle.
23.10 Strain Energy of Triangular Shell Element Inserting (23.49), (23.61), and (23.62) into (23.48), we obtain Π=
1 T 1 1 1 ε K ε ε + (εεT K εϑ ϑ + ϑT K Tεϑε ) + ϑ T K ϑ ϑ + C TΓ Γ , 2 2 2 2 ε = {ε1 ε2 ε3 }T , ϑ = {ϑ23 ϑ12 ϑ21 ϑ13 ϑ22 ϑ11 }T , D N C dV, K ε = D N dV, K εϑ = zD V
z2C T D N C dV, V
(23.68) (23.69)
V
Kϑ =
(23.67)
⎛ ⎞ ⎜⎜⎜ ⎟⎟⎟ ⎜ ⎟ C TΓ = ⎜⎜⎜⎜ G T dV ⎟⎟⎟⎟ H Δ k , ⎝ ⎠
⎡ ⎤ ⎢⎢⎢ k1 0 0 ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢ k = ⎢⎢⎢⎢ 0 k2 0 ⎥⎥⎥⎥ . ⎢⎢⎣ ⎥⎥⎦ 0 0 k3
(23.70)
V
(23.71)
Here the shear correction factors ki are calculated by formulas (23.59) and (23.60) and C is a 3 × 6 matrix whose nonzero entries are given by C11 = [1 + η11(L3 − L2 )]l−1 1 ,
C12 = −[1 + η11(L2 − L3 )]l−1 1 ,
C23 = [1 + η21(L1 − L3 )]l−1 2 ,
C24 = −[1 + η21(L3 − L1 )]l−1 2 ,
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C35 = [1 + η31(L2 − L1 )]l−1 3 ,
C36 = −[1 + η31(L1 − L2 )]l−1 3 .
(23.72)
Assuming that elastic properties are constant over the area of the element, we write (23.69) and (23.70) as ⎛ h/2 ⎞ h/2 ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ D N dz⎟⎟⎟⎟ C F , Kε = F D N dz, K εϑ = F ⎜⎜⎜ zD (23.73) ⎟⎠ ⎝ −h/2
Kϑ = F
−h/2
⎛ h/2 ⎞ ⎜⎜⎜ ⎟⎟⎟ ⎜ ⎟ C T ⎜⎜⎜⎜⎜ z2 D N dz⎟⎟⎟⎟⎟ C dF, ⎝ ⎠ −h/2
⎛ h/2 ⎞ ⎜⎜⎜ ⎟⎟⎟ ⎜ ⎟ C TΓ = F ⎜⎜⎜⎜⎜ G T dz⎟⎟⎟⎟⎟ H Δ k , ⎝ ⎠
(23.74)
−h/2
⎡ ⎤ ⎢⎢⎢ l−1 −l−1 0 0 0 0 ⎥⎥⎥ 1 ⎢⎢⎢ 1 ⎥⎥⎥ −1 0 ⎥⎥⎥ . C F = ⎢⎢⎢⎢ 0 0 l−1 −l 0 ⎥⎥⎥ 2 2 ⎢⎢⎣ −1 ⎦ 0 0 0 0 l−1 −l 3 3
(23.75)
The integral over the area of the element in the expression for K ϑ is calculated exactly using three Gaussian points. Using Eq. (23.62) and relation (23.36)2 , we rewrite the last term in (23.67) as C TΓ Γ = ϑT K Γ ϑ,
(23.76)
where ⎡ ⎤ ⎢⎢⎢ CΓ1 0 0 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ M = ⎢⎢⎢⎢ 0 CΓ2 0 ⎥⎥⎥⎥ , ⎢⎢⎣ ⎥⎥⎦ 0 0 CΓ3
KΓ = WT MW , ⎡ ⎤ ⎢⎢⎢ η12 η12 0 0 0 0 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ W = ⎢⎢⎢⎢ 0 0 η22 η22 0 0 ⎥⎥⎥⎥ . ⎢⎢⎣ ⎥⎥⎦ 0 0 0 0 η32 η32
In view of (23.76), the strain energy (23.67) becomes ⎡ ⎤ ⎢⎢⎢ K ε ⎥⎥⎥ K 1 T εϑ ⎥⎥⎥ , Π = u (1) K u (1) , K = ⎢⎢⎢⎣ T 2 K εϑ K ϑ + K Γ ⎦ uT(1) = {ε1 ε2 ε3 ϑ23 ϑ12 ϑ21 ϑ13 ϑ22 ϑ11 } .
(23.77)
(23.78) (23.79)
where the components of u (1) are given by (23.49) and (23.56).
23.11 Variations of the Strain Energy and Algorithm of Solution To formulate an iterative method for determining equilibrium states of the shell, it is necessary to evaluate the first and second variations of the total energy. Since the finite-element relations considered above are highly nonlinear, straightforward
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derivation of the variations leads to extremely cumbersome analytical expressions. An effective technique for evaluating the variations is based on the chain rule where three “levels” of varied parameters are introduced. The first level is represented by the vector (23.79) and the other two levels are represented by the vectors 3 4T u (2) = e∗1 e∗2 e∗3 ψ∗23 ψ∗12 ψ∗21 ψ∗13 ψ∗22 ψ∗11 , 3 4T u (3) = q T1 q T2 q T3 ,
3 4T q i = x∗1i x∗2i x∗3i ω1i ω2i
(i = 1, 2, 3).
(23.80) (23.81)
Variations in the vector u (3) are the degrees of freedom of the element. For each level, the first and second variations of the strain energy of the shell finite element can be written in terms of variations of the components of vectors (23.79), (23.80), and (23.81): δΠ = δuuT(m) g (m) ,
δ2 Π = δuuT(m) H (m) δuu(m) ,
(23.82)
where g (m) and H (m) are the gradient and the Hessian matrix corresponding to the mth variation level, respectively. The desired values of g = g (3) and H = H (3) are computed by the recursive formulas (summation over s): g(m+1) = u(m) g (m) ,
H (m+1) = u (m) H (m) uT (m) + g(m)s u(m)s
(m = 1, 2; s = 1, . . ., 9). (23.83) Here g(m)s are the components of the vector g(m) and u (m) and u (m)s are the matrices composed of the first and second partial derivatives of the components of the mth level with respect to the components of the (m + 1)th variation level, respectively. Formulas for calculating the matrices u (m) and u (m)s are given by Kuznetsov and Levyakov [12]. The computation process (23.83) begins for the initial values g (1) = K u (1) ,
H (1) = K .
(23.84)
Once the gradient g = g (3) and Hessian matrix H = H (3) have been calculated for individual elements, they are assembled into global vectors and matrices of the finiteelement assemblage using a standard technique [5]. To find a deformed state of the shell as a finite-element assemblage, we use the arc-length control method [19]. For a certain loading step, the Newton–Raphson system of equations has the form H p−1 δqq p + w p−1 δλ p + g p−1 = 0,
(23.85)
where H and g are total Hessian matrix and gradient of the assemblage, respectively, w is the vector composed of the derivatives of the form ∂2 W/∂qi ∂λ, W is the total potential energy of the discrete shell model, λ is the loading parameter, δqq p is the unknown vector of the increments in the generalized coordinates and the superscript p denotes the iteration number.
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23.12 Numerical Testing Below, we give results of numerical testing of the proposed finite element. The verification problems are chosen to test the capability of the finite element to adequately describe the transverse shear effects in the cases of linear and geometrically nonlinear deformations. Special attention is given to very thin plates and shells to show that the finite-element solution converges to the solutions based on the Kirchhoff– Love assumptions. The accuracy and mesh convergence of the numerical solutions are studied.
23.12.1 Linear Analysis of Orthotropic Square Plates Under Uniformly Distributed Transverse Load To study shear locking free behavior of the finite element proposed, we consider an orthotropic single-layer square plate of side length L = 0.305 m and thickness h subjected to uniformly distributed load q. The material characteristics are E 11 = 2.068 · 1010 Pa, E 22 = 0.8825 · 1010 Pa, G 12 = 0.2551 · 1010 Pa, and ν12 = 0.32. The material axes are assumed to be directed along the plate sides. Owing to symmetry of the problem, a quarter of the plate was discretized using mesh types shown in Fig. 23.3. Table 23.1 lists the dimensionless values of the central deflections of very thin and moderately thick plates. The present solution converges rapidly with the number of finite elements and agrees with the data available in the literature. For thin plate where L/h = 104 , the present solution is in excellent agreement with the value of central deflection wc = 7.492qL4/(D11 h3 ) calculated by the analytical formulas of Timoshenko and Woinowsky-Krieger [16]. For thick plate with the spanto-thickness ratio L/h = 8.696, good agreement is observed with the finite-element solution wc = 8.375qL4/(D11 h3 ) obtained by Dau et al. [7].
Fig. 23.3 Square plate and mesh types
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Table 23.1 Central deflection of a simply supported orthotropic plate subjected to uniform loading wc D11 h3 /(qL4 )
Mesh
Mesh type
2×2
A B C
7.259 7.299 7.233
8.086 8.073 8.058
4×4
A B C
7.436 7.443 7.430
8.315 8.302 8.302
8×8
A B C
7.475 7.476 7.474
8.359 8.354 8.354
16 × 16
A B C
7.484 7.485 7.484
8.368 8.367 8.367
32 × 32
A B C
7.487 7.487 7.487
8.370 8.370 8.370
L/h
= 104
L/h = 8.696
23.12.2 Linear Analysis of Laminated Square Plates Under Uniformly Distributed Transverse Load A two-layer angle-ply square plate with clamped edges is subjected to uniformly distributed transverse load q. Each layer is made of an orthotropic material with the following characteristics [20]: E11 = 4 · 107 , E 22 = 106 , ν12 = 0.25, and G 12 = G 23 = G13 = 5 · 105. The plate dimensions are L = 20 and h = 0.02. It is worth noting that since the plate is very thin (L/h = 1000), transverse shear effects are negligible. We study the effect of fiber orientation on the laminate deflection using union-jack meshes. Due to asymmetry of deformation, the entire plate is discretized. Table 23.2 illustrates mesh convergence for the central deflections wC for various fiber angles. One can see that the convergence rate somewhat decreases as the angle of fiber orientation increases. Calculation results are compared with the results of Zhang and Kim [20] obtained by triangular element with 18 degrees of freedom.
23.12.3 Pure Bending of a Plate A cantilever plate of length l = 1 m, width b = 1 m, and thickness h = 0.001 m is loaded by bending moment M. This is a popular problem for testing the ability of nonlinear finite elements to reproduce behavior of flexible structures under finite rotations and zero membrane strains. To investigate nonlinear capabilities of the proposed finite element, we consider simple two- and eight-element models shown in Fig. 23.4. The material of the plate is assumed to be isotropic with Young’s
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Table 23.2 Central deflection of a two-layer fully clamped plate under uniform pressure for various fiber angles 1000
Fiber angle
E22 h3 wc qL4
4×4
8×8
16 × 16
32 × 32
±5◦
a) 1.1754 b) 1.242
1.0775 1.098
1.0526
1.0464
±15◦
a) 1.7617 b) 1.793
1.9493 1.950
1.9789
1.9863
±25◦
a) 1.8964 b) 2.148
2.4524 2.472
2.5531
2.5785
±35◦
a) 1.8282 b) 2.335
2.6652 2.729
2.8280
2.8696
a) 1.7879 2.7244 2.9109 2.95887 b) 2.394 2.808 Remark: a) present shell FE; b) plate element LDT18 of Zhang and Kim [20]
±45◦
Fig. 23.4 Plate under pure bending and mesh types
modulus E = 2 · 1011 Pa and Poisson’s ratio ν = 0. The load-displacement curves are compared with the exact theoretical solution in Fig. 23.5. One can see that even the two-element model with 10 degrees of freedom (mesh A in Fig. 23.4) provides accurate solution for finite bending of the plate where the tip rotation reaches approximately 57◦ for the load parameter M/D < 1, where D = Eh3 /12. For higher loads, however, nonsymmetric deformation of the two-element model becomes pronounced because of coarse mesh and divergence of the solution occurs when the tip rotation reaches 97◦ for M/D = 1.7. The numerical solution obtained by the eight-element model (mesh B in Fig. 23.4) is in excellent agreement with theoretical results for up to 360 degrees rotations where the plate is rolled up into a full circular cylinder.
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Fig. 23.5 Moment versus tip displacements for the plate subjected to pure bending
23.12.4 Nonlinear Analysis of Laminated Square Plates Under Uniformly Distributed Transverse Load A simply supported four-layer symmetric cross-ply square plate is subjected to uniformly distributed load q. Material properties are characterized by the following ratios [20]: E11 /E 22 = 25, G12 /E22 = 0.5, G 23 /E22 = 0.2, ν12 = 0.25, and G 13 /G12 = 1, where E 11 = 172.3689 · 109 Pa. Due to symmetry, a quarter of the plate is modeled by union jack meshes. The boundary conditions for the model are such that all displacements and rotations about the vector normal to the plate edge are set equal to zero, the only nonzero rotation being about the edge. Tables 23.3 and 23.4 list the central deflections as functions of the load for thin and thick plates, respectively. The solution converges rapidly with the mesh refinement. For a coarse 2 × 2 mesh, the error in calculating the central deflection does not exceed 5%. The results are in agreement with the data obtained by LDT18 triangular element of Zhang and Kim [20] using 4 × 4 mesh but somewhat softer.
23.12.5 Snap-through of a Hinged Cylindrical Laminated Roof Figure 23.6 shows a shallow cylindrical three-layer roof subjected to a central force. Two longitudinal edges of the shell are restrained against all translations and the other two edges are free. The data are as follows [14]: R = 2540, L = 254, θ = 0.1, h = 6.35, E11 = 3300, E 22 = 1100, G 12 = 660, ν12 = 0.25, and G13 = G23 = 660. Two laminate sequences are considered: [0◦ /90◦ /0◦ ] and [90◦ /0◦/90◦ ]. A ply is of
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Table 23.3 Central defection wc /h for square simply supported [0◦ /90◦ /90◦ /0◦ ] plate under uniformly distributed load (L/h = 40) qL4 E22 h4
Zhang&Kim [20]
50
Present solution 2×2
4×4
8×8
0.2829
0.2788
0.2902
0.2933
100
0.4513
0.4443
0.4597
0.4638
150
0.5678
0.5567
0.5746
0.5790
200
0.6577
0.6430
0.6626
0.6674
250
0.7317
0.7137
0.7347
0.7398
Table 23.4 Central defection wc /h for square simply supported [0◦ /90◦ /90◦ /0◦ ] plate under uniformly distributed load (L/h = 10) qL4 E22 h4
Zhang&Kim [20]
50
Present solution 2×2
4×4
8×8
0.3317
0.3537
0.3632
0.3657
100
0.4922
0.5088
0.5198
0.5225
150
0.6002
0.6113
0.6232
0.6263
200
0.6834
0.6904
0.7030
0.7062
250
0.7521
0.7559
0.7691
0.7724
0◦ if its fibers are parallel to the longitudinal direction of the shell. All plies in the same laminate are equal in thickness. As in the most studies of this popular problem available in the literature, we consider doubly symmetric deformation. Owing to symmetry, one-quarter of the roof is modeled using an 8 × 8 union-jack mesh. Figures 23.7 and 23.8 show the load versus the deflections of two points A and B. One can see that the present solution is in excellent agreement with the benchmark solution of Sze et al. [14] shown by dark circles.
Fig. 23.6 Hinged cylindrical roof subjected to a central vertical load
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Fig. 23.7 Load versus deflection for the cylindrical roof: laminate [90◦ /0◦ /90◦ ]
Fig. 23.8 Load versus deflection for the cylindrical roof: laminate [0◦ /90◦ /0◦ ]
23.12.6 Circular Ring Pinched by Four Loads To demonstrate the capabilities of the finite element in the presence of arbitrarily large spatial displacements and rotations, we consider a thin circular ring of radius R = 0.05 m pinched by four radial loads P. The cross section of the ring is rectangular with width b = 0.01 m and wall thickness h = 0.0005 m. The material is isotropic with Young’s modulus E = 2.0 · 1011 Pa and Poisson’s ratio ν = 0. The loaded nodes are allowed to displace only in the radial direction, which implies that rigid-body
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Fig. 23.9 Load versus radial deflection for the pinched ring
Fig. 23.10 Planar and spatial equilibrium states of the pinched ring
displacements of the ring are excluded. The ring was modeled using a union-jack mesh with two elements in the width direction and 96 elements in the circumferential direction. Figure 23.9 shows the dimensionless load versus the dimensionless deflection w/R at a loaded node. Solid curves refer to stable states of equilibrium and dashed curves to unstable. At the fundamental path, the ring assumes four-lobe symmetric configurations, which are stable only at the initial stage of loading. Upon attainment of the bifurcation point B1 , the ring buckles and becomes nearly oval but still doubly symmetric. At point B2 , new branch bifurcates from the fundamental path. This branch of unstable twisted configurations of the ring contains bifurcation points B3 and B4. At point B4 , this branch intersects a branch that refers to new planar configurations of the ring. It is interesting to note that, for P = 0, the ring is deployed into a ring of radius R/3. Some planar and spatial configurations marked in Fig. 23.9 are shown in Fig. 23.10. The spatial twisted configurations can be reproduced experimentally using a simple celluloid model compressed with the help of four threads (Fig. 23.11).
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Fig. 23.11 Photographs of the spatial equilibrium states of the celluloid model
23.13 Concluding Remarks A nonconventional approach has been applied to develop a curved triangular finite element for linear and geometrically nonlinear analysis of shear deformable composite shells. The approach is based on invariants expressed in terms of normal stresses and strains in the direction of the element edges which are referred to as natural stresses and strains. The computational benefits of this approach are (1) concise formulation of the element with algorithmic and easily programmable formulas and (2) reduction of computations owing to the fact that no local coordinates are required. The main results of this study can be summarized as follows: • explicit relations between Cartesian and natural components of the tensors have been derived; • simple and elegant formulas have been obtained for the invariants in terms of natural components of the tensors; • the strain energy of a composite shell has been written in terms of the invariants, which is essential to formulate algorithmic technique for computing finiteelement parameters; • a curved triangular finite element with 15 degrees of freedom has been developed. Numerical testing has been performed to investigate accuracy, mesh convergence, shear locking behavior, and nonlinear capabilities of the shell element. Calculation results show that the element has excellent nonlinear bending characteristics, but its shortcoming is that membrane and transverse shear strains are constant. The finite element is shear locking free and can be used in the analysis of thin and moderately thick shells undergoing large elastic displacements and rotations. Acknowledgements The author wishes to express gratitude to Dr. V. V. Kuznetsov for useful discussions and suggestions.
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References 1. Argyris, J., Tenek, L.: Linear and geometrically nonlinear bending of isotropic and multilayered composite plates by the natural mode method. Computer Methods in Applied Mechanics and Engineering 113, 207–251 (1994) 2. Argyris, J., Tenek, L.: Natural mode method: A practicable and novel approach to the global analysis of laminated composite plates and shells. Applied Mechanics Review 49, 381–399 (1996) 3. Argyris, J., Tenek, L., Olofsson, L.: TRIC: a simple but sophisticated 3-node triangular element based on 6 rigid-body and 12 straining modes for fast computational simulations of arbitrary isotropic and laminated composite shells. Computer Methods in Applied Mechanics and Engineering 145, 11–85 (1997) 4. Argyris, J.H., Dunne, P.C., Malejannakis, G.A., Schelkle, E.: A simple triangular facet shell element with applications to linear and non-linear equilibrium and elastic stability problems. Computer Methods in Applied Mechanics and Engineering 10, 371–403 (1977) 5. Bathe, K.J.: Finite Element Procedures. Prentice Hall, New Jersey (1996) 6. Bucalem, M.L., Bathe, K.J.: Finite element analysis of shell structures. Arch. Comput. Meth. Engng. 4(1), 3–61 (1997) 7. Dau, F., Polit, O., Touratier, M.: C1 plate and shell finite elements for geometrically nonlinear analysis of multilayered structures. Computers and Structures 84, 1264–1274 (2006) 8. Gal, E., Levy, R.: Geometrically nonlinear analysis of shell structures using a flat triangular shell finite element. Arch. Comput. Meth. Engng. 13(3), 331–388 (2006) 9. Kuznetsov, V.V.: Shell theory based on invariants. Journal of Applied Mechanics and Technical Physics 32(5), 779–783 (1991) 10. Kuznetsov, V.V., Levyakov, S.V.: Kinematic groups and finite elements in deformable body mechanics. Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela 3, 67–82 (1994). In Russian 11. Kuznetsov, V.V., Levyakov, S.V.: Phenomenological invariant-based finite-element model for geometrically nonlinear analysis of thin shells. Computer Methods in Applied Mechanics and Engineering 196, 4952–4964 (2007) 12. Kuznetsov, V.V., Levyakov, S.V.: Geometrically nonlinear shell finite element based on the geometry of a planar curve. Finite Elements in Analysis and Design 44, 450–461 (2008) 13. Kuznetsov, V.V., Levyakov, S.V.: Phenomenological invariants and their application to geometrically nonlinear formulation of triangular finite elements of shear deformable shells. International Journal of Solids and Structures 46, 1019–1032 (2009) 14. Sze, K.Y., Liu, X.H., Lo, S.H.: Popular benchmark problems for geometric nonlinear analysis of shells. Finite Elements in Analysis and Design 40, 1551–1569 (2004) 15. Timoshenko, S., Goodier, J.N.: Theory of Elasticity. McGraw-Hill, New York (1970) 16. Timoshenko, S., Woinowski-Krieger, S.: Theory of plates and shells, 2 edn. McGraw-Hill Inc., New York (1959) 17. Vlachoutsis, S.: Shear correction factors for plates and shells. International Journal for Numerical Methods in Engineering 33, 1537–1552 (1992) 18. Yang, H.T.Y., Saigal, S., Masud, A., Kapania, R.K.: A survey of recent shell finite elements. International Journal for Numerical Methods in Engineering 47, 101–127 (2000) 19. Yang, Y.B., Shieh, M.S.: Solution method for nonlinear problems with multiple critical points. AIAA Journal 28, 2110–2116 (1990) 20. Zhang, Y.X., Kim, K.S.: A simple displacement-based 3-node triangular element for linear and geometrically nonlinear analysis of laminated composite plates. Computer Methods in Applied Mechanics and Engineering 194, 4607–4632 (2005) 21. Zienkiewicz, O.C., Taylor, R.L.: The finite element method. Butterworth, Oxford (2000)
Chapter 24
Consistency Issues in Shell Elements for Geometrically Nonlinear Problems Teodoro Merlini and Marco Morandini
Abstract Some singular concepts and non-standard practices in the FEM solution of geometrically nonlinear shell problems are highlighted and discussed. In particular, four issues are addressed. (i) The question of the drilling rotation: a shell is essentially a non-polar medium in its tangent plane, so the drilling rotation is a redundant d.o.f. to be defined by an extra stress field, and the latter ought to hold as a primary unknown field of the surface mechanics. It is shown that a proper constitutive characterization and a sound variational formulation lead to a full micropolar setting of the shell mechanics with a true three-parametric rotation tensor. (ii) The interpolation of the orientation field on the shell surface. It is shown that an interpolation scheme firmly abiding by the rules of the SO(3) group leads naturally to frame-invariant and path-independent finite elements. (iii) The linearization of the virtual functional. Again, an approach fully complying with the special orthogonal group allows an easy and correct resolution of the mixed virtual-incremental variation variables that issue in nonlinear variational formulations involving finite rotations. (iv) The question of a good discrete representation of curved surface geometries. It is shown that a pole-based kinematics built on an integral orthogonal oriento-position field leads to a fair approximation of curved geometries and allows to build low-order finite elements that are naturally locking-free. Keywords Micropolar variational mechanics · Non-polar finite elasticity · Drilling rotations · Finite rotations and rototranslations · Multiplicative interpolation of rotations · Dual numbers and tensors · Helicoidal modeling · Nonlinear shell elements
T. Merlini (B) · M. Morandini Politecnico di Milano, Dipartimento di Ingegneria Aerospaziale, Via La Masa 34, 20156 Milano, Italy e-mail:
[email protected],
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 24, © Springer-Verlag Berlin Heidelberg 2011
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24.1 Introduction Consistency is an asset for successful discrete approximations of continuum mechanics problems. An early advice for consistency is found in a short note by an eminent scientist in the field [23]: ... it may be of advantage, in the approximate solution of problems by the direct methods of the calculus of variations, to satisfy moment as well as force equilibrium equations approximately only, instead of satisfying one set exactly and the other approximately.
We find this sentence by Reissner (1965) true and topical. It raises a warning against the customary axiom that the stress tensor is symmetrical (indeed, this is true for common non-polar materials in absence of external body couples). When a finite element approximation takes this assumption implicitly, it is fulfilling exactly the equilibrium of the moments while is solving the forces equilibrium in weak form. Moreover, this consistency suggestion, advanced by Reissner in the years the first linear FEM codes were appearing, explicitly stresses the concern of treating consistently the whole equilibrium set—hence the relevant work-conjugate displacement and rotation fields—right when passing from the continuum variational mechanics to its numerically discrete approximations. Reissner himself proposed to apply such concept “to the derivation of approximate stress-strain relations for thin elastic shells”, being aware that any proposed through-the-thickness kinematics is the first discrete approximation in a solid shell element. The advice of Reissner was listened by several people that wrote important papers in the Nineties, e.g. [9, 24, 26, 27], often concerning shell mechanics. In those years, our approach to shell mechanics began by addressing first the 3D solid mechanics to get acquainted with the micropolar description. A constitutive characterization of the common non-polar medium was established and the relevant variational principles including the rotation field and a workless stress field were settled and exploited to build a first successful nonlinear solid element [13], so following Reissner’s advice. Then, in order to achieve a consistent description of discrete surface elements, a new modeling of the continuum was invented, where the particle position and orientation fields are coupled together. The geometrical representation of this so-called helicoidal modeling became affordable after discovering the dual tensor algebra [1]. The 3D variational principles were rewritten, a consistent interpolation established and a new solid element with nodal oriento-positions developed and successfully tested [15–17]. With the proper tools at hand, the computational shell mechanics was then addressed. Two levels of approximation are concerned in such an extreme geometry for a solid, a former one across the thickness [19] and a latter one on the reference surface [20]. The 3D variational mechanics developed for the non-polar medium is exploited in the former approximation, whose parameters include 2D strains and the parameters of the workless stress field. The outcome of this first step are the 2D constitutive equation and the relevant tangent map, to be used in the next step. Then, the latter approximation is accomplished using the interpolation developed in the 3D case and produces a solid shell element with four nodal oriento-positions
24 Consistency Issues in Shell Elements for Geometrically Nonlinear Problems
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and one workless stress parameter. In either approximations we consistently exploit the helicoidal modeling [21]. The aim of this contribution is to highlight and discuss a few singular concepts and non-standard practices in the nonlinear shell element formulations, that have been elaborated along with the course of our research work in this area. They concern: in Sect. 24.2, the variational framework that yields a thorough micropolar setting of the material surface mechanics and consistently includes the drilling rotation; in Sect. 24.3, the consistent approximation of the angular part of the kinematic field on a surface element by an objective and path-independent interpolation scheme; in Sect. 24.4, the consistent linearization of the variational functionals to carry out before the discrete approximation process; in Sect. 24.5, the consistent modeling of the displacement field on curved surfaces by the proposed integral kinematics. As far as concerns the notation, boldface characters (aa, A , α , ...) are used for 1st and 2nd -order tensors, calligraphic bold capitals (A, ...) for 3rd -order tensors and blackboard bold capitals (A, ...) for 4th -order tensors. A superscript ( )S means either a symmetric tensor or the symmetric part of a tensor. The default tensor notation is index-free, but when necessary Latin indexes are used for the range 1–3 and Greek indexes for the range 1–2. The Einsteinian rule of implicit summation over repeated indexes is understood.
