Multiscale Deformation and Fracture in Materials and Structures

MULTISCALE DEFORMATION AND FRACTURE IN MATERIALS AND STRUCTURES

SOLID MECHANICS AND ITS APPLICATIONS Volume 84 Series Editor:

G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI

Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

Multiscale Deformation and Fracture in Materials and Structures The James R. Rice 60th Anniversary Volume Edited by

T.-J. Chuang National Institute of Standards & Technology, Gaithersburg, U.S.A. and

J. W. Rudnicki Northwestern University, Evanston, Illinois, U.S.A.

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOST ON, DORDRECHT, LONDON, MOSCOW

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Editors’ Preface

The work of J. R. Rice has been central to developments in solid mechanics over the last thirty years. This volume collects 21 articles on deformation and fracture in honor of J.R. Rice on the occasion of his 60th birthday.Contributors include students (P. M. Anderson, G. Beltz, T.-J. Chuang, W.J. Drugan, H. Gao, M. Kachanov, V. C. Li, R. M. McMeeking, S. D. Mesarovic, J. Pan, A. Rubinstein, and J. W. Rudnicki), post-docs (L. B. Sills, Y. Huang, J.Yu, J.-S. Wang), visiting scholars (B. Cotterell, S. Kubo, H. Riedel) and co-authors (R. M. Thomson and Z. Suo). These articles provide a window on the diverse applications of modern solid mechanics to problems of deformation and fracture and insight into recent developments. The last thirty years have seen many changes to the practice and applications of solid mechanics. Some are due to the end of the Cold War and changes in the economy. The drive for competitiveness has accelerated the need to develop new types of materials without the costly and time-consuming process of trial and error. An essential element is a better understanding of the interaction of macroscopic material behavior with microscale processes, not only mechanical interactions, but also chemical and diffusive mass transfer. Unprecedented growth in the power of computing has made it possible to attack increasingly complex problems. In turn, this ability demands more sophisticated and realistic material models. A consistent theme in modern solid mechanics, and in this volume, is the effort to integrate information from different size scales. In particular, there is an increasing emphasis on understanding the role of microstructural and even atomistic processes on macroscopic material behavior. Despite the great advances in computational power, current levels do not approach that needed to employ atomic level formulations in practical applications. Consequently, idealized problems that link behavior at small, even atomic, size scales to macroscopic behavior remain essential. It would be presumptuous to hope that the articles here are as original, rigorous, clear and as strongly connected to observations as the work of the man they are meant to honor. Nevertheless, we hope that they do reflect the high standards that he has set. That they do is in no small measure a consequence of the interaction, both formal and informal, of the authors with J. R. Rice and the inspiration that his work has provided. The articles in this volume are grouped into sections on Deformation and Fracture although, obviously, there is some overlap in these topics. As is evident by reading the titles, the scope and subjects of the articles are diverse. This reflects not only the extensive impact of Rice’s work but also the broad applicability of certain fundamental tools of solid mechanics.

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Arguably, Rice’s most well-known contribution is the FRACTURE: introduction of the J-integral in 1968 and its application to problems of fracture. Because of its path-independent property, the integral has become a standard tool of fracture mechanics that makes it possible to link processes at the crack-tip to applied loads. Three of the papers in the Fracture section discuss this J-integral (and several others use it). Kubo gives a concise catalog of various versions of the integral and related extensions. Li discusses applications of the J-integral to characterization and tailoring of cementitious materials. A special feature of these materials is the presence of fibers or aggregate particles that transmit tractions across the crack-faces behind the tip. In his 1968 paper, Rice showed that the J-integral is equal to the energy released per unit area of crack advance for elastic materials. Consequently, this energy or the value of J could be used as criterion for fracture. Haug and McMeeking use the J-integral to study the effect of an extrinsic surface charge on the energy release rate for a piezoelectric compact tension specimen. They find that the presence of the free charge diminishes the effect of the electric field and suggest that this will complicate attempts to infer the portions of the crack tip singularity that are due to stress and to the electric field. A related path-independent integral, the M-integral, is used by Banks-Sills and Boniface to determine the stress intensity factors for a crack on the interface between two transversely isotropic materials. A finite element analysis is used to determine the asymptotic near-field displacements needed to evaluate the M-integral. Interpretation of the J-integral as an energy release is rigorous only for nonlinear elastic materials. But much of its usefulness arises from applications to elastic-plastic materials whose response, for proportional loading paths, is indistinguishable from a hypothetical nonlinear elastic one. For significant deviations from proportional loading, the interpretation of J in terms of fracture energy is approximate. Cotterell et al. present a method for accounting for the extra work arising from deviations from proportional loading due to significant crack growth in elastic plastic materials. Crack growth is affected not only by mechanical loading (or coupled piezoelectric loading as considered by Haug and McMeeking) but also by chemical processes. Numerical simulations by Tang et al. show that the presence of chemical activity at the crack tip can lead to blunting, stable steady crack growth or unstable sharpening of the crack tip. In the steady state regime, the computed crack velocity as a function of applied load agrees qualitatively with experiments but uncertainties in material parameters make quantitative comparison difficult. Consistent with previous studies, Tang et al. find the existence of a threshold stress level that leads to sharpening and fracture, but, contrary to previous studies, this threshold depends not only on the mechanical driving force, but also on the chemical kinetics. A classic problem of material behavior is to delineate the conditions for which materials fail ductilely or brittlely. Rice and Thomson addressed this problem by considering the interaction of a dislocation with a sharp crack-tip and arguing that ductile behavior occurred when the energetics of the interaction favored emission of a dislocation. In a concise analysis, Beltz and Fischer extend this formulation to consider the effect of the T-stress, that is , the non-singular portion of the crack-tip stress field. They show that the

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effect of this stress can be significant for small cracks, with lengths on the order of 100 atomic spacings. Klein and Gao present an innovative approach to the problem of dynamic fracture instability. They suggest that the discrepancy between predictions and observations could be resolved by including non-linear deformations near the crack-tip. They do this by a cohesive potential model that bridges the gap between continuum scale and atomistic scale calculations. Using as a measure of failure the loss of strong ellipticity, they suggest that crack branching may be associated with a loss of stiffness in biaxial stretching near the crack-tip. Several pioneering papers by Rice have considered the problem of determining the stress and deformation fields near the tip of a crack in a ductile material. The chapter by Drugan extends consideration to the case of a crack propagating along the interface of two ductile (elastic-ideally plastic) materials. An interesting by-product of the analysis for anti-plane deformation of bimaterials is a family of admissible solutions for homogeneous materials (including the well-known Chitaley -McClintock solution). Analysis reveals that beyond a certain level of material mismatch (ratio of yield stresses) a single term of the asymptotic expansion is not sufficient to characterize accurately the near-tip field. This suggests that the number of terms required will depend on some microstructural distance. Yu and Cho present detailed observations of the crack-tip fields in plastically deforming copper single crystals and compare them with fields predicted by Rice (Mechanics of Materials, 1987). They suggest that discrepancies could be due to absence of latent hardening in the elastic ideally plastic model analyzed by Rice. Rubinstein presents the results of numerical calculations based on a complex variable formulation for a variety of micromechanical models of composites. Though the calculations are elastic, they take explicit account of various reinforcing fibers, particles, etc. and, as a result the solutions depend on the ratio of fiber size to spacing, an important design variable. Another major contribution of Rice has been the DEFORMATION: development of shear localization theory as a model of failure in ductile materials. In contrast to fracture, where the stress intensification caused by acute geometry plays a dominant role, the approach of shear localization is based on the constitutive description of homogeneous deformation. The constitutive relation developed by Gurson, under Rice’s direction, has seen much application in this context because it includes softening due to the nucleation and growth of micro-voids, an important microscale feature of ductile metal deformation. Chen et al. discuss modifications of the Gurson model that are necessary to describe the anisotropy of aluminum sheets. A related chapter by Chien et al. uses a three dimensional finite element analysis of a unit cell to confirm the accuracy of a phenomenological anisotropic yield condition for porous metal and apply the phenomenological condition to analyze failure in a fender forming operation. The chapter by Rudnicki discusses shear localization of porous materials in a quite different context: the effects of coupling between pore fluid diffusion and deformation on the development of shear localization in geomaterials.

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Although the constitutive model developed by Gurson and those used by Chien et al., Chen et al. and Rudnicki are more complex than classic elastic-plastic relations, they include microstructural information simply by means of the void volume fraction or porosity. The paper by Riedel and Blug presents an example of the type of sophisticated constitutive model needed for implementation in a finite element code to model a complex technology, solid state sintering. Application of the model to silicon carbide demonstrates the level of detail and accuracy this kind of material modelling combined with finite element analysis can bring to technological processes. Elastic-plastic contact is an example of the fruitful application of continuum mechanics to microscale processes. Applications include indentation hardness testing, atomic force microscopy, powder compaction, friction and wear. Mesarovic reviews and summarizes the current understanding in this area and identifies a number of problems in need of further work. Recent computational advances have improved understanding but further work is needed in several areas. Hydrogen is an element whose presence on an interface or at a crack-tip can lead to embrittlement. In an elegant analysis that combined thermodynamics and fracture mechanics and extended the introduction of surface energy into fracture analysis by Griffith, Rice showed how the presence and mobility of segregants can alter the surface energy.Wang reviews the analysis of Rice and co-workers and shows that the predictions are consistent with observations of hydrogen embrittlement in iron single crystals. Anderson and Xin address the classic problem of the stress needed to drive a dislocation. In particular, they examine how this stress is affected by a welded interface using a model that allows them to vary independently the unstable stacking fault energy gus, the peak shear strength and the slip at peak shear. Using a numerical solution, they find that the critical resolved shear stress increases with gus, but is relatively insensitive to the maximum shear strength. Suo and Lu present a model for the growth of a two-phase epilayer on an elastic substrate. By means of a linear perturbation analysis and numerical computations, they show that the competition between phase coarsening, due to phase boundary energy, and phase refining, due to concentration dependent surface stress, can lead to a variety of growth patterns, including a stable periodic structure. The chapter by Kachanov et al. gives a complete solution for the problem of translation and rotation of ellipsoidal inclusions in an elastic space. Although they do not pursue applications of the solution, the solution is relevant to deformation around hard particles in a matrix, motion of embedded anchors, etc. Thomson et al. present a percolation theory approach to addressing the inevitable inhomogeneous deformation on the microscale. They show how it can be used to construct stress/ strain response and give insight into processes of microlocalization. We consider it an honor and privilege to have had the opportunity to edit this volume. In the preparation of the biography, H. Gao, W. Drugan and Y. Ben-Zion provided extra needed information. Jim himself provided autobiographical source material and helped proofread it to assure its correctness and completeness. We are grateful to the individual authors for their contributions and timely cooperation, and to the technical review

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board members who enhanced the quality of the volume by providing critical reviews on the articles. Our special thanks are due to Kluwer Academic Publishers, Dordrecht Office and its professional staff for their editing and production, and for their agreement to publish the Volume given even when it was still unwritten, but existed simply as a proposal in the form of a list of authors and titles. Financial support and encouragement from NIST management team, S. Freiman, G. White and E. R. Fuller, Jr. are gratefully acknowledged. Finally, we would like to express our appreciation to Drs. W. Luecke, X. Gu and J. Guyer for their help in the editing of this book.