24.2 Thorough Micropolar Setting of the Surface Mechanics It should be noted that shells, unlike beams, are hybrid structured solids: in fact a shell, though capable of withstanding bending and torquing, is essentially a nonpolar medium in its tangent plane. So the pretense of withstanding elastically drilling rotations seems at least questionable. Rather, the drilling rotation is a redundant degree-of-freedom whose definition would entail an extra stress field. Our approach to achieve a complete and consistent micropolar setting of the 2D shell mechanics descends from the 3D solid mechanics formulated in the context of a full micropolar description.
24.2.1 Constitutive Characterization of the 3D Non-Polar Medium The characterization of the 3D non-polar medium, as proposed in [13] and restated in [17], is summarized here for the reader’s convenience. Start from considering first a hyperelastic micropolar medium, which features a strain energy w function of two strain tensor parameters [12], namely a linear strain ε and an angular strain β ; these parameters are often referred to as stretch or extension tensor and wryness or distortion tensor, respectively. The derivatives of the strain energy w(εε, β ) with respect to the strain parameters define the stress parameters Tˆ (εε , β ) = w/εε ˆ (εε, β ) = w/ββ ; such relations represent the (direct) hyperelastic constitutive and M
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ˆ (εε, β ) equations of the micropolar medium. Inversion of functions Tˆ (εε, β ) and M ˆ ) and β (Tˆ , M ˆ ) and allows to introduce gives the inverse constitutive equations ε (Tˆ , M ˆ ) by means of a Legendre transform such the complementary-energy function v(Tˆ , M ˆ ) = εε, Tˆ + ββ, M ˆ , with , the notation of the scalar product. that w(εε , β ) + v(Tˆ , M ˆ are all working parameters for a micropolar Strains ε and β and stresses Tˆ and M medium. So we sketch: micropolar
w(εε, β ) ˆ) v(Tˆ , M
working params
ε and β ˆ Tˆ and M
Then, consider the existence of a particular micropolar medium whose strain energy w does not depend on the angular strain β . This constitutive postulate allows ˆ = w/ββ . We refer to such a medium as a pseudoto discard the couple-stress M polar medium. In the following sketch relevant to the pseudo-polar case, the linear strain and stress parameters have been split for convenience into their symmetric S and skew-symmetric parts, ε = ε S + (ax ε )× and Tˆ = Tˆ + τˆ ×,1 where ax() means the extraction of the axial vector of a tensor and τˆ denotes the axial vector of tensor Tˆ . So we write: pseudo-polar
εS
w (εε , ax ε , S β) S v (Tˆ , τˆ , )
working params
workless params
εS
ε and ax ε S Tˆ and τˆ
vanishing params
β ˆ →0 M
Finally, consider the existence of a particular pseudo-polar medium whose complementary energy v does not depend on the skew-symmetric part τˆ × of the stress Tˆ . This further constitutive postulate allows to discard vector ax ε = 12 v/τˆ and to S retain the symmetric part of the inverse constitutive equation as ε S (Tˆ ) = v S . /Tˆ
S ε S (Tˆ )
S
(εε S )
Inversion of function for the direct constitutive equation Tˆ allows to recover the strain-energy function w(εεS ) by means of a Legendre transform such S S S that w(εε S ) + v(Tˆ ) = εεS , Tˆ , and to write Tˆ (εεS ) = w/εεS . The strain energy of this medium depends on a symmetric stretch tensor (6 scalar parameters) like in classical elasticity, and this medium coincides with the common non-polar medium: non-polar
εS
w (εε , , S β) S v (Tˆ , τAˆ , )
working params
workless params
vanishing params
εS
β
ax ε → 0 ˆ →0 M
ε S Tˆ
τˆ
The foregoing constitutive characterization of the non-polar medium has important consequences in 3D continuum mechanics. With reference again to micropolar mechanics, consider the notation introduced in Table 24.1. The strain and stress 1
The notation v × denotes a skew-symmetric tensor and v is its axial vector. Tensor v × transforms w. a vector w into the cross-product vector v ×w
24 Consistency Issues in Shell Elements for Geometrically Nonlinear Problems
359
Table 24.1 Micropolar mechanical quantities and equations linear fields
angular fields
Geometry in the reference configuration: x , particle’s position vector
α , particle’s orientation tensor k a j , three angular curvature vectors
g j = x , j , three base vectors I = gj
⊗ g j,
the identity tensor
αT k a j )×) (from α T α , j = (α j k a = k a j ⊗ g , the angular curvature tensor
dxx dxx
ν , the boundary outward normal vector Geometry in the current configuration: x , particle’s position vector g j
=
x,
j,
α , particle’s orientation tensor k a j , three angular curvature vectors
three base vectors
F = g j ⊗ g j , the deformation gradient Particle’s kinematics: u = x − x , the displacement
dxx dxx
k a
αT k a j )×) (from α T α , j = (α = k a j ⊗ g j , the angular curvature tensor
Φ = α α T , the rotation
χ = F − Φ I , the linear strain
ω a = k a − Φ k a , the angular strain
Statics: f , t , the external force densities T´ , the stressa
c , m , the external couple densities ´ , the couple-stressa M
divT´ + f = 0, body linear balance T´ ν = t, boundary linear balance a
´ + 2ax(T´ F T ) + c = 0, body angular balance div M ´ ν = m , boundary angular balance M
analogs of the 1st Piola–Kirchhoff stress tensor
parameters are related to the mechanical fields of the kinematic deformation and of the stress and couple-stress by ε = ΦTχ = ΦT F − I , Tˆ = Φ T T´ ,
β = Φ T ω a = Φ T k a − k a , ˆ = ΦT M ´; M
(24.1a) (24.1b)
refer to [22] for a comprehensive analysis. Tensor Φ T F = I + ε is usually referred to as the Cosserat deformation tensor in micropolar elasticity. In the case of non-polar medium, a first noteworthy outcome follows from the ´ = ΦM ˆ : the angular condition w/ββ = 0, hence the vanishing of the couple-stress M T ´ equilibrium equations are reduced to the algebraic form 2ax(T F ) + c = 0 and to m = 0. Therefore, the non-polar medium allows for external body couples c which are balanced by the skew-symmetric part of the analogue of the Cauchy stress tensor (det F )−1 T´ F T , but it cannot be loaded by external couples m at the boundary. A second important outcome follows from the condition v/ˆτ = 0, i.e. the vanishing of the skew-symmetric part of the strain parameter ε : the compatibility condition (24.1a)1 splits into its symmetric and skew-symmetric parts and is reduced to
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T. Merlini and M. Morandini
ΦT F − I )S , and an equation that forces to a condition on symmetric tensors, ε S = (Φ ΦT F ) = 0. The latter equazero the right-hand-side of the skew-symmetric part, ax(Φ tion represents a statement of the polar decomposition theorem of the deformation gradient and holds as a constitutive equation of the foregoing model of non-polar medium under micropolar description, as it constitutes the definition of the material rotation—a redundant d.o.f. in classical elasticity. A third important feature is the presence of a workless stress field, vector τˆ . Tensor Tˆ from (24.1b)1 is called the Biot stress tensor [2, 6] and is in general a non-symmetric tensor—apart from the case of isotropic media, for which a symmetrical Biot stress is expected in absence of external body couples [6]. However, it is commonly recognized that the strain energy of classical non-polar media depends on the six components of a symmetric strain tensor, so the stress parameter must be symmetrical. In the proposed characterization of the non-polar medium, just the S symmetric part Tˆ of the Biot stress works elastically, whereas its axial vector τˆ , hereafter referred to as the Biot-axial, must be retained as a workless stress field, as its contribution to the equilibrium may be significant. The primary unknown fields of the minimal formulation (the so-called one-field, or displacement formulation) of the non-polar medium are the displacement u , the parameter ϕ of the rotation (we assume the rotation vector as parameter, such that ϕ×)) and the Biot-axial τˆ . The problem is governed by the field equations Φ = exp(ϕ divT´ + f = 0,
2ax(T´ F T ) + c = 0,
ΦT F ) = 0, ax(Φ
(24.2)
and by the natural boundary conditions T´ ν = t and m = 0. The weak form of the problem follows by standard variational procedure and results in the internally constrained Principle of Virtual Work Πδ = Πint δ + Πextδ = 0, with the internal virtual work given by % & Πintδ = δ w + ˆτ×, ΦT F dV. (24.3) V
ΦT F
In (24.3) the contribution ˆτ ×, Φ F from the internal constraint adds to the strain energy. It should be noted, however, that this term descends in a natural way from the constitutive characterization of the non-polar medium and is not introduced as an appended constraint, as e.g. in [26].
24.2.2 A Linear Example When the current configuration is so close to the reference configuration that the deformation can be taken as infinitesimal, and linear stress-strain relations are assumed as well, a linear problem follows. Equations (24.2) of the so-called one-field formulation are reduced to the equations T S − curlττ + f = 0, divT
2ττ + c = 0,
ϕ = 0, curl u − 2ϕ
(24.4)
24 Consistency Issues in Shell Elements for Geometrically Nonlinear Problems
361
T S + τ ×)νν = t . For isotropic materials, the and to the natural boundary condition (T only symmetric stress tensor is given by T S = λ (tr ε S )II + 2μ ε S , with λ and μ the Lam´e moduli and ε S = gradS u . As an example, consider the disk of Fig. 24.1. It is restrained along the circular boundary and is loaded with body fields of tangential force f and planar couple c . Assume a centered triad I = i j ⊗ i j as absolute reference frame so to measure the position vectors as x = x j i j . With reference to a convected system of three dimensionless coordinates ξ j in the radial (0 ≤ ξ1 ≤ 1), circumferential (0 ≤ ξ2 < 2π) and axial (−1 ≤ ξ3 ≤ 1) directions, respectively, the absolute coordinates read x1 = Rξ1 cos ξ2 , x2 = Rξ1 sin ξ2 , x3 = hξ3 and the covariant base vectors are given by g 1 = R(ii1 cos ξ2 + i 2 sin ξ2 ), g 2 = Rξ1 (−ii1 sin ξ2 + i 2 cos ξ2 ) and g 3 = h i 3 . As external loads, a uniform force field and a linearly varying couple field are considered, with densities f = f0 (−ii1 sin ξ2 + i 2 cos ξ2 ),
c = ((1 − ξ1)c0 + ξ1 c1 ) i 3 ,
(24.5)
respectively. They result in an overall couple of value 43 πR2 h(R f0 + 12 c0 + c1 ) and are balanced by a tangential reaction of density t = t (−ii1 sin ξ2 + i 2 cos ξ2 ), with t = − 13 (R f0 + 12 c0 + c1 ). Equation (24.4)2 is algebraically solved for the stress-axial τ = − 12 c. Then, the problem for the displacement u is governed by the differential equation (24.4)1 together with the natural boundary condition and the boundary constraint u = 0 at ξ1 = 1. The solution of type u = v(ξ1 ) g 2 leads to the ordinary differential problem ξ1 v,11 +3v,1 +3L = 0, where L=
with v = 0 and v,1 = −L for ξ1 = 1, R f0 − 12 (c1 − c0 ) 3μ
is a dimensionless load parameter. Function v = L(1 − ξ1 ) satisfies the equations, and the rotation field is then recovered from (24.4)3 as ϕ = 12 curluu. The final result is
Fig. 24.1 Disk under volume force and couple as external loads.
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T. Merlini and M. Morandini
u = u (−ii1 sin ξ2 + i 2 cos ξ2 ),
u = LRξ1 (1 − ξ1 ),
ϕ = ϕ i3,
ϕ = L(1 − 32 ξ1 ),
τ = τii3 ,
τ=
(24.6)
− 12 ((1 − ξ1)c0 + ξ1 c1 ).
Strain and stress are recovered as T S = 2μ ε S ,
% & 1 ε S = − Lξ1 cos 2ξ2 (ii1 ⊗ i 2 + i 2 ⊗ i 1 ) + sin2ξ2 (ii2 ⊗ i 2 − i 1 ⊗ i 1 ) . 2
Plots of the solution (24.6) are shown in Fig. 24.2. In absence of body couples (left plot), the response is a tangential displacement u as expected and a rotation ϕ. The latter decreases linearly from the disk center and changes sign at ξ1 = 23 . The stress-axial τ is null everywhere. When only body couples are present, instead (right plot), the disk still deforms in general with both a tangential displacement u and a rotation ϕ but the stress-axial τ is never null. With c0 = 240 and c1 = 0 (L = 0.04) we obtain the same response as in the left plot; however τ is not null, in this case. In the particular case of uniform couple ( f0 = 0 and c1 = c0 , i.e. L = 0), displacement and rotation are null everywhere and the stress-axial is uniform. It is worth noting that we could face the same example problem from the classical displacement method, forcing the linear balance as in (24.4)1 and the angular balance by means of (24.4)2 to account for the body couples. We would obtain the same solution as in (24.6) for the unknowns u and τ, but we could never determine the rotation ϕ. However, it may be observed that such an approach would look somewhat inconsistent after considering that in a variational context the natural work-conjugate multipliers of equations (24.4)1 , (24.4)2 and (24.4)3 are virtual displacements, rotations and stress-axials, respectively. In fact, if we face the problem from a classical point of view, we are making use of just (24.4)1 and (24.4)2 whereas on) i 0i
d 0r 0d
K
i 0−Ri
K
rn)
ldi
X
rnd
l 0i
X
lii
bi l−i
dn)
−Ri
mdn)
U
mrnd
dno
di
li
U
dnd
mrn) d
−ii
dng dn 0 r
Y
dnR
ri
ia−
-
0
0 0s p
ond
−di d 0rno
i l 0i bi l−i li −Ri
iaR iab 0 l
Y
di lii
ia
ldi l
Fig. 24.2 Disk data: R = 100, h = 0.5, μ = 1000. Solution under the force alone (left plot, L = 0.04) and under the couple alone (right plot, L from 0.04 to 0.). Markers denote FE results, see Sect. 24.2.4.
24 Consistency Issues in Shell Elements for Geometrically Nonlinear Problems
363
we are retaining as unknowns the displacement and the stress-axial, work-conjugate respectively to (24.4)1 and (24.4)3 , instead. So this example stresses the consistency of the constitutive characterization of the non-polar medium sketched in Sect. 24.2.1 and of the relevant continuum mechanics based on three primary unknown fields.
24.2.3 Micropolar Surface Mechanics Shells are 2D structured solids within a 3D space. They are able to withstand outof-plane distortions by opposing couple-stresses, but if made of non-polar material they are unable, like plates, to withstand in-plane distortions. So, they behave essentially in a non-polar fashion in their tangent plane. Now, the proposed constitutive characterization of the 3D non-polar medium leads to a micropolar formulation of continuum mechanics that involves the rotation as unknown field and is able to withstand body couples. Hence, an analogous concept is expected to yield a complete micropolar formulation of the 2D shell mechanics that consistently involves a full three-parametric rotation unknown. Of course, such a formulation shall involve an extra surface-stress field in charge of defining the drilling degree-of-freedom. As usual in shell modeling, two approaches are envisaged. The direct approach tends to ignore the mechanics within the shell thickness and applies the principles of the 3D realm to formulate the intrinsic mechanics of the material surface. Formulations of this kind are often referred to as Cosserat surface models. Direct approaches to micropolar surface mechanics with unrestrained rotations are used e.g. in the early paper by Zhilin [31], endowed with a modern notation for rotational kinematics, and in the finite-element formulation by Sansour and Bednarczyk [24], based on the Kirchhoff–Love shell theory. No mention to a workless surface-stress field is found in literature except for [14], where a constitutive characterization of a nonpolar-type surface on the steps of Sect. 24.2.1 is outlined and a surface Biot-axial is consistently introduced. However, the usefulness of such an approach is questionable, as the interpretation of the surface Biot-axial itself rises new difficulties that add to the lack of constitutive laws proper to direct approaches. Alternatively, the solid shell approach is recommended. First, a 3D mechanical formulation is applied to the shell body, then the problem is reduced to a 2D model by variational means with integration over the thickness. Several solid shell approaches to micropolar surface mechanics with unrestrained rotations have appeared in literature, e.g. [7, 10, 25]. In general, solid shell approaches do not resort to a workless surface stress to define the drilling degree-of-freedom, but noteworthy exceptions are the papers by Wisniewski and Turska [27–30] and by Badur and Pietraszkiewicz [3]. In the latter paper a Kirchhoff–Love shell theory is assumed and a 2D workless stress field is introduced as Lagrange multiplier of the constraint on the symmetry of the strain. In the authors’ solid shell approach [19], the minimal formulation for non-polar materials governed by equations (24.2) in the unknowns u , ϕ and τˆ is assumed for the 3D mechanics within the thickness. A variational setting leads to the internal
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T. Merlini and M. Morandini
virtual work (24.3), and the 2D shell model is obtained after integrating over the thickness the integrand in (24.3). This reduction is performed in a finite-element-like sense and implies the choice of approximate fields for both the kinematic variables (displacement and rotation) and the Biot-axial stress vector. Different shell theories can be incorporated according to the hypotheses of the approximate kinematic field, including the classical Kirchhoff–Love and Reissner–Mindlin kinematics. In [19] a new shell theory is developed and adopted (see Sect. 24.5.2) and a linear Biot-axial field across the thickness is assumed. Of course, the parameters of the throughthe-thickness Biot-axial field hold as primary unknowns of the 2D shell mechanics besides the appropriate kinematic unknown fields. The 2D model ensues straightforwardly equipped with the shell constitutive equations, which relate the 2D strains and Biot-axial parameters to the relevant work-conjugate fields in a coupled way. Thus, a thorough nonlinear micropolar formulation is established for the material surface mechanics, including drilling rotations.
24.2.4 A Nonlinear Example The example disk of Sect. 24.2.2 is analyzed by the helicoidal shell element developed in [20]. The element geometry is a curved quadrilateral surface whose approximate configuration is interpolated among the oriento-positions of four corner nodes (see Sects 24.3 and 24.5); the formulation is based on 4 × 6 nodal kinematic unknowns and a unique Biot-axial vector parameter. The disk is modeled by an axisymmetrical mesh with 8 elements in the radial direction and 32 elements in the circular direction (the elements of the internal circle degenerate into triangles). First, a De Saint Venant–Kirchhoff material with Lam´e moduli λ = 1500 and μ = 1000 (i.e. E = 2600 and ν = 0.3) is used to match the linear analysis of Sect. 24.2.2. Force and couple densities are set at the nine quadrature points of each element to reproduce the loads (24.5). We ran two nonlinear computations (5 iterations to converge) with data f0 = 1.2, c0 = c1 = 0 and f0 = 0, c0 = 240, c1 = 120 respectively. The results are marked in Fig. 24.2 and agree well with the analytical ones. Then, a full nonlinear computation is addressed using a constitutive model of Neo-Hookean type, referred to as Pseudo-Hookean hyperelastic law in [18]. It descends from the strain-energy function 1 U ) = λ(ln det U )2 − 2μ ln detU U + 2μ tr (U U − I ), w(U 2 with U = I +εεS the symmetric Cosserat deformation tensor. A nearly incompressible material is chosen with moduli λ = 4999000 and μ = 1000, that means a Poisson ratio ν = 0.4999. A variable radial element size with a geometric progression of common ratio 1.2 allows to refine the mesh in the center zone. The target load is a drilling body couple linearly varying along the radius from c0 = 4800 to c1 = 0. A clipping of the deformed configuration is displayed in Fig. 24.3 and plots of the
24 Consistency Issues in Shell Elements for Geometrically Nonlinear Problems
365
Fig. 24.3 Nearly incompressible disk. Undeformed mesh (straight radial lines) and deformed mesh under drilling body couples at c0 = 4800, c1 = 0 Table 24.2 Nearly incompressible disk. Planar displacements and rotations (degrees) under drilling body couples at c0 = 4800, c1 = 0 Radial position
u
v
ϕ
0.00
0
0
72.81
6.06
-4.29
4.62
67.18
13.33
-8.33
9.94
57.65
22.06
-11.35
15.60
45.89
32.54
-12.56
20.76
32.41
45.10
-11.43
23.98
18.47
60.18
-8.05
23.23
4.33
78.28
-3.60
16.44
-8.43
100.00
0
0
-25.33
element thicknesses and Biot-axials in the radial direction are given in Fig. 24.4. Displacement and rotation values are listed in Table 24.2.
]
3
D
]15
5
−15
−
3
[ i −]]]
i
−
o !oooo 2]]] ]
−15
4
3
]15 315
] ]
3]
.] D] i
d]
2 −]]
Fig. 24.4 Nearly incompressible disk. Thickness and Biot-axial at c0 = 4800, c1 = 0
2
] ]
3]
K
5] o ooooos i
r]
Fig. 24.5 Nearly incompressible disk. Drill load-deflection curve
(]
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T. Merlini and M. Morandini
The load-rotation curve of Fig. 24.5 is drawn in 20 equal load steps with 5–6 Newton–Raphson iterations per step. This problem is not highly nonlinear and the target load can also be reached quickly in four load steps for a total of 45 iterations. The helicoidal shell element works without resorting to reduced integration nor other numerical expedients widely adopted in shell analysis like incompatible modes, assumed natural strains, enhanced assumed strains.
24.3 Consistent Approximation of the Element Kinematic Field It is common practice in finite-element technology to interpolate positions in the Euclidean vector space by looking for the weighted average position vector from the equation N W J (xx − x J ) = 0 . (24.7) J=1
Solution of (24.7) for normalized weights yields x = NJ=1 W J x J in closed form. This interpolation scheme allows to build the metric within the element in the reference configuration and the position vector x and the deformed metric in the current configuration. Thereby, in the linear vector space, the unknown displacement u = x − x can be straightforwardly interpolated from the nodal displacements as u=
N
WJ u J ,
(24.8)
J=1
leading to the isoparametric concept. All in all, the interpolation of the position field is a rather plain task. The interpolation of the orientation field, instead, is a demanding task. Orientations can be represented by the orthogonal tensors α = α j ⊗ i j that orient local orthonormal triads α j with respect to the self-reciprocal absolute reference frame I = i j ⊗ i j . Orientations belong to the special orthogonal group SO(3) and compose multiplicatively: the current orientation α is the reference orientation α rotated by tensor Φ , i.e. α = Φ α . Being an orthogonal tensor, the rotation is defined by three independent parameters; for simplicity, in the following discussion we rest on the vectorial ϕ×). parameterization defined by the exponential map Φ = exp (ϕ The simplest and straightest interpolation scheme is to interpolate the total nodal rotation vectors ϕ J just like nodal displacements are interpolated in (24.8), i.e. ϕ ϕ×) and recover the current = NJ=1 W J ϕ J , then build the rotation tensor Φ = exp (ϕ orientation as α = Φ α . It is immediate to verify that such an interpolation scheme is not frame-invariant [16]. Other approaches propose to interpolate the incremental rotation vector ϕ ∂ = N J=1 W J ϕ J∂ from the incremental nodal rotation vectors ϕ J∂ computed at the th ϕ∂ ×) α k . k configuration and recover the current orientation as α k+1 = exp(ϕ th The k configuration is either defined as the last converged configuration or the
24 Consistency Issues in Shell Elements for Geometrically Nonlinear Problems
367
configuration obtained so far during the iterative solution process. It is possible to design this type of interpolation schemes in such a way that no spurious strains are added for a rigid motion [11]. However, as first explicitly noted in [8], this type of interpolation is path-dependent. In fact, this type of interpolation naturally leads to an approximate numerical integration of the angular strain β k+1 = β k + ∂ββ (a leading strain in shell elements), for finite incremental rotation vectors ϕ ∂ . As a proof, one could design a complex load sequence in such a way to reach a final configuration coincident with the reference one, with null final stresses and strains; still the simulation will end with a final configuration that differs from the initial one [8, 11]. Of course, path-dependence can be an acceptable feature in computational mechanics, as a high precision level can be always achieved by raising the number of load steps, thus reducing the error implied in the angular strain integration. However, it is much preferable to avoid path-dependence altogether. These drawbacks, either path-dependence or frame-non-invariance, affect the above mentioned interpolation schemes mainly because they were designed keeping in mind a well-established scheme that works so charmingly when interpolating positions. One can indifferently interpolate the current positions (Eq. (24.7) applied to the current configuration) or the displacements (Eq. (24.8)) just because positions belong to a linear vector space, i.e. because u = x − x . With isoparametric elements, this fact introduces some ambiguity when defining the unknowns of the problem: the problem is often stated assuming as unknown the displacement from the reference configuration, while in the authors’ opinion the deformed configuration should be intended as the real unknown instead. This distinction is almost academic when interpolating positions, but is important when dealing with orientations: one cannot use the same interpolation scheme for the reference orientation field α, the current orientation field α and the rotation field Φ, because the interpolated orientation α will differ, in general, from Φ α [16] (Fig. 24.6). In this sense, interpolating orientations and rotations with the same scheme is inconsistent. uuJ ≡ x − x
α1
x1 u1
x1
ΦΦJ = α αT
x xJ uuJ
xxJ
x2 u2
Φ1
α1 x2
Φα
α αJ
α2
ΦΦJ
ααJ
Φ2
α2
Fig. 24.6 Interpolation of the configuration vs. interpolation of the change of configuration: displacement (left) and rotation (right). The notation a aa J means a as interpolated from the a J (J = 1, ... N).
Consistency can be achieved after removing definitely the pretense to interpolate rotations: build an interpolation scheme for either orientation fields and a posteriori recover the rotation as Φ = α α T . Just like Eq. (24.7) stems from the additive
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composition in the linear vector space, the sought interpolation scheme should properly account for the multiplicative composition of rotations. The analogue of (24.7) for the orientation field is thus [16] N
αα TJ ) = 0 . W J log (α
(24.9)
J=1
The implicit Eq. (24.9) does not yield in general the sought weighted average orientation α in closed form, but gives a nonlinear problem to solve numerically [16]. Then, the angular curvature is obtained as well, and the local rotation Φ = α α T and the angular strain β = Φ T k a − k a are recovered by comparing the interpolated quantities in the current and in the reference configurations. This leads to the correct equivalent of an isoparametric element for rotations: the reference and the current orientation fields (but not the rotation field) are approximated with the same interpolation scheme. The proposed interpolation is frame-invariant and path-independent at the same time. Equation (24.9) averages relative orientations (just like Eq. (24.7) averages relative distances), and this fact ensures frame invariance. The mere fact of interpolating orientations instead of rotations ensures path independence.
24.4 Consistent Linearization and Discrete Approximation Despite a large attention paid, in the finite-element literature, to linearize consistently the variational principles, a careful account of the properties of the special orthogonal group still raises some consistency issues also in this field. Usually, the linearization is performed on the previously approximated principle, however let us address the linearization of a virtual functional in its continuous form. For the present purpose, we may limit ourselves to focus on just the contribution ) to the variational principle from the strain energy, namely the term Πw δ = V δw dV in Eq. (24.3). The strain energy w(εεS ) is a function of the symmetric strain parameS ter; its first and second variations define the stress parameter Tˆ (εε S ) and the elastic S S SS SS tensor Eˆ (εεS ) from δw = δεεS : w/εεS = δεεS : Tˆ and ∂ Tˆ = Eˆ : ∂εεS . In the present minimal formulation, the strain parameter is a short name for the relevant kinematic ΦT F − I )S . Therefore, the virtual functional Πw δ and its strain to match, i.e. εS ≡ (Φ (incremental) variation ∂Πw δ can be written )
S
ΦT F ) : Tˆ dV, δ(Φ ) S SS ΦT F ) : Eˆ : ∂(Φ ΦT F ) : +∂δ(Φ ΦT F ) : Tˆ dV. ∂Πw δ = V δ(Φ Πw δ =
V
(24.10)
24 Consistency Issues in Shell Elements for Geometrically Nonlinear Problems
369
A consistent development of (24.10) leads to the following expressions [13], Πw δ =
V
⎧ ⎪ ⎪ ⎨ F : ϕδ· ⎪ δF ⎪ ⎩
⎫ ⎪ ⎪ ⎬ dV, S T ⎪ ⎭ ˆ ΦT F ) ⎪ −2ax (Φ Φ Tˆ
S
(24.11a)
⎡ ⎤⎧ ⎫ SS ⎢⎢⎢ Φ Eˆ SS Φ T ⎪ ⎪ ΦEˆ Φ T : F ×T321 ⎥⎥⎥⎥ ⎪ FT ⎪ −Φ ⎨ : ∂F ⎬ ⎢ ⎥⎥⎦ ⎪ ∂Πw δ = dV F : ϕ δ · ⎢⎢⎣ δF ⎪ SS T SS T ⎪ × × ×T321 ⎩ ⎭ ˆ ˆ F : ΦE Φ F : ΦE Φ : F −F · ϕ∂ ⎪ V ⎡ ⎤⎧ ⎫ ⎥⎥⎥ ⎪ ˆ S )×T231 ⎢⎢⎢ ⎪ ⎪ Φ FT ⎪ O (Φ T ⎨ : ∂F ⎬ ⎢ ⎥ ⎢ ⎥ + F : ϕ δ · ⎢⎣ δF ⎪ S ×T132 S T S S T ⎥⎦ ⎪ ⎪ ⎪ dV ⎩ ˆ ˆ ˆ ΦT ) ΦT F ) − I tr (Φ ΦT F ) · ϕ∂ ⎭ (Φ (Φ V ⎧ ⎫ S ⎪ ⎪ ⎪ ⎪ Φ Tˆ ⎨ ⎬ + dV, (24.11b) F : ϕ ∂δ · ⎪ ∂δF ⎪ S ⎪ T ⎩ ⎭ ˆ ΦT F ) ⎪ −2ax (Φ V
A · g j ) × ⊗ g j denotes a 3rd -order tensor of skew-symmetric nature where A × = (A built from tensor A , and ( )Txyz denotes the transposition of the dyadic legs x, y and z in the order they appear. O is the 4th -order null tensor. The development leading to (24.11) is consistent in the sense that the variations of orthogonal tensors are performed respectfully of the SO(3) environment. It is worth noting that the expressions of the rotation variations used in (24.11) are parameterization-free. Independently of any parameterization, the differential of an orthogonal tensor is characterized by a differential rotation vector, and the skewsymmetric parts of subsequent differentials are characterized by further differential rotation vectors independent of the preceding ones. Independently of any parameterization, we like to denote such characteristic vectors by the symbol of the rotation vector itself with a subscript that recalls the variation in hand. Thus [15], ΦΦ T = ϕ δ ×, ∂Φ ΦΦ T = ϕ ∂ ×, ∂δΦ ΦΦ T = ϕ ∂δ × + δΦ
( 1' ϕ∂ × ϕ δ × , (24.12) ϕ δ × ϕ ∂ × +ϕ 2
with ϕ δ , ϕ ∂ and ϕ ∂δ independent differential rotation vectors. It follows that there is no means to solve directly the second differential rotation vector ϕ ∂δ for the first ones ϕ δ and ϕ ∂ . Moreover, no approximation of the unknown fields has been introduced in (24.11) yet. So Eqs (24.11) actually belong to the continuum mechanics realm and not to the finite-element technology: the contribution Πw δ + ∂Πw δ to the linearized variational principle is provided by Eqs (24.11) independently of the discrete approximation chosen to formulate a finite element. However, the nucleuses of the residual and the tangent matrix to exploit in an iterative solution process are recognizable in Eqs (24.11a) and (24.11b), respectively; the latter nucleus comes with the separate contributions from the elastic tensor and from the pre-stress state. The discrete approximation of the unknown fields is often set as a linear function of the global (nodal) variables. In (24.11) a linear interpolation is appropriate when computing the deformation gradient F that belongs to the Euclidean vector space,
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T. Merlini and M. Morandini
but not for the rotation field that is subject to the SO(3) rules, as discussed in Sect. 24.3. To be used within (24.11), a nonlinear interpolation must be linearized itself in turn. If we resort to the scheme governed by Eq. (24.9), this linearization process yields the interpolation formulas ϕδ =
N
Y J · ϕ Jδ , ϕ ∂ =
J=1
N K=1
Y K · ϕ K∂ , ϕ ∂δ =
N J=1
Y J · ϕ J∂δ +
N N
Y JK : ϕ Jδ ⊗ ϕ K∂ ,
J=1 K=1
(24.13) as shown in [16] (where the curvature variation variables are also given). In (24.13) the local variation variables are linear function of the global variation variables, and it is seen that the mixed differential rotation vector ϕ ∂δ is controlled by the virtual and incremental nodal rotation vectors ϕ Jδ and ϕ K∂ and by the mixed virtualincremental nodal rotation vectors ϕ J∂δ . Application of Eqs (24.13) to Eqs (24.11), together with appropriate linear interF , ∂F F , ∂δF F , and integration over the element domain gives polation formulas for δF the contribution to the discrete linearized variational principle in the form ⎤ ⎧ ⎧ ⎫T ⎧ ⎫ ⎧ ⎫T ⎡ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ δuu J ⎪ ⎬ ⎪ ⎨ FuJ ⎪ ⎬ ⎪ ⎨ δuu J ⎪ ⎬ ⎢⎢⎢⎢ K uu JK K u ϕ JK ⎥⎥⎥⎥ ⎪ ⎨ ∂uu K ⎪ ⎬ ⎥⎥⎦ · ⎪ Πw δ + ∂Πw δ = ⎪ ·⎪ +⎪ · ⎢⎢⎣ T + ϕ J∂δ · F ϕ J , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ϕ ⎭ ⎩ F ϕJ ⎭ ⎩ ϕ ⎭ ⎩ ⎭ K K ϕ ϕϕ JK Jδ Jδ K∂ uϕ JK (24.14) that involves the virtual and incremental variations of the nodal displacements u J . It is seen that no mixed virtual-incremental displacement variations are there, as we are linearizing Πw δ with respect to the discrete unknowns u J , not the independent F = 0.) The mixed virtual-incremental rodiscrete field δuu J . (That also means ∂δF tation vectors ϕ J∂δ are there instead, and must be solved for ϕ Jδ and ϕ J∂ before assembling the problem matrix. The resolution of ϕ J∂δ is possible at each single node, where the rotation vector ϕ J is an actually free variable. As explained in [16, 17], using the differential map ϕ J , the resolution formula of the rotation and discarding of course the term with ∂δϕ is easily obtained, ϕJ + Γ J Γ −123 : ϕ Jδ ⊗ ϕ J∂ , ϕ J∂δ = Γ J · ∂δϕ IIIJ −123
(24.15)
ϕ J ) and Γ IIIJ (ϕ ϕ J ) are the first and second differential mapping tensors2 where Γ J (ϕ ϕ J ×). Note associated with the exponential map of the total nodal rotation Φ J = exp (ϕ ϕ that the local analogue of (24.15) gives the expression for the mixed variation ∂δϕ −123 −1 = Γ ϕ ∂δ − Γ III : ϕ δ ⊗ ϕ ∂ , which is not null, of course. Using (24.15), Eq. (24.14) becomes a linear function of just the virtual and incremental variation variables, with a symmetrical contribution to the tangent matrix. It is worth noting that an attempt to arrive at (24.14) by first discretizing Eq. (24.11a) by means of Eq. (24.13) and then linearizing the result obtained, would be a quite hard and very insidious task. Also note that the foregoing discussion 2
−123
Each Γ IIIJ is a 3rd -order tensor symmetric in the rightmost dyadic legs 2 and 3.
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applies as is to 3D variational mechanics. In the case of shell elements, the same concepts should apply first to the integration across the thickness to compute the shell constitutive equations and the relevant tangent map, then to the integration on the element surface. Note that the latter integration will involve also the angular curvatures and the relevant variation variables, a fact that in turn involves a third variation of the rotation tensor [15] to be appended to Eqs (24.12) and unavoidably complicates things further.
24.5 Consistent Modeling of the Displacement Field on Curved Surfaces Consider a low-order rectangular shell element with four corner nodes equipped with rotational d.o.f.’s. In the reference configuration the element is flat, with two vectors of the nodal triads aligned with the element sides and the third vector normal to the element plane. At a certain converged solution, the current configuration is represented by the corner nodes still lying on a plane with orientations as shown if Fig. 24.7 (left). Using the interpolation scheme of Eqs (24.7) and (24.9) with bilinear
Fig. 24.7 Shell element kinematic interpolation. Uncoupled interpolation of positions and orientations (left) vs. helicoidal interpolation (right).
weights, an internal triad is positioned and oriented as in the left picture. However, the nodal orientations made it clearly understood that the actual configuration of this piece of surface would be closer to the right picture than to the left one. Note that the interpolated orientation is identical in both pictures, whereas the interpolated position is different. In the picture on the right, the interpolated position is “driven” by the nodal orientations. Is there another consistency issue? Let us explain how we drew the picture on the right. We need dual tensors: a sort of complex tensors that obey the algebra of dual numbers [1]. A dual tensor (like a complex number) is composed as the sum of a primal part and a dual part, the latter being multiplied by the dual unit E. The dual unit is a nilpotent element such that E2 = E3 = . . . = 0 in any computations. Consider for instance the arm operator X = I + E x ×, a dual tensor made of the identity as primal part and of the cross product
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of the position vector as dual part. X is a special dual tensor, in fact it is orthogonal, X −1 = X T . A way of describing by a single geometric entity the position and the orientation of a frame is by means of the so-called oriento-position tensor [15], a dual tensor A = X α = α + E x × α made of the orientation tensor itself as primal part and of its moment with respect to the origin as dual part. Note that A is an orthogonal dual tensor itself and represents the rototranslation of the frame from the absolute reference frame I . When a shell particle moves from its reference oriento-position A to the current oriento-position A , it undergoes a rototranslation H = A A T = X Φ X T , another orthogonal dual tensor. Thus, rototranslations compose multiplicatively just like rotations do. Therefore, an interpolated oriento-position on the element domain can be built in a very consistent way by a straightforward extension of the interpolation equation (24.9) to the dual case [16], N
A A TJ ) = 0 . W J log (A
(24.16)
J=1
Note that positions and orientations are coupled together in this interpolation scheme, so the nodal orientations do affect the interpolated position. The picture on the right of Fig. 24.7 was drawn this way.
24.5.1 Helicoidal Parameterization and Helicoidal Modeling A consistent modeling of the position field on curved surfaces (Fig. 24.7) was the motivation to further deepen the representation of motion of the particles of a deforming structured solid in discrete approximations. The rototranslation is an orthogonal dual tensor depending on six parameters. A widely used parameterization decomposes the rototranslation into two separate and independent motions, displacement and rotation; the latter is in turn parameterized mostly by means of the eigenvector of the rotation tensor (additive parameterizations like subsequent Euler angles are generally dismissed in computational mechanics). However, the rototranslation is found to be represented as the exponential map H = exp (ηη×) of a skew-symmetric dual tensor built on the dual vector η = ϕ + E $ , called the helix [5]. Hence, it is immediately seen that the helix is the natural eigenvector of the rototranslation, and as such is a favorite candidate for an appropriate integral parameterization. The helicoidal parameterization of motion is a rightful alternative to the classical additive decomposition into subsequent displacement and rotation: these motions become coupled together into a synchronous motion. Rototranslation and helicoidal parameterization inherit all the properties of the rotation and its vectorial parameterization; in particular, the analogs of Eqs (24.12) for the dual case hold true as well, and we may observe that differential helices control the tangent space of rototranslations.
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The same parameterization may apply to the modeling of the structured solid itself. In fact, the oriento-positions of any two particles of a body in a given configuration “differ” by a rototranslation. Considering the particles belonging to an infinitesimal 3D neighborhood of a point, we may define three (generalized) curvature dual vectors k j along three coordinate lines by means of the relevant differential AT k j )×. These are the component vectors of a curvature helices, such that A T A , j = (A j dual tensor k = k j ⊗ g , which actually controls the tangent space of the orientoα grad (α αT x ), position field. In the explicit expression of the dual curvature, k = k a +Eα the primal part is the angular curvature itself (see Table 24.1) and the dual part is the co-rotational gradient of the position vector. The latter part couples intimately the spatial derivatives of the orientation with those of the position: that is, the orientation field affects the evaluation of neighboring positions. This so-called helicoidal modeling [15] is appropriate for a pole-based setting of 3D continuum mechanics. The dual mechanical quantities and statics laws of a micropolar solid are outlined in Table 24.3, where the remarkably plain structure of the equilibrium equations is in evidence.3 Note that the self-based version of the dual Table 24.3 Elements of pole-based micropolar mechanics Geometry in the reference configuration: A = α+Ex×α particle’s oriento-position tensor αT x), j AT k j )×) kj = k a j + E α (α three curvature dual vectors (from A T A , j = (A αT x ) k = k j ⊗ g j = k a + E α grad (α curvature dual tensor Geometry in the current configuration: A = α + E x × α
particle’s oriento-position tensor αT x ), j AT k j )×) k j = k a j + E α (α three curvature dual vectors (from A T A , j = (A j T α x ) curvature dual tensor k = k j ⊗ g = k a + E α grad (α
Particle’s kinematics: H = A A T = Φ + E h × Φ ω = k − Φ k = ωa + E Φ grad (Φ ΦT h)
rototranslation (hh = x − Φ x , the translation vector) dual strain tensor
Statics: b s S´ divS´ + b = 0 S´ ν = s
= f + E (cc + x × f ) m + x × t ) = t + E (m ´ + x × T´ ) = T´ + E ( M
body load dual density boundary load dual density dual stress tensor (of the 1st Piola–Kirchhoff kind) body balance boundary balance
´ , gather the angular and χ and X T S´ = T´ +E M strain and stress tensors, X T ω = ω a +Eχ linear variables of the classical micropolar mechanics (see Table 24.1). Then, the strain and stress parameters are related to the mechanical variables as in (24.1), i.e. ΦT X T ω ) and Tˆ = primal(Φ ΦT X T S´ ). In the case of hyperelastic non-polar ε = dual(Φ 3
The pole is made to coincide with the absolute origin for convenience.
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media, the strain energy is function of just the symmetric part of the strain parameter, S ΦT X T ω )S , ax dual(Φ ΦT X T ω ) = 0, w = w(εεS ) and Tˆ = w/εεS . Hence, ε S = dual(Φ T T ´ Φ X S ) = w/εεS + τˆ ×. primal(Φ The helicoidal modeling allows to complete the interpolation scheme (24.16) as detailed in [16] and to integrate it within the formulation of finite elements [17] according to the concepts discussed in Sect. 24.3: the local rototranslation H = A A T and the kinematic dual strain ω = k − H k are recovered by comparing the interpolated quantities in the current and in the reference configurations. Note that the helicoidal interpolation proposed in [16] may be considered as the multi-coordinate version of successful multiplicative interpolations published in the Nineties for the 1D case of 2-node beam elements [4, 8]; however, it is worth stressing that the consistent design leading to Eqs (24.9) and (24.16) yields a general interpolation scheme that is ready as is for higher-order elements as well.
24.5.2 The Helicoidal Shell Element Starting from the internally constrained principle of virtual work tailored to hold the helicoidal modeling concept, a working shell element was developed in [19,20]. The proposed modeling allows a reliable representation of the position field on curved and curving surfaces, and in particular provides a faithful discrete approximation on the element domain. Moreover, in [19] the helicoidal modeling has been applied as well to the 3D mechanics across the thickness, that leads to the shell constitutive equations to be used in [20]. Note that, given the close affinity between rotations and rototranslations, all the issues concerning interpolation, linearization and discrete approximation, discussed above with reference to the rotation field alone, apply straightaway to the rototranslation field as well. So, adopting the helicoidal modeling in shell finite elements does not introduce significant complications. Rather, in a sense, it simplifies things, as the Euclidean part of motion is discarded in favor of one integral orthogonal field. The advantage of modeling helicoidally a curved surface becomes evident with the lowest-order shell elements. Lowest-order elements rely by nature on the tangent space of the underlying modeling: so, in a Euclidean context they are “flat” elements—e.g. 2D or 3D solid elements and membrane elements. The lowest-order helicoidal shell element, instead, is a truly curved element, featuring orientationdriven positions: the nodal rotations control both the internal rotations and the internal displacements. Numerical tests evidence that the element, though not equipped with widespread techniques as incompatible modes, assumed natural strains nor enhanced assumed strains, does not suffer from locking [20]. This welcome behavior can be likely ascribed to the consistent approximate kinematic field faithful to the characteristics of the underlying helicoidal modeling; presently however, an analytic proof that this element is locking-free is still lacking.
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24.6 Conclusion Four consistency issues in the formulation of proper shell elements for geometrically nonlinear problems have been discussed. Each of the first three issues may be hopefully of interest to people resting on uncoupled position and orientation fields. Firstly, the question of the drilling rotation is addressed in Sect. 24.2 and answered to by a variationally consistent approach. Essentially, the drilling rotation is a redundant d.o.f. for a material surface that behaves in a non-polar fashion in its tangent plane, so its definition entails an extra stress field to retain as a primary unknown of the surface mechanics. A sound variational formulation based on a proper constitutive characterization allows to achieve a genuine and thorough micropolar setting of the shell mechanics, endowed with a full three-parametric rotation tensor. Secondly, the interpolation of the orientation field on the shell surface is addressed in Sect. 24.3. It is shown that an interpolation scheme consistent with the rules of the special orthogonal group is the key to arrive at a frame-invariant and path-independent finite element. Thirdly, the linearization of the virtual functional is dealt with in Sect. 24.4. Again, an approach totally respectful of the SO(3) properties leads to a consistent resolution of the mixed virtual-incremental variation variables that exist in a nonlinear variational formulation involving finite rotations. The fourth consistency issue (Sect. 24.5) concerns a new way of addressing finite element kinematics. Here, the independent position and orientation fields are coupled together into an orthogonal oriento-position integral field that inherits every features pertaining to the orientation field alone, so the discussions in Sects 24.3 and 24.4 immediately apply to this case as well. The prominent feature of this approach, which was explicitly designed for consistent kinematic approximations, is the capability it offers to build low-order shell elements that naturally do not suffer from membrane and shear locking.
References 1. Angeles, J.: The application of dual algebra to kinematic analysis. In Computational Methods in Mechanical Systems (Edited by Angeles, J. and Zakhariev, E.), Springer-Verlag, Heidelberg, Vol. 161, 1–31 (1998) 2. Atluri, S.N.: Alternate stress and conjugate strain measures, and mixed variational formulations involving rigid rotations, for computational analyses of finitely deformed solids, with application to plates and shells - I, Theory. Comput. Struct. 18, 93–116 (1984) 3. Badur, J., Pietraszkiewicz, W.: On geometrically non-linear theory of elastic shells derived from Pseudo-Cosserat continuum with constrained micro-rotations. In: W. Pietraszkiewicz (ed.) Finite rotations in structural mechanics, pp. 19–32. Springer, Berlin (1986) 4. Borri, M., Bottasso, C.: An intrinsic beam model based on a helicoidal approximation – Part I: Formulation. Int. J. Numer. Meth. Engng 37, 2267–2289 (1994) 5. Borri, M., Trainelli, L., Bottasso, C.L.: On representations and parameterizations of motion. Multibody Syst Dyn 4, 129–193 (2000) 6. Bufler, H.: The Biot stresses in nonlinear elasticity and the associated generalized variational principles. Ing. Arch. 55, 450–462 (1985)
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7. Chr´os´cielewski, J., Makowski, J., Stumpf, H.: Genuinely resultant shell finite elements accounting for geometric and material nonlinearity. Int. J. Numer. Meth. Engng 35, 63–94 (1992) 8. Crisfield, M., Jelenic, G.: Objectivity of strain measures in the geometrically exact three dimensional beam theory and its finite-element implementation. Proceedings: Mathematical, Physical and Engineering Sciences 455(1983), 1125–1147 (1999) 9. Hughes, T.J.R., Brezzi, F.: On drilling degrees of freedom. Comp. Meth. Appl. Mech. Engng 72, 105–121 (1989) 10. Ibrahimbegovi´c, A.: Stress resultant geometrically non-linear shell theory with drilling rotations – Part I. A consistent formulation. Comp. Meth. Appl. Mech. Engng 118, 265–284 (1994) 11. Ibrahimbegovi´c, A., Taylor, R.L.: On the role of frame-invariance in structural mechanics models at finite rotations. Comp. Meth. Appl. Mech. Engng 191(45), 5159–5176 (2002) 12. Kafadar, C.B., Eringen, A.C.: Micropolar media – I. The classical theory. Int. Jnl Engng Sci. 9, 271–305 (1971) 13. Merlini, T.: A variational formulation for finite elasticity with independent rotation and Biotaxial fields. Computat. Mech. 19, 153–168 (1997) 14. Merlini, T.: Variational formulations for the helicoidal modeling of the shell material surface. Scientific Report DIA-SR 08-06, Aracne Editrice, Roma (2008). ISBN: 978-88-548-1887-3 15. Merlini, T., Morandini, M.: The helicoidal modeling in computational finite elasticity. Part I: Variational formulation. Int. J. Solids Struct. 41, 5351–5381 (2004) 16. Merlini, T., Morandini, M.: The helicoidal modeling in computational finite elasticity. Part II: Multiplicative interpolation. Int. J. Solids Struct. 41, 5383–5409 (2004). Erratum on Int. J. Solids Struct. 42, p. 1269 (2005) 17. Merlini, T., Morandini, M.: The helicoidal modeling in computational finite elasticity. Part III: Finite element approximation for non-polar media. Int. J. Solids Struct. 42, 6475–6513 (2005) 18. Merlini, T., Morandini, M.: Helicoidal shell theory. Scientific Report DIA-SR 08-07, Aracne Editrice, Roma (2008). ISBN: 978-88-548-1889-7 19. Merlini, T., Morandini, M.: Computational shell mechanics by the helicoidal modeling. Part I: Theory. Jo. Mech. Mater. Struct. 6 (to appear) (2011) 20. Merlini, T., Morandini, M.: Computational shell mechanics by the helicoidal modeling. Part II: Shell element. J. Mech. Mater. Struct. 6 (to appear) (2011) 21. Merlini, T., Morandini, M.: The helicoidal modelling in the approximation of shell structures mechanics. In: W. Pietraszkiewicz, I. Kreja (eds.) Shell Structures: Theory and Applications, vol. 2. CRC Press, Taylor & Francis, London pp. 269-272 (2010) 22. Pietraszkiewicz, W., Eremeyev, V.A.: On natural strain measures of the non-linear micropolar continuum. Int. J. Solids Struct. 46, 774–787 (2009) 23. Reissner, E.: A note on variational principles in elasticity. Int. J. Solids Struct. 1, 93–95 (1965) 24. Sansour, C., Bednarczyk, H.: The Cosserat surface as a shell model, theory and finite-element formulation. Comp. Meth. Appl. Mech. Engng 120, 1–32 (1995) 25. Sansour, C., Bufler, H.: An exact finite rotation shell theory, its mixed variational formulation, and its finite element implementation. Int. J. Numer. Meth. Engng 34, 73–115 (1992) 26. Simo, J.C., Fox, D.D., Hughes, T.J.R.: Formulation of finite elasticity with independent rotations. Comp. Meth. Appl. Mech. Engng 95, 277–288 (1992) 27. Wisniewski, K.: A shell theory with independent rotations for relaxed Biot stress and right stretch strain. Computat. Mech. 21, 101–122 (1998) 28. Wisniewski, K., Turska, E.: Kinematics of finite rotation shells with in-plane twist parameter. Comp. Meth. Appl. Mech. Engng 190, 1117–1135 (2000) 29. Wisniewski, K., Turska, E.: Warping and in-plane twist parameters in kinematics of finite rotation shells. Comp. Meth. Appl. Mech. Engng 190, 5739–5758 (2001) 30. Wisniewski, K., Turska, E.: Second-order shell kinematics implied by rotation constraintequation. Jnl Elasticity 67, 229 – 246 (2002) 31. Zhilin, P.A.: Mechanics of deformable directed surfaces. Int. J. Solids Struct. 12, 635–648 (1976)
Chapter 25
An Algorithm for the Automatisation of Pseudo Reductions of PDE Systems Arising from the Uniform-Approximation Technique Patrick Schneider and Reinhold Kienzler
Abstract One way to develop theories for the elastic deformation of two- or one-dimensional structures (like, e.g., shells and beams) under a given load is the uniform-approximation technique (see [2] for an introduction). This technique derives lower-dimensional theories from the general three-dimensional boundary value problem of linear elasticity by the use of series-expansions. It leads to a set of power series in one or two characteristic parameters, which are truncated after a given power, defining the order of the approximating theory. Finally, a so-called pseudo reduction of the resulting PDE system in the unknown displacement coefficients is performed, as the last step of the derivation of a consistent theory. The aim is to find a main differential equation system (at best a single PDE) in a few main variables (at best only one) and a set of reduction differential equations, which express all other unknown variables in terms of the variables of the main differential equation system, so that the original PDE system is identically solved by inserting the reduction equations, if the main variables are a solution of the main differential equation system. To find a valid pseudo reduction by inserting the PDEs of the original system into each other is a complicated and very time-consuming task for higher-order theories. Therefore, an structured algorithm seeking all possibilities of valid pseudo reductions (to a given number of PDEs in a given number of variables) is presented. The key idea is to reduce the problem to finding a solution of a linear equation system, by treating each product of different powers of characteristic parameters with the same variable as formally independent variables. To this end, all necessary equations, which can be build from the original PDE system, have to be identified and added to the system a-priori. Keywords Uniform-approximation technique · Pseudo reduction · Partial differential equations · Higher-order theory · Linear elasticity P. Schneider (B) · R. Kienzler Department of Production Engineering, University of Bremen, Germany e-mail:
[email protected],
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 25, © Springer-Verlag Berlin Heidelberg 2011
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25.1 Introduction 25.1.1 The Uniform-Approximation Technique One way to develop theories for the elastic deformation of two- or one-dimensional structures (disks, plates, shells, bars, beams and arcs) under a given load is the uniform-approximation technique (see [2] or [4] for an introduction). This technique derives equivalent two- or one-dimensional formulations to the general three-dimensional boundary value problem of linear elasticity by the use of seriesexpansions (see [6] for details). These formulations are sets of countably many PDEs, which are power series in one or two characteristic parameters. For example: In the case of a plate theory this characteristic parameter is the second power of the so-called plate parameter which describes the relative thickness of the plate and is assumed to be a small quantity since plates are defined as being thin. In the case of a beam, we have two parameters which are assumed to be small, since the two cross-sectional measurements are assumed to be small against the measurement in beam length direction. Because of the smallness of the characteristic parameters, an approximative theory in a finite number of PDEs in a finite number of unknown quantities, which are the series coefficients of the sought displacement field, could be generated by the truncation of this power series after a certain power. The highest power which is not neglected is called the approximation order of the derived theory.
25.1.2 The General Aim of the Pseudo-Reduction Process Since a set of many PDEs in many unknown variables is not suitable for analytical discussions, as a last step of the derivation of a consistent theory, a so-called pseudo reduction of the resulting PDE System is performed. The aim is to find a main differential equation system (at best a single PDE) in a few main variables (at best only one) and a set of reduction differential equations, which express all other unknown variables in terms of the main variables, so that the original PDE system is identically solved by inserting the reduction equations, if the main variables are a solution of the main differential equation system. (See the derivation of Kirchhoff’s theory in [2] for a simple example of a pseudo reduction.) Furthermore, the reduction equations must allow us to eliminate all variables which are not main variables from all stress resultants for a complete description of the theory in terms of the main variables. More precisely: Not every variable has to be expressable independently by the reduction equations, but it is crucial that they allow us to eliminate all variables which are not main variables from all stress resultants and all non-main PDEs. This means: If unknown variables always occur in specific linear combinations of derivatives in the original PDE system and in the stress resultants, it is sufficient to express these specific forms in terms of the main variables. (See [4], Sects 3.6 and 3.7 for an
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example.) If we find a set of PDEs (main PDE system and reduction equations) satisfying this condition, we call them a correct pseudo reduction. The word ”pseudo” indicates that only the system of the main PDEs with the reduction equations can be equivalent to the original PDE system.