T-J. CHUANG, Gaithersburg, MD 25 August 2000

J. W. RUDNICKI, Evanston, IL

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TABLE OF CONTENTS

v

Editors’ Preface Biography of James R. Rice T.-J. Chuang and J. W. Rudnicki List of Publications by James R. Rice List of Contributors

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PART I: DEFORMATION

Approximate Yield Criterion for Anisotropic Porous Sheet Metals and its Applications to Failure Prediction of Sheet Metals under Forming Processes W. Y. Chien, H.-M. Huang, J. Pan and S. C. Tang

1

A Dilatational Plasticity Theory for Aluminum Sheets B. Chen, P. D. Wu, Z. C. Xia, S. R. MacEwan, S. C. Tang and Y. Huang

17

Internal Hydrogen-Induced Embrittlement in Iron Single Crystals J.-S. Wang

31

A Comprehensive Model for Solid State Sintering and its Application to Silicon Carbide H. Riedel and B. Blug

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Mapping the Elastic-Plastic Contact and Adhesion S. Dj. Mesarovic

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The Critical Shear Stress to Transmit a Peierls Screw Dislocation across a Non-Slipping Interface P. M. Anderson and X.J. Xin

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Self-Organizing Nanophases on a Solid Surface Z. Suo and W. Lu

107

Elastic Space Containing a Rigid Ellipsoidal Inclusion Subjected to Translation and Rotation M. Kachanov, E. Karapetian, and I. Sevostianov

123

Strain Percolation in Metal Deformation R. M. Thomson, L. E. Levine and Y. Shim

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Diffusive Instabilities in Dilating and Compacting Geomaterials J. W. Rudnicki

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PART II: FRACTURE

Fracture Mechanics of an Interface Crack between a Special Pair of Transversely Isotropic Materials L. Banks-Sills and V. Boniface

183

Path-Independent Integrals Related to the J-Integral and Their Evaluations S. Kubo

205

On the Extension of the JR Concept to Significant Crack Growth B. Cotterell, Z. Chen and A. G. Atkins

223

Effect of T-Stress on Edge Dislocation Formation at a Crack Tip under Mode I Loading G. E. Beltz and L. L. Fischer

237

Elastic-Plastic Crack Growth along Ductile/Ductile Interfaces W. J. Drugan

243

Study of Crack Dynamics Using Virtual Internal Bond Method P. A. Klein and H. Gao

275

Crack Tip Plasticity in Copper Single Crystals J. Yu and J. W. Cho

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Numerical Simulations of SubCritical Crack Growth by Stress Corrosion in an Elastic Solid Z. Tang, A. F. Bower and T.-J. Chuang

331

Energy Release Rate for a Crack with Extrinsic Surface Charge in a Piezoelectric Compact Tension Specimen A. Haug and R. M. McMeeking

349

Micromechanics of Failure in Composites -An Analytical Study A. A. Rubinstein

361

J-Integral Applications to Characterization and Tailoring of Cementitious Materials V. C. Li

385

Author Index

407

Subject Index

415

James R. Rice

Biography of James R. Rice

James Robert Rice (JRR) was born on 3 December 1940 in Frederick, Maryland to Donald Blessing Rice and Mary Celia (Santangelo) Rice. Located some 50 miles northwest of the nation’s capital, Frederick was then a small city of about 20,000 people, set in a rural, farming area. Commemorated in Whittier’s poem about Dame Barbara Fritchie’s patriotism, Frederick was a crossroads for troop movements during the Civil War (1861-1865) and the birthplace of Francis Scott Key who wrote the American National Anthem. JRR’s mother Mary was the child of a Sicilian immigrant family and now resides in Adamstown, Maryland. The family of JRR’s father, Donald, had long lived in that part of the USA. Donald, who died in 1987, operated a gasoline station, served 3 terms as alderman and a term as mayor of Frederick City in the early 1950s, later founded a successful tire company, and, like Mary, was highly active in Frederick community affairs. JRR was raised in Frederick, and was the second of three children. His older brother, Donald Blessing Rice Jr., served as corporate CEO of several companies (such as the RAND Corporation) in the private sector and one term as Secretary of the U.S. Air Force under the Bush Administration. He now resides in Los Angeles. JRR’s younger brother, Kenneth Walter Rice, continues to live in Frederick and runs the business started by his father. JRR attended primary and secondary school at St. John’s Literary Institute, a local parish school in Frederick. He played baseball and basketball, worked part-time delivering newspapers and in his father’s businesses, and read a lot. Influenced by his high school teachers of math and physics, recruited from Fort Dieterich, a local army base, JRR’s early interest in auto mechanics gradually evolved into an interest in mechanical engineering. Armed with several scholarships, he began undergraduate studies in that subject at Lehigh University in Bethlehem, PA, in 1958, one year after the launch of Sputnik propelled the U.S. into a keen competition in outer space with the then-USSR. During his undergraduate studies at Lehigh, JRR realized his particular interest was in theoretical mechanics, especially fluid and solid mechanics, and applied mathematics. Under the influence of inspiring teachers including Ferdinand Beer, Fazil Erdogan, Paul Paris, Jerzy Owczarek, George Sih, and Gerry Smith, he did his subsequent studies in the engineering mechanics and applied mechanics programs. Paul Paris has said that for the courses JRR took from him, half of Paul’s preparation for each lecture consisted of answering the questions JRR had posed during the previous class meeting. Because of his proficiency in math and physics, JRR earned all his academic degrees, from B.S. to Ph.D in only six years (1958-1964), the shortest time in Lehigh’s record. Ferdinand Beer directed JRR’s M.S. and Ph.D. theses on stochastic processes, specifically on the statistics of highly correlated noise. The results were summarized in 1964 in his Ph.D. thesis, entitled “Theoretical Prediction of Some Statistical Characteristics of Random Loadings Relevant

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to Fatigue and Fracture”. At the same time, he continued working with George Sih on the subject of his undergraduate research project, elastic stress analysis of cracks along a bimaterial interface. He independently developed a simple elastic-plastic crack model, which turned out to be the same as D. S. Dugdale had already published, and then extended the model to the case of cyclic loads. His work on “The Mechanics of Crack Tip Deformation and Extension by Fatigue” was published in ASTM STP 415 in 1967, and was awarded the ASTM Charles B. Dudley Medal in 1969. In the late 1950s, fracture mechanics was still in the early stages of development. Egon Orowan of MIT and George Irwin of Naval Research Laboratory were beginning to advocate using stress analysis of cracks to solve fracture and fatigue problems in conventional metals and metal alloys. Motivated by the problems encountered while working at Boeing in the summers, Paul Paris was especially keen to work in this field. Together Paris, George Sih and Erdogan offered the first graduate course on fracture mechanics, which JRR took in his senior year. In addition, they recruited bright graduate students, including JRR, to do thesis research in this area. This environment cultivated JRR’s interest in fracture mechanics, which became a major focus of his teaching and research. After JRR’s graduation from Lehigh in 1964, his advisor, Ferdinand Beer, suggested he accept an offer from Daniel C. Drucker to be a post-doctoral research fellow in the Solid Mechanics Group of the Division of Engineering at Brown University. Brown was (and still is) well known internationally in the solid mechanics community. At that time many world-renowned researchers in solid mechanics were members of the faculty. They included, among others, Daniel C. Drucker, Morton E. Gurtin, Harry Kolsky, Joseph Kestin, Alan C. Pipkin, Ronald S. Rivlin, Richard T. Shield, and Paul S. Symonds. At Brown, JRR, armed with enthusiasm, energy, and innovative ideas, pursued his research on many critical fronts in fracture mechanics. He continued to collaborate with his former professors on the unfinished work from Lehigh, including characterization of fatigue loadings, plastic yielding at a crack tip and stress analysis of cracks and notches in elastic and work-hardening plastic materials under longitudinal shear loading. At Lehigh, he had also obtained some results for determining energy changes due to material removal, such as cracking or cavitation, in a linear elastic solid. At Brown, Drucker opened his eyes to the importance of generalizing these results to the widest possible class of materials; thus, JRR developed this work into a procedure for calculating energy changes in a general class of solids. This work led to JRR’s discovery of the well-known J-integral a few years later. With these impressive achievements, he was offered a tenure-track faculty job as Assistant Professor in 1965. As an assistant professor at Brown, JRR devoted his energy and efforts not only to research but also to teaching. He always believed that a good professor must excel in teaching and research. He offered many courses in applied mechanics. He developed his own lecture notes in each course without relying on specific text books. During lecturing in a typical class, he memorized every important piece of information and used the blackboard to convey the concepts to students. He was an excellent and effective

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communicator. Students were always welcome and encouraged to ask questions or engage in discussions. Copies of his lecture notes highlighting the key information including methods of derivations and final resulting formulae were distributed to his students. In research, he obtained federal funding from agencies such as NSF, DARPA, NASA, ONR, and the DOE to support project initiatives on mechanics of deformation and fracture. At this time, fracture mechanics was still in the early stages of development. JRR seized the opportunity to work out many unsolved problems in stress and deformation fields around a crack in various materials systems, mostly in 2D. Some examples are: elasticplastic mechanics of crack extension, stresses in an infinite strip containing a semi-infinite crack, plane-strain deformation near a crack in a power-law hardening material (with G.F. Rosengren), energy changes in stressed bodies due to void and crack growth (with D.C. Drucker), a path independent integral and the approximate analysis of strain concentration by notches and cracks. At the invitation of H. Liebowitz, this work was summarized in a classic review article entitled “Mathematical Analysis in the Mechanics of Fracture”, which appeared in 1968 as Chapter 3, in Volume 2, Mathematical Fundamentals of Fracture, of the book series, Fracture: An Advanced Treatise. Of particular significance was the discovery of a path-independent integral resulting from his prior probe into energy variations due to cracking of a nonlinear elastic solid. He named this particular integral the “J-Integral” with the upper case letter “J” inadvertently coinciding with his nickname “big Jim” respectfully used by his students. This integral turned out to coincide with a 2D version of the general 3D energy momentum tensor proposed by J. D. Eshelby in England in 1956. A similar concept was also developed by Cherepanov in Russia at about the same time as Rice’s J-integral, but JRR exploited the integral’s usefulness more fully in fracture analysis, especially by focusing on aspects relating to path-independence. Because of its path independence, the J-integral is a powerful tool to evaluate energy release due to cracking, bypassing the difficulties arising from strain concentration at the crack-tip. Using the procedure he developed with Drucker, JRR showed that the J-Integral is identical to the rate of reduction of potential energy with respect to crack extension. In addition, JRR, together with the late Göran F. Rosengren, showed in 1968 that the J-integral plays the role of a single unique parameter that governs the amplitude of the nonlinear deformation and stress fields inside the plastic zone near a crack tip. This result established criticality of the J-integral as a criterion for fracture even for an elastic-plastic material and made possible its use for practical engineering applications. Simultaneously, John Hutchinson at Harvard also derived a similar result. Based on their studies, the nonlinear stress distribution in the crack tip zone is now referred to as the “Huchinson-Rice-Rosengren” or “HRR” field. Over the next decade, criticality of the JIntegral was adopted as the major design criterion against failure. It is used in the ASME Pressure Vessel and Piping Design Code, and in general purpose finite element codes such as ABAQUS and ANSYS. JRR’s paper on the J-integral, which appeared in the Journal of Applied Mechanics in 1968, received the ASME Henry Hess Award in 1969 and has become a classic, attracting more than 1000 citations and references. The J-Integral forms an essential part of the subject matter contained in any textbook on fracture mechanics.

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Because of this and other contributions, JRR was promoted to Associate Professor in Engineering in 1968 and received the ASME Pi Tau Sigma Gold Medal Award for outstanding achievement in mechanical engineering within 10 years following graduation in 1971. As Associate Professor at Brown, JRR extended his research interests from mechanics to the physics and thermodynamics aspects of fracture phenomena. He worked with his student N. Levy on the prediction of temperature rise by plastic deformation at a moving or stationary crack-tip. When applied to a set of aluminum and mild steel alloys, this work helped to explain the experimentally observed relationship between the temperaturedependent toughness and the loading rate. Other accomplishments included his work with his student Dennis Tracey on the ductile void growth in a triaxial stress field. This work clarified the mechanism of void growth under applied stress in ductile metals. The role of large crack tip geometry changes in plane strain fracture was quantified in a paper with M. Johnson. He also actively participated in the development of formulations for finite element computations. He directed Ph.D. thesis research in computational fracture mechanics by Dennis Tracey. He interacted with Pedro Marcal, a faculty colleague and the founding developer of the MARC finite element code, and with Dave Hibbitt, Marcal’s graduate student and the co-developer of the ABAQUS code. Together, they developed an appropriate numerical algorithm to compute large strains and large displacements in the finite element code. This scheme has been implemented in many general purpose finite element codes such as MARC, ABAQUS and ANSYS. With another faculty colleague, Joseph Kestin, JRR worked on the application of thermodynamics to strained solids. For example, although the chemical potential is welldefined in fluids, the proper definition in solids is not clear. A paper by Kestin and Rice helped to clarify the concept and served as a starting point to extend JRR’s developing interest in high temperature fracture, namely, creep and creep rupture. In 1970, JRR was promoted to Full Professor of Engineering. With financial support from federal funding agencies such as the National Aeronautic and Space Administration (NASA), Office of Naval Research (ONR), DARPA, National Science Foundation (NSF) and Atomic Energy Commission (AEC, the predecessor of ERDA and the Department of Energy (DOE)), he was directing a research team of 7 Ph.D. graduate students. The team participated in the Materials Research Laboratory, a large-scale, interdisciplinary research program, funded by DARPA and NSF, and in a program of the AEC Basic Sciences Division directed by Joseph Gurland. JRR’s students worked in a wide range of areas in the mechanics of solids and fracture: Dennis Tracey, Dave Parks, and Bob McMeeking in (1) theoretical and computational fracture mechanics; Art Gurson in (2) constitutive relationships in metals and metallic alloys; Glenn Brown and (Jerry) T.- j. Chuang in (3) creep and creep rupture in the high temperature range; and Mike Cleary in (4) mechanics of geomaterials. Representative work in (1) included an alternative formulation of Bueckner’s (1970) weight function method to evaluate the stress intensity factor K I of a given 2D linear elastic cracked solid subject to arbitrary loading, based on any known solution to the same geometry; a finite element analysis of small scale yielding near

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a crack in plane-strain (with N. Levy, P.V. Marcal and W.J.Ostergren); an approximate method for analysis of a part-through surface crack in an elastic plate (with N. Levy); and 3D elastic-plastic stress analysis for fracture mechanics (with N. Levy and P. V. Marcal). In (2) JRR worked out the fundamental structure for the time-dependent stress-strain relationship of a metal in the plastic deformation range and proposed an internal variable theory for the inelastic constitutive relations in metal plasticity. In 1971-72, JRR took a year of sabbatical leave with support from a NSF Senior Postdoctoral Fellowship. He spent the year at the Department of Applied Mathematics and Theoretical Physics of the University of Cambridge, where he was affiliated with Churchill College under the support of a Churchill College Overseas Fellowship. At Cambridge, he worked with a number of people, including Rodney Hill, one of the pioneers in classical plasticity, Andrew C. Palmer in soil mechanics, and John Knott and his student Rob Ritchie on elastic-plastic fracture. With Hill, JRR developed a general structure of inelastic constitutive relations assuming the existence of elastic potentials, and gave a special implementation for elastic/plastic crystals at finite strain. In the latter case, crystallographic slip along a set of active slip planes was considered as the sole deformation mechanism responsible for the inelastic behavior. This theory successfully explained various aspects of plasticity such as strain hardening, the existence of a flow rule and normality. With Knott and Ritchie, JRR proposed a relationship between the critical tensile stress and the fracture toughness of mild steel. The analysis predicts the observed temperature dependence of K IC in the brittle to ductile transition range. With Andrew Palmer, JRR used his newly developed J-integral to develop a mode-II “shear crack” model for the growth of slip surfaces in over-consolidated clay slopes. Returning to Brown in 1972, JRR continued to pursue research on many aspects of fracture mechanics. John Landes and Jim Begley of the Westinghouse R&D Center became keen advocates of using the J-Integral as a design criterion in the nuclear energy business, and in a paper with Landes and Paul Paris, JRR developed an elegantly simple procedure to estimate the value of J-Integrals from experiments. Eventually, this procedure became the ASTM standard and part of the ASME Pressure Vessels and Piping design code. Besides analysis on the continuum level, JRR strongly felt that there was a need to study fracture at the microstructural level in order to bridge the atomic and engineering scales. One important area that required such a treatment is high temperature creep and creep rupture where mass transport plays an important role. At that time, a group at Harvard led by Mike Ashby was also interested in this topic. As a result, there was much interaction between Harvard and Brown during 1972-74: JRR and Ashby and their students made frequent mutual visits to give seminars and to exchange ideas. One important result, jointly developed in 1973 with his student, T.- J. Chuang, was the discovery of creep crack-like cavity shapes induced by surface diffusion. This type of cavity, referred to as a Chuang-Rice crack-like cavity, is frequently observed at the grain boundaries of a ruptured tensile specimen. This work defines the boundary conditions at the cavity apex and satisfactorily explained non-linear stress dependence on cavity growth rate. The degree of non-linearity depends on the deformability of the grains, and JRR obtained solutions for the stress

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dependence on creep cavity growth in rigid grains (with Chuang, Kagawa, Sills and Sham), in elastic grains (with Chuang) and in plastic grains (with Needleman). The predicted stress dependence was verified experimentally by Bill Nix and his students at Stanford in the late 1970s using implanted water vapor cavities at grain boundaries in pure silver and nickel-tin alloys. Later in the 1980s and 90s, this work was used by many researchers to predict cavity growth induced by electromigration in aluminum interconnect wires. In 1973, JRR was offered a Chair by the Brown President, Donald Hornig with the title L. Herbert Ballou Professor of Theoretical and Applied Mechanics. This privileged title is an honor comparable to a University Professorship, which is the highest rank of teaching professors at Brown. In physical metallurgy, it had become well-known that dislocations at the atomic level are fully responsible for the room temperature plastic behavior in metals. Since the early 1960s, many researchers (such as Hirth, Lothe, Mura and Weertman) devoted their efforts to this area and helped to build the foundation of dislocation theory. JRR was among those cutting edge scholars who excelled in mathematical dislocation theory. In 1972, he met Robb Thomson of SUNY-Stony Brook at a conference and they puzzled over the ductile versus brittle transition phenomenon in crystals. Since dislocation movement leads to ductility and rapid crack growth leads to catastrophic failure, they believed the interactions of both must play a dominant role in ductile/brittle behavior. They proposed that the ability to emit dislocations from a pre-existing sharp crack tip is the source of ductility in metals. On the other hand, the resistance of a crack tip to dislocation emission leads to brittleness in ionic or covalent crystals like ceramics. By analyzing the energetic forces between a dislocation and a crack, they derived an important parameter that governs the ductility. If this parameter, which is shear modulus times Burgers vector over surface energy, exceeds 8.5 to 10, then the crystal exhibits intrinsically brittle behavior. If less, it is generally ductile. The Rice -Thomson theory has become a classic in the Science Citation Index with more than 200 citations. In the late 1970’s, Mike Ohr of Oak Ridge National Laboratory provided direct experimental evidence for the theory by observing emission of dislocations from the crack tip in a variety of metal specimens in situ under TEM. In another noteworthy work, JRR helped his student Art Gurson to develop in 1975 the plasticity theory of porous media, in which yield criteria and flow rules were predicted in stress space using 2D or 3D unit cell models. The model predicts the effect of porosity on the plastic behavior of ductile materials and has come to be known as the “Gurson” model. It is well-known in the metallurgy and mechanics communities and is one of the major yield criteria adopted in the commercial general purpose finite element codes for assessing inelastic behavior of metallic materials. Motivated by his studies of shear bands with Andrew Palmer, JRR became interested in the fundamental question of why deformation would localize in a narrow zone. A basic premise of fracture mechanics, going back to the ideas of Griffith, is that the presence of flaws in a material causes a local elevation of the stress and leads to propagation of the flaw and, eventually, to failure. Although this process provides a satisfactory explanation of failure in many materials, it does not explain why macroscopically uniform

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deformation should give way to localized deformation in very ductile materials or under conditions of compressive stress that suppress flaw propagation. Based on antecedents in the work of Hadamard, Hill, Thomas and Mandel, JRR and his student Rudnicki treated the initiation of localized deformation as a bifurcation from homogeneous deformation and showed that its onset was promoted by certain subtle features of the constitutive behavior. This work, which was published in the Journal of the Mechanics and Physics of Solids in 1975, received the Award for Outstanding Research in Rock Mechanics from the U. S. National Committee on Rock Mechanics in 1977. Although this work was originally intended to describe fault formation in rock, JRR extended the approach to consider localized necking in thin sheets (with S. Storen), strain localization in ductile single crystals (with R. J. Asaro), and limits to ductility in sheet metal forming (with A. Needleman). He summarized the state of the subject in a keynote lecture on “The Localization of Plastic Deformation” at the 14 th International Congress on Theoretical and Applied Mechanics in Delft in 1976. The printed version of this lecture is a widely-cited classic. In the early 1970’s, there were many reports of observations precursory to earthquakes that were attributed to the coupling of deformation with the diffusion of pore fluid. A series of papers, by JRR with students (Cleary and Rudnicki) and Don Simons, an Assistant Professor at Brown, analyzed the effects of this coupling on models for earthquake instability and for quasi-statically propagating creep events. One of these papers (“Some basic stress-diffusion solutions for fluid-saturated elastic porous media with compressible constituents”, with M. P. Cleary, Rev. Geophys. Space Phys., 14, pp. 227-241, 1976) reformulated, in a particularly insightful way, the equations first derived by Biot for a linear elastic, porous, fluid-infiltrated solid. This version of the equations has proven so advantageous that it is now the standard form. The models of the earthquake instability formulated to study these effects were among the first in which the instability was not postulated but arose in a mechanically consistent way from the interaction of the fault zone material behavior and the surroundings. JRR’s interest in the mechanics of earthquakes proved durable and became a major branch of his work. With Florian Lehner and Victor Li, he worked on time-dependent effects due to coupling of the shallow, elastic portion of the Earth’s lithosphere with deeper viscoelastic portions. This work was based on a generalization of an earlier thin plate model by Elsasser. This work demonstrated that the viscous deformation of the lower crust and upper mantle following large earthquakes could affect surface deformation for decades and provided a new model for the interpretation of increasingly detailed surface deformation measurements. In the early 90s, JRR used the finite element code ABAQUS together with Yehuda Ben-Zion, Renata Dmowska, Mark Linker, and Mark Taylor to explore the behavior of this model in 3D and to compare model predictions with geophysical observations. JRR’s growing interest in the mechanics of earthquakes complemented nicely the interests of his spouse, Renata Dmowska, a seismologist. Together, Renata's analysis of data and JRR’s mathematical models have been combined in several papers on aspects of earthquakes, particularly in subduction zones. JRR’s interest in the mechanics of earthquakes soon led to a study of frictional

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stability. Stick-slip is a widely observed phenomenon and has long been regarded as a physical analog for the earthquake instability. But the standard constitutive description, static and dynamic friction, was inconsistent with the steady sliding often observed and contained no mechanism for restrengthening that would allow repeated events on the same surface. Based on experimental observations of Dieterich at the U.S. Geological Survey, JRR and his student Andy Ruina formulated a rate- and state-dependent constitutive relationship for sliding on a frictional surface. By examining the stability of a one degree-offreedom system with this relationship, they were able to predict the variety of behaviors observed in rock friction experiments: steady sliding, damped oscillations, stick-slip and sustained periodic oscillations. Other papers with Tse and Gu examined the dynamics and nonlinear stability of these systems. JRR and his student Tse showed that when this type of relationship was applied on a surface between two elastic solids and modified to include a depth dependence appropriate for the temperature and pressure dependence in the earth, the calculations produced periodic events with a depth dependence remarkably similar to that of observed earthquakes. In the late 1970s and early 1980s, JRR also continued to work on many aspects of inelasticity and fracture. With Joop Nagtegaal and Dave Parks, he developed a numerical scheme to improve the accuracy of finite element computations in the fully plastic range. With Bob McMeeking, he worked out the proper finite element formulation in the large elastic-plastic deformation regime. With a colleague at Brown, Ben Freund, and a student, Dave Parks, he helped solve the problem of a running crack in a pressurized pipeline. In materials science, he studied stress corrosion and hydrogen embrittlement problems. He also orchestrated a remarkable multidirectional attack on the problem of quasistatic crack growth in elastic-plastic materials. This began with a paper with Paul Sorensen in 1978 that proposed an elegant way of using near-tip elastic-plastic fields to derive theoretical predictions for crack growth resistance curves (J R curves). Then, he and his student Walt Drugan derived asymptotic analytical elastic-ideally plastic solutions for the stress and deformation fields near a plane strain growing crack which showed the necessity of an elastic unloading sector in the near-tip field. [Independent work by L. I. Slepyan in the then-USSR and Y. C. Gao in China also addressed this problem, for incompressible material and steady-state conditions.] The detailed numerical finite element elastic-plastic growing crack solutions of JRR’s student T-L. Sham confirmed the analytical predictions, and in a 1980 paper with Drugan and Sham, JRR combined the method proposed earlier with Sorensen, with the new analytical asymptotic solutions and Sham’s numerical results, to produce a comprehensive and fundamentals-based model of stable ductile crack growth and predictions of plane strain crack growth resistance curves. Then, with Lawrence Hermann, JRR conducted and analyzed “plane strain” crack growth tests and showed that this theory was indeed capable of describing the experimentally-measured crack growth resistance curves under contained yielding conditions. The asymptotic analysis of elastic-ideally plastic growing crack fields, involving the assembling of different possible types of near-tip solution sectors into complete near-tip solutions, prompted JRR and Drugan to inquire more fundamentally about what continuity