25.1.3 The Aim of the Presented Algorithm In former papers (compare [2], [3], [5] and [6]) the pseudo reduction of the PDE systems was performed by differentiating and inserting the PDEs of the original system into another. It is a complicated and very time-consuming task to find a sequence of operations that actually leads to a correct pseudo reduction. Furthermore, there was in the end no possibility to show whether the resulting pseudo reduction was the only possible one, or there are degrees of freedom. (Of course a reduction with only one main PDE of the least possible order would be the most desirable one.) Therefore we intend to find an algorithm which detects all possible pseudo reductions to a given PDE system, or shows that there are no possible pseudo reductions.
25.1.4 The Main Idea of the Presented Algorithm First of all one selects the main variables (the variables of the aimed for main differential equation system) and the PDEs which shall be used to generate the main differential equation system by elimination of the other variables. In practice there is always a first generic choice, given by the type of theory which shall be derived. Consider a plate theory for example (compare [2], and [6]). Since Kirchhoff theory is a single PDE in only one variable, which is the displacement of the mid-plane in thickness direction, say w, one would first try to develop a theory in one PDE only formulated in w. So the main variable would be w. The main differential equation would be the PDE which has the constant surface load as a right-hand side, since this is the dominating load case. In addition the triangular structure of the PDE systems, which is characteristic for PDE systems arising from the uniform-approximation technique, due to the truncation of the series (compare [2], [3], [5] and [6]), predicts this equation to be a generic choice for the main differential equation, since it seams intuitive to set the PDEs into one another from the bottom to the top of the PDE system. Or considered in another way: Since the top PDE is also the one with the most variables we will have to derive a complete set of reduction equations to eliminate all other variables and therefore we will have good chances to eliminate all variables in the stress resultants with the gained set of reduction equations. In the case of a beam theory the main PDE would also be the one with the constant surface load and the main variable would be the transverse displacement of the neutral axis. In both cases, the pseudo reduction to a single PDE in one variable turns out to be possible even for second-order theories (see [4] or [6] for the cases of a plate / the case of a
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beam will be published in a forthcoming paper). So the step of choosing the main variables and the PDEs which shall be used to generate the main differential system is in practice easier than it might look at first. After this choice, the problem of pseudo reduction is reduced to find for each main differential equation and each stress resultant a linear combination of differentials of the set of the other PDEs, so that they build the same term in nonmain variables as the corresponding equation. Then we can obtain a formulation expressed only in main variables by building the differences. If we would deal with exact linear equations (no differentials), all possibilities would be found by applying the Gauss algorithm (symbolic LU-decomposition). Our situation is more complex, since we are dealing with partial differential equations, which are furthermore truncated power series. Since all PDEs are linear in the unknown variables the differential-operators could simply be treated as factors (compare Sect. 25.3.4 below). On the other hand, the fact that the PDEs are truncated power series results in the problem that the characteristic parameters are not treatable as normal factors (see Sect. 25.3.1 below). We overcome this problems by treating each product of different powers of characteristic parameters with the same variable as a formally independent variables. Of course interpreted in these replacement variables (later called cd-variables, compare 25.3.2 below) the original PDE system does not provide enough equations to achieve a pseudo reduction. Our approach is to add ”all” (compare 25.3.5 below) equations that can be build from the equations of the original PDE system by multiplications and divisions with characteristic parameters a-priori to the system, before investigating all reduction possibilities by the application of the Gauss algorithm to this system (interpreted in cd-variables).
25.2 The Setting At first we will have a closer look on the setting of a PDE system which is generated by the uniform-approximation technique and the ”rules” arising from this setting for the reduction process.
25.2.1 The One-Dimensional Case We will deal with the case of a PDE system for deriving a one-dimensional theory. In this case we have two characteristic parameters which we will call c and d and only derivatives with respect to one coordinate, we will call x. The case of a PDE system for the derivation of a two-dimensional theory is just a simple special case of the situation treated here concerning the implementation of the algorithm, since there is only one characteristic parameter in this case, which simplifies the generation of additional equations (compare 25.3.5 below). The fact that there are derivatives with
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respect to two coordinates does not require any special treatment, besides the fact that there are two different differential factors. (compare 25.3.4 below)
25.2.2 Homogeneous Material Besides some general modeling assumptions required to develop equivalent lower-dimensional theories (see [6] for a detailed discussion of the modeling assumptions.), the materials under consideration are assumed to be linear elastic and homogeneous (but possibly non-isotropic), which means that the elements of the forth order elasticity tensor E are constant material parameters: ∀(i, j, k, l) ∈ {1, 2, 3}4 : Ei jkl = const.
25.2.3 Regularity In general we will not discuss any regularity questions, which simply means that all functions are assumed to be sufficiently smooth.
25.2.4 The Original PDE System In the one-dimensional case, the original PDE system is actually an ODE system of M ∈ N ODEs in N ∈ N sought displacement coefficients {uk : I → R| k ∈ {1, ..., N}} being scalar functions depending on one coordinate, we call x ∈ I ⊂ R. In the following, we will call this displacement coefficients the original variables. Each ODE was generated by the truncation of an infinite power series in the characteristic parameters c and d, neglecting every summand with a factor ci d j where i + j > a. As mentioned before, we call a the approximation order of the theory. Each ODE has the same specific form. For each μ ∈ M we have a n n=0 i=0
i n−i
cd
N 2
aμnikl E μnikl Dlx (uk (x)) = Kcrμ d sμ Pμ (x), x ∈ I,
(25.1)
k=1 l=0
with Pμ : I → R being a load resultant, assumed to be a given scalar function and rμ , sμ ∈ N ∪ {0} fulfilling the condition rμ + sμ ≤ a − 1. Furthermore, K ∈ R and for all (μ, n, i, k, l) in 3 4 T := (μ, n, i, k, l) ∈ {1, ..., M} × {0, ..., a}2 × {1, ..., N} × {0, 1, 2}|i ≤ n
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we have aμnikl ∈ R and E μnikl is a specific constant of the forth-order elasticity tensor E. D x denotes the differential operator with respect to x, with the convention D0x (•) = •. In addition most of the Pμ are usually neglected because they describe load cases, which are not very common in practice (compare [6] p. 52), or they are of higher order (compare Sect. 25.3.3 below). Most of the aμnikl equal 0 and the ODE system has furthermore a characteristic triangular structure: The number of the aμnikl 0 decreases for a rising μ. (See [1], pp. 72-73, [2], p. 237 and [6], pp. 53-55, for explicit examples of PDE systems of this form.)
25.2.5 Differentiability The triangular structure of the ODE system generated by the uniform-approximation approach suggest that variables could successively be eliminated, by inserting the eventually differentiated equations into each other, beginning at the bottom of the system (μ = M / see [2] for a simple example). It is clear that we have to assume that the equations keep their magnitude (concerning the characteristic parameters) during differentiation, for having a chance to reduce the system, which implies that all unknown displacement coefficients uk have to keep their magnitude during differentiation. This assumption is of course mathematically inappropriate, since we do not have any a-priori indications about the structure of the solution. Indeed this is not a pure theoretical problem. For example in second-order plate theories it is known that the rate of twist has derivatives that are of higher magnitude than the original function (compare [3]). This results in the fact that a correct pseudo reduction might not give a physically meaningful theory without further a-posteriori extensions (as discussed in [6] in detail). However, these problems are not subject of this paper, since nevertheless, a valid pseudo reduction remains an important step in the derivation of a consistent theory.
25.2.6 Multiplication with Characteristic Parameters Since all equations are power series in ci d j truncated after a certain common power i + j = a, an equation witch is multiplied with a product of certain powers of the characteristic parameters cn dm has to be truncated again after multiplication, (neglect all summands with factors cn+i d m+ j where n + m + i + j > a) to give an equation which is of the desired accuracy and therefore usable for the uniform-approximation technique.
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25.2.7 Division by Characteristic Parameters Division by a certain product of powers of characteristic parameters cn d m results in a loss of accuracy of the resulting equation. To get an equation which is appropriate to generate a consistent theory of the desired order a, the original equation has to be computed with sufficiently higher accuracy a + n + m. The generation of additional higher-order terms is not a problem, since one has closed form representations of the infinite power serieses (compare [6]).
25.3 The Algorithm In this section we will describe the algorithm for the case of the pseudo reduction of a PDE system for deriving a one-dimensional theory. The case of a PDE system for the derivation of a two-dimensional theory is just a simple special case of the situation treated here concerning the implementation of the algorithm, as already mentioned. Furthermore, we will describe the algorithm for one main PDE for linguistic simplicity. For each additional main PDE and each stress resultant the described procedure has to be performed in a complete analog manner.
25.3.1 Selection of Main Variable and Main PDE As a first step the main variables and the main PDE is selected. The result of the algorithm depends on this choice. If the result of the canonical choice (compare 25.1.4 above) is that there are no pseudo reductions possible, the next higher (concerning the degree of the corresponding basis polynomial; see [2] or [6] for details) displacement coefficients are to be declared as main variables.
25.3.2 cd-Variables As a consequence of 25.2.6 and 25.2.7, every occurrence of a product of an original variable uk with powers of characteristic parameters in an input PDE for the Gauss algorithm has to be treated as an independent variable to prevent the Gauss algorithm to perform operations that change the approximation order a of the equations. For example: One summand %appearing &in the left hand side of (25.1) could be c2 d2 3E 1111 D1x u3 (x) = 3E1111 D1x c2 d 2 u3 (x) and we would treat c2 d2 u3 (x) as a variable independent from, e.g., c2 u3 (x). Otherwise, the Gauss algorithm could, e.g., divide our example equation by d 2 , leading to an equation with missing terms of order a − 1 and a. We call the new replacement variables the cd-variables.
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Consequently (right / with regard of 25.2.6 and 25.2.7) multiplications or divisions by products of powers of characteristic parameters cn d m generate formally new equations, which might be linear independent (by means of cd-variables) of the original equations.
25.3.3 Right-Hand Sides There is no need to treat the load-resultants Pμ appearing at the right-hand sides of the equations (25.1) in an analog manner like the variables uk by introducing cd-load-resultants, because they are assumed to be given and not to be eliminated. Therefore, they do not appear in the equation system for the derivation of the linear combination which finally leads to the pseudo reduction (compare derivation of x in Sect. 25.3.6 below). The load-resultants are assumed to have a magnitude 1 concerning the common power n = i + j in ci d j themselves (in contrast to the variables uk ) which is calculable by examining a zeroth-order theory (compare [2] Sect. 4.1). This magnitude has to be considered when a truncation of the power-serieses is performed. (compare conditions for rμ and sμ and the comment on the order of Pμ in Sect. 25.2.4)
25.3.4 D-Factors Since the equations of the original PDE system are linear in the original variables uk and the load resultants Pμ and their prefactors are constant with respect to x (compare (25.1)) and Gauss algorithm only builds linear combinations of the lines of an input matrix, we could simply treat D x as a symbolic factor in our special case. Valid factors for multiplications of lines concerning the Gauss algorithm are therefore in the field 3 4 K = R ∪ {D x } ∪ Ei jkl |(i, j, k, l) ∈ {1, 2, 3}4 , (25.2) allowing the Gauss algorithm to perform integrations and differentiations of our PDEs. This coincides with the normal behavior of a computer algebra system which is able to perform symbolic LU-decompositions (e.g., Maple or Mathematica) if the E i jkl and D x are declared as symbols, allowing straight forward implementations of the presented algorithm.
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25.3.5 Generating Additional Equations Written in cd-variables and D-factors the original PDE System is merely a linear equation system with a given right-hand side. Because the Gauss algorithm can not perform valid multiplications or divisions by characteristic parameters, all necessary equations have to be generated a-priori. In most cases a pseudo reduction can be found without considering any equations, which can be build from the PDEs of the original system by divisions of characteristic parameters. Nevertheless we consider these equations for our algorithm. In a first execution of the algorithm the generation of additional equations by divisions can be omitted, if the task is only to find any valid pseudo reduction. The number of PDEs which can be build from the PDEs of the original system by divisions by products of powers of characteristic parameters cn d m is infinite, so that we can not really consider all of them. But since the aim is, finally, to generate a given linear combination of cd-variables, there are two types of these equations which are potentially not useful. The main PDEs and stress resultants only contain positive powers of c and d (compare [2], [3] and [6]). Therefore, equations with cd-variables of negative powers are potentially not useful. Furthermore there is not really a physically interpretation for these variables, because of the used seriesexpansions with only positive powers (compare [2], [3] and [6]). The second type of equations which are potentially of no use are those, which contain variables, which are not in the set of the original variables {uk |k ∈ {1, . . . , N}}, since the use of these equations for the elimination of variables would always add new variables which are also to eliminate. If we decide not to consider these two types of equations, we end up with a finite set of PDEs which can be build by divisions.
25.3.5.1 A Simple Approach for Generating All Additional Equations A simple approach for generating all additional equations would be: Make a list of all PDEs of the original system. Traverse the list and build for each PDE the new PDE which is generated by multiplication with c (with regard of 25.2.6). If it is not identical zero and not already in the list, add it to a new list. Do the same with the factor d. Then traverse the list and build for each PDE the new PDE which is generated by division by c (with regard of 25.2.7). If it does not contain negative powers of c, does not contain new variables (not in the set {uk |k ∈ {1, . . . , N}}) and is not already present, add it to the new list. Do the same with the quotient d. If the new list is not empty, append it to the list and begin again, otherwise stop. The procedure indeed always terminates. We will give a more detailed presentation of another approach in the following. Our approach calculates the equations which are to generate a-priori. This omits the testing procedure for each different equation which is build and the computation of higher-order terms for each division, which is the main reason why the approach has a higher performance, if it is implemented. Furthermore, it reveals a greater insight into the structure of the problem of generating additional equations.
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25.3.5.2 Extension by Divisions by Characteristic Parameters The PDEs which can be build by divisions by characteristic parameters among the considered set of equations can be generated by the following procedure, which is to be performed for each PDE of the original PDE system. First compute the PDE μ ∈ M with increasing approximation order b > a until the next higher approximation order b + 1 would contain terms in variables that are not in the original PDE system {uk |k ∈ {1, . . ., N}}, leading to an PDE of the form b n n=0 i=0
i n−i
cd
N 2
aμnikl E μnikl Dlx (uk (x)) = Kcrμ d sμ Pμ (x), x ∈ I.
(25.3)
k=1 l=0
Than compute the maximal excludable powers of c and d of this PDE (so that no negative powers of c and d occur after excluding). 3 ex c power(μ) := min i | (μ, n, i, k, l) ∈ T 4 with aμnikl 0 and Eμnikl 0 in (25.3) 3 ex d power(μ) := min n − i | (μ, n, i, k, l) ∈ T 4 with aμnikl 0 and Eμnikl 0 in (25.3) Traverse all products which can be build of powers of c and d up to the common power ex c power + ex d power: 3 4 c j dn− j |n = 1, ..., (ex c power + ex d power), j = 0, ..., n . If the specific factor c j d n− j is excludeable (( j ≤ ex c power) and (n − j ≤ ex d power)) and the PDE can be computed with a sufficient high accuracy to give a PDE of approximation order a after the division by the factor without adding new variables to the system (n ≤ b − a); compute the PDE and add it to the system. This gives us nearly all equations which can be build from the original PDE by divisions by powers of characteristic parameters; we only omitted to build the equations where (n > b − a). In this case it is not possible to exclude the factor without adding new variables to the system for a general factor, unless it is cn or dn . Since the maximal accuracy the PDE can be computed with, without adding new variables is b, we can avoid this by multiplying the equation with the other characteristic parameter of a power of at least a + n − b. By doing so, the algorithm so far will give us at least all ”lowest” PDEs which can be build by divisions. This means precisely all PDEs which can not be divided by a power of c or d without either containing cdvariables of negative powers or adding new variables to the system, which are not in the set of the variables of the original PDE system {uk |k ∈ {1, . . ., N}}. All ”higher” ones will be generated by the next extension step. The special case of a factor cn or d n could be treated simultaneous as table 25.1 shows. In pseudo code the procedure reads as follows.
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PDE_system:=original_PDE_system basevariables:=compute_all_variables_of(original_PDE_system) for all PDE of original_PDE_system do b:=a basevarcheck:=true while basevarcheck=true do b:=b+1 higher_order_PDE:=compute_with_order_N(PDE,b) if not(compute_all_variables_of(higher_order_PDE)
The procedure above terminates obviously. If we had decided to consider equations with negative cd-variables or new basic variables (also resulting in new cd-variables which are not present in the original PDE system) we would not be able to treat the problem with an algorithm that is guaranteed to terminate, because of the infinite number of equations we would have to consider. This is the main reason for not considering these equations. Theoretically, they might offer unrevealed chances for pseudo reductions, since they might be used to build linear combinations where all new cd-variables vanish and which are furthermore linear independent from all considered equations.
Table 25.1 simultaneous treatment of the special cases c• j
+n − 2 j +
(n− j) n
d• (−b + a) −n + 2 j + nj (−b + a)
j=0
a+n−b > 0
−n < 0
j=n
−n < 0
a+n−b > 0
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25.3.5.3 Extension by Multiplications by Characteristic Parameters As a next step we will extend the PDE system by all equations which can be build by multiplications with characteristic parameters. The number of the equations in C has raised ( M C > M) due to the extension by divisions by characthe PDE system M teristic parameters, while the typical form of all equations is still as in (25.1). If we C with a product c j d i of a sufficiently high common power, the multiply an PDE μ ∈ M PDE is identical zero after the truncation of all terms with higher common power than the approximation order a (compare Sect. 25.2.6). Therefore, we can simply build all PDEs which can be build by multiplications with products c j d i with rising common power n = i + j, until n = a − mcp(μ), with mcp(μ) being the minimal C common power of all factors c j di appearing in the specific PDE μ ∈ M 3 mcp(μ) := min n | (μ, n, i, k, l) ∈ T
4 with aμnikl 0 and Eμnikl 0 in (25.1) .
Since we already did the extension of the PDE system by divisions of characteristic parameters, we only add the so build new PDEs to the system if they are not already present. In pseudo code this procedure reads as follows. for all PDE of PDE_system do mcp:=minimal_common_power_of(PDE) for n from 1 to a-mcp do for j from 0 to n do new_PDE:=truncate_after_common_power(cˆj*dˆ(n-j)*PDE,a) if not(member_of(new_PDE,PDE_system)) then add_to(new_PDE,PDE_system) end if end do end do end do
25.3.6 Computation of Pseudo Reduction Possibilities So far we have added all equations to the PDE system, which can be build by multiplications and/or divisions by characteristic parameters, without either containing cd-variables of negative powers or adding new variables to the system. Not all of them will be linear independent from each other (written in cd-variables and D-factors). Therefore, we build the n × m-matrix A, where each of the n lines corresponds to one PDE of the extended PDE system without the main PDE and each of the m rows corresponds to one cd-variable appearing in the extended PDE system with exception of one row corresponding to the negative right-hand sides. Note that lines in A can only be linear dependent if they have the same magnitude in right-hand sides including the factor cr d s , since all other rows do not contain a factor in c or d, so that an introduction in cd-load-resultants was not necessary. With
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application of the Gauss algorithm we can eliminate all linear dependent equations, C with rank(A) = rank(A) C = n˜ < n. resulting in a n˜ × m-matrix A n˜ ×m−1−k Now we build the matrix B ∈ K where each of the n˜ lines corresponds to one PDE of the reduced PDE system of the last step and each of the m − 1 − k rows corresponds to one cd-variable which is not in the set of the cd-variables which can be generated from variables declared as main variables. Furthermore, build the vector b ∈ Km−1−k×1 , where each entry corresponds to the prefactor in the main PDE of the cd-variables which are not in the set of the cd-variables which can be generated from variables declared as main variables, in the same order as used during the generation of B. Now seek x ∈ Kn˜ ×1 solving BT x = b, for example by applying the Gauss algorithm (symbolic LU-decomposition). If there is no solution, there is no possible pseudo reduction. If there is a solution space generated by span (x1 , ..., xr ) the main PDE in terms of main variables is ⎛ ⎞ r ⎜⎜ 1 ⎟⎟⎟ T⎜ vC ⎜⎜⎝⎜ ⎟⎟⎠⎟ = 0, with x = ti xi , ti ∈ K, −x i=1 where v is a 1 × 1 + k line vector of the main cd-variables with 1 as a first entry; C is the 1 + n˜ × 1 + k-matrix, where the first line corresponds to the main PDE and each further line corresponds to a PDE of the reduced PDE system, in the same order as before, and the first row corresponds to the right-hand sides and each further row corresponds to the main cd-variables in the same order as in v. The ti ∈ K (remember (25.2)!) are r ∈ N free parameters describing all possible pseudo reductions. Because of the application of the Gauss algorithm on A before, every different choice of the parameters actually leads to a different main PDE, so that r ∈ N is actually a measure for the degrees of freedom of possible pseudo reductions. Remember that the procedure is to proceed for each main PDE and all stress resultants. The set of reduction equations can be generated by successively proceeding the equations of the linear combination x by the same algorithm.
25.4 Summary A structured algorithm seeking possibilities of pseudo reductions to a given PDE system was presented. Since there are infinite possibilities to generate new equations by divisions by characteristic parameters, only equations with positive cd-variables and variables present in the original system are considered, resulting in an algorithm that is guaranteed to terminate. The regarded set of equations is the biggest set which allows an implementation of an algorithm with guaranteed termination. Furthermore the presented algorithm finds all possibilities of pseudo reductions, or shows that there are no possible pseudo reductions (in both cases among the set of
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considered equations), ending up with a main PDE system with free selectable parameters. In all up-to-now published cases (see [2], [3] and [6]) a pseudo reduction is possible without considering any equations generated by divisions by characteristic parameters. In the case of a second-order beam theory (of a forthcoming paper) a pseudo reduction is possible among the considered set of equations, which can be generated by divisions by characteristic parameters. Therefore, the algorithm should be sufficient for the treatment of all pseudo reduction problems arising from the uniform-approximation technique. Because of the big dimensions of the generated matrices an implementation using a computer algebra system seems to be necessary.
References 1. Kienzler, R.: Eine Erweiterung der klassischen Schalentheorie; Der Einfluss von Dickenverzerrungen und Querschnittsverw¨olbungen. Phd Thesis, Technische Hochschule Darmstadt, Darmstadt (1980) 2. Kienzler, R.: On consistent plate theories. Archive of Applied Mechanics, 72, 229-247 (2002) 3. Kienzler, R.: On Consistent Second-Order Plate Theories. In: Theories of plates and shells: critical review and new applications, 85-96. Springer-Verlag, Berlin Heidelberg (2004) 4. Bose, D.K.: Erhaltungss¨atze der Kontinuumsmechanik f¨ur eine konsistente Plattentheorie. Phd Thesis, Universit¨at Bremen, Bremen (2004) 5. Bose, D..K. and Kienzler, R.: On material conservation laws for a consistent plate theory. Archive of Applied Mechanics, 75, 607-617 (2006) 6. Schneider, P.: Eine konsistente Plattentheorie zweiter Ordnung f¨ur monotropes Material. Diploma thesis, Universit¨at Bremen, Bremen (2010)
Chapter 26
Recent Improvements in Hu-Washizu Shell Elements with Drilling Rotation Krzysztof Wi´sniewski and Ewa Turska
Abstract Enhanced four-node shell elements based on the Hu-Washizu (HW) functional are developed for the Green strain and constitutive laws modified by the zero-normal-stress condition. They have six dofs/node, i.e. the drilling rotation is included. The key features of the approach are as follows: 1. For the membrane part of HW shell elements, we use the 7-parameters stress, 9-parameter strain and 2-parameter EADG enhancement, which are selected as optimal for the 2D elements with drilling rotations. This ensures excellent accuracy and robustness to shape distortions for coarse meshes and very good performance for finite deformations. 2. The assumed representations of stress and strain are defined in skew coordinates, which not only improves accuracy, but also implies, as checked for 2D elements, that the homogenous equilibrium equations and the compatibility condition are satisfied point-wise for an arbitrary shape of an element. The skew coordinates are also easy to implement. 3. The bending/twisting and transverse shear parts are treated either by the HW functional or by the potential energy functional, in order to select the formulation which performs best for a reduced number of parameters. The performance of the shell HW elements is illustrated on several numerical examples and compared to the EADG elements in such aspects as: accuracy, radius of convergence, required number of iterations of the Newton method, and time of computation. K. Wi´sniewski (B) Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawi´nskiego 5B, 02-106 Warsaw, Poland e-mail:
[email protected] E. Turska Polish Japanese Institute of Information Technology, Koszykowa 86, 02-008 Warsaw, Poland e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 26, © Springer-Verlag Berlin Heidelberg 2011
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Keywords Drilling rotation · Hu-Washizu functional method · Four-node shell element
·
Finite element
26.1 Introduction Hu-Washizu functional with rotations. The standard 3D Hu-Washizu (HW) functional is mixed and involves two additional fields, strains and stresses, where the stresses play the role of the Lagrange multipliers. For shells, the standard way of applying the Reissner hypothesis to the HW functional does not yield a mixed functional w.r.t. the rotations used in this hypothesis. However, if we first construct an extended 3D HW functional, with the Rotation Constraint (RC) added via the Lagrange multiplier method, then the resulting functional will be mixed and the subsequent use of the Reissner hypothesis does not change this fact. We treat the tangent and normal components of the RC differently. 1. If we enforced the tangent components of the RC then the transverse shear strains would be equal to zero, similarly as for the Kirchhoff kinematics, see [19]. As the transverse shear strains are important, hence, we omit these two constraints. 2. We enforce the normal component of the RC as then the drilling rotation becomes the additional rotational variable and, together with the two components introduced by the Reissner hypothesis, we have three rotational components in the formulation, which is beneficial in numerical implementations. This consistent way of introducing the drilling rotation furnishes an alternative to the so called Allman shape functions used for three- and four-node elements. It is more accurate and can be used to develop elements with arbitrary number of nodes, see nine-node elements with 6 dofs/node in [10]. Mixed elements without or with drilling rotation. We shall distinguish between the mixed elements without and with the drilling rotation, for two reasons: (a) the optimal assumed representations are different for these two cases, and (b) the implementation of the drilling RC involves several difficulties. 1. In the class of mixed elements without the drilling rotation, the standard of accuracy is set by the Pian-Sumihara (PS) element of 1984, formulated for the Hellinger-Reissner (HR) functional and the 5-parameter representation of stress in natural coordinates, see [11]. It is slightly more accurate for coarse distorted meshes than the assumed strain element with the 4-parameter enhancement. Several mixed elements developed afterwards perform slightly better for coarse distorted meshes than the PS element, including the HW element of [12] with 16 additional parameters and our HW14-S element using 14 parameters and skew coordinates, [22]. In [22], we established that at least a 7-parameter representation of the assumed strain should correspond to the 5-parameter assumed stress, as implied by the inverse constitutive equation, see eq. (1) therein. However, the best performance
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is provided by the 9-parameter strain, which is used in the HW14-S element. This element is simple, as it is a purely mixed element with no enhancement, and very accurate. While in [22], the covariant strain representations was used, in [23] we tested the contravariant strain representation. For the HW element based on the 7-parameter strain representation, HW12-S, the results for the contravariant assumed strain have improved. For the HW14-S element, based on the 9-parameter assumed strain, with all terms being linear in skew coordinates, the results for both types of the assumed strain are identical. It is still our best HW element. 2. In the class of mixed elements with the drilling rotation, to develop a HW element, we have to solve several additional questions, particularly for shells. a. The first step is to implement the drilling RC, which is a challenging task, involving difficulties comparable with these encountered for the in-plane shear strain. In our approach, the drilling rotation is introduced by the perturbed Lagrange multiplier method using the local (elemental) Lagrange multipliers, which are discontinues across element’s boundaries. Besides, for the equalorder bi-linear approximations of displacements and the drilling rotation, the drill RC is incorrectly approximated, hence we have to omit one term and stabilize one spurious mode, see [17]. Here, we describe an additional refinement of the stabilization involving the use of the gamma-stabilization. b. The next step is to select suitable assumed stress and strain representations for the membrane part. We note that the enhancement of the deformation gradient F is needed to obtain good element’s performance because this enhancement also affects the drilling RC. In consequence, the assumed representations which are optimal for 2D+drill elements are different than those for 2D elements. In particular, the 5-parameter stress, which was proposed for the 2D Pian-Sumihara element and was used in our two-dimensional Hellinger-Reissner and HW elements but in skew coordinates, is replaced by the 7-parameter stress. c. For shells, the potential energy functional depends on membrane strains, bending/twisting strains and transverse shear strains, and it is not obvious whether we should construct the HW functional for all parts of the strain energy or only for some of them. This has to be decided taking into account the accuracy and speed as well as radius/rate of convergence in non-linear problems of several possible formulations. Despite these difficulties, the HW shell elements have the following advantageous features, compared to elements based on other formulations: 1. In linear problems, displacements and stresses for coarse distorted meshes have a slightly higher accuracy, 2. In nonlinear problems, they have better convergence properties,
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see [15], [7] and [22]. As the parameters of the assumed fields are eliminated on the element’s level, hence, externally, the HW elements have the form of standard elements and do not require any changes in the FE codes. Scope of paper. In this paper, we develop several four-node shell elements from the enhanced HW functional with the drilling rotation. All of them use have an identical membrane part, based on the 7-parameter representation of stress, the 9-parameter representation of strain, and the 2-parameter enhancement of the displacement gradient. They differ in the treatment of the bending/twisting part and the transverse shear part. As the starting point, we take our HW47 shell element, in which the bending/twisting part is analogous to the membrane part, and the transverse shear part is treated by the HW functional. It involves the following number of parameters: • 18 parameters for the membrane stress, strain, and enhancement (7+9+2=18), • 3 parameters for the Lagrange multiplier for the drill RC, • 18 parameters for the bending/twisting stress, strain, and enhancement (7+9+2 =18), • 8 parameters for transverse shear stress, strain (4+4=8), i.e. totally 47 parameters. We use this element for reference, and try to construct other HW elements, which perform similarly well using a smaller number of parameters. Three such shell elements are developed and tested, and they have 39, 31 and 29 parameters. The performance of the developed shell HW elements is illustrated on several numerical examples and compared to the EADG elements in such aspects as: the accuracy, the radius of convergence, the required number of iterations of the Newton method, and the time of computation.