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and jump conditions are required across quasi-statically propagating surfaces in elasticplastic materials by the fundamental laws of continuum mechanics and broad, realistic constitutive constraints (such as the maximum plastic work inequality). Their resulting restrictions (published in the D. C. Drucker Anniversary Volume), and the later generalization of these to dynamic conditions by Drugan and Shen, have been utilized repeatedly in elastic-plastic crack growth studies. Not surprisingly, perhaps the most important applications of these discontinuity results are due to JRR himself, in his fundamental studies of stationary and growing crack fields in ductile single crystals, wherein JRR showed that a precise understanding of possible discontinuity types is absolutely essential in deriving correct solutions. Beginning in 1985 with his student R. Nikolic on the anti-plane shear crack problem, and in a landmark, pioneering 1987 paper on plane strain tensile cracks, JRR produced fascinating analytical solutions for the near-tip fields in elastic-ideally plastic ductile single crystals. These fields differ dramatically from crack fields in isotropic (i.e., polycrystalline) ductile materials, being characterized by discontinuous displacements and stresses for stationary cracks, discontinuous velocities for quasi-statically growing cracks, and, in another fascinating paper with Nikolic in 1988, JRR showed that the near-tip field for a dynamically propagating anti-plane shear crack in a ductile single crystal must involve shock surfaces across which stress and velocity jump. JRR and his student M. Saeedvafa generalized the stationary crack ductile single crystal solutions to incorporate Taylor hardening, revealing even more complex near-tip behavior. Other major work in the late 1970s and early 1980s included two important papers with visiting faculty members: one on the crack tip stress and deformation fields for a crack in a creeping solid, with Hermann Riedel; and another heavily-cited paper on crack curving and kinking in elastic materials, with Brian Cotterell. For his significant contributions to sciences and engineering, JRR was elected to Fellow grade of the American Academy of Arts and Sciences in 1978, Fellow of the American Society of Mechanical Engineers and Membership in the National Academy of Engineering in 1980, and membership in the National Academy of Sciences in 1981. The next move was to Harvard University in September 1981. A Gordon McKay Chaired Professorship in Engineering Sciences and Geophysics was created for JRR, jointly in the Division of Applied Sciences and the Department of Earth and Planetary Sciences. He further expanded the scope of his research activities along two major branches in mechanics, namely, fracture of engineering materials and geological materials. At Harvard, he recruited many bright students from all over the world to work on topical fracture problems in engineering and geology. He directed Peter Anderson to study constrained creep cavitation and the Rice-Thomson model, supervised Huajian Gao on three dimensional crack problems, worked with Jwo Pan, Ruzica Nikolic and Maryam Saeedvafa on inelastic behaviors of cracks in single crystal metals, and collaborated with Renata Dmowska, Victor Li, Paul Segall, Andy Ruina, Yehuda Ben-Zion, G. Perrin, J.-c. Gu, Mark Linker, Simon T. Tse , G. Zheng, and F. K. Lehner in developing friction laws and shear crack models of geological faults as related to earthquake events in seismology. JRR’s recent work on earthquakes has focused on several important aspects of the

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process. One issue is the origin of earthquake complexity, that is, the distribution of events of various sizes, as described by the well-established Gutenberg-Richter relationship. One previous explanation was that fault slip, as modeled by friction between two elastic solids, was an inherently chaotic process. In a series of papers that combine elegant analysis and prodigious calculations, JRR and his post-doc, Yehuda Ben-Zion, showed that the chaotic behavior predicted in these models was the subtle result of numerical discretization and oversimplification of the frictional constitutive relation. Other work was motivated by observations that slip during an earthquake does not propagate in the fashion predicted by classical dynamic fracture mechanics with most of the surface slipping for the entire duration of the event. Instead, slip is pulse-like and any point on the surface slips only for a short time. Papers with Zheng and Perrin showed that only certain types of frictional constitutive relations were consistent with these observations. Another, very influential paper, “Fault Stress States, Pore pressure Distributions and the Weakness of the San Andreas Fault” addresses a long-standing paradox in earthquake mechanics: A variety of measurements indicate that the San Andreas fault in southern California is much weaker, both in an absolute sense and relative to the surrounding crust, than would be expected from a straightforward interpretation of laboratory friction experiments. JRR showed that the discrepancy could be resolved by high fluid pressures within the fault zone and summarized a variety of evidence for this possibility. Another mechanism that can explain the discrepancy and produce slip in a pulse-like form is dynamic rupture along a bi-material interface. JRR has been studying this problem recently together with his student K. Ranjith and Post-Doc A. Cochard, following earlier works of Weertman, Adams, and Andrews and Ben-Zion, thus returning to a subject he investigated statically as an undergrad at Lehigh. In the mid-1980’s, JRR and other faculty members including John Hutchinson and Bernie Budiansky formed a joint research team with Tony Evans at the University of California at Santa Barbara to study mechanical behavior and toughening mechanisms of ceramics. Between 1988 and 1994, faculty and students at Harvard regularly visited and exchanged ideas with Tony Evans and his research group at UCSB. The Harvard-UCSB collaboration generated tremendous research output. During this period, JRR worked with John Hutchinson, Jian-Sheng Wang, Mark E. Mear and Zhigang Suo on crack growth on or near a bi-material interface. With Jian-Sheng Wang, he developed a model of interfacial embrittlement by hydrogen and solute segregation. This model has been referred to as the Rice-Wang Model which provided a basis for the materials community in pursuit of better design of steels. Between 1989 and 1995, JRR worked with Glenn Beltz, Y. Sun and L. Truskinovsky to reformulate the Rice-Thomson model in terms of interactions between a crack and a Peierls dislocation being emitted from the crack tip. This study eliminated the need to define a core cut-off radius for dislocations and instead established unstable stacking fault energy as the new physical parameter governing the intrinsic ductility of crystals. Rice’s new model caused an instant sensation among materials scientists and physicists and is now used as the new paradigm for understanding brittle-ductile transition of crystals. Separate from his other activities at Harvard, JRR began to develop a growing interest in three dimensional crack problems, starting around 1984. Together with Huajian

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Gao, the first of his graduate students at Harvard to work on 3-D crack problems, he developed a series of ingenious methods of analysis based on the idea of 3-D weight functions, generalizing a 2-D concept he and Hans Bueckner had developed in the early 1970’s. These methods were used to study configurational stability of crack fronts, crack interaction with dislocation loops and transformation strains, and trapping of crack fronts by tough particles. In 1987, he began to work with K. S. Kim, who spent a year of sabbatical at Harvard, to generalize these methods to model dynamically propagating 3-D crack fronts. This then led to a burst of his interests in the following years in the spontaneous dynamics of 3-D tensile crack propagation and of slip ruptures in earthquake dynamics. He directed a number of graduate students, post-docs and visiting scientists on those areas, including K. S. Kim, Yehuda Ben-Zion, G. Perrin, G. Zheng, Phillipe Geubelle, A. Cochard, J. W. Morrissey, and Nadia Lapusta. He also encouraged other leading scientists such as John Willis and Daniel Fisher to work in this field. An example of significant discoveries coming out of these activities is a new kind of wave which propagates along the crack front at a velocity different from the usual body and surface elastic wave speeds. JRR continues today to lead an international research effort in crack and fault dynamics. Needless to say, the output of his research group is of the highest quality and generates significant impact on the engineering, materials science and geophysics communities. As a result of his contributions to science and engineering, JRR received numerous awards and recognitions by professional societies and academic institutions. In 1981, he was elected to Fellow of AAAS. Next year in 1982, he received the George R. Irwin Medal from ASTM Committee E-24, shared with John Hutchinson, for “significant contributions to the development of nonlinear fracture mechanics”. In 1985, he was one of the recipients of an Honorary Doctor of Science Degree at his alma mater, Lehigh University. In 1988, he was elected Fellow of the American Geophysical Union, and received the William Prager Medal from the Society of Engineering Science for his “outstanding achievements in solid mechanics”. Two years later, he was elected Fellow of the American Academy of Mechanics and the Royal Society of Edinburgh. In 1992, he received an award from AAM for “Distinguished Service to the Field of Theoretical and Applied Mechanics”. The following year he served as Francis Birch Lecturer on “Problems on Earthquake Source Mechanics” at the American Geophysics Union. The next year he received the ASME Timoshenko Medal with the following citation: “for seminal contributions to the understanding of plasticity and fracture of engineering materials and applications in the development in the computational and experimental methods of broad significance in mechanical engineering practice”. In 1996, he was elected as a Foreign Member of the Royal Society of London for his work on “earthquakes and solid mechanics” and received an honorary degree from Northwestern University. In addition, he received the ASME Nadai Award for major contributions to the fundamental understanding of plastic flow and fracture processes in engineering and geophysical materials and for the invention of the JIntegral which forms the basis for the practical application of nonlinear fracture mechanics to the development of standards for the safety of structures. He also received the Francis J. Clamer Medal from the Franklin Institute for Advances in Metallurgy with the citation: “for

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development of the J-Integral for the accurate prediction of elastic-plastic fracture behavior in metal from easily obtained data”. In 1997, he received an honorary Doctor of Science degree from Brown University. In 1998, a donation from David Hibbitt and Paul Sorensen of HKS, Inc. established the Rice Professorship at Brown in his honor. Recently, he was awarded the Blaise Pascal Professorship by the Region Ile-de-France for the 1999 calendar year for research on “Rupture Dynamics in Seismology and Materials Physics”, and he was the recipient of an Honorary Doctoral Degree at the University of Paris VI in March 1999. He was elected a Foreign Member (Associé Étrager) of the French Academy of Sciences in April 2000. There is no need to place complimentary words here on the impact of his work. The recognitions described in the previous paragraph speak for themselves. His standards of scholarship and intellectual honesty are the highest. He is always ready to appreciate the good work of other colleagues, and to give them proper credit. On the other hand, he does not hesitate to dispense candid criticism of inconsistent or misguided thinking, though in a gentle rather than harsh manner -- as some oral comments in conferences or written book reviews testify. A man is as young as he thinks. JRR enjoys long walks, whether in urban or mountain settings, reads broadly in science, history and social commentary, and likes listening to classical and folk music in his spare time.. He has an excellent sense of humor, a razor-sharp wit and a cheerful disposition. His wife Renata Dmowska, in addition to being a regular and important scientific collaborator, is an excellent influence on Jim. Renata is an enthusiastic polymath with a warm and cheerful personality and a seemingly endless array of interests. She insists that he take much-deserved breaks from his research to attend concerts, to visit art museums, to travel, to read literature, and to socialize with their large circle of friends. JRR is increasingly active in his research, full of curiosity, creativity and persistence. As his students can attest, he is also an excellent teacher in the classroom. He gives lectures in a humorous, but comprehensive way that can be easily digested by his audience. As a thesis advisor, he defines the scope of a research area in which he sees the potential for advancement. He inspires and encourages, but does not push his students. When a student heads in a wrong direction or reaches a dead end, he wastes no time to steer him or her back to the right track. His good qualities as an advisor were recognized by his recent Excellence in Mentoring Award conferred by the Graduate Student Council of Harvard University in April 1999. JRR recently returned from his full year sabbatical leave (January 1999 to January 2000) in Paris, France, working in the Département Terre Atmosphère Océan of École Normale Supérieure, and also part time at École Polytechnique in Paliseau. His flow of publications shows no sign of diminishing and his friends and colleagues surely will hope that the short legend “J. R. Rice” will appear again and again in the scientific literature for many years to come. TZE-JER CHUANG Gaithersburg, MD

JOHN W. RUDNICKI Evanston, IL

List of Publications by James R. Rice

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G. C. Sih and J. R. Rice, “The Bending of Plates of Dissimilar Materials with Cracks”, Journal of Applied Mechanics, 31, (1964), pp. 477-482. J. R. Rice and E. J. Brown, “Discussion of ‘Random Fatigue Failure of a Multiple Load Path Redundant Structure’ by Heller, Heller and Freudenthal”, in Fatigue: An Interdisciplinary Approach (eds. J. Burke, N. Reed and V. Weiss), Syracuse University Press, (1974), pp. 202-206. J. R. Rice and F. P. Beer, “On the Distribution of Rises and Falls in a Continuous Random Process”, Transactions ASME (Journal of Basic Engineering), 87D, (1965), pp. 398-404. J. R. Rice and G. C. Sih, “Plane Problems of Cracks in Dissimilar Materials”, Journal of Applied Mechanics, 32, (1965), pp. 418-423. J. R. Rice, F. P. Beer and P. C. Paris, “On the Prediction of Some Random Loading Characteristics Relevant to Fatigue”, in Acoustical Fatigue in Aerospace Structures (eds. W. Trapp and D. Forney), Syracuse University Press, (1965), pp. 121-144. J. R. Rice, “Starting Transients in the Response on Linear Systems to Stationary Random Loadings”, Journal of Applied Mechanics, 32, (1965) pp. 200-201. J. R. Rice, “Plastic Yielding at a Crack Tip”, in Proceedings of the 1st International Conference on Fracture, Sendai, 1965 (eds. T. Yokobori, T. Kawasaki, and J. L. Swedlow), Vol. I, Japanese Society for Strength and Fracture of Materials, Tokyo, (1966), pp. 283-308. J. R. Rice, “An Examination of the Fracture Mechanics Energy Balance from the Point of View of Continuum Mechanics”, in Proceedings of the 1st International Conference on Fracture, Sendai, 1965 (eds. T. Yokobori, T. Kawasaki, and J. L. Swedlow),Vol. I, Japanese Society for Strength and Fracture of Materials, Tokyo, (1966) pp. 309-340. J. R. Rice and F. P. Beer, “First Occurrence Time of High Level Crossings in a Continuous Random Process”, Journal of the Acoustical Society of America, 39, (1966) pp. 323-335. J. R. Rice, “Contained Plastic Deformation Near Cracks and Notches Under Longitudinal Shear”, International Journal of Fracture Mechanics, 2, (1966) pp. 426-447. J. R. Rice and D. C. Drucker, “Energy Changes in Stressed Bodies due to Void and Crack Growth”, International Journal of Fracture Mechanics, 3, (1967) pp. 19-27. J. R. Rice, “Stresses due to a Sharp Notch in a Work Hardening Elastic-Plastic Material Loaded by Longitudinal Shear”, Journal of Applied Mechanics, 34, (1967), pp. 287-298.

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LIST OF PUBLICATIONS BY J. R. RICE J. R. Rice, “The Mechanics of Crack Tip Deformation and Extension by Fatigue”, in Fatigue Crack Propagation, Special Technical Publication 415, ASTM, Philadelphia, (1967), pp. 247-311. J. R. Rice, “Discussion of ‘Stresses in an Infinite Strip Containing a Semi-Infinite Crack’ by W.G. Knauss”, Journal of Applied Mechanics, 34, (1967), pp. 248-250. J. R. Rice, “A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks”, Journal of Applied Mechanics, 35, (1968), pp. 379-386. J. R. Rice and G. F. Rosengren, “Plane Strain Deformation Near a Crack in a Power Law Hardening Material”, Journal of the Mechanics and Physics of Solids, 16, (1968), pp. 1-12. J. R. Rice, “The Elastic-Plastic Mechanics of Crack Extension”, International Journal of Fracture Mechanics, 4, (1968), pp. 41-49 (also published in International Symposium on Fracture Mechanics, Wolters-Noordhoff Publ., Groningen, 1968,41-49). J. R. Rice, “Mathematical Analysis in the Mechanics of Fracture”, Chapter 3 of Fracture: An Advanced Treatise (Vol. 2, Mathematical Fundamentals) (ed. H. Liebowitz), Academic Press, N.Y., (1968), pp. 191-311. J. R. Rice and N. Levy, “Local Heating by Plastic Deformation at a Crack Tip”, in Physics of Strength and Plasticity (ed. A. S. Argon), M.I.T. Press, Cambridge, Mass., (1969), pp. 277-293. J. R. Rice and D. M. Tracey, “On the Ductile Enlargement of Voids in Triaxial Stress Fields”, Journal of the Mechanics and Physics of Solids, 17, (1969), pp. 201-217. D. C. Drucker and J. R. Rice, “Plastic Deformation on Brittle and Ductile Fracture”, Engineering Fracture Mechanics, 1, (1970), pp. 577-602. J. Kestin and J. R. Rice, “Paradoxes in the Application of Thermodynamics to Strained Solids”, in A Critical Review of Thermodynamics (eds. E.G. Stuart, B. Gal-Or and A.J. Brainard), Mono Book Corp., Baltimore, MD (1970), pp. 275298. H. D. Hibbitt, P. V. Marcal and J. R. Rice, “A Finite Element Formulation for Problems of Large Strain and Large Displacement”, International Journal of Solids and Structures, 6, (1970), pp. 1069-1086. J. R. Rice, “On the Structure of Stress-Strain Relations for Time-Dependent Plastic Deformation in Metals”, Journal of Applied Mechanics, 37, (1970), pp. 728-737. J. R. Rice and M. A. Johnson, “The Role of Large Crack Tip Geometry Changes in Plane Strain Fracture”, in Inelastic Behavior of Solids (eds. M. F. Kanninen, et al.), McGraw-Hill, N.Y., (1970), pp. 641-672. N. Levy, P. V. Marcal, W. J. Ostergren and J. R. Rice, “Small Scale Yielding Near a Crack in Plane Strain: A Finite Element Analysis”, International Journal of Fracture Mechanics, 7, (1971), pp. 143-156.

LIST OF PUBLICATIONS BY J. R. RICE 27. 28.

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J. R. Rice and N. Levy, “The Part-Through Surface Crack in an Elastic Plate”, Journal of Applied Mechanics, 39, (1972), pp. 185-194. N. Levy, P. V. Marcal and J. R. Rice, “Progress in Three-Dimensional ElasticPlastic Stress Analysis for Fracture Mechanics”, Nuclear Engineering and Design, 17, (1971), pp. 64-75. J. R. Rice, “Inelastic Constitutive Relations for Solids: An Internal Variable Theory and Its Application to Metal Plasticity”, Journal of the Mechanics and Physics of Solids, 19, (1971), pp. 433-455. J. R. Rice, “Some Remarks on Elastic Crack Tip Stress Fields”, International Journal of Solids and Structures, 8, (1972), pp. 571-578. J. R. Rice and D. M. Tracey, “Computational Fracture Mechanics”, in Numerical and Computer Methods in Structural Mechanics (eds. S. J. Fenves et al.), Academic Press, N.Y., (1973), pp. 585-623. B. Budiansky and J. R. Rice, “Conservation Laws and Energy-Release Rates”, Journal of Applied Mechanics, 40, (1973), pp. 201-203. J. R. Rice and M. A. Chinnery, “On the Calculation of Changes in the Earth’s Inertia Tensor due to Faulting”, Geophysical Journal of the Royal Astronomical Society, 29, (1972), pp. 79-90. R. J. Bucci, P. C. Paris, J. D. Landes and J. R. Rice, “J Integral Estimation Procedures”, in Fracture Toughness, Special Technical Publication 514, Part 2, ASTM, Philadelphia, (1972), pp. 40-69. J. R. Rice, “The Line Spring Model for Surface Flaws”, in The Surface Crack: Physical Problems and Computational Solutions (ed. J.L. Swedlow), ASME, N.Y., (1972), pp. 171-185. R. Hill and J. R. Rice, “Constitutive Analysis of Elastic/Plastic Crystals at Arbitrary Strain”, Journal of the Mechanics and Physics of Solids, 20, (1972), pp. 401-413. J. R. Rice, “Elastic-Plastic Fracture Mechanics (Remarks for Round Table Discusison on Fracture at the 13th International Congress of Theoretical and Applied Mechanics, Moscow, 1972)“, Engineering Fracture Mechanics, 5, (1973), pp. 1019-1022. A. C. Palmer and J. R. Rice, “The Growth of Slip Surfaces in the Progressive Failure of Overconsolidated Clay”, Proceedings of the Royal Society of London, A 332, (1973), pp. 527-548. J. R. Rice, “Plane Strain Slip Line Theory for Anisotropic Rigic/Plastic Materials”, Journal of the Mechanics and Physics of Solids, 21, (1973), pp. 63-74. J. R. Rice, P. C. Paris and J. G. Merkle, “Some Further Results of J-Integral Analysis and Estimates”, in Progress in Flaw Growth and Fracture Toughness Testing, Special Tech. Publication 536, ASTM, Philadelphia, PA (1973), pp. 231245.

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LIST OF PUBLICATIONS BY J. R. RICE R. Hill and J. R. Rice, “Elastic Potentials and the Structure of Inelastic Constitutive Laws”, SIAM Journal of Applied Mathematics, 25, (1973), pp. 448461. J. R. Rice, “Continuum Plasticity in Relation to Microscale Deformation Mechanisms”, in Metallurgical Effects at High Strain Rate (eds. R.W. Rohde et al.), Plenum Press, (1973), pp. 93-106. J. R. Rice, “Elastic-Plastic Models for Stable Crack Growth”, in Mechanics and Mechanisms of Crack Growth (ed. M.J. May), British Steel Corporation Physical Metallurgy Centre Publication, April 1973 (issued 1975), pp. 14-39. T.- J. Chuang and J. R. Rice, “The Shape of Intergranular Creep Cracks Growing by Surface Diffusion”, Acta Metallurgica, 21, (1973), pp. 1625-1628. R. O. Ritchie, J. F. Knott and J. R. Rice, “On the Relationship Between Critical Tensile Stress and Fracture Toughness in Mild Steel”, Journal of the Mechanics and Physics of Solids, 21, (1973), pp. 395-410. L. B. Freund and J. R. Rice, “On the Determination of Elastodynamic Crack Tip Stress Fields, International Journal of Solids and Structures, 10, (1974), pp. 411417. J. R. Rice, “Limitations to the Small Scale Yielding Approximation for Crack Tip Plasticity”, Journal of the Mechanics and Physics of Solids, 22, (1974), pp. 17-26. J. R. Rice and R. M. Thomson, “Ductile vs. Brittle Behavior of Crystals”, Philosophical Magazine, 29, (1974), 73-97. J. R. Rice, “The Initiation and Growth of Shear Bands”, in Plasticity and Soil Mechanics (edited by A. C. Palmer), Cambridge University Engineering Department, Cambridge, (1973) pp. 263-274. J. C. Nagtegaal, D. M. Parks and J. R. Rice, “On Numerically Accurate Finite Element Solutions in the Fully Plastic Range”, Computer Methods in Applied Mechanics and Engineering, 4, (1974) pp. 153-177. J. R. Rice, “Continuum Mechanics and Thermodynamics of Plasticity in Relation to Microscale Deformation Mechanisms”, Chapter 2 of Constitutive Equations in Plasticity (ed. A. S. Argon), M.I.T. Press, (1975), pp. 23-79. R. M. McMeeking and J. R. Rice, “Finite-Element Formulations for Problems of Large Elastic-Plastic Deformation”, International Journal of Solids and Structures, 11, (1975), pp. 601-616. J. R. Rice, “On the Stability of Dilatant Hardening for Saturated Rock Masses”, Journal of Geophysical Research, 80, (1975), pp. 1531-1536. J. R. Rice, “Discussion of ‘The Path Independence of the J-Contour Integral’ by G. G. Chell and P. T. Heald”, International Journal of Fracture, 11, (1975), pp. 352353. J. W. Rudnicki and J. R. Rice, “Conditions for the Localization of Deformation in Pressure-Sensitive Dilatant Materials”, Journal of the Mechanics and Physics of Solids, 23, (1975) pp. 371-394.