26.2 Three-Dimensional Formulation with Rotations Extended configuration space. The classical configuration space of the non-polar Cauchy continuum is defined as: C {χ : B → R3 }, where χ is the deformation function defined on the reference configuration of the body B. In the present work, we consider an extended configuration space, which is defined as follows: Cext {(χ, Q) : B → R3 × S O(3)
∧
χ ∈ C},
(26.1)
where χ is required to belong to the classical C, see e.g. [1]. The rotations Q ∈ S O(3) are constrained by the Rotation Constraint (RC) equation skew(QT F) = 0,
(26.2)
where F ∇χ is the deformation gradient. This equation is used in the present work to introduce the rotations into the 3D formulation; various forms of governing
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functionals modified by the RC are presented in [18]. There exists also a Cosserattype approach, in which the rotations remain unconstrained, see e.g. [4] and [5]. 3D HW functional with rotations. Our formulation is based on the 2nd PiolaKirchhoff stress S and the Green strain E 12 (FT F − I). The independent fields of stress and strain of the HW functional are designated as S∗ and E∗ . To incorporate rotations into 3D formulations, we constrain the governing HW functional by the weak form of the Rotation Constraint of eq. (26.2). Consider the classical form of the three-field Hu-Washizu (HW) functional, , . /F HW (u, S∗ , E∗ ) W(E∗ ) + S∗ · E(∇u) − E∗ dV − Fext , (26.3) V
where W(E∗ ) is the strain energy expressed by the independent strain E∗ , and the independent stress S∗ plays the role of the Lagrange multiplier of the relation between the independent strain E∗ and the Green strain E(∇u). The strain energy density W is a function of the independent strain E∗ . Fext is the potential of the external loads, the body force, and the displacement boundary conditions. To obtain the functional with rotations, the Lagrange multiplier method is applied to append the Rotation Constraint of eq. (26.2) to eq. (26.3). Then, we obtain the five-field functional, 3 4 . / ∗ ∗ F 5 (u, Q, S , E , T) W(E∗ ) + S∗ · E(∇u) − E∗ + T∗ · skew(QT F) dV − F ext , V
(26.4) where T∗ is the Lagrange multiplier for the RC equation. If this functional is regularized in T∗ , then we obtain another five-field functional, D E . / ∗ ∗ ∗ ˜ F5 (u, Q, S , E , T ) W(E∗ ) + S∗ · E(∇u) − E∗ + FRC dV − F ext , (26.5) V
where the underlined term related to the Rotation Constraint of eq. (26.2) has the so-called Perturbed Lagrange (PL) form, F RC (u, Q, T∗ ) T∗ · skew(QT F) −
1 ∗ ∗ T ·T , 2γ
(26.6)
and γ ∈ (0, ∞) is the regularization parameter. Remark. To derive the HW functional for shells, we can use the functional (26.5) as a starting point but then we have to make additional assumptions about the ζ-dependence of the stress S∗ and strain E∗ . However, we can also proceed in another way: first, express the 3D strain energy W(E) in terms of shell strains ε and κ and then construct the shell HW functional of the form analogous to F˜ 5 of eq. (26.5). This form will naturally include the shell strain fields ε∗ and κ∗ instead of the 3D strain E∗ as well as the shell force and couple fields N∗ and M∗ instead of the 3D stress S∗ .
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26.3 Shell with Drilling Rotation Reissner’s shell kinematics. The position vector of an arbitrary point of a shell in the initial configuration is expressed as y(ζ) = y0 + ζ t3 , where y0 is a position of the reference surface and t3 is the shell director, normal to the reference surface. Besides, ζ ∈ [−h/2, +h/2] is the coordinate in the direction normal to the reference surface, where h denotes the initial shell thickness. z, x3 director t3 0
ta
reference surface
y
a
x
y0
i3 0
i1
i2
Fig. 26.1 Normal coordinates at a shell cross-section for initial configuration
In the current configuration, the position vector is expressed by the Reissner’s kinematical hypothesis, x(ζ) = x0 + ζ Q0 t3 , (26.7) where x0 is a position of the reference surface and Q0 ∈ S O(3) is a rotation tensor, which is parameterized by the canonical rotation vector ψ as follows Q0 (ψ) I +
sin ω ˜ 1 − cosω ˜ 2 ψ+ ψ , ω ω2
(26.8)
√ where ω = ψ = ψ · ψ ≥ 0 and the skew-symmetric ψ˜ ψ × I. The rotation tensor Q0 is constant over ζ. Note that all the vectors and tensors used above are functions of the natural coordinates ξ, η ∈ [−1, +1], which parameterize the reference (middle) surface locally, i.e. within a single finite element. The deformation function χ : x = χ(y) maps the reference configuration of the shell onto the current (deformed) one, and the deformation gradient is obtained as ∂x ∂x ∂y F = ∂y ∂ξk ∂ξk
−1
,
k = 1, 2, 3,
(26.9)
where ξ1 ξ, ξ2 η, ξ3 = 2ζ/h, and ξk ∈ [−1, +1]. We can split the deformation gradient F into the constant and linear parts in ζ as follows: F = F0 + ζF1 . Then,
26 Recent Improvements in Hu-Washizu Shell Elements with Drilling Rotation
FT F = FT0 F0 + ζ (FT0 F1 + FT1 F0 ) + ..., ;<=> ;<=> to ε to κ
397
(26.10)
where we indicated the parts of the Green strain to which these terms contribute. Hence, for shells, the 3D Green strain is split as E(ζ) = ε + ζκ, where ε and κ are the shell strains. The Green strain can be transformed from the global reference basis to the local Cartesian basis at the element’s center {tck }, 1 E (RT0c FT F R0c − I), 2
(26.11)
where R0c ∈ S O(3) is the orientation of the vectors tck , i.e. at the element’s center. Drilling Rotation Constraint. The drilling rotation ω is defined as the rotation about the vector normal to the reference surface t3 , see Fig. 26.2, so the rotation vector is ψ ωt3 .
w t3 Fig. 26.2 Drilling rotation: the axis of rotation is perpendicular to the reference surface
Other features of this rotation are a result of the definition by which it is linked to displacements. We define it using one component of the RC of eq. (26.2) and neglecting the terms which depend on tangent components of the rotation vector. Then, we obtain a single drilling Rotation Constraint (drill RC), 1 c (x0,1 · a2 − x0,2 · a1 ) = 0, 2
(26.12)
where aα Q0 (ω) tα (α = 1, 2) and tα are the initial tangent vectors. A physical interpretation of the drilling rotation ω can be established by considering a planar deformation of a pair of ortho-normal vectors, tα . Each of these vectors is rotated and stretched, t∗1 = Ft1 = λ1 Q1 (β1 )t1 ,
t∗2 = Ft2 = λ2 Q2 (β2 )t2 ,
(26.13)
where λ1 , λ2 > 0 are scalar stretch parameters, and Q1 , Q2 are rotation tensors, each depending on one rotation angle, βα , see Fig. 26.3. Using eq. (26.13) and the drill RC equation, we obtain 1 ω ≈ (β1 + β2 ) + kπ, 2
k = 0, . . ., K,
(26.14)
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-b2 l2 t2
t*2 t*1
b1
t1
Fig. 26.3 Deformation of a pair of basis vectors t1 and t2
l1
for cos ω 0, λ1 c1 + λ2 c2 0 and λα ≈ 1, where sα sin βα and cα cos βα . Hence, the drilling angle ω is an average of rotations of vectors tα . This interpretation is valid also for shell, for which ω is a rotation about the director t3 and tα are the vectors tangent to the reference surface, for details see [18], p. 26.
26.4 Features of Enhanced HW Shell Elements 26.4.1 Skew Coordinates To define the assumed representations of stress and strain in the HW four-node elements, we use the skew coordinates instead of the natural ones, see [21] and [22]. This modification slightly improves accuracy of mixed elements and renders that the homogenous equilibrium equations and the compatibility condition are satisfied point-wise for elements of arbitrary shape. Natural basis at element’s center. The position vector of a four-node element in the initial configuration is approximated as follows: y(ξ, η) =
4
NI (ξ, η) yI ,
I=1
1 NI (ξ, η) (1 + ξI ξ) (1 + ηI η), 4
(26.15)
where NI are the standard bi-linear shape functions, ξ, η ∈ [−1, +1] are the natural coordinates and {ξI , ηI } are the coordinates of nodes I = 1, ..., 4. The vectors of the natural basis are defined as ∂y(ξ, η) ∂y(ξ, η) , g2 (ξ, η) , (26.16) ∂ξ ∂η
and, at the element’s center, gc1 g1
ξ,η=0 and gc2 g2
ξ,η=0 . In general, gc1 and gc2 are neither unit nor mutually orthogonal, see Fig. 26.4. g1 (ξ, η)
26 Recent Improvements in Hu-Washizu Shell Elements with Drilling Rotation
399
In the reference Cartesian basis {i1 , i2 }, the position vector is expressed as y = xi1 + yi2 . The Jacobian matrix is defined as ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ ∂x ∂x ⎥⎥⎥ ⎢⎢⎢ g1 · i1 g2 · i1 ⎥⎥⎥ ∂ξ ∂η ⎥⎥⎥ , J ⎢⎢⎢⎢⎣ ∂y ∂y ⎥⎥⎥⎥⎦ = ⎢⎢⎢⎢⎣ (26.17) ⎥⎦ g · i g · i 1 2 2 2 ∂ξ ∂η and, at the element’s center, Jc J|ξ,η=0 . h 1 c 2
g
x c(0,0)
Fig. 26.4 Natural basis at the element’s center, and the ’fictitious’ parallelogram (dotted line) obtained for y¯ = ξ gc1 + η gc2
A(1,1)
g1c
1
Skew coordinates. It is a common belief that the natural coordinates ξ, η ∈ [−1, +1] are associated with the natural basis at the element’s center {gc1 , gc2 }, see Fig. 26.4. However, this is true only for parallelograms but not for irregular (trapezoidal) shapes. The coordinates which are associated with {gc1 , gc2 } are designated as “skew” and are derived below. Consider the position vector relative to the element’s center, and express it as y¯ y − yc = xS gc1 + yS gc2 ,
(26.18)
where xS , yS are the skew coordinates. Below we express the skew coordinates in terms of the natural coordinates. First, we find the relation between the skew coordinates and the Cartesian coordinates {x, y} associated with the reference basis {i1 , i2 }. The position vector can be written in the two basis as follows: x¯ i1 + y¯ i2 = xS gc1 + yS gc2 .
(26.19)
Taking a scalar product of this equation with the vectors i1 and i2 , we obtain two equations, which can be solved for the skew coordinates, ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ xS ⎪ ⎬ −1 ⎨ x¯ ⎬ = J , (26.20) ⎪ ⎪ c ⎪ ⎪ ⎪ ⎪ ⎩ yS ⎪ ⎭ ⎩ y¯ ⎪ ⎭ where Jc is the Jacobian of eq. (26.17) at the element’s center.
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Next, we find the relation between the skew coordinates and the natural coordinates. By approximations of eq. (26.15), we have ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x¯ ⎪ ⎬ ⎪ ⎨ x − a0 ⎪ ⎬ ⎪ ⎨ a1 ξ + a2η + a3 ξη ⎪ ⎬ ⎪ =⎪ , (26.21) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y¯ ⎪ ⎭ ⎪ ⎩ y − b0 ⎪ ⎭ ⎪ ⎩ b1 ξ + b2η + b3 ξη ⎪ ⎭ where the coefficients ai , bi are functions of the nodal positions, and a0 , b0 are coordinates of the element’s center. Using this in eq. (26.20), we obtain ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ xS ⎪ ⎬ ⎪ ⎨ ξ + A ξη ⎪ ⎬ =⎪ , (26.22) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ yS ⎭ ⎩ η + B ξη ⎪ ⎭ where A and B are functions of ai , bi . Finally, the coefficients A and B can be expressed in terms of the determinant of the Jacobian J of eq. (26.17), which, in the current notation, is ⎡ ⎤ ⎢⎢⎢ a1 + a3 η a2 + a3 ξ ⎥⎥⎥ ⎥⎥⎥⎥ . J = ⎢⎢⎢⎢⎣ (26.23) ⎦ b1 + b 3 η b 2 + b2 ξ The determinant of the Jacobian matrix can be expanded as follows: det J = jc + ( j,ξ )c ξ + ( j,η )c η,
(26.24)
where jc = a1 b2 − a2 b1 , ( j,ξ )c = a1 b3 − a3 b1 , and ( j,η )c = a3 b2 − a2 b3 . It can be verified that the coefficients A and B can be expressed as A=
( j,η )c , jc
B=
( j,ξ )c , jc
(26.25)
see [21] for details. Finally, the skew coordinates are expressed in terms of the natural coordinates by Eqs (26.22) and (26.25). The skew coordinates can be easily computed and improve accuracy for the mixed (HW and HR) elements of irregular shape. Remark. For parallelograms, ( j,ξ )c = ( j,η )c = 0, so A = B = 0, and, then, the skew coordinates are equal to the natural coordinates. If we apply a formula with the natural coordinates, i.e. y¯ = ξ gc1 + η gc2 , to an irregular trapezoidal element then a ‘fictitious’ parallelogram shown in Fig. 26.4 is obtained. On the other hand, eq. (26.18), which uses the skew coordinates, reproduces the element’s shape exactly.
26 Recent Improvements in Hu-Washizu Shell Elements with Drilling Rotation
401
26.4.2 Assumed Stress and Strain Let take as the starting point, the shell counterpart of the functional of eq. (26.6), with the shell strains ε∗ and κ∗ instead of the 3D strain E∗ as well as the shell force and couple fields N∗ and M∗ instead of the 3D stress S∗ . We treat all the fields marked by asterisks as the assumed fields, N∗ → Na ,
M∗ → Ma ,
ε∗ → εa ,
κ ∗ → κa ,
(26.26)
which is indicated by the superscript “a”. The assumed fields are defined as follows. We define the assumed stress and strain in the natural basis at the element’s center {gck }, σa = σkl gck ⊗ gcl ,
εa = εkl gck ⊗ gcl ,
k, l = 1, 2,
(26.27)
where σkl and εkl are the contra-variant components of stress and strain, respectively. The representations of these components are assumed as polynomials of the skew coordinates xS , y s and local (elemental) parameters qi . Denote the matrices of assumed components as σξ [σkl ], and εξ [εkl ]. They are transformed to the Cartesian reference basis using the transformation rule for contravariant components, σre f = Jc σξ JTc , εre f = Jc εξ JTc , (26.28) where Jc is the Jacobian matrix J of eq. (26.17) at the element’s center. Remark. In [22], the components of strain were assumed in the co-basis {gkc }, εa = εkl gkc ⊗ glc ,
k, l = 1, 2,
(26.29)
where the co-vectors gkc are defined in the standard way by gkc · gcl = δkl . The matrix of assumed components, εξ [εkl ], was transformed to the Cartesian reference basis as follows: −1 εre f = J−T (26.30) c εξ Jc , which is the transformation rule for covariant components. For the contravariant assumed stress and the covariant assumed strain, the scalar product of them does not depend on J, σre f · εre f = tr (J σξ JT J−T εTξ J−1 ) = σξ · εξ ,
(26.31)
or, in other words, is identical regardless of the element’s shape. Then, the accuracy: (a) deteriorates for 7p or 5p strain representations, and (b) is unchanged for the 9p strain representation, see [23]. Hence, in the sequel, we use the contra-variant rule, which is suitable for all tested strain representations. The following representations were selected for the shell elements developed and tested in this paper. They all are expressed in terms of the skew coordinates xS , yS .
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1. For the assumed stress, we use the representation, ⎡ ⎢⎢⎢ q1 + q2 yS q5 + q 6 x S + q7 y S ξ 7p: σ ⎢⎢⎢⎢⎣ sym. q3 + q4 xS
⎤ ⎥⎥⎥ ⎥⎥⎥ . ⎥⎦
(26.32)
Comparing to the well-known 5p PS representation of [11], there are two differences: the off-diagonal terms are not constant but linear in xS , yS and it is written in the skew coordinates. Another 7p representation of stress is, ⎡ ⎤ ⎢⎢⎢ q1 + q2 yS + q6 xS q5 − q7 xS − q6 yS ⎥⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ , (26.33) ⎢⎣ sym. q3 + q4 xS + q7 yS ⎦ in which the parameters q6 and q7 appear in the diagonal and off-diagonal terms. In [21], the linking transformations for several forms of this representation existing in the literature is derived. As indicate our tests for the elements with drilling rotation, the representation of eq. (26.33) is less accurate than the representation of eq. (26.32), which we use. 2. For the assumed strain, we use two representations: ⎡ ⎢⎢⎢ q1 + q2 yS + q3 xS q7 + q 8 x S + q9 y S 9p: εξ ⎢⎢⎢⎢⎣ sym. q4 + q 5 x S + q6 y S ⎡ ⎤ ⎢⎢⎢ q1 + q2 yS ⎥⎥⎥ q5 ⎢ ⎥⎥⎥ . ξ 5p: ε ⎢⎢⎢⎣ ⎥ sym. q3 + q4 xS ⎦
⎤ ⎥⎥⎥ ⎥⎥⎥ , ⎥⎦
(26.34)
(26.35)
In the 9p representation, all components are linear in xS , yS , while the 5p representation is analogous to this used in [11] for stress, but expressed in the skew coordinates. Remark. We verify the assumed representations as follows. For the assumed representations of stress, we check whether they satisfy the homogenous equilibrium equations, ∂σxy ∂σyy ∂σxx ∂σxy + = 0, + = 0, (26.36) ∂x ∂y ∂x ∂y while for the assumed representations of strain, we check whether they satisfy the compatibility condition, ∂2ε xy ∂2 ε xx ∂2 εyy + = 2 . ∂x ∂y ∂y2 ∂x2
(26.37)
We verify these equations symbolically, see [21] and [22] for details. For representations in the skew coordinates, they are satisfied point-wise, even for irregular trapezoidal element’s shapes. For the same representations but in the natural coordinates, these equations are point-wise satisfied only for parallelograms.
26 Recent Improvements in Hu-Washizu Shell Elements with Drilling Rotation
403
26.4.3 Enhanced Assumed Displacement Gradient (EADG) Method For the elements with drilling rotation, it is more suitable to enhance the displacement gradient than the strain, because the former affects also the drill RC. The gradient of compatible displacements uc is enhanced additively by the ˜ so the deformation gradient becomes matrix H, ˜ F ∇χ + H,
(26.38)
˜ is constructed using the transforwhere ∇χ = I + ∇uc . The enhancing matrix H mation −1 jc ˜ H J c G Jc , (26.39) j where G is the matrix of assumed enhancing modes and j det J. The EADG method is a generalization of the Incompatible Displacement method of [16] and [14], and was proposed for 2D elements in [13]. We find it beneficial for elements with the drilling rotation. For shells, the transformation (26.39) is modified, for details see [24]. The matrix G of the assumed enhancing modes corresponds to the selected 7p representation of the assumed stress, and, in the current paper, it is as follows: ⎡ ⎤ ⎢⎢⎢ 0 η q1 ⎥⎥⎥ ⎥⎥⎥⎥ . 2p, EADG2: G = ⎢⎢⎢⎢⎣ (26.40) ξ q2 0 ⎦ Besides, we also use 4p, EADG2k:
⎡ ⎤ ⎢⎢⎢ ⎥⎥⎥ 0 η (q + ζq ) 1 2 ⎥⎥⎥ , G = ⎢⎢⎢⎢⎣ ⎥⎦ ξ (q3 + ζq4) 0
(26.41)
which is a sum of the EADG2 enhancement plus an analogous enhancement depending on the coordinate ζ ∈ [−h/2, +h/2].
26.4.4 Approximation of Drilling RC For the equal-order bi-linear approximations of displacements and the drilling rotation, the drill RC is incorrectly approximated. We have already discussed this issue in [17], but here we provide an additional refinement of the method. Consider a 2D problem with drilling rotation, and denote the coefficients of the bilinear shape function of the tangent displacements and the drilling rotation as ui , vi , ωi (i = 0, . . ., 3). These coefficients are functions of nodal values of respective components.
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The linearized form of the drill RC for a bi-unit square element is as follows: c ω + 12 (u,η − v,ξ ) = 0, and, for the equal-order bilinear approximations, it has the form 1 1 1 c ω0 + (u2 − v1 ) + ω1 + u3 ξ + ω2 − v3 η + ω3 ξη = 0. (26.42) 2 2 2 We see that the ξη-term contains only a rotational parameter but not displacement ones, so the displacement and rotational parameters are not linked, which is incorrect. For this reason, we omit this term using a linear expansion of the drill RC at the element’s center. The sampling at mid-side points of element’s edges can also be used, see [18], p. 318.
+1
-1 Mode Q2
Fig. 26.5 Spurious mode Θ2 for drilling rotation
-1
+1
Omitting of the ξη-term renders that the tangent matrix for the drill RC has one spurious zero eigenvalue, which we eliminate using the stabilization function in the penalty form, 1 Θ2 (ω1 − ω2 + ω3 − ω4 ), 4
P2 = 10−3 G V Θ22 ,
(26.43)
where V is the element volume, Θ2 is the mode shown in Fig. 26.5. This stabilization function was proposed in [9] for the Allman-type quadrilaterals, and it provides the stabilization matrix which has 1 non-zero eigenvalue. This simple stabilization is sufficient for rectangular elements, but for irregular shapes we can improve it as follows. Let us re-write the spurious mode as 1 Θ2 h ωTI , 4
(26.44)
where ωI {ω1 , ω2 , ω3 , ω4 } is the vector of nodal drilling rotations, and h {1, −1, 1, −1} is the hourglass vector. Then, instead of 14 h we can use the γ-vector, identical as this used in the gamma-stabilization of the one-integration point elements. The gamma-stabilization method stems from [8], was proposed in [6] and, afterwards, refined in numerous papers. Using the gamma-stabilization method, the mode is re-defined as follows: Θ2 γ ωTI ,
γ
1 h − (hS1 ) b1 − (hS2 ) b2 , 4
(26.45)
26 Recent Improvements in Hu-Washizu Shell Elements with Drilling Rotation
405
where b1 =
1 (ηS2 ) ξ − (ξS2 ) η , 4A ξ {−1, 1, 1, −1},
b2 =
1 −(ηS1 ) ξ + (ξS1 ) η , 4A
η {−1, −1, 1, 1}.
Besides, Sα are the vectors of projections of nodal relative position vectors on tα and A = 4 jc . This modification of the stabilization is relatively simple and improves performance for irregular elements.
26.4.5 Approximation of Lagrange Multiplier for Drill RC The Lagrange multiplier of the functional FRC of eq. (26.6) is assumed in the natural basis of the element center, {gck }, and transformed to the local ortho-normal basis {tck } using the transformation rule for contra-variant components ⎡ ⎤ ⎢⎢⎢ 0 T a ⎥⎥⎥ ⎢ ⎥⎥⎥ JT , T = J Lc ⎢⎢⎢⎣ ⎥ Lc a −T 0 ⎦ a
(26.46)
where the assumed representation for the Lagrange multiplier is as follows: T a (ξ, η) q15 + ξ q16 + η q17 ,
(26.47)
where J Lc is the local Jacobian at element’s center. This representation was selected in [17], as optimal for non-linear in-plane bending.