LIST OF PUBLICATIONS BY J. R. RICE 56. 57.

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S. Storen and J. R. Rice, “Localized Necking in Thin Sheets”, Journal of the Mechanics and Physics of Solids, 23, (1975), pp. 421-441. J. R. Rice, “Some Mechanics Research Topics Related to the Hydrogen Embrittlement of Metals” (discussion appended to paper by J. P. Hirth and H. H. Johnson); Corrosion, 32, (1976), pp. 22-26. J. R. Rice and M. P. Cleary, “Some Basic Stress-Diffusion Solutions for FluidSaturated Elastic Porous Media with Compressible Constituents”, Reviews of Geophysics and Space Physics, 14, (1976), pp. 227-241. J. R. Rice, “Hydrogen and Interfacial Cohesion”, in Effect of Hydrogen on Behavior of Materials (eds. A.W. Thompson and I.M. Bernstein), Metallurgical Society of AIME, (1976), pp. 455-466. L.B. Freund, D.M. Parks and J. R. Rice, “Running Ductile Fracture in a Pressurized Line Pipe”, in Mechanics of Crack Growth, Special Technical Publication 590, ASTM, Philadephia, (1976), pp. 243-262. J. R. Rice, “The Localization of Plastic Deformation”, in Theoretical and Applied Mechanics (Proceedings of the 14th International Congress on Theoretical and Applied Mechanics, Delft, 1976, ed. W.T. Koiter), Vol. 1, North-Holland Publishing Co., (1976), 207-220. J. R. Rice and D. A. Simons, “The Stabilization of Spreading Shear Faults by Coupled Deformation-Diffusion Effects in Fluid-Infiltrated Porous Materials”, Journal of Geophysical Research, 81, (1976), pp. 5322-5334. J. R. Rice, “Elastic-Plastic Fracture Mechanics”, in The Mechanics of Fracture (ed. F. Erdogan), Applied Mechanics Division (AMD) Volume 19, American Society of Mechanical Engineers, New York, (1976), pp. 23-53. J. R. Rice, “Mechanics Aspects of Stress Corrosion Cracking and Hydrogen Embrittlement”, in Stress Corrosion Cracking and Hydrogen Embrittlement of Iron Base Alloys_ (eds. R. W. Staehle et al.), National Association of Corrosion Engineers, Houston, (1977), pp. 11-15. A. P. Kfouri and J. R. Rice, “Elastic/Plastic Separation Energy Rate for Crack Advance in Finite Growth Steps”, in Fracture 1977 (eds. D.M.R. Taplin et al.), Vol. 1, Solid Mechanics Division Publication, University of Waterloo, Canada, (1977), pp. 43-59. R. J. Asaro and J. R. Rice, “Strain Localization in Ductile Single Crystals”, Journal of the Mechanics and Physics of Solids, 25, (1977), pp. 309-338. J. R. Rice, “Pore Pressure Effects in Inelastic Constitutive Formulations for Fissured Rock Masses”, in Advances in Civil Engineering Through Engineering Mechanics (Proceedings of 2nd ASCE Engineering Mechanics Division Specialty Conference, Raleigh, N.C., 1977), American Society of Civil Engineers, New York, (1977), pp. 295-297.

xxxii 68.

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LIST OF PUBLICATIONS BY J. R. RICE J. R. Rice, “Fracture Mechanics Model for Slip Surface Propagation in Soil and Rock Masses”, in Advances in Civil Engineering Through Engineering Mechanics (Proceedings of 2nd ASCE Engineering Mechanics Division Specialty Conference, Raleigh, N.C., 1977), American Society of Civil Engineers, New York, NY (1977), pp. 373-376. J. R. Rice, J. W. Rudnicki and D. A. Simons, “Deformation of Spherical Cavities and Inclusions in Fluid-Infiltrated Elastic Materials”, International Journal of Solids and Structures, 14, (1978), pp. 289-303. A. Needleman and J. R. Rice, “Limits to Ductility Set by Plastic Flow Localization”, in Mechanics of Sheet Metal Forming (Proceedings of General Motors Research Laboratories Symposium, October 1977, ed. D.P. Koistinen and N.-M. Wang), Plenum Press, (1978), pp. 237-267. J. R. Rice, “Some Computational Problems in Elastic-Plastic Crack Mechanics”, in Numerical Methods in Fracture Mechanics (Proceedings of the First International Conference on Numerical Methods in Fracture Mechanics, Swansea, Wales, 1978; eds. A. R. Luxmoore and D. R. J. Owen), Department of Civil Engineering, University College of Swansea, Wales, (1978), pp. 434-449. J. R. Rice, “Thermodynamics of the Quasi-Static Growth of Griffith Cracks”, Journal of the Mechanics and Physics of Solids, 26, (1978) pp. 61-78. J. R. Rice and E. P. Sorensen, “Continuing Crack Tip Deformation and Fracture for Plane-Strain Crack Growth in Elastic-Plastic Solids”, Journal of the Mechanics and Physics of Solids, 26, (1978), pp. 163-186. B. Budiansky and J. R. Rice, “On the Estimation of a Crack Fracture Parameter by Long-Wavelength Scattering”, Journal of Applied Mechanics, 45, (1978), pp. 453454. V. N. Nikolaevskii and J. R. Rice, “Current Topics in Non-elastic Deformation of Geological Materials”, in High-Pressure Science and Technology: Sixth AIRAPT Conference, Volume 2: Applications and Mechanical Properties (ed. K.D. Timmerhaus and M.S. Barber), Plenum Press, New York, NY (1979), pp. 455464. J. R. Rice, “Theory of Precursory Processes in the Inception of Earthquake Rupture”, in Proceedings of the Symposium on Physics of Earthquake Sources (at General Assembly of International Association of Seismology and Physics of the Earth’s Interior, Durham, England, August 1977), Gerlands Beitrage zur Geophysik, 88, (1979), pp. 91-127. J. R. Rice and J. W. Rudnicki, “Earthquake Precursory Effects due to Pore Fluid Stabilization of a Weakening Fault Zone”, Journal of Geophysical Research, 84, (1979), pp. 2177-2193. J. B. Walsh and J. R. Rice, “Local Changes in Gravity Resulting from Deformation”, Journal of Geophysical Research, 84, (1979) pp. 165-170.

LIST OF PUBLICATIONS BY J. R. RICE 79.

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T.- J. Chuang, K. I. Kagawa, J. R. Rice and L. B. Sills, “Non-equilibrium Models for Diffusive Cavitation of Grain Interfaces”, Acta Metallurgica, Overview Paper No. 2, 27, (1979), pp. 265-284. J. R. Rice, R. M. McMeeking, D. M. Parks and E. P. Sorensen, “Recent Finite Element Studies in Plasticity and Fracture Mechanics”, in Proceedings of the FENOMECH '78 Conference (Stuttgart, edited by K.S. Pister et al.), NorthHolland Publ. Co., Vol. 2, (1979), pp. 411-442; also, Computer Methods in Applied Mechanics and Engineering, 17/18, (1979), pp. 411-442. W. Kohn and J. R. Rice, “Scattering of Long Wavelength Elastic Waves form Localized Defects in Solids”, Journal of Applied Physics, 50, (1979), pp. 33463353. J. R. Rice, “The Mechanics of Quasi-static Crack Growth”, in Proceedings of the 8th U.S. National Congress of Applied Mechanics (at U.C.L.A., June 1978; ed. R. E. Kelly), Western Periodicals Co., North Hollywood, California, (1979), pp. 191216. B. Budiansky and J. R. Rice, “An Integral Equation for Dynamic Elastic Response of an Isolated 3-D Crack”, Wave Motion, 1, (1979), pp. 187-192. J. R. Rice, “Plastic Creep Flow Processes in Fracture at Elevated Temperature”, in Time-Dependent Fracture of Materials at Elevated Temperature (ed. S.M. Wolf), U.S. Department of Energy Report CONF 790236 UC-25 (June 1979), pp. 130-145. B. Budiansky, D. C. Drucker, G. S. Kino and J. R. Rice, “The Pressure Sensitivity of a Clad Optical Fiber”, Applied Optics, 18, (1979), pp. 4085-4088. B. Cotterell and J. R. Rice, “Slightly Curved or Kinked Cracks”, International Journal of Fracture, 16, (1980), pp. 155-169. A. G. Evans, J. R. Rice and J. P. Hirth, “The Suppression of Cavity Formation in Ceramics: Prospects for Superplasticity”, Journal of the American Ceramic Society, 63, (1980), pp. 368-375. J. R. Rice, “The Mechanics of Earthquake Rupture”, in Physics of the Earth’s Interior (Proc. International School of Physics ‘Enrico Fermi’, Course 78, 1979; (ed. A. M. Dziewonski and E. Boschi), Italian Physical Society and North-Holland Publ. Co., (1980), pp. 555-649. J. R. Rice, “Discussion on ‘Outstanding Problems in Geodynamics: Mechanisms of Faulting"', in Physics of the Earth’s Interior (Proc. International School of Physics ‘Enrico Fermi’, Course 78, 1979; ed. A. M. Dziewonski and E. Boschi), Italian Physical Society and North-Holland Publ. Co., (1980) pp. 713-716. J. R. Rice and J. W. Rudnicki, “A Note on Some Features of the Theory of Localization of Deformation”, International Journal of Solids and Structures, 16, (1980), pp. 597-605. H. Riedel and J. R. Rice, “Tensile Cracks in Creeping Solids”, in Fracture Mechanics: Twelfth Conference (ed. P.C. Paris), Special Technical Publication 700, ASTM, Philadelphia, (1980), pp. 112-130.

xxxiv 92.

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LIST OF PUBLICATIONS BY J. R. RICE J. R. Rice, W. J. Drugan and T. L. Sham, “Elastic-Plastic Analysis of Growing Cracks”, in Fracture Mechanics: Twelfth Conference (ed. P. C. Paris), Special Technical Publication 700, ASTM, Philadelphia, PA (1980), pp. 189-221. A. Needleman and J. R. Rice, “Plastic Creep Flow Effects in the Diffusive Cavitation of Grain Boundaries”, Acta Metallurgica, Overview Paper No. 9, 28, (1980), pp. 1315-1332. J. P. Hirth and J. R. Rice, “On the Thermodynamics of Adsorption at Interfaces as it Influences Decohesion”, Metallurgical Transactions, 11A, (1980), pp. 15011511. L. Hermann and J. R. Rice, “Comparison of Theory and Experiment for ElasticPlastic Plane-Strain Crack Growth”, Metal Science, 14, (1980), pp. 285-291. J. R. Rice, “Pore-Fluid Processes in the Mechanics of Earthquake Rupture”, in Solid Earth Geophysics and Geotechnology (ed. S. Nemat-Nasser), American Society of Mechanical Engineers, Appl. Mech. Div. Volume 42, New York, NY (1980), pp. 81-89. J. R. Rice, “Elastic Wave Emission from Damage Processes”, Journal of Nondestructive Evaluation, 1, (1980), pp. 215-224. J. R. Rice and T.- J. Chuang, “Energy Variations in Diffusive Cavity Growth”, Journal of the American Ceramic Society, 64, (1981), pp. 46-53. J. R. Rice, “Creep Cavitation of Grain Interfaces”, in Three-Dimensional Constitutive Relations and Ductile Fracture (ed. S. Nemat-Nasser), North-Holland Publ. Co., (1981), pp. 173-184. J. R. Rice, “Constraints on the Diffusive Cavitation of Isolated Grain Boundary Facets in Creeping Polycrystals”, Acta Metallurgica, 29, (1981), pp. 675-681. F. K. Lehner, V. C. Li and J. R. Rice, “Stress Diffusion along Rupturing Plate Boundaries”, Journal of Geophysical Research, 86, (1981), pp. 6155-6169. J. R. Rice, “Elastic-Plastic Crack Growth”, in Mechanics of Solids: The Rodney Hill 60th Anniversary Volume (ed. H.G. Hopkins and M.J. Sewell), Pergamon Press, Oxford and New York, (1982), pp. 539-562. W. J. Drugan, J. R. Rice and T.-L. Sham, “Asymptotic Analysis of Growing Plane Strain Tensile Cracks in Elastic-Ideally Plastic Solids”, Journal of the Mechanics and Physics of Solids, 30, 1982, pp. 447-473; erratum, 31, (1983), p. 191. J. R. Rice and A. L. Ruina, “Stability of Steady Frictional Slipping”, Journal of Applied Mechanics, 50, (1983), pp. 343-349. J. Pan and J. R. Rice, “Rate Sensitivity of Plastic Flow and Implications for Yield Surface Vertices”, International Journal of Solids and Structures, 19, (1983), pp. 973-987. V. C. Li and J. R. Rice, “Pre-seismic Rupture Progression and Great Earthquake Instabilities at Plate Boundaries”, Journal of Geophysical Research, 88, (1983), pp. 4231-4246.

LIST OF PUBLICATIONS BY J. R. RICE 107.

108. 109. 110. 111.

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V. C. Li and J. R. Rice, "Precursory Surface Deformation in Great Plate Boundary Earthquake Sequences", Bulletin of the Seismological Society of America, 73, (1983), pp. 1415-1434 J. R. Rice and J.-c. Gu, "Earthquake Aftereffects and Triggered Seismic Phenomena", Pure and Applied Geophysics, 121, (1983), pp. 187-219. J. R. Rice, "Constitutive Relations for Fault Slip and Earthquake Instabilities", Pure and Applied Geophysics, 121, (1983), pp. 443-475. J. R. Rice, "On the Theory of Perfectly Plastic Anti-Plane Straining", Mechanics of Materials, 3, (1984), pp. 55-80. W. J. Drugan and J. R. Rice, "Restrictions on Quasi-Statically Moving Surfaces of Strong Discontinuity in Elastic-Plastic Solids", in Mechanics of Material Behavior (the D.C. Drucker anniversary volume, ed. G.J. Dvorak and R.T. Shield), Elsevier, (1984), pp. 59-73. J. R. Rice, "Shear Instability in Relation to the Constitutive Description of Fault Slip", in Rockbursts and Seismicity in Mines (ed. N.C. Gay and E.H. Wainwright), Symp. Ser. No. 6, S. African Inst. Mining and Metallurgy, Johannesburg, (1984), pp. 57-62. J.-c. Gu, J. R. Rice, A. L. Ruina and S.T. Tse, "Slip Motion and Stability of a Single Degree of Freedom Elastic System with Rate and State Dependent Friction", Journal of the Mechanics and Physics of Solids, 32, (1984), pp. 167196. J. R. Rice, "Comments on 'On the Stability of Shear Cracks and the Calculation of Compressive Strength' by J.K. Dienes", Journal of Geophysical Research, 89, (1984), pp. 2505-2507. J. R. Rice, "Shear Localization, Faulting and Frictional Slip: Discusser’s Report", in Mechanics of Geomaterials (Proc. IUTAM W. Prager Symp., Sept. 1983, ed. Z.P. Bazant), J. Wiley and Sons Ltd., (1985), Chp. 11, pp. 211-216. J. R. Rice, "Conserved Integrals and Energetic Forces", in Fundamentals of Deformation and Fracture (Eshelby Memorial Symposium), ed. B.A. Bilby, K.J. Miller and J.R. Willis, Cambridge Univ. Press, (1985) pp. 33-56. P. M. Anderson and J. R. Rice, "Constrained Creep Cavitation of Grain Boundary Facets", Acta Metallurgica, 33, (1985), pp. 409-422. J. R. Rice, "First Order Variation in Elastic Fields due to Variation in Location of a Planar Crack Front", Journal of Applied Mechanics, 52, (1985), pp. 571-579. S. T. Tse, R. Dmowska and J. R. Rice, "Stressing of Locked Patches along a Creeping Fault", Bulletin of the Seismological Society of America, 75, (1985), pp. 709-736. J. R. Rice and R. Nikolic, "Anti-plane Shear Cracks in Ideally Plastic Crystals", Journal of the Mechanics and Physics of Solids, 33, (1985), pp. 595-622. J. R. Rice, "Three Dimensional Elastic Crack Tip Interactions with Transformation Strains and Dislocations", International Journal of Solids and Structures, 21, (1985), pp. 781-791.

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LIST OF PUBLICATIONS BY J. R. RICE J. R. Rice (Editor), Solid Mechanics Research Trends and Opportunities (Report of the Committee on Solid Mechanics Research Directions of the Applied Mechanics Division, American Society of Mechanical Engineers), Applied Mechanics Reviews, 38, (1985), pp. 1247-1308; published simultaneously as AMD-Vol. 70, ASME Book No. I00198. J. R. Rice, "Fracture Mechanics", in Solid Mechanics Research Trends and Opportunities, ed. J. R. Rice, Applied Mechanics Reviews, 38, (1985), pp. 12711275; published simultaneously in AMD-Vol. 70, ASME Book No. I00198. J. R. Rice and S. T. Tse, "Dynamic Motion of a Single Degree of Freedom System following a Rate and State Dependent Friction Law", Journal of Geophysical Research, 91, (1986), pp. 521-530. R. Dmowska and J. R. Rice, "Fracture Theory and Its Seismological Applications", in Continuum Theories in Solid Earth Physics (Vol. 3 of series "Physics and Evolution of the Earth's Interior"; ed. R. Teisseyre), Elsevier and Polish Scientific Publishers, (1986), pp. 187-255. H. Gao and J. R. Rice, "Shear Stress Intensity Factors for a Planar Crack with Slightly Curved Front", Journal of Applied Mechanics, 53, (1986), pp. 774-778. S. T. Tse and J. R. Rice, "Crustal Earthquake Instability in Relation to the Depth Variation of Frictional Slip Properties", Journal of Geophysical Research, 91, (1986), pp. 9452-9472. P. M. Anderson and J. R. Rice, "Dislocation Emission from Cracks in Crystals or Along Crystal Interfaces", Scripta Metallurgica, 20, (1986), pp. 1467-1472. J.-S. Wang, P.M. Anderson and J. R. Rice, "Micromechanics of the Embrittlement of Crystal Interfaces", in Mechanical Behavior of Materials - V (Proceedings of the 5th International Conference, Beijing, 1987; ed. M.G. Yan, S.H. Zhang and Z.M. Zheng), Pergamon Press, (1987), pp. 191-198. J. R. Rice, "Mechanics of Brittle Cracking of Crystal Lattices and Interfaces", in Chemistry and Physics of Fracture (proceedings of a 1986 NATO Advanced Research Workshop; edited by R.M. Latanision and R.H. Jones), Martinus Nijhoff Publishers, Dordrecht, (1987), pp. 22-43. P. M. Anderson and J. R. Rice, "The Stress Field and Energy of a ThreeDimensional Dislocation Loop at a Crack Tip", Journal of the Mechanics and Physics of Solids, 35, (1987), pp. 743-769. H. Gao and J. R. Rice, "Somewhat Circular Tensile Cracks", International Journal of Fracture, 33, (1987), 155-174. J. R. Rice, "Two General Integrals of Singular Crack Tip Deformation Fields", Journal of Elasticity, 20, (1988), pp. 131-142. H. Gao and J. R. Rice, "Nearly Circular Connections of Elastic Half Spaces", Journal of Applied Mechanics, 54, (1987) pp. 627-634. R. Hill and J. R. Rice, "Discussion of 'A Rate-Independent Constitutive Theory for Finite Inelastic Deformation' by M.M. Carroll", Journal of Applied Mechanics, 54, (1987), pp. 745-747.

LIST OF PUBLICATIONS BY J. R. RICE 136. 137. 138.

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147. 148.

149. 150.

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V. C. Li and J. R. Rice, "Crustal Deformation in Great California Earthquake Cycles", Journal of Geophysical Research, 92, (1987), pp. 11,533-11,551. J. R. Rice, "Tensile Crack Tip Fields in Elastic-Ideally Plastic Crystals", Mechanics of Materials, 6, (1987), pp. 317-335. J. W. Hutchinson, M. E. Mear and J. R. Rice, "Crack Paralleling an Interface Between Dissimilar Materials", Journal of Applied Mechanics, 54, (1987), pp. 828832. J. R. Rice and M. Saeedvafa, "Crack Tip Singular Fields in Ductile Crystals with Taylor Power-Law Hardening, I: Anti-Plane Shear", Journal of the Mechanics and Physics of Solids, 36, (1988), pp. 189-214. J. R. Rice, "Elastic Fracture Mechanics Concepts for Interfacial Cracks", Journal of Applied Mechanics, 55, (1988), pp. 98-103. R. Dmowska, J. R. Rice, L.C. Lovison and D. Josell, "Stress Transfer and Seismic Phenomena in Coupled Subduction Zones During the Earthquake Cycle", Journal of Geophysical Research, 93, (1988), pp. 7869-7884. J. R. Rice, "Crack Fronts Trapped by Arrays of Obstacles: Solutions Based on Linear Perturbation Theory", in Analytical, Numerical and Experimental Aspects of Three Dimensional Fracture Processes (eds. A. J. Rosakis, K. Ravi-Chandar and Y. Rajapakse), ASME Applied Mechanics Division Volume 91, American Society of Mechanical Engineers, New York, (1988), pp. 175-184. J. Yu and J. R. Rice, "Dislocation Pinning Effect of Grain Boundary Segregated Solutes at a Crack Tip", in Interfacial Structure, Properties and Design (eds. M.H. Yoo, W.A.T. Clark and C.L. Briant), Materials Research Society Proc. Vol. 122, (1988), pp. 361-366. R. Nikolic and J. R. Rice, "Dynamic Growth of Anti-Plane Shear Cracks in Ideally Plastic Crystals", Mechanics of Materials, 7, (1988), pp. 163-173. J. R. Rice, "Weight Function Theory for Three-Dimensional Elastic Crack Analysis", in Fracture Mechanics: Perspectives and Directions (Twentieth Symposium), Special Technical Publication 1020, eds. R. P. Wei and R. P. Gangloff, ASTM, Philadelphia, (1989), pp. 29-57. H. Gao and J. R. Rice, "Application of 3D Weight Functions - II. The Stress Field and Energy of a Shear Dislocation Loop at a Crack Tip", Journal of the Mechanics and Physics of Solids, 37, (1989), pp. 155-174. J. R. Rice and J.-S. Wang, "Embrittlement of Interfaces by Solute Segregation", Materials Science and Engineering, A107, (1989), pp. 23-40. M. Saeedvafa and J. R. Rice, "Crack Tip Singular Fields in Ductile Crystals with Taylor Power-Law Hardening, II: Plane Strain", Journal of the Mechanics and Physics of Solids, 37, (1989), pp. 673-691. H. Gao and J. R. Rice, "A First Order Perturbation Analysis of Crack Trapping by Arrays of Obstacles", Journal of Applied Mechanics, 56, (1989), pp. 828-836. J. R. Rice, D. E. Hawk and R. J. Asaro, "Crack Tip Fields in Ductile Crystals", International Journal of Fracture, 42, (1990), pp. 301-321.