26.5 Numerical Tests In this section, we describe numerical tests of 4 enhanced HW four-node shell elements with 6 dofs/node listed in Table 26.1. They are developed from the enhanced HW functional and use the assumed representations in terms of skew coordinates, see Sect. 26.4.2. In the HW element’s indicator, the total number of elemental parameters is included. In the sequel, “parameter” is abbreviated to “p”. All the tested HW elements have an identical membrane part, with 7p Na , 9p and the 2p enhancement EADG2, which we selected as optimal for the 2D HW elements with the drilling rotation. The motivation for this form of the membrane part stems from the Cook’s membrane example, see Sect. 26.5.1. This membrane part is different than for the elements without drilling rotation! εa
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Table 26.1 Characteristics of tested shell elements. Drill RC by PL with γ = G Element
Assumed stresses
Assumed strains
Enhancement
7p Na , 7p Ma , 4p Qa
9p εa , 9p κa , 4p εaα3
4p EADG2k
a
9p εa , 5p κa , 4p εaα3
2p EADG2
9p εa , 5p κa
2p EADG2
Enhanced HW in skew coord. HW47
a
a
HW39
7p N , 5p M , 4p Q
HW31
7p Na , 5p Ma
HW29
7p Na ,
9p εa ,
4p Qa
4p εaα3
2p EADG2
Enhanced PE EADG4
-
-
4p EADG4
EADG5A
-
-
5p EADG5A
The HW47 element has all the features included, i.e. its bending/twisting part is fully analogous to the membrane part, i.e. 7p Ma , 9p κa and the 2p EADG2 ζ-dependent enhancement. Besides, its transverse shear part is treated by the HW functional with 8 (4 stress and 4 strain) parameters. It is a very good element, which can be used for reference when the accuracy and convergence properties are considered. The other 3 elements have a reduced number of parameters and differ from the HW47 element in the bending/twisting and/or transverse shear parts. 1. The HW39 element has a simplified bending/twisting part, based on 5p Ma and 5p κa . No enhancement is used for this part. The transverse shear part is identical as in the HW47 element. 2. The HW31 element has the bending/twisting part identical as the HW39 element, but the transverse shear part is standard, i.e. derived from the (non-enhanced) potential energy. 3. The HW29 element has a standard bending/twisting part, i.e. derived from the (non-enhanced) potential energy. The transverse shear part is identical as in the HW47 element. For comparison, we use the EADG4 and EADG5A elements developed from the potential energy (PE) with the enhancement of the displacement gradient implemented as in Eqs (26.38) and (26.39), where ⎡ ⎢⎢⎢ q1 ξ ⎢ G ⎢⎢⎢⎢ ⎣ q2 ξ + q5 ξη
⎤ q3 η + q5 ξη ⎥⎥⎥⎥ ⎥⎥⎥ . ⎥⎦ q η
(26.48)
4
The underlined bi-linear mode is used only in the EADG5A element and it improves performance in the pinched hemisphere example of Sect. 26.5.2. In the EADG4 element, this mode is not used. In all (HW and EADG) elements, an identical form of the Perturbed Lagrange (PL) method is used to enforce the drill RC. It has the following features: (1) the 3-parameter Lagrange multiplier T is used, and (2) gamma-stabilization is applied to eliminate one spurious mode, see Sect. 26.4.4.
26 Recent Improvements in Hu-Washizu Shell Elements with Drilling Rotation
407
All elements use the Green strain, and the multipliers of additional modes are eliminated on the element’s level and updated by the scheme U2, see [18]. The 2 × 2 Gauss integration is used in all elements. For the transverse shear strains, the ANS technique of [2] is used. All tested elements have a correct number of zero eigenvalues and pass the constant strain patch tests, also with unconstrained boundary conditions for the drilling rotation, see [20].
26.5.1 Cook’s Membrane
P
44
44
A
16
This test is essential in evaluating the quality of a membrane part of a shell element, as in this test the shear deformation dominates and the elements are skew and tapered. The membrane is clamped at one end, while at the other end, the uniformly distributed tangent load P = 1 is applied, see Fig. 26.6. The data is as follows: E = 1, ν = 1/3, and the thickness h = 1. This test was proposed in [3]. Two meshes are used in computations; a coarse 2 × 2-element mesh and a fine 32 × 32-element mesh. The regularizing parameter γ = G is used.
y
Fig. 26.6 Cook’s membrane. Initial geometry and load
x 48
For the 2D+drill and 2D elements, the vertical displacement at point A is compared in Table 26.2, and the conclusions are as follows: 1. Comparing the 2D+drill elements to the 2D elements, we can evaluate the quality of implementation of the drilling RC, which should not disturb the displacements, as stipulated by eq.(26.1). We see that all the elements with the drilling rotation are slightly locked comparing to their 2D counterparts. If we used the 5p stress for the 2D+drill elements then the results would be much more locked than the results presented here for the 7p stress and the EADG2 enhancement. Note that we do not manipulate the value of the regularizing parameter γ; the
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Table 26.2 Cook’s membrane. Linear test of 2D and 2D+drill elements Element
Number of
Displacement uy at A
characteristics
strain modes
Mesh 2 × 2
Mesh 32 × 32
HW, 2D+drill,
9
21.317
23.936
7p stress + EADG2
7
20.722
23.932
5
20.713
23.931
9
21.353
23.940
7
21.177
23.939
5
20.751
23.936
EADG4, 2D+drill
4
21.015
23.936
EADG4, 2D
4
21.050
23.940
HW, 2D, 5p stress
lower value would improve the results but we would not be able to evaluate the quality of approximations. 2. Comparing 2D+drill HW elements to the reference 2D+drill EADG4 element, i.e. with the enhanced displacement gradient element with 4 modes, we see that only the HW element using the 9p assumed strain is more accurate than the EADG4 element. For this reason, we use 9p assumed strain also to the membrane part of the HW shell elements. For the shell elements, the results of the linear analysis are presented in Table 26.3, where the vertical displacements and the drilling rotation at point A are given. All the tested HW shell elements perform identically and displacements are more accurate than by the EADG elements. Note that the drilling rotation converges from the above for both types of elements, which can be attributed to the Perturbed Lagrange method used to the drill RC. Table 26.3 Cook’s membrane. Linear test of shell HW elements. γ = G Element
Mesh 2 × 2
Mesh 32 × 32
uy
ω
uy
ω
HW, all
21.317
0.925
23.936
0.891
EADG5A
21.017
0.908
23.936
0.891
EADG4
21.015
0.908
23.936
0.891
To compare the time of generation of the tangent matrix for the tested elements (without solving the equations), we used a fine 300 × 300-element mesh, which yields 90 thousands elements and about 542 thousands unknowns. The times obtained for particular elements divided by the time for the EADG4 element are presented in Table 26.4, and they indicate that the HW elements are relatively slow. We checked that the condensation procedure used during the
26 Recent Improvements in Hu-Washizu Shell Elements with Drilling Rotation
409
generation of the tangent matrix consumes up to 15% of time for the HW47 element. Hence, even with a very effective condensation of the elemental parameters, the element still will be more than 2 times slower than the EADG4 element. Table 26.4 Cook’s membrane. Time of generation of tangent matrix for 300 × 300-element mesh Element Time
HW47
HW39
HW31
HW29
EADG5A
EADG4
2.56
2.08
1.90
1.88
1.06
1.00
26.5.2 Pinched Hemisphere with Hole The hemispherical shell with an 18o hole is loaded by two pairs of equal but opposite external forces, see Fig. 26.7. The shell undergoes an almost in-extensional deformation, and a membrane locking can destroy the solution. For the shown FE mesh, the elements are flat and trapezoidal. Due to the double symmetry, only a quarter of the shell is modeled, and the 16 × 16-element mesh is used. The data is as follows: E = 6.825 × 107, ν = 0.3, the thickness h = 0.01. The shell is very thin, h/R = 0.001. 18 R=10 symm. symm.
z x
y
P. P.
free
Fig. 26.7 Pinched hemisphere. a) Geometry and boundary conditions. b) Deformed shape
This example was computed using our four-node shell elements, and the nonlinear analysis was performed using the Newton method. The solution curves for the inward displacement under the force are shown in Fig. 26.8. We see that the curves for the HW47, HW39, HW31 and HW29 element almost coincide. The EADG4 element yields a locked solution but the EADG5A elements is as accurate as the HW elements. Note that the difference between the EADG4 and EADG5 elements is caused by the bi-linear terms in matrix G of eq. (26.48), because in all other aspects these elements are identical.
410
K. Wi´sniewski and E. Turska Green strain, mesh 16x16, h=0.01 5 HW47 HW39 HW31 HW29 EADG5A EADG4
4
Load
3
2
1
0 0
1
2
3
4
5
6
7
8
9
Inward displacement at force P
Fig. 26.8 Pinched hemispherical shell. Inward displacement at force
Another question is the overall effectiveness (speed and robustness) of these elements in non-linear analyzes, which depends not only on the time of a single iteration but also on the radius and rate of convergence. To illustrate the difference between the elements, we perform two simple analysis: 1. One non-linear step. In this test, we seek the maximum load for which the Newton method converges, checking every P = 0.05. The results are presented in Table 26.5, where the maximum load and the number of Newton’s iterations are given. We see that the HW elements converge much faster and in a much smaller number of iterations than the EADG elements. The worst of the HW elements is HW31. Table 26.5 Pinched hemisphere with hole. One non-linear step Element Max P to converge No. of iterations
HW47
HW39
HW31
HW29
EADG5A
EADG4
0.80
0.80
0.55
0.80
0.40
0.40
10
10
10
9
19
20
2. Several non-linear steps up to the load P = 8.8. In this test, we account for the fact that the elements converge for different load increments. Thus, we perform 11 Newton’s steps for the elements which converge for P = 0.80, 16 steps for this which converges for P = 0.55, and 22 steps for these which converge for P = 0.40. The final load is P = 8.8 for all elements. The time divided by the time for the EADG4 element and the total number of iterations to convergence are presented in Table 26.6. We see that all
26 Recent Improvements in Hu-Washizu Shell Elements with Drilling Rotation
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HW elements are about two times faster than the EADG4 element, and HW29 is the fastest. Table 26.6 Pinched hemisphere with hole. Several non-linear steps up to P = 8.8 Element P No. of steps No. of iterations Time
HW47
HW39
HW31
HW29
EADG5A
EADG4
0.80
0.80
0.55
0.80
0.40
0.40
11
11
16
11
22
22
62
62
89
61
204
193
0.65
0.55
0.71
0.47
1.11
1.00
26.6 Final Remarks Four HW shell elements, with the same membrane part but differing in the bending/twisting part and/or the transverse shear part, are developed and tested. The results of numerical tests indicate that: i The comparison of the HW elements among themselves, shows that the HW elements are equally accurate but differ in efficiency; the fastest is the element with the smallest number of modes, HW29. The least efficient is HW31. ii The comparison of the EADG elements, shows that EADG5A is more accurate than EADG4, e.g. as in the pinched hemisphere example. The set of shell examples presented in [15] and [7] indicates that the HW elements outperform the EAS elements in the class of shell elements without the drilling rotation. The examples presented in the current paper for the shell elements with the drilling rotation also seem to support this conclusion. However, a broader set of linear and nonlinear tests is needed, which will be a subject of our next paper, [24]. Acknowledgements This research was partially supported by the Polish Committee for Scientific Research (KBN) under grant No. N501-290234.
References 1. Badur J., Pietraszkiewicz W.: On geometrically non-linear theory of elastic shells derived from pseudo-Cosserat continuum with constrained micro-rotations. In: Pietraszkiewicz W. (ed.) Finite Rotations in Structural Mechanics, 19–32, Springer, Berlin, 1986 2. Bathe K-J., Dvorkin E.N.: A four-node plate bending element based on Mindlin-Reissner plate theory and mixed interpolation. Int. J. Num. Meth. Engng., Vol. 21, 367–383 (1985)
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3. Cook R.D.: A plane hybrid element with rotational d.o.f. and adjustable stiffness. Int. J. Num. Meth. Engng., Vol. 24, 1499–1508 (1987) 4. Chr´os´cielewski J., Makowski J., Stumpf H.: Genuinely resultant shell finite elements accounting for geometric and material nonlinearity. Int. J. Num. Meth. Engng., Vol. 35, 63–94 (1992) 5. Chr´os´cielewski J., Makowski J., Pietraszkiewicz W.: Statics and dynamics of multi-segmented shells. Nonlinear theory and finite element method. IFTR PAS Publisher, Warsaw, 2004 (in Polish) 6. Flanagan D.P., Belytschko T.: A uniform strain hexahedron and quadrilateral with orthogonal hourglass control. Int. J. Num. Meth. Engng., Vol.17, 679–706 (1981) 7. Gruttmann F., Wagner W.: Structural analysis of composite laminates using a mixed hybrid shell element, Comput. Mech., Vol.37, 479–497 (2006) 8. Kosloff D., Frazier G.A., Treatment of hourglass pattern in low order finite element codes. Int. J. Numer. Analyt. Meths. Geomech. Vol.2, 57–72 (1978) 9. MacNeal R.H., Harder R.L.: A refined four-noded membrane element with rotational degrees of freedom. Computers & Structures. Vol. 28, No. 1, 75–84 (1988) 10. Panasz P., Wisniewski K.: Nine-node shell elements with 6 dofs/node based on two-level approximations. Finite Elements in Analysis and Design, Vol. 44, 784–796 (2008) 11. Pian T.H.H., Sumihara K.: Rational approach for assumed stress finite elements. Int. J. Num. Meth. Engng., Vol.20, 1685–1695 (1984) 12. Piltner R., Taylor R.L.: A quadrilateral mixed finite element with two enhanced strain modes. Int. J. Num. Meth. Engng., Vol.38, 1783–1808 (1995) 13. Simo J.C., Armero F.: Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int. J. Num. Meth. Engng., Vol. 33, 1413–1449 (1992) 14. Taylor R.L., Beresford P.J., Wilson E.L.: A non-conforming element for stress analysis. Int. J. Num. Meth. Engng., Vol. 10, 1211–1220 (1976) 15. Wagner W., Gruttmann F.: A robust nonlinear mixed hybrid quadrilateral shell element. Int. J. Num. Meth. Engng., Vol.64, 635-666 (2005) 16. Wilson E.L., Taylor R.L., Doherty W.P., Ghaboussi J.: Incompatible displacement models. In: Fenves S.J., Perrone N., Robinson A.R., Schnobrich W.C. (eds.) “Numerical and Computer Methods in Finite Element Analysis”. Academic Press, New York, 43–57 (1973) 17. Wisniewski K.: Recent Improvements in formulation of mixed and mixed/enhanced shell elements. SHELL STRUCTURES: THEORY AND APPLICATIONS. Proceedings of 9th Conference SSTA’2009, Jurata, October 14-16, 2009. (W. Pietraszkiewicz, I. Kreja, Eds) General Lecture, p.35-44, ISBN 978-0-415-54883-0, Taylor & Francis, 2010 18. Wisniewski K.: Finite Rotation Shells. Basic Equations and Finite Elements for Reissner Kinematics. CIMNE-Springer, 2010 19. Wisniewski K., Turska E.: Kinematics of finite rotation shells with in-plane twist parameter. Comput. Methods Appl. Mech. Engng., Vol. 190, No. 8-10, 1117–1135 (2000) 20. Wisniewski K., Turska E.: Enhanced Allman quadrilateral for finite drilling rotations. Comput. Methods Appl. Mech. Engng., Vol. 195, No. 44-47, 6086–6109 (2006) 21. Wisniewski K., Turska E.: Improved four-node Hellinger-Reissner elements based on skew coordinates. Int. J. Num. Meth. Engng., Vol.76, 798–836 (2008) 22. Wisniewski K., Turska E.: Improved four-node Hu-Washizu elements based on skew coordinates. Computers & Structures, Vol.87, 407–424 (2009) 23. Wisniewski K., Wagner W., Turska E., Gruttmann F.: Four-node Hu-Washizu elements based on skew coordinates and contravariant assumed strain. Computers & Structures, Vol.88, 1278–1284 (2010) 24. Wisniewski K., Turska E.: Four-node Hu-Washizu shell elements with drilling rotation. In preparation, 2011
Part V
Engineering Design
Chapter 27
Dynamic Analysis of Debonded Sandwich Plates with Flexible Core – Numerical Aspects and Simulation Vyacheslav N. Burlayenko and Tomasz Sadowski
Abstract Although significant work has been done in modeling sandwich panels, models for debonded sandwich plates with a flexible core, especially the vibration analysis, are at their infancy, and it will be the main focus of this paper. This study deals with a finite element (FE) analysis of vibrations of flexible core sandwich plates that are weakened by damage embedded along the face sheet-to-core interface. The FE model developed is based on a refined general-purpose sandwich panel theory, where the first order shear deformation theory and assumptions of the 3-D elasticity theory are used for modeling the face sheets and the core, respectively. The FE mesh contains continuum shell elements for each of the face sheet layers and 3-D brick elements for the core. The comparison of the FE predictions to those known experimental and analytical results allow us to estimate the accuracy of the FE model developed, as well as to find the influence of the geometrical nonlinearity of the flexible core in vibrating and the contact nonlinearity caused by debonding on dynamics of sandwich plates. Keywords Sandwich plate · Debonding · Vibration · Finite element method
27.1 Introduction Composite sandwich panels are increasingly being utilized as load-carrying components in aircraft, aerospace and marine structures. A standard sandwich panel is commonly composed of two strong outer faces (face sheets) and a low-strength inner V. N. Burlayenko (B) Lublin University of Technology, 40 Nadbystrzycka str., Lublin, 20-618, Poland and National Technical University KhPI, 21 Frunze str., 61002, Ukraine e-mail:
[email protected] T. Sadowski Lublin University of Technology, 40 Nadbystrzycka str., Lublin, 20-618, Poland e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 27, © Springer-Verlag Berlin Heidelberg 2011
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core. In a continuing effort for attaining higher stiffness and strength to weight ratios, the constituent sandwich materials have evolved from metallic face sheets and aluminum honeycomb core to composite materials for the face sheets, and non-metallic honeycombs and plastic foams for the core [1]. Thereby, advanced sandwich-type constructions imply the presence of a thick orthotropic core with bonded anisotropic composite laminated face sheets, where the ratio of Young’s moduli of the face sheets to the core lies between 500 and 1000. According with the sandwich structural concept, the top and bottom face sheets carry the in-plane and bending loads and are interacting through the core. The core should be strong enough in order to keep the desired distance between the face sheets and to prevent their sliding with respect to each other [2]. This ability of the core is achieved by its transverse shear strength. However, the novel sandwich panels made of the soft cores are flexible in the through-the-thickness direction. This flexibility conduces to the core compressibility and, as a result, to the change in the height of the core, Fig. 27.1. Such a transverse compression may arise due to a locally distributed load on the face sheets or unequal displacements of the face sheets from bending. Therefore, a normal transverse stress will appear apart from transverse stresses in the core. So, the plane section of the core takes on a nonlinear pattern, and a sandwich structure manifests a nonlinear response to a variety of load conditions. Thereby, the classical sandwich panel theories are insufficient to capture existing nonlinearities. One of the topics, playing a major role in the design of flat and curved composite sandwich-type panels is their vibration behavior. A good understanding of the vibration behavior of such structures is of crucial importance toward a reliable prediction of their dynamic response to time-dependent external excitations, prevention of the occurrence of the resonance, and for optimal design from the vibrational point of view [3]. Furthermore, during the service life sandwich panels often encounter low-velocity impacts, such as tool drop, runway stones, etc. Even though a visual examination of the impacted surface may reveal very little damage, known as barely visible impact damage, significant damage might exist between the face sheet and the core [4]. Manufacturing defects such as an incomplete bonding of the constituent layers or the air entrapping into the resin reach layer during the curing process may also lead to debonding existence as well. This type of damage invokes a
h1
h1/2 facesheet 1
hc/2
core undeformed section facesheet 2
hc/2
h2
h2/2
j01
u01 z1 ,w01
u1
w1
deformed section
x, u0c
zc ,w0c
uc
u02 z2 ,w02
u2 j02
wс
w2
Fig. 27.1 Cross section deformation pattern of a sandwich plate with a flexible core
27 Dynamic Analysis of Debonded Sandwich Plates with Flexible Core
417
substantial reduction of the compressive and bending strengths of the sandwich construction [5]. To ensure the structural integrity after the local striking or with the local faults, the ability of the sandwich structure to retain the load carrying capacity in the presence of the imperfections must be examined and quantified. In general, to analyze sandwich structures with a perfect interface, one of two approaches are commonly used, that is either the equivalent single layer (ESL) approach or the layer-wise (LW) one. Theories based on ESL are a straightforward extension on a sandwich panel regarded as a 2-D equivalent single layer of either the Kirchhoff’s classical theory (CPT) or the Reissner-Mindlin’s first-order shear deformation theory (FSDT) or those higher order theories (HSDT), assuming the displacements in form of continuously differentiable function of the thickness coordinate [6]. However, though ESL models are able to predict well the global dynamic behavior of thin and moderately thick sandwich panels, they still cannot account for discontinuities in displacements field and stresses at the face sheet-to-core interface. In this case the analysis of sandwich panels requires the use of layer-wise theories, where each layer is considered separately. By assuming a unique displacement field for each layer and fulfilling the interface kinematic continuity conditions, a sandwich panel is analyzed [7]. Recently, special theories, such as the high-order sandwich panel theory (HSAPT) [8] and the mixed layer-wise theory [9] have been developed. Herewith, in these theories, the formulation uses either CPT or FSDT, or HSDT for the face sheets and a 3-D elasticity theory or equivalent one for the core. Different such theories are required to model any kind of flat, cylindrical and spherical sandwich panels, because of using equilibrium approach [10]. Nevertheless, as expected, analytical modeling of sandwich-type panels with discontinues such that debonding is much more intricate than that of perfectly bonded sandwich structures. In contrast to the case of perfect sandwich panels, for which the assumptions are postulated for the structure as a whole, in the case of a sandwich panel with an imperfect interface such assumptions have to impose separately on perfect and debonded its parts and involve both compatibility conditions between those parts and contact conditions associated with local interactions in the debonding zone. Moreover, the dynamic analysis of sandwich panels with a flexible core and containing debonding is complicated by the presence of various types of nonlinearities, some of them are: (i) asymmetry with respect to the mid-surface of the panel inducing global bending-stretching coupling; (ii) contact-impact problem, which induces a local structural behavior along with a global one, and (iii) stress concentration in the vicinity of the discontinuous promoting the debonding growth. Thus, the actual dynamic behavior of such panels can be examined only by nonlinear analysis techniques. A comprehensive summary on modeling techniques used to perform a dynamic analysis of debonded (or delaminated in the case of laminated structures) composite beams, plates and shells is done by Della and Shu [11]. In line with the authors’ comments, from a mathematical point of view, analytical solutions are possible only for a limited class of idealized dynamic problems such 1-D sandwich beams or 2-D plates with through-the-width debonding. Moreover, the nonlinear effects are usually neglected because of linear models applied in most of those studies. There
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have been only few papers studying dynamics of debonded beams including the nonlinearity effects, e.g. [12]. The complexity of the problem have progressively led to the predominance of numerical models based mainly on the finite element method (FEM). Nonlinear FE dynamic analyses of laminated beams have been accomplished, e.g. in [13, 14], where beam elements and 2-D plate elements were utilized, respectively. An analytical approach and FE model of a three-layered sandwich beam accounting for intermittent contact in the modal analysis have been developed in [15]. A further contribution to the nonlinear dynamic behavior of sandwich plates with imperfections can be found in [16], where the case of sandwich plates containing post-impact zone was considered. In this study, a FE modeling approach is used within the commercial package ABAQUS, to develop models of both intact and damaged by debonding sandwich plates with a flexible core and simulate their dynamic responses. The models proposed utilize orthotropic shell/plate elements for the face sheets and continuum either isotropic or anisotropic solid elements for the core. The presence of the transverse deformation and general material orthotropy, coupled with the contact-impact interaction of detached surfaces, makes the dynamic FE analysis rather computationally complex. In this regard, the use of the explicit integration rule provides a benefit for solving the dynamic problem in a computationally cost effective manner. Thereby, to simulate dynamic responses of sandwich plates involving contact-impact conditions in the detached segments of the debonding zone, the ABAQUS/Explicit code [17] is exploited.
27.2 FE Model Developments In fact, in sandwich panels with a flexible core stresses have an inherent threedimensional nature, consequently, from the point of view of accuracy, usually the best approach in terms of FE modeling is to utilize three-dimensional (3-D) elements for each sandwich layer (faces and core). However, this approach will lead to severe problems of aspect ratio, and require an extremely refined mesh and, as a result huge computational efforts. In practice such models are limited only to small and particular tasks [18]. An alternative to such an approach is the employment of finite elements based on partially 3-D models, where the core is represented by solid elements and the face sheets by plate or shell elements [19]. In this case reasonable computational cost is combined with the fidelity of strain-stress states associated with HSAPT that provides a good trade-off between the level of accuracy and computational efficiency. Thus, 3-D models were developed utilizing shell elements, employing the Reissner-Mindlin’s hypotheses in conjunction with the laminated plate theory on the face sheets and solid brick elements, based on the 3-D elasticity theory on the core, Fig. 27.2.
27 Dynamic Analysis of Debonded Sandwich Plates with Flexible Core
Nx
Nxy
Nxy
Ny
Nxy+Nxy,ydy
top face
s 23
419
Nx+Nx,xdx Ny+Ny,ydy Nxy+Nxy,xdx
f
sf22 tf12 sc22 tc12
tf12 t 12 c
s
f s 33 s 13 f
s 33 c
f 11
s 13 c
Mxy
s 23 c
Mx My
Mxy
My +My,ydy
Mxy + Mxy,xdx
sc22
Mxy + Mxy,ydy Mx +Mx,xdx
core
(a)
(b)
Fig. 27.2 Details of the stress state at the face sheet-to-core interface: (a) core and face sheet stresses; (b) force and moment resultants for face sheet
27.2.1 Face Sheet FE Model In general, composite sandwich plates studied in this paper are made of two composite laminated face sheets and a soft core. The faces are considered to be arbitrary thickness and consist of an elastic orthotropic material. For the face sheets it was utilized 8-node continuum shell reduced integrated SC8R finite elements. These elements are positioned on the upper and lower core sides and are directly connected to the core through their share nodes. Fig. 27.3 shows the node numbering, integration point and faces of the used finite element. The SC8R continuum shell elements discretize an entire three-dimensional body, unlike conventional shells which discretize a reference surface. These elements have displacement degrees of freedom u f = (u1 , v1 , w1 , . . . , u8 , v8 , w8 )T only, use linear interpolation, and allow mechanical loading for static and dynamic procedures. In what follows, the usual definitions of the FEM are throughout used. The displacement vector u at any point of the face sheet as a shell-like structure may be expressed uf =
8
f
f
Ni [I]ui ,
(27.1)
i=1 f
ui are the displacement vector corresponding to the ith node of the SC8R element, f Ni are the shape functions associated with node i, and [I] is the 3×3 identity matrix. For simplicity herein and further such equations may be written as Fig. 27.3 Node numbers, faces and an integration point of a SC8R element
face 2 face 6
4 5
face 5
8
7 3
x1
6 1 face 1
2 face 3
face 4
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u f = [N] f u f
(27.2)
From a modeling point of view continuum shell elements look like threedimensional continuum solids, but their kinematic and constitutive behavior is similar to conventional shell elements. It is important to notice that the SC8R elements are general-purpose elements that can provide solutions to both thin and thick shell problems. Thereby, the hypotheses of Reissner-Mindlin’s shell theory, including the effects of transverse shear deformation are assigned to the element’s kinematics as the shell thickness increases, and the Kirchhoff’s assumptions are fulfilled within the elements as the thickness decreases, herewith the transverse shear deformation tends to be very small. Moreover, these elements account for finite membrane strains, arbitrary large rotation, and allow for a thickness change, making them suitable for nonlinear geometric analysis. In doing so, computation of the change in thickness (is only valid in geometrically nonlinear analysis) is based on the element nodal displacements, which in turn are computed from an effective elastic modulus defined at the beginning of an analysis [17]. If the displacement vector u is known at all points within the elements, the strain vector ε at any point can be calculated in terms of nodal variables as follows ε f = [B] f u f , (27.3) where [B] f are the differential operators associated with the strain-displacement relationships mentioned earlier. In general, each of the orthotropic face sheets is of composite laminate with an arbitrary lay-up. When the material and reference coordinate systems coincide, the constitutive relationship for the mth face sheet orthotropic lamina is represented by σ (m) = [Q](m)ε (m) ,
(27.4)
where σi j and εi j are the components of the stress and strain vectors, respectively, and Qi j represents the stiffness matrix with nine independent material constants and depend on the material properties and fiber orientation of the mth lamina. In accordance with the FSDT, the five components of the strain vector are taken into account such as the in-plane normal and shearing strains (ε1 , ε2 and γ12 ) varying linearly, and the transverse shearing strains ( γ13 and γ23 ) being constant trough the thickness of the laminate. This assumption necessitates the use of shear correction factors. The transverse shear stiffness is computed by matching the shear response for the shell to that of a three-dimensional solid for the case of bending about one axis and assuming a parabolic variation of transverse shear stress in shell each layer. Then the shear strain energy, expressed in terms of section resultants and strains, is equated to the strain energy of this distribution of transverse shear stresses [17]. Moreover, the stress in the thickness direction may not be zero in the continuum shell element and may cause additional strain beyond that due to Poisson’s effect. This stress is computed by penalizing the effective thickness strain with a constant ’thickness modulus’ [17]. The shell elements use bending strain measures that are approximations to those of Koiter-Sanders shell theory, i.e. the displacement field normal to the shell surface
27 Dynamic Analysis of Debonded Sandwich Plates with Flexible Core
421
does not produce any bending moments. The section force and moment resultants per unit length in the normal basis directions in a given laminate section of thickness h can be defined on this basis as (N1 , N2 , N12 ) = (M1 , M2 , M12 ) =
+ h2
− h2 + h2 − h2
(V1 , V2 ) =
(σ1 , σ2 , τ12 )dz
(σ1 , σ2 , τ12 )zdz
+ h2 − h2
(27.5)
(τ13 , τ23 )dz
This leads to the constitutive relationships related the resultants and the generalized strain vectors ⎤⎧ f ⎫ ⎧ f ⎫ ⎡ ⎪ ⎢⎢⎢ A B 0 ⎥⎥⎥ ⎪ ⎪ N ⎪ ⎪ ⎪ ⎪ ε ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥⎥⎥ ⎪ ⎪ f⎬ ⎪ ⎢⎢ ⎪ pf ⎪ ⎪ ⎨ ⎨ ⎬ ⎢ ⎥ ⎢ ⎥ = (27.6) M B D 0 ε ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪, ⎢ ⎥ b⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Vf ⎪ ⎭ ⎢⎣ 0 0 H ⎥⎦ ⎪ ⎩εf ⎪ ⎭ s
where ε p , ε b and ε s are the in-plane, bending and transverse shear strain vectors and Ai j , Bi j , Di j and Hi j are the components of the in-plane, bending-in-plane, bending and transverse shear stiffness matrices of this section and are given by %
& A i j , B i j , Di j = Hi j =
+ h2
−h +2 h 2 − h2
% & 2 Qm i j 1, z, z dz,
(i, j = 1, 2, 3) (27.7)
Qm αβ ki k j dz,
(i, j = 1, 2, α, β = i + 4, j + 4)
Here the k1 , k2 parameters are the shear correction coefficients as defined above. If there are n laminae in the lay-up, we can rewrite the above equations as a summation of integrals over the n laminae. The material coefficients will then take the form Ai j =
n
Qm i j (hm − hm−1 )
m=1 n
% & 1 2 2 Qm i j hm − hm−1 2 m=1 n & 1 m% 3 Di j = Qi j hm − h3m−1 3 m=1 n Hi j = Qm αβ (hm − hm−1 ) ki k j , Bi j =
(27.8)
m=1
where the hm and hm−1 in these equations indicate that the mth lamina is bounded by surfaces z = hm and z = hm−1 . In the matrix notations the constitutive relationships (27.6) can be written as
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σ f = [E] f ε f ,
(27.9)
where σ f and ε f are generalized stress and strain vectors, respectively, and [E] f is the matrix of the elastic material constants associated with equations defined above. It should be mentioned that in the case of laminated structures, continuum shell elements can be also stacked to provide more refined through-the-thickness responses of transverse shear stresses and forces [17].
27.2.2 Core FE Model Because, continuum shell elements can be connected directly to first-order continuum solids without any kinematic transition, 8-node isoparametric linear solid ’brick’ elements with incompatible mode C3D8I are chosen for core modeling. In addition to the standard displacement degrees of freedom (three translations in each a node), incompatible deformation modes are added internally to the elements. The primary effect of these modes is to eliminate the parasitic shear stresses that cause the response of the regular first-order displacement elements to be too stiff in bending [17]. The incompatible mode elements use full integration and, thus, have no hourglass modes. An effective (equivalent) core material is considered as a homogeneous continuum either isotropic for plastic foams or orthotropic for non-metallic honeycomb structures. In the latter, the material constants of the equivalent material, related to the cell geometry and actual parent material properties of the cells were calculated by using either the analytical homogenization formulae [20] or numerical FEM techniques, e.g. [21]. Although brick elements provide accurate results just for the cases when transverse shear effects are predominant and when the normal stress cannot be ignored, they have a deficiency in modeling the transverse shear stress through the thickness. It should be acknowledged in the context of the stress/displacement solution that the transverse shear stresses in solid elements usually do not vanish at the free surfaces of the structure and are usually discontinuous at layer interfaces. Isoparametric interpolation adopted for the elements C3D8I is defined in terms of the isoparametric element coordinates (g, h, r) shown in Fig. 27.4. These are material coordinates. They each span the range -1 to +1 in an element. For instance, the interpolation function defined the translation in the first direction is as follows [17] u = 18 (1 − g)(1 − h)(1 − r)u1 + 18 (1 + g)(1 − h)(1 − r)u2 + 18 (1 + g)(1 + h)(1 − r)u3 + 18 (1 − g)(1 + h)(1 − r)u4 + 18 (1 − g)(1 − h)(1 + r)u5 + 18 (1 + g)(1 − h)(1 + r)u6 + 18 (1 + g)(1 + h)(1 + r)u7 + 18 (1 − g)(1 + h)(1 + r)u8,
(27.10)
27 Dynamic Analysis of Debonded Sandwich Plates with Flexible Core face 2
423
face 5
8
7
face 6 r 5 h
3 4 x
3
4
x
face 4
g 4
x
1
1
2
3
x
Z
2 1
2
face 1
Y X
face 3
Fig. 27.4 Node ordering, face numbering and integration points on each the faces for brick elements
where ui , i = 1, . . . , 8 are nodal translations in the first direction. Analogously, one can write the interpolation functions for the translations in other directions (v, w). In general, the translations through the interpolation functions and nodal variables can be expressed as follows uc = [N]c uc (27.11) Here the matrix [N]c includes the isoparametric shape functions Ni associated with node i and uc is the vector of the nodal displacements. In the core all the six components of the strain vector are taken into account. The strain-displacement equations allowing for finite strains and rotations in largedisplacement analysis in conjunction with the displacement fields are applied to calculate these components as ε c = [B]c uc , (27.12) where [B]c is the corresponding differential operator. The stress-strain relationships based on the 3-D elasticity theory give the components of the stress vector as follows σ c = [E]cε c ,
(27.13)
where [E]c is the matrix of the elastic material constants associated with either an isotropic or orthotropic homogeneous material.
27.2.3 General 3-D FE Model The dimensions of the elements for the 2-D continuum shell and 3-D brick elements are identical on the panel plane directions. For the case of perfect bonding the links between them are provided by means of share nodes, since both the solid elements and shell elements have only the translation degree-of-freedom (DOF). Thus, the deformation compatibility between the faces and the core elements is fulfilled. In the case of the debonding presence an actual small gap is introduced
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Fig. 27.5 3-D modeling of a sandwich plate with a circular debonding zone
in a certain zone between the face sheet and core finite elements. Because it was assumed that debonding may be placed arbitrarily within the face-to-core interface, the finite elements of the general 3-D FE mesh were generated by partition of the total model onto several parts, which were connected with each other through share nodes, Fig. 27.5. An additional adjustment was not used. The general mesh contained three different zones meshed by applying different mesh methods and mesh densities such as a fine mesh for the debonding zone, the next zone surrounding debonding with decreased mesh density, and coarse mesh that was introduced to minimize a CPU time in calculations, Fig. 27.5. No artificial adjustment of either the material or geometrical properties was made at the debonding region to ensure as close as possible a physically real case.
27.3 Aspects of FE Modeling A finite element model based on the aforementioned assumptions is derived to analyze dynamics of sandwich plates. In order to be consequent, a brief presentation of the main steps of this finite element procedure is described in the following subsections.
27.3.1 FE Equations of Motion A solution of the problem, to which solid mechanics is applied, implies finding displacements, deformations, stresses, forces, and other variables in a solid body, which have to obey the equilibrium requirement for both force and moment at all times over any arbitrary volume of the body. The FEM is based on approximating this equilibrium requirement by replacing it with a weaker requirement, that equilibrium must be maintained in an average sense over a finite number of divisions of the volume of the body [22]. This equilibrium statement is usually derived in the form of the virtual work statement for later reduction to the approximate form of equilibrium used in a finite element model. For a finite element as a body at a time instance t, occupying in the current configuration a volume V with the surface S, bounding this volume and subjected at any
27 Dynamic Analysis of Debonded Sandwich Plates with Flexible Core
425
point on S to the force t per unit of current area and at any point within V to the body force f per unit of current volume the equilibrium statement, in the form of the virtual work principle, can be written as σ · ·δεεdV = tT · δudS + fT · δudV, (27.14) V
S
V
where t, f, and Cauchy stress matrix σ are an equilibrium set ∂ t = n · σ, · σ + f = 0, σ = σ T , ∂x T ∂δu and δεε = 12 ∂δu + is the virtual strain, and δu is an arbitrary vector-valued ∂x ∂x admissible function, compatible with all kinematic constraints. To discuss the dynamic behavior, the d’Alembert force is introduced into the overall equilibrium equation. Then, body force at a point f, can be written as an externally prescribed body force F, and a d’Alembert force, i.e. f = F − ρu¨ where ρ is the current density of the material at this point and u¨ is the acceleration filed of the point. The virtual work equation (27.14) can be rewritten as T T σ · ·δεεdV = t · δudS + F · δudV − ρu¨ T · δudV (27.15) V
S
V
V
In the general case, the finite element interpolator can be written as u = [N] u,
(27.16)
where [N] are interpolation functions that depend on some material coordinate system and u are nodal variables of the finite element. The virtual field δu must be compatible with all kinematic constraints. Introducing the above interpolations the displacement will have a certain spatial variation, then, δu must also have the same spatial form δu = [N] δu
(27.17)
Thereby, the continuum variational statement is approximated by a variation over the finite set of nodal variables δu. Then, the virtual material strain associated with δu is δεε = [B] δu (27.18) Following the well-known FEM procedure the equilibrium equation, related to as the equations of motion for each a finite element will be approximated as )
. e/ . / M u¨ + Ke u = Fe ,
(27.19)
where [M ] = V ρ0 [N] [N] dV0 is the mass matrix of the element, 0 ) [Ke ] = V [B]T [E] [B] dV0 is the stiffness matrix of the element, [E] is the elasticity e
0
T
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V.N. Burlayenko and T. Sadowski
matrix in stress-strain relationships, and V0 is the element volume in its reference state. Including into the equations of motion the Rayleigh damping, defined through a damping matrix such that it is a linear combination of the mass and the stiffness matrices, the finite element equations of motion one can read as follows . e/ . / . / M u¨ + Ce u˙ + Ke u = Fe
(27.20)
Thereafter the assembling matrix procedure, usual in the FEM is performed and the final discretized equations of motion are the following ¨ + [C] U(t) ˙ + [K] U(t) = F(t), [M] U(t)
(27.21)
where [M], [C] and [K] are the global mass, damping and stiffness matrices obtained ˙ and U(t) ¨ are the global vectors of unknown by the assembly procedure, U(t), U(t) nodal displacements, velocities and accelerations, respectively, F(t) is the global vector of nodal external forces. The boundary conditions are imposed to the initial boundary (its position is known), giving the statement of the boundary problem at the given initial conditions of the equations of motion consistently closed. Thus, the position of any point of the body, including the boundary will be known once the system of equations is solved.
27.3.2 Contact Model In general, the vibration of a structure containing detached fragments is accompanied by intermittent contact between them. The dynamic response of the structure depends on such a contact which is a function of both space and time domains. Therefore, the extent of the contact region and the contact pressure arising between the contactable interfaces should be also determined as part of the solution. To address the contact-impact problem, the contact analysis between two deformable bodies has to be incorporated into the general FE formulation. For a body with a contactable interface Γ, the two surfaces called Γ + and Γ − are assumed, Fig. 27.6. In fact, the interaction between them is associated with the appropriate constraints. The contact constraints add the following Kuhn-Tucker conditions: the surfaces cannot interpenetrate, and the contact traction (contact pressure) must satisfy momentum conservation on the interface [23]. Using finite element discretization, one of two contacting surfaces is defined as the slave and another one is called as the master. The non-penetration conditions, usually represented by the normal component of displacement and stress, are given a gap (or ‘overclosure’) function g N (x) defining the length of the orthogonal projection of a typical slave node s on the master surface: ' ( g N (x, t) = n · x s − x ,
(27.22)
27 Dynamic Analysis of Debonded Sandwich Plates with Flexible Core Fig. 27.6 FE contact discretization
slave surface s G-
n x
xs
427
contact force
gN
master surface
G+
s penetration
where n is the outward unit vector orthogonal to the master segment, while x s and x are the vectors identifying the current positions of the salve node s and the closest nodes on the master segment, respectively. The expression has to be checked for all candidate contact nodes s, which are elements of the finite set Γ − . The normal contact pressure acting on the node s is tN = tc · n, where tc is a contact traction vector. For gN ≤ 0 the constraint equation for a node s becomes active, that is x s ∈ Γ + ∩ Γ − and tN ≥ 0, otherwise the constraint is inactive, i.e. x s ∈ Γ − and the surfaces are separated. The relationship between the gap function and contact pressure is referred to as a constitutive contact model. Analogously, the tangential contact constraints can be enforced, where a tangential gap function, called as a slip function gT (x, t) and a tangential traction tT = tc · (I − n ⊗ n) can be related by the classical Coulomb friction law [24]. The enforcement of the contact constraints is achieved by appending the additional term into the virtual work principle (27.15). The virtual work done by the contact forces including the contribution from both the normal and tangential contact responses can be written as δWc = δ (tN g N + tT gT )dΓ (27.23) Γ
Herewith, the integral over the contact surface Γ consists of two surface integrals over Γ + and Γ − . The gap functions g N and gT are functions of the displacement vector u(x,t), and following the FE approximation (27.16), they are functions of the nodal variables u. The contact traction components can be expressed to be proportional to surface penetrations (gap functions) in the penalty method, or treated as additional unknowns in the Lagrange multipliers method. These two methods can also be combined in the augmented Lagrangian formulation. Thereby, the discrete weak form of the dynamic problem involving contact-impact conditions in terms of the global nodal displacements has been formed as follows ¨ + [C] U(t) ˙ + [K] U(t) = F(t) + FN (t) + FT (t), [M] U(t)
(27.24)
where F N (t) and FT (t) are the generalized global normal and shear contact force vectors, respectively, generated so that eliminate all having penetrations between surfaces of the detached segments during oscillations, Fig. 27.6. It should be noticed that the contact forces do not depend on the shape functions [N], but instead on the approximation of the surface, that is the gap functions approximation.
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27.3.3 Integration Rule The most computationally efficient way for solving the discrete equations of motion (27.24) is the use of the explicit integration rule together with diagonal lumped element mass matrices. The explicit central difference integration operator used can be written in the following form ˙ (i+ 2 ) = U ˙ (i− 2 ) + Δt U 1
1
(i+1) + Δt(i)
2
¨ (i) , U
˙ U (i+1) = U(i) + Δt(i+1) U
(i+ 12 )
, (27.25)
% & % & where the superscript (i) refers to the increment number and i − 12 and i + 12 refer to midincrement values. The explicit procedure requires no iterations and no tangent stiffness matrix, because the accelerations at the beginning of the each increment can be calculated quite simple by the inversion of the lumped mass matrix diagonalized in advance as follows % & ¨ (i) = [M]−1 F(i) − I(i) , U (27.26) where F(i) is the vector of external nodal forces, and I(i) is the general vector of internal nodal forces. In the case of dynamic contact existing along with global dynamic motion, the general vector of internal nodal forces will include contact forces as well, when the surfaces come into contact at a defined time increment (i). The explicit procedure integrates the equation (27.24) through time by using many small time increments. The time increment used in the analysis must be smaller than the stability limit of the central-difference operator applied. An approximation to the stability limit is defined by the smallest transit time of a dilatational wave across any of the elements in the mesh as the following [17] le Δt = min , (27.27) cd where le is the characteristic element dimension and cd is dilatational wave speed of the material.
27.3.4 FE Analysis Using the FE code ABAQUS static bending, free vibration, dynamic transient and dynamic steady-state analysis of sandwich plates containing the debonding zone along the face sheet-to-core interface were carried out. The static and free vibration analysis were run by using ABAQUS/Standard with a linear model for debonded sandwich plates developed in [25]. The other two analysis exploited the nonlinear finite element model based on the abovementioned modeling approach and were run by utilizing ABAQUS/Explicit. It is important to notice that the nonlinear FE model does not assume the advance of the debonding zone during oscillations.
27 Dynamic Analysis of Debonded Sandwich Plates with Flexible Core Fig. 27.7 Graphical interpretation of the pressureoverclosure relationship used
429 Contact pressure
tN = 0 for gN = 0 for
gN < 0 tN > 0
(open), and (closed)
Any pressure possible when in contact
No pressure when no contact
Clearance
To provide during the dynamic forced analysis the contact-impact behavior of the face sheet and the core within the debonding zone of the vibrating sandwich plate, the interaction option was activated in the FE code [17]. The contactable parts were simulated by using their surface-to-surface discretization in terms of slave and master surfaces. Such a contact formulation enforces contact conditions in an average sense over the regions nearby slave nodes rather than only at an individual slave node. Thereby, surface-to-surface contact will provide more accurate contact stress resultants, which are being used to form with greater precision the right hand side of the equations of motion (27.24). The relative motion of the interacting surfaces in the contact simulation was described with finite sliding kinematic assumptions that are the most general case, and which allow any arbitrary motion of the surfaces involving their separation, sliding and rotation. The constitutive behavior of the surfaces coming into contact in the normal direction was assumed to be governed by the ’hard contact’ model, whereas, for the sake of reducing the computational cost and because the sliding in the contacting surfaces is small, frictionless conditions were accepted between them in the tangential direction. It is worthwhile to notice that ‘hard contact’ implies that the interacting surfaces transmit no contact pressure unless the nodes of the slave surface contact the master surface and no penetration is allowed at each constraint location, Fig. 27.7. Thus, the hard contact model will minimize a measure of overclosure of nodes of the slave surface into the master surface at the constraint locations in the best way. To resolve in the dynamic analysis the normal contact constraints imposed by the physical pressure-overclosure relationship corresponding to the hard contact model applied, the penalty constraint enforcement algorithm is used. This method was chosen because it does not increase the cost of the analysis compared with the Lagrange multipliers algorithm.
27.4 Numerical Results The developed finite element model has been employed for the analysis of both perfect and debonded sandwich plates with a flexible core. Both static and dynamic loading cases have been considered. The numerical results, obtained with ABAQUS
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V.N. Burlayenko and T. Sadowski
were compared to known analytical or numerical solutions whenever it was possible. Some new results for soft-cored sandwich plates are below presented as well.
27.4.1 Validation In order to demonstrate the accuracy and performance of the developed FE model, several examples of undamaged sandwich composite plates subjected to static and dynamic loading have been analyzed using the package ABAQUS and the results obtained have been compared with the results published in the literature. Example 27.1. A symmetric simply supported square sandwich plate with planar dimensions a × b = 200 × 200 mm2 subjected to bi-directional sinusoidal transverse load on its top surface with the intensity of 100 Nmm−2 has been considered. The 3-D elasticity solution obtained by Pagano [26] has been used for comparison of the FEM results. The total thickness of the plate is h = 50 mm, the thickness of each face sheet is one tenth of the total thickness. The performance of several different models of discretization was assessed. A mesh-sensitivity analysis was performed for all models. A suitable mesh was established once a plateau with a deviation less than 2% in the calculated values of deflections and stresses was found. As many as 80 elements along the length and 10 elements across the width were used. One SC8R element per ply across the thickness of the face sheet with seven cross-section integration points and eight C3D8I elements across the thickness of the core were utilized, as it is important to keep these first-order solid elements cubic in shape. The materials of the elements were modeled as orthotropic solids with constants listed in Table 27.1. The comparative results of the analysis are shown in Table 27.2, where the following normalization coefficients were utilized: f
(σ ¯ 11 , σ ¯ 22 , σ¯ 12 ) =
100E 2 h3 h2 h (σ ) (¯ ) (τ ) , σ , σ , τ , τ ¯ = , τ , w ¯ = w 11 22 12 13 23 13 23 q0 a q 0 a2 q0 a4
The stresses were calculated at the element integration points, but the values of stresses in the output file were determined at the nodes by extrapolating the stresses from the integration points [17]. As can be seen from Table 27.2, the results of the present ABAQUS model are in a good agreement with the analytical results. Fig. 27.8 showing through-the-thickness variation of the maximum normalized deflection has also presented to pay attention the necessity of using the layer-wise model for the analysis of the sandwich plate with a low strength core material. One can see the differences in the value of deflections between the top and bottom face sheets that means the change of the core height. As well the through-the-thickness % & a b variation of normalized transverse shear stresses calculated at the points 12 , 2,z % & b and a2 , 12 , z , respectively, have been presented in Figs 27.9a and 27.9b to have an
27 Dynamic Analysis of Debonded Sandwich Plates with Flexible Core
431
Table 27.1 Material properties of sandwich plates Example Ref. Components Face sheet
1
E1 , GPa
E2 = E3 , GPa
G 12 = G 13 , GPa G23 , GPa
ν12 = ν13 ν23
172.4
6.89
3.45
0.25
0.25
[26] Core E1 = E2
E3
G13 = G 23
G 12
ν12
ν13 = ν23
0.276
3.45
0.414
0.1104
0.25
0.1
Face sheet, ρ = 1627 kgm
2
1.378
−3
E1 , GPa
E2 = E3 , GPa
G 12 = G 13 , GPa G23 , GPa
ν12 = ν13 ν23
131.0
10.34
6.895
0.22
6.205
0.49
[9] Core, ρ = 97 kgm−3 E1 = E2
E3
G13 = G 23
G 12
ν12
ν13 = ν23
6.89 · 10−3
6.89 · 10−3
3.45 · 10−3
3.45 · 10−3
0.0
0.0
Face sheet, ρ = 4400 kgm−3
3
E1 = E2 = E3 , GPa
G 12 = G 13 = G23 , GPa
ν12 = ν13 = ν23
36.0
14.0
0.3
E1 = E2 = E3 , GPa
G 12 = G 13 = G23 , GPa
ν12 = ν13 = ν23
50.0 · 10−3
21.0 · 10−3
0.29
[15] Core, ρ = 52 kgm
−3
Table 27.2 Deflection and stresses in square sandwich plate under bi-directional sinusoidal load % & % & % & % & Source σ¯ 11 a2 , b2 , z σ ¯ 22 a2 , b2 , z τ¯ 13 0, b2 , 0 τ¯ 23 a2 , 0, 0 τ¯ 12 (0, 0, z) w¯ + h2
− h2
+ h2c
− h2c
+ h2
− h2
+ h2
− h2
[26] 1.556 -1.512 -0.233 0.196 0.259 -0.253
0.239
0.107
-0.144 0.148 7.425
FEA 1.268 -1.302 -0.255 0.219 0.233 -0.282
0.237
0.113
-0.137 0.140 7.367
insight into the strength of sandwich plates and demonstrate the interaction between the strong face sheet and the low density core. Example 27.2. For further validation, the free vibration analysis of sandwich panel with a flexible core is carried out using the ABAQUS software. A 20 mm thick, fivelayer (0/90/core/0/90) simply supported square of 200 mm by 200 mm sandwich plate is herein analyzed. The plate was made of composite anisotropic face sheets and an isotropic core, where the ratio of the face sheet thickness to the core thickness was 10. Material properties of the sandwich plate are given in Table 27.1. In a FE mesh one SC8R element per each ply across the thickness of the face sheet with five cross-section integration points and five C3D8I elements across the thickness of the core were used. The planar dimensions of the plate were discretized by 50 elements along each edge.
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Fig. 27.8 Through-thethickness variation of the maximum normalized deflection
0.5 FEM
Thickness
0.25
Pagano
0
-0.25
-0.5 6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
0.5
0.5
0.25
0.25
Thickness
Thickness
Normalized deflection
0
-0.25
-0.25
-0.5
-0.5 0
(a)
0
0.05
0.1
0.15
0.2
0.25
Normalized shear stress in direction 1
0
(b)
0.03
0.06
0.09
0.12
Normalized shear stress in direction 2
Fig. 27.9 Through-the-thickness variation of the normalized transverse shear stresses
The computed with ABAQUS results are compared with those predictions obtained 2 in [9]. Natural frequencies have been normalized by using the relation a4 ρ f Ω=ω , where h is the total plate thickness and ω is natural cyclic frequency. 2 f h E2
The comparative results of the normalized natural frequencies are presented in Table 27.3. A good agreement between the results one can be observed. To make evident the effect of ’soft’ core on the considered kinematics of the sandwich plate, the frequency spectrum containing fifty its natural frequencies have been analyzed. The results showed that high frequencies of the plate exhibit symmetric out-of-plane movements of the face sheets, so-called pumping modes, relating to the changes of the plate height. Several of such modes are presented in Fig. 27.10, where the core was excluded from the contour plot for the sake of clearness. Therefore, the linear assumptions for modeling sandwich plates with a flexible core are
Table 27.3 Comparison of non-dimensional natural frequencies for the intact sandwich plates Source
(1,1)
(1,2)
(1,3)
(2,2)
(2,3)
(3,3)
[9]
1.8480
3.2196
5.2234
4.2894
6.0942
7.6721
FEA
1.7952
3.1445
5.2235
4.1985
5.9945
7.5653
27 Dynamic Analysis of Debonded Sandwich Plates with Flexible Core
433
suitable only for describing low vibration modes, and the current model is capable to capture such new nonclassical effects.
(a)
(b)
(c)
(d)
Fig. 27.10 Contour plots of the pumping modes
Example 27.3. The free vibration analysis of a sandwich beam with a damaged region was considered in [15], where natural frequencies and relevant mode shapes were analytically calculated. That model was developed with ABAQUS to make a comparison between the predictions and assess the applicability of the FE model developed for taking into account an interfacial damage. A 300 mm long sandwich beam cored by a foam with a rectangular cross-section of 20 mm by 21 mm, containing damage of a 20 mm long at the middle span of the beam was simulated by 60 elements along the beam length and eight elements through its thickness. One element per thickness of each the face sheet and six elements across the core thickness were used. The thickness of the upper face sheet was of 0.5 mm and the lower one was 1 mm. Material properties of the sandwich beam were the same as described in [15] and they are listed in Table 27.1. The comparative values of the six first natural frequencies are shown in Table 27.4. As one can be seen, the close results are demonstrated. Table 27.4 Comparison of natural frequencies (Hz) between the intact and debonded sandwich plates Intact
Debonded
Mode
[15]
FEA
[15]
FEA
1
289.3
293.46
288.98
293.07
2
683.3
707.09
388.32
433.67
3
1096.9
1106.7
1093.2
1093.2
4
1151.6
1495.8
1146.9
1132.0
5
1778.2
1818.7
1771.3
1769.9
6
1895.3
1918.1
1842.2
2080.2
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Table 27.5 Material properties of the sandwich plate Source
Components Face sheet, ρ = 1650 kgm−3
[27]
E1 = E3 , GPa
E2 , GPa
G12 = G23 , GPa G13 , GPa
ν12
ν13 = ν23
3.48
19.3
1.65
0.05
0.25
7.7
Core, ρ = 52 kgm−3 E1 = E2 = E3 , GPa
G12 = G13 = G 23 , GPa
ν12 = ν13 = ν23
92.1 · 10−3
27.3 · 10−3
0.42
27.4.2 Free Vibration Analysis A free vibration of a sandwich plate containing a debonding zone of the circular shape between the upper face sheet and the core is further studied. The plate was simply supported with planar dimensions of 270 mm by 180 mm and was made of a 50 mm thick foam core and a 2.4 mm thick each of the face sheets. The debonding zone’s diameter, d was of 30 mm. The material properties of the constituent layers of the plate are showed in Table 27.5. Pursuing the purpose, to assess the influence of the debonding presence on the modal parameters of the sandwich plate, the free vibration analysis of the same intact sandwich plate was performed too. The natural frequencies obtained with ABAQUS for these two plates are presented in Table 27.6. The comparative results demonstrate that debonding, firstly, changes the order of frequencies in the spectrum and, primarily, leads to reduction of magnitudes of the natural frequencies for bending modes in the low part of the frequency spectrum. Secondly, it does not almost change the natural frequencies associated with in-plane modes. Finally, debonding significantly affects the high frequencies making their values even higher than those for the intact plate, so-called thickening phenomenon appears. The above results imply that the debonding region exerts specific effects on the relevant mode shapes. Several numerically calculated mode shapes are shown in Fig. 27.11. For the better presentation the core was excluded from the contour plots.
Table 27.6 Comparison of natural frequencies (Hz) between the intact and debonded sandwich plates Bending INTa DBDb Bending INT modes modes
a
DBD In-plane INT modes
DBD Pumping INT modes
DBD
(1,1)
1078.5 995.4
(3,3)
3424.4 3461.6
1
1772.1 1771.7
1
4012.3 4035.3
(1,2)
1938.5 1818.6
(4,3)
3844.9 3895.8
2
2637.8 2637.4
–
–
–
(2,1)
1600.5 1567.2
(2,4)
3907.9 3946.7
3
3517.5 3517.1
–
–
–
(2,2)
2274.9 2267.1
(3,4)
4183.9 4220.1
–
–
–
–
–
b
is the intact sandwich plate; is the debonded sandwich plate
–
27 Dynamic Analysis of Debonded Sandwich Plates with Flexible Core
(a)
(b)
435
(c) (e)
(d)
(f)
Fig. 27.11 Contour plots of the free vibration modes of the debonded sandwich plate
One can see that debonding is visualized as a local separation between the constitutive layers. Herewith, these local interactions can be seen in all mode shapes, but they in certain modes are larger than in others. Hence, the effect of debonding on the sandwich plate’s free response will be more pronounced for such modes. Moreover, the mode shapes in the frequency spectrum will include both only a local deformation within the detached zone of the face sheet (Figs 27.11c and 27.11d) and mixed modes, which are a combination of debonding zone deformations along with the globally deformed pattern of the sandwich plate (Figs 27.11e and 27.11f). Thereby, one may conclude that the permutation in the order of modes, inversion of the mode shapes and thickening phenomenon are a result of complex local interaction phenomenon within the sandwich plate, induced by the debonding region.
27.4.3 Forced Vibration Analysis The flexibility of the low strength core affects the overall structural behavior and may lead to stress concentration in the vicinity of the discontinuities and support regions, in terms of shear and transverse normal stresses at the face sheet-to-core. To provide a deeper insight into dynamics of plates sandwiched by a flexible core and containing debonding, their general dynamic response is examined. Herewith, contact-impact conditions for describing interactions of surfaces coming into contact during oscillations are taken into account. The forced vibration analysis of sandwich plate with a flexible core is carried out using the explicit solver of the ABAQUS commercial software, as it was described in Sect. 27.3.3. Simplifying assumptions were formulated and adopted for calculations as the following • Elastic deformation was present. • Debonding was assumed exists before vibration started and was constant during oscillations. The transient response of the debonded sandwich plate, the same as in previous Sect. 27.4.2 was studied. The simply supported rectangular sandwich plate was
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(a)
(b)
(c)
(d)
(e)
(f)
Fig. 27.12 Contour plots of deformed shapes of the debonded sandwich plate subjected to the impulse force: (a) t = 1 ms; (b) t = 3 ms; (c) t = 5 ms; (d) t = 7 ms; (e) t = 9 ms; (f) t = 10 ms
excited by an impulse force at the central point of the bottom face sheet to simulate a hammer hit. The duration of the applied force was much shorter than the analysis time, i.e. ⎧ ⎪ ⎪ ⎨ F 0 , 0 ≤ t ≤ t0 F(t) = ⎪ ⎪ ⎩ 0, t > t0 with t0 = 1 ms and F 0 = 1 kN. Fig. 27.12 presents the deformed forms of the sandwich plate during its transient movement at the different discrete time moments for first 10 ms. For the sake of clearness half the plate is presented. It can be clearly seen the intermittent contact of the detached surfaces in the vibrating sandwich plate. The comparison of results calculated for both the debonded sandwich plate and the same intact plate gives a way to assess an influence of debonding on the sandwich plate’s transient response. The finite element results such as displacements, strains and stresses can be computed at different points as time histories. As one can see from Fig. 27.13 the displacement time histories, calculated at the central points A and B are quite different for the intact (INT) and debonded (DBD) plates. In the case of the debonded plate the displacements of both the face sheet (DBD-f) and core (DBD-c) at the point A are shown, Fig. 27.13a. It is important to notice that the local interaction in the detached segments, located between the upper face sheet and the core of the plate changes significantly the transient response of the sandwich plate at its lower points, Fig. 27.13b. Such different deformation patterns in the top and bottom face sheets are expected due to the compressibility of the soft core. While the flexibility of the low strength core affects the overall structural behavior, it also leads to stress concentrations in the vicinity of the discontinuity. Fig. 27.14 demonstrates the contour plot of the Mises stress at the face sheet-to-core interface separately for the core and the face sheet at such a time moment (t = 3 ms), when it is maximum. One can see that the maximum value of the stress has been attained, as expected, on the boundary points of the debonding zone. As mentioned above, the degraded interface will generate higher in-plane stresses in the face sheet and transverse shear and normal stresses in the core, which may
27 Dynamic Analysis of Debonded Sandwich Plates with Flexible Core
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make the damaged face sheet-to-core interface vulnerable to debonding propagation. For analyzing the stress states in the interface with% a time, variations of the & b hc stresses at a boundary point of the debonding zone, C a+d , , calculated for 2 2 2 both the intact and debonded plates are compared in Fig. 27.15. It can be seen that the transient stress time histories of the intact and debonded plates are obviously different. Besides, the amplitudes of the stress variations for the debonded sandwich plate are significantly larger than those for the intact one. Thus, the assumption of debonding not propagation during oscillations is doubtful and may be used for the sake of model simplification as preliminary strength information. Although this assertion should be considered in more details involving the crack mechanics approaches and it may be an objective of our future investigations.
5
6 INT DBD-f DBD-c
INT
Deflection, mm
Deflection, mm
4 2 0 -2 -4
0
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0
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Fig. 27.13 Transient displacement time histories at the point: (a) A
0.006
0.008
0.01
Time, s
(b)
%
a b hc 2, 2,+ 2
(a) C
(b)
Fig. 27.14 Contour plot of Mises stress within: (a) the core; (b) the face sheet
&
; (b) B
%
a b h 2, 2,−2
&
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Stress t23, MPa
20
Stress t13, MPa
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0
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0 -4 -8
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-5
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Stress s22, MPa
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60 30 0
-30
3 0 -3 -6
-60 0
0.002
0.004
0.006
Time, s
(e)
0.008
0.01
0
0.002
0.004
0.006
Time, s
(f)
Fig. 27.15 Transient stress time histories of the: (a) transverse shear stress in direction 1; (b)transverse shear stress in direction 2; (c) transverse normal stress; (d) in-plane normal stress in direction 1; (e) in-plane normal stress in direction 2; (f) in-plane shear stress
27.5 Conclusions In the paper is explored a way of predicting the dynamic behavior of debonded sandwich plates with flexible core using free vibration analysis and dynamic transient analysis, and is examined the influence of the soft core and debonding on modal characteristics and transient response of the sandwich plates. The FE model
27 Dynamic Analysis of Debonded Sandwich Plates with Flexible Core
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involving continuum shell elements for each of the face sheet layers and 3-D brick elements for the core is developed within the ABAQUS code. The free vibration analysis is performed with an implicit version of ABAQUS whereas the dynamic transient analysis is carried out by using its explicit version. Contact conditions, which are available in ABAQUS/Explicit only, are applied in detached segments of the debonded sandwich plates to avoid a physical unreal penetration of the surfaces coming into contact during oscillations and to simulate as close as possible the dynamic behavior of the sandwich plates in order to gain a better understanding of their performance. From results obtained in the paper the following conclusions can be drawn: • The core flexibility in the through-the-thickness direction leads to highly nonlinear responses of sandwich plates under both static and dynamic loading. Unequal deflections of the top and bottom face sheets and jumps of the transverse shear stresses at the face sheet-to-core interfaces were observed in the case of plate bending. The appearance of the pumping modes in the frequency spectrum resulted from the free vibration analysis. • Natural frequencies of a debonded sandwich plate usually decreases, compared to those for the same intact plate. The associated mode shapes along with global deformed forms contain local deformation modes induced by debonding and such local deformations are highly dependent on the mode number. As well, the high natural frequencies and mode shapes are more disturbed by the debonding presence. • Taking into account contact interactions in detached fragments for modeling debonded sandwich plates is very important to properly describe their dynamic forced response. The neglect of contact-impact conditions in predicting of their dynamic behavior will lead to the incorrect results. • The intermittent contact in detached segments of the debonding region significantly changes the dynamic transient response of the sandwich plate. Finite element results calculated have shown completely different deformed shapes of the debonded plate in comparison with those for the same intact sandwich plate. • The flexible core produces the high stress level at the boundary points of the debonding region such that may promote debonging to the propagation. Although this conclusion demands of a further study. Acknowledgements The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007–2013), FP7 - REGPOT - 2009 - 1, under grant agreement No: 245479 and the support by Polish Ministry of Science and Higher Education, Grant No 1471-1/7PR UE/2010/7, is also acknowledged.
References 1. Librescu L, Hause T (2000) Recent developments in the modeling and behavior of advanced sandwich constructions: a survey. Compos Struct 48:1–17 2. Vinson JR (2001) Sandwich structures. Appl Mech Rev 54:201–214
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3. Hause T, Librescu L (2006) Flexural free vibration of sandwich flat panels with laminated anisotropic face sheets. J Sound Vib 297:823–841 4. Bernard ML, Lagace PA (1987) Impact resistance of composite plates. In: Proceedings of the American Society for Composites. 2nd Technical Conference 167–176 5. Akay M, Hanna R (1990) A comparison of honeycomb-core and foam-core carbonfibre/epoxy sandwich panels. Composites 21:325–231 6. Altenbach H (1998) Theories for laminated and sandwich plates: a review. Int Applied Mech 34:243–252 7. Ferreira AJM (2005) Analysis of composite plates using a layerwise shear deformation theory and multiquadrics discretization. Mech Adv Mater Struct 12:99–112 8. Frostig Y, Thomsen OT (2004) High-order free vibration of sandwich panels with a flexible core. Int J Solids Struct 41:1697–1724 9. Rao MK, Desai YM (2004) Analytical solutions for vibrations of laminated and sandwich plates using mixed theory. Compos Struct 63:361–373 10. Frostig Y, Thomsen OT, Vinson JR (2004) High-order bending analysis of unidirectional curved soft sandwich panels with disbonds and slipping layers, J Sandwich Struct Mater 6:167–194 11. Della CN, Shu D (2007) Vibration of delaminated composite laminates: a review. Applied Mechanics Reviews 60:1–20 12. Lu X, Lestari W, Hanagud S (2001) Nonlinear vibrations of a delaminated beam. J Vib Control 7:803–831 13. Ju F, Lee HP, Lee KH (1994) Dynamic response of delaminated composite beams with intermittent contact in delaminated segments. Compos Eng 4:1211–1224. 14. Kwon YW, Lannamann DL (2002) Dynamic numerical modeling and simulation of interfacial cracks in sandwich structures for damage detection. J Sandwich Struct Mater 4:175–199 15. Schwarts-Givli H, Rabinovitch O, Frostig Y (2008) Free vibrations of delaminated unidirectional sandwich panels with a transversely flexible core and general boundary conditions - a high-order approach. J Sandwich Struct Mater 10:99–131 16. Burlayenko VN, Sadowski T (2011) A numerical study of the dynamic response of sandwich plates initially damaged by low velocity impact. Comput Mater Sci doi:10.1016/j.commatsci.2011.01.009 17. ABAQUS Version 6.9-1 EF User’s Manual (2009) Dassault Systems Simulia Corp., Providence, RI, USA 18. Burton WS, Noor AK (1996) Assessment of continuum models for sandwich panel honeycomb cores. Comput Meth Appl Mech145:341–360 19. Nabarrete A, De Almeida SFM, Hansen JS (2003) Sandwich plate vibration analysis: three layer quasi three-dimensional finite element model. AIAA J 41:1547–1555 20. Gibson LJ, Ashby MF (1988) Cellular solids: structure and properties. Oxford, Pergamon Press 21. Burlayenko VN, Sadowski T (2010) Effective elastic properties of foam-filled honeycomb cores of sandwich panels. Compos Struct 92:2890–2900 22. Zienkiewicz OC (1977) The finite element method. 3rd edn. McGraw-Hill, London 23. Kikuchi N, Oden JT (1988) Contact problems in elasticity: a study of variational inequalities and finite element methods. SIAM, Philadelphia 24. Laursen TA, Simo JC (1993) A continuum-based finite element formulation for the implicit solution of multi-body large deformation frictional contact problems. Int J Numer Meth Eng 36:3451–3485 25. Burlayenko VN, Sadowski T (2010) Influence of skin/core debonding on free vibration behavior of foam and honeycomb cored sandwich plates. Int J Non-Linear Mech 45:959–968 26. Pagano NJ (1970) Exact solutions for rectangular bi-directional composites and sandwich plates. J Compos Mater 4:20–35 27. Shipsha A, Hallstr¨om S, Zenkert D (2003) Failure mechanisms and modelling of impact damage in sandwich beams - a 2D approach: part I - experimental investigation. J Sandwich Struct Mater 5:7–31
Chapter 28
On Elasto-Plastic Analysis of Thin Shells with Deformable Junctions ´ Jacek Chr´os´cielewski, Violetta Konopinska and Wojciech Pietraszkiewicz
Abstract The non-linear equilibrium conditions for irregular thin shells are formulated from the appropriate form of the principle of virtual displacements. 2D constitutive relations of elasto-plastic behaviour of thin shells are established by dividing the shell into n layers and then integrating the corresponding 3D constitutive relations throughout all layers at each step of non-linear incremental solution by FEM. As example, deformation and stress states in the casing of pressure measuring devise are calculated taking into account deformability of the junctions. Keywords Thin shell · Plasticity · Deformable junction · Finite element method · Casing
28.1 Introduction In the recent paper by Chr´os´cielewski et al. [1] we have proposed a methodology of the non-linear elasto-plastic analysis of thin shells with deformable junctions. The regular parts of the shell have been modelled by dividing the shell into n layers assumed to be in the plane stress state. The 3D incremental constitutive equations of each layer are described by the generalized elasto-plastic law of PrandtlReuss for small strains, with the associated flow rule and plasticity condition of Huber-Mises-Hencky with linear combination of isotropic and kinematic hardening. J. Chr´os´cielewski (B) · V. Konopi´nska Gda´nsk University of Technology, Department of Structural Mechanics and Bridge Strctures, ul. G. Narutowicza 11/12, 80-952 Gda´nsk, Poland e-mail:
[email protected],
[email protected] W. Pietraszkiewicz Institute of Fluid-Flow Machinery of the Polish Academy of Sciences, ul. Gen. J. Fiszera 14, 80-952 Gda´nsk, Poland e-mail:
[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 28, © Springer-Verlag Berlin Heidelberg 2011
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The 2D incremental constitutive equations for the shell stress resultants and stress couples are established by direct through-the-thickness integration throughout all layers of 3D relations mentioned above. The irregular shell with junctions are modelled according to Makowski et al. [2, 3] by a union of regular surface elements joined together along curvilinear surface edges, along which appropriate forms of 1D strain energy densities have been proposed. The methodology has been illustrated by numerical results of deformation and stress states in the casing of pressure measuring devise having two circular junctions between axisymmetric parts of different thickness. Only the limit cases of stiff and simply supported junctions have been analysed in [1]. In this report we first briefly remind basic non-linear relations of thin irregular shells with deformable junctions analysed within the elasto-plastic range of deformation. Then we present supplementary to [1] numerical results of deformation and stress states in the casing, where some intermediate cases of elastic junction behaviour defined by prescribed stiffness parameters are discussed.
28.2 Notation and Basic Relations According to [2, 3], the consistent field equations and jump conditions of thin irregular shell structures can be derived using two postulates. In the kinematic one, the deformation of the irregular shell is assumed to be determined by stretching and bending of the irregular surface-like material continuum being a union of regular smooth surface elements M (k) , k = 1, 2, ..., K, joined together along spatial curvilinear surface edges ∂M (k) , which in the reference configuration are together denoted as M and Γ, respectively. Then the equilibrium conditions are required to be derivable from the principle of virtual displacements (PVD) involving only dynamic fields associated with the assumed kinematics of M. Such PVD is postulated in the form G ≡ Gi − G e − GΓ = 0 ,
(28.1)
where G i means the internal virtual work, G e is the external virtual work, and G Γ accounts for an additional virtual work of forces and couples acting only along all singular curves modelling the shell junctions. Let u denote symbolically the translation field u inside all M (k) ∈ M, the translation and rotation-like fields (u, ϕ) and (uΓ , ϕΓ ) along regular parts of ∂M (k) and Γ, respectively, the translation ui at each corner Pi ∈ Γ, and the translation ub at each corner Pb ∈ ∂M. Then after complex transformations given in detail in [3] we obtain
28 On Elasto-Plastic Analysis of Thin Shells with Deformable Junctions
G(u; δu) = − +
∂M f
(Divs T + l) · δuda M\Γ
, (pν − p∗ + k) · δu + (h − h∗) δϕ ds
3 4 [f · δu]b − f ∗b · δub
+
Pb∈∂ M¯ f
+
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∂Md
,
(pν + k) · δu + h δϕ ds + [f · δu]b
(28.2)
Pb∈∂ M¯ d
, + ([pν + k] − f Γ ) · δuΓ + ([h] − hΓ) δϕΓ ds Γ , + [f ]i − f i · δui = 0 . Pi ∈Γ¯
In (28.2), the compound tensor field T is defined in [3] through the surface symmetric stress resultant and stress couple tensors N, M, the compound vector l, pν , p∗ , k and scalar h, h∗ fields are defined in [3] through the external surface force p and ¯ (k) but measured per moment h resultant vectors, acting on each deformed surface M (k) unit area of M , as well as through the external boundary force t∗ and moment ¯f resultant h∗ vectors, prescribed along regular parts of the deformed boundary ∂ M ∗ but measured per unit length of ∂M f . Additionally, f b are the external concentrated forces prescribed at each singular point Pb ∈ ∂ M¯ f , [...] means the jump of (...) along each regular part of Γ, [...]i is the jump of (...) at each singular point of Γ, [...]b means the jump of (...) at each singular point of ∂M, while δ is the symbol of variation and Divs is the surface divergence operator. For any kinematically admissible virtual displacement δu the fields δu and δϕ identically vanish along ∂Md , so that the fourth row of (28.2) identically vanishes as well. Then the transformed PVD requires the equilibrium equations, the static boundary and corner conditions as well as the corresponding jump conditions along Γ to be satisfied. In such formulation the kinematic relations, the material and junction characterisation by the constitutive relations as well as the kinematic boundary conditions should additionally be specified. The whole set of shell relations constitute the highly non-linear boundary value problem (BVP) in terms of translations and their surface gradients as the only independent field variables. This complex BVP can effectively be solved only by numerical methods applying some incremental-iterative solution procedure. The procedure is usually based on approximation of the non-linear BVP by series of linearised BVPs. For the Lagrangian non-linear theory of thin, regular elastic shells (without junctions) such solution procedure was worked out in [4], where the general structure of incremental shell equations and corresponding buckling shell equations were explicitly derived. However, in case of highly non-linear irregular elasto-plastic shell problems it is more efficient to apply the numerical incremental-iterative procedure directly to the incremental variational functional (28.2), not to the field equations following from it.
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Let us briefly recall some statements of [4] and extend them to elasto-plastic shell problems with deformable junctions. Notice that components of the external loads in M, along ∂M f and along Γ may be specified entirely independently, in general, by now 18 dimensionless parameters λ p ∈ Λ ⊂ R18 . Then the non-linear BVP for a thin irregular shell generated by (28.2) can be presented symbolically as F(u, λ p ) = 0, where the non-linear continuously differentiable operator F is defined on the product space C(M, E 3 ) × R18 with values in the Banach space, where C(M, E 3 ) is a set of all components of u and its surface gradients up to the 4th order. In engineering applications all external loads are usually specified by a single common parameter λ ∈ Λ ⊂ R. In this case the solution u(λ) of the BVP form a one-dimensional submanifold in C(M, E 3 ) usually called the equilibrium path. u(λ) is called the weak solution if G (u(λ); δu(λ)) = 0 for all kinematically admissible δu(λ). For finding the weak solution u(λ) one usually applies the Newton-Kantorovich method [5] based on successive approximations to the exact solution at some λm+1 through solving a series of linearised BVPs following from linearisation of G (u(λ); δu(λ)) = 0 about a λm close to λm+1 , % & % & G u(i) ; δu + ΔG u(i) ; Δu(i+1) , δu = 0 . (28.3) m+1 m+1 m+1 The first term in (28.3) represents the value of G at the approximation u(i) m+1 . Since this approximation may not belong to the equilibrium path, the first term in (28.3) does not vanish, in general, and allows one to calculate the unbalanced force vector at the configuration corresponding to u(i) m+1 . The second term in (28.3) linear with regard to Δu(i+1) allows one to calculate the tangent stiffness matrix at u(i) of the m+1 m+1 non-linear BVP. If um+1 corresponds to the regular solution point then the successive approximations u(i+1) m+1 established by this method converge to um+1 with velocity of geometrical progression, provided that the initial approximation u(0) m+1 is sufficiently close to um+1 . The incremental shell relations following from (28.3) are valid for unrestricted translations, rotations, strains and/or bendings of the shell irregular base surface, arbitrary configuration-dependent external static loading, an arbitrary combination of work-conjugate boundary conditions, and arbitrary incremental constitutive relations of the shell and the junctions.
28.3 Constitutive Elasto-Plastic Modelling in Thin Shells Analysis in the elasto-plastic range of deformation of thin irregular shells with deformable junctions can be performed by the finite element method with C 1 elements worked out by Nolte and Chroscielewski [6], which has been extended here to account deformability of the shell junctions. The shell is first divided into n layers and the plane stress state is assumed within each layer. The 3D incremental constitutive equations of each layer are described by the generalized elasto-plastic law of Prandtl-Reuss for small strains, with the associated flow rule and plasticity condition
28 On Elasto-Plastic Analysis of Thin Shells with Deformable Junctions
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of Huber-Mises-Hencky (HMH) with linear combination of isotropic and kinematic hardening. In particular, in our constitutive shell model we apply the following relations: 1. Additive decomposition of differential increment of the Green strain tensor e into elastic and plastic parts, de = deE + deP . (28.4) 2. The overstress tensor, Σ = s−α ,
Σ = s − α ,
Σ)I , Σ = Σ − 13 tr(Σ
(28.5)
where s is the 2nd Piola-Kirchhoff stress tensor, s = s − 13 tr (s) I is its deviatoric part, α is the corresponding back stress tensor, and I is the identity tensor of the 3D vector space. 3. The Huber-Mises-Hencky (HMH) yield condition, 8 ∂f Σ, ε¯ P ) = σ¯ − σY (ε¯ P ) = 0 , σ f (Σ ¯ = 32 Σ : Σ , ≡r, (28.6) Σ ∂Σ where σ¯ is the HMH effective stress and σY is the yield stress in uniaxial tension. 4. The associated plastic flow rule and evolution equation, r : C E : de de = (dλ) r , dλ = , H + r : CE : r α = (1 − β)H deP , dσY = βH d ε¯ P , dα P
d ε¯ = P
8
2 3
deP : deP ,
(28.7)
where CE is the 4th -order tensor of elastic moduli, ε¯ P is the accumulated effective plastic strain, H is the strain hardening parameter, and β ∈ [0, 1] is the material parameter determining proportion between isotropic and kinematic hardening. 5. The incremental constitutive relation of the elasto-plastic continuum, ds = C EP : de ,
(28.8)
where C EP is the instantaneous tangent 4th -order tensor of the elasto-plastic material behaviour given by CEP = CE −
(CE : r) ⊗ (r : CE ) . H + r : CE : r
(28.9)
In this approach, as the hardening function we can also take the multi-segment approximation of experimental curves following from material tests in tension, if necessary. The stress increments corresponding to the strain increments are calculated from velocity relations using the Euler method of forward integration with correction following from the plasticity condition. Then the incremental constitutive equations for the shell stress resultants and stress couples are established by direct through-the-thickness integration throughout all layers of 3D relations mentioned above. All matrix relations for the finite element are calculated numerically
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using 3-point Gauss integration within the element, and up to n = 10 integration points across the shell thickness are applied.
28.4 Deformation and Stress States in Axisymmetric Casing The axisymmetric casing of measuring device being a part of a pressure installation consists of three regular thin shells of revolution: the circular cylindrical part of thickness hc , length H and diameter D, and two toroidal parts of thickness h s , inner boundary diameter dz and radius r. These dimensions are related by D = dz + 2r, see Fig. 28.1.
hs
dw
hs
hs r
dz
D
r dz
H
r
sym.
A
sym.
A
dz
hc r A-A
Fig. 28.1 The axisymmetric casing: geometry
The toroidal parts are connected with the cylindrical one by welding, while at the upper and lower ends the toroidal parts are connected with rigid parts by welding or screw joints. Thus, in this example we have different technological inaccuracies at the junctions associated with welding (or screw joints) and with change of thickness. The force P(a) acting at the inner and lower toroidal boundaries, see Fig. 28.2, comes from pressure difference applied to rigid parts of the casing and is calculated according to & 1 % (28.10) P(a) = qπ dz2 − dw2 . 4
28 On Elasto-Plastic Analysis of Thin Shells with Deformable Junctions
hs
dw /2
s
w(s)
dz /2
r
H/2
hc sym.
f
u(s)
r (b)
[jG (b)]
H/2 (c)
X
sym.
sym.
(a)
sym.
sym.
[jG (a)]
P(a)
sym.
dz /2
hs
hs
Y
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Fig. 28.2 The axisymmetric casing: scheme of analysis
In the numerical analysis of this example we use the program MINIMOD [6, 7] with the axisymmetric two-node RING element based on the theory of thin shells with finite rotations proposed in [8]. In this element two translation components u, w are approximated using the Hermite interpolation with C 1 interelement continuity of the form ⎛ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫⎞ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u˜ ⎪ ⎬ ⎜⎜⎜⎜ 0 ⎪ ⎨ uk ⎪ ⎬ ⎨ u, sk ⎪ ⎬⎟⎟⎟⎟ 1 ⎜ ⎟, = H + H (28.11) ⎪ ⎪ ⎪ ⎪ ⎪ ⎜ sk ⎪ ⎪ ⎪ ⎝ k⎪ ⎩ w˜ ⎪ ⎭ ⎩ wk ⎪ ⎭ ⎩ w, sk ⎪ ⎭⎟⎠ k=1 1 where Hk0 and H sk are shape functions in the Hermite interpolation. In the analysis the following numerical data have been used: h s = 1 mm, H = 50 mm, D = 100 mm, dz = 10 mm, dw = 5 mm, r = 45 mm. Within th