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LIST OF PUBLICATIONS BY J. R. RICE J. R. Rice, "Summary of Studies on Crack Tip Fields in Ductile Crystals", in Yielding, Damage, and Failure of Anisotropic Solids (ed. J. P. Boehler), Mechanical Engineering Publications (London), (1990), pp. 49-52. P. M. Anderson, J.-S. Wang and J. R. Rice, "Thermodynamic and Mechanical Models of Interfacial Embrittlement", in Innovations in Ultrahigh-Strength Steel Technology (eds. G. B. Olson, M. Azrin and E. S. Wright), Sagamore Army Materials Research Conference Proceedings, Volume 34, (1990), pp. 619-649. J. R. Rice, Z. Suo and J.-S. Wang, "Mechanics and Thermodynamics of Brittle Interfacial Failure in Bimaterial Systems", in Metal-Ceramic Interfaces (eds. M. Rühle, A. G. Evans, M. F. Ashby and J. P. Hirth), Acta-Scripta Metallurgica Proceedings Series, Volume 4, Pergamon Press, (1990), pp. 269-294. Y. Sun, J. R. Rice and L. Truskinovsky, "Dislocation Nucleation Versus Cleavage in Ni3Al and Ni", in High-Temperature Ordered Intermetallic Alloys (eds. L. A. Johnson, D. T. Pope and J. O. Stiegler), Materials Research Society Proc. Vol. 213, (1991) pp. 243-248. G. E. Beltz and J. R. Rice, "Dislocation Nucleation Versus Cleavage Decohesion at Crack Tips", in Modeling the Deformation of Crystalline Solids (eds. T. C. Lowe, A. D. Rollett, P. S. Follansbee and G. S. Daehn), The Minerals, Metals and Materials Society (TMS), Warrendale, Penna., (1991), pp. 457-480. H. Gao, J. R. Rice and J. Lee, "Penetration of a Quasistatically Slipping Crack into a Seismogenic Zone of Heterogeneous Fracture Resistance", Journal of Geophysical Research, 96, (1991), 21535-21548 J. R. Rice, "Fault Stress States, Pore Pressure Distributions, and the Weakness of the San Andreas Fault", in Fault Mechanics and Transport Properties in Rocks (eds. B. Evans and T.-F. Wong), Academic Press, (1992), pp. 475-503. J. R. Rice, "Dislocation Nucleation from a Crack Tip: An Analysis Based on the Peierls Concept", Journal of the Mechanics and Physics of Solids, 40, (1992), pp. 239-271. G. E. Beltz and J. R. Rice, "Dislocation Nucleation at Metal/Ceramic Interfaces", Acta Metallurgica et Materiala, 40, Supplement, (1992), pp. s321-s331. J. R. Rice, G. E. Beltz and Y. Sun, "Peierls Framework for Analysis of Dislocation Nucleation from a Crack Tip", in Topics in Fracture and Fatigue (ed. A. S. Argon), Springer Verlag, (1992), Chapter 1, pp. 1-58. M. Saeedvafa and J. R. Rice, "Crack Tip Fields in a Material with Three Independent Slip Systems: NiAl Single Crystal", Modelling and Simulation in Materials Science and Engineering, 1, (1992), pp. 53-71. Y. Ben-Zion, J. R. Rice and R. Dmowska, "Interaction of the San Andreas Fault Creeping Segment with Adjacent Great Rupture Zones, and Earthquake Recurrence at Parkfield, Journal of Geophysical Research, 98, (1993), pp. 21352144.

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167. 168.

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175. 176.

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J. R. Rice, "Mechanics of Solids", section of the article on "Mechanics", in Encyclopaedia Britannica (1993 printing of the 15th edition), volume 23, pp. 734747 and 773, (1993). J. R. Rice, "Spatio-temporal Complexity of Slip on a Fault", Journal of Geophysical Research, 98, (1993), pp. 9885-9907. Y. Sun, G. E. Beltz and J. R. Rice, "Estimates from Atomic Models of TensionShear Coupling in Dislocation Nucleation from a Crack Tip", Materials Science and Engineering A, 170, (1993), pp. 67-85. Y. Ben-Zion and J. R. Rice, "Earthquake Failure Sequences Along a Cellular Fault Zone in a 3D Elastic Solid Containing Asperity and Non-Asperity Regions", Journal of Geophysical Research, 98, (1993), pp. 14,109-14,131. J. R. Rice and G. E. Beltz, "The Activation Energy for Dislocation Nucleation at a Crack", Journal of the Mechanics and Physics of Solids, 42, (1994), pp. 333-360. J. R. Rice, Y. Ben-Zion and K.-S. Kim, "Three-Dimensional Perturbation Solution for a Dynamic Planar Crack Moving Unsteadily in a Model Elastic Solid", Journal of the Mechanics and Physics of Solids, 42, (1994), pp. 813-843. G. Perrin and J. R. Rice, "Disordering of a Dynamic Planar Crack Front in a Model Elastic Medium of Randomly Variable Toughness", Journal of the Mechanics and Physics of Solids, 42, (1994), pp. 1047-1064. Y. Ben-Zion and J. R. Rice, "Quasi-Static Simulations of Earthquakes and Slip Complexity along a 2D Fault in a 3D Elastic Solid", in The Mechanical Involvement of Fluids in Faulting, Proceedings of June 1993 National Earthquake Hazards Reduction Program Workshop LXIII, USGS Open-File Report 94-228, Menlo Park, CA, (1994), pp. 406-435. Y. Ben-Zion and J. R. Rice, "Slip Patterns and Earthquake Populations along Different Classes of Faults in Elastic Solids", Journal of Geophysical Research, 100, (1995), pp. 12959-12983. G. Perrin, J. R. Rice and G. Zheng, "Self-healing Slip Pulse on a Frictional Surface", Journal of the Mechanics and Physics of Solids, 43, (1995), pp. 14611495. P. H. Geubelle and J. R. Rice, "A Spectral Method for Three-Dimensional Elastodynamic Fracture Problems", Journal of the Mechanics and Physics of Solids, 43, (1995), pp. 1791-1824. J. R. Rice, "Text of Timoshenko Medal Speech", in Applied Mechanics Newsletter (ed. B. Moran), American Society of Mechanical Engineers, (Spring 1995), pp. 23. P. Segall and J. R. Rice, "Dilatancy, Compaction, and Slip Instability of a Fluid Infiltrated Fault", Journal of Geophysical Research, 100, (1995), pp. 22155-22171. J. R. Rice and Y. Ben-Zion, "Slip Complexity in Earthquake Fault Models", Proceedings of the National Academy of Sciences USA, 93, (1996), pp. 38113818.

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183. 184. 185.

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LIST OF PUBLICATIONS BY J. R. RICE R. Dmowska, G. Zheng and J. R. Rice, "Seismicity and Deformation at Convergent Margins due to Heterogeneous Coupling", Journal of Geophysical Research, 101, (1996), pp. 3015-3029. M. A. J. Taylor, G. Zheng, J. R. Rice, W. D. Stuart and R. Dmowska, "Cyclic Stressing and Seismicity at Strongly Coupled Subduction Zones", Journal of Geophysical Research, 101, (1996), pp. 8363-8381. G. Zheng, R. Dmowska and J. R. Rice, "Modeling Earthquake Cycles in the Shumagins Subduction Segment, Alaska, with Seismic and Geodetic Constraints", Journal of Geophysical Research, 101, (1996), pp. 8383-8392. G. E. Beltz, J. R. Rice, C. F. Shih and L. Xia, "A Self-Consistent Model for Cleavage in the Presence of Plastic Flow", Acta Materiala, 44, (1996), pp. 39433954. M. F. Linker and J. R. Rice, "Models of Postseismic Deformation and Stress Transfer Associated with the Loma Prieta Earthquake", in U. S. Geological Survey Professional Paper 1550-D: The Loma Prieta, California, Earthquake of October 17, 1989 - Aftershocks and Postseismic Effects, (1997), pp. D253-D275. A. Cochard and J. R. Rice, "A Spectral Method for Numerical Elastodynamic Fracture Analysis without Spatial Replication of the Rupture Event", Journal of the Mechanics and Physics of Solids, 45, (1997), pp. 1393-1418. Y. Ben-Zion and J. R. Rice, "Dynamic Simulations of Slip on a Smooth Fault in an Elastic Solid", Journal of Geophysical Research, 102, (1997), pp. 17771-17784. J. W. Morrissey and J. R. Rice, "Crack Front Waves", Journal of the Mechanics and Physics of Solids, 46, (1998), pp. 467-487. M. A. J. Taylor, R. Dmowska and J. R. Rice, "Upper-plate Stressing and Seismicity in the Subduction Earthquake Cycle", Journal of Geophysical Research, 103, (1998), pp. 24523-24542. G. Zheng and J. R. Rice, "Conditions under which Velocity-Weakening Friction allows a Self-healing versus a Cracklike Mode of Rupture", Bulletin of the Seismological Society of America, 88, (1998), pp. 1466-1483. K. Ranjith and J. R. Rice, "Stability of Quasi-static Slip in a Single Degree of Freedom Elastic System with Rate and State Dependent Friction", Journal of the Mechanics and Physics of Solids, 47, (1999), pp. 1207-1218. J. R. Rice, "Foundations of Solid Mechanics", in Mechanics and Materials: Fundamentals and Linkages (eds. M. A. Meyers, R. W. Armstrong, and H. Kirchner), Chapter 3, Wiley, in press, (1999). J. W. Morrissey and J. R. Rice, "Perturbative Simulations of Crack Front Waves", Journal of the Mechanics and Physics of Solids, in press.

List of Contributors

Professor Peter M. Anderson, Department of Materials Science and Engineering, The Ohio State University, 2041 College Road, Columbus, OH 43210-1179 U. S. A. Professor A. G. Atkins, Department of Engineering, University of Reading, Reading, BG6 6AY, UK Professor Leslie Bank-Sills, The Dreszer Fracture Mechanics Laboratory, Department of Solid Mechanics, Materials and Structures, The Fleischman Faculty of Engineering, Tel Aviv University, 69978 Ramat Aviv, Israel Dr. B Blug, Fraunhofer-Institut für Werkstoffmechanik,Wöhlerstr. 11,79108 Freiburg, Germany Dr. Vinodkumar Boniface, The Dreszer Fracture Mechanics Laboratory, Department of Solid Mechanics, Materials and Structures, The Fleischman Faculty of Engineering, Tel Aviv University, 69978 Ramat Aviv, Israel Professor Allan F. Bower, Division of Engineering, Brown University, Providence, RI 02912, U.S.A. Dr. B. Chen,. Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, IL 61801 Dr. Z. Chen, Institute of Materials Research and Engineering, 3 Research Link, Singapore 117602 Dr. W. Y. Chien, Department of Mechanical Engineering and Applied Mechanics, The University of Michigan, Ann Arbor, MI 48109, USA Dr.J. W. Cho, Technical Center, Deawoo Heavy Industries Co, Inchun, Korea Dr. T.-J. Chuang, Ceramics Division, National Institute of Standards and Technology, Gaithersburg, MD 20899-8521, U. S. A. Dr. Brian Cotterell, Institute of Materials Research and Engineering, 3 Research Link, Singapore 117602

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LIST OF CONTRIBUTORS

Professor Walter W.Drugan, Department of Engineering Physics, University of Wisconsin, Madison, 1500 Engineering Drive, Madison, WI 53706 Professor Glenn E. Beltz, Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, CA 93106-5070, USA Professor Huajian Gao, Division of Mechanics and Computation, Department of Mechanical Engineering, Stanford University, Stanford, CA 94305-4040 Mr. Anja Haug, Materials Department, University of California, Santa Barbara, California 93106 USA Professor Young Huang, Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, IL 61801 Mr. H.-M. Huang, Department of Mechanical Engineering and Applied Mechanics, The University of Michigan, Ann Arbor, MI 48109, USA Professor Mark Kachanov, Department of Mechanical Engineering, Tufts University, Medford, MA 02155 Dr. E. Karapetian, Department of Mechanical Engineering, Tufts University, Medford, MA 02155, U.S.A. Dr.Patrick A Klein, Sandia National Laboratories, Mail Stop 9161,P.O. Box 0969, Livermore, CA 94551 Professor Shiro Kubo, Department of Mechanical Engineering and Systems, Graduate School of Engineering, Osaka University, 2-1, Yamadaoka, Suita, Osaka 565-0871 Japan Dr. L. L. Fischer, Department of Mechanical and Environmental Engineering, University of California,Santa Barbara, CA 93106-5070, USA Dr. L. E. Levine, Maaterials Science and Engineering Lab., National Institute of Standards and Technology, Gaithersburg, MD 20899 Professor Victor C. Li, Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI, 48109-2125 Mr. W. Lu, Mechanical and Aerospace Engineering Department and Materials Institute, Princeton University, Princeton, NJ 08544

LIST OF CONTRIBUTORS

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Dr. S. R. MacEwen, Alcan International Ltd., P.O. Box 8400, Kinston, Ontario, K7L 5L9, Canada Professor Robert M. McMeeking, Department of Mechanical and Environmental Engineering, University of Californi, Santa Barbara, California 93106, USA Professor Sinisa Dj. Mesarovic, Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22903 U. S. A. Professor Joe Pan, Department of Mechanical Engineering and Applied Mechanics, The University of Michigan, Ann Arbor, MI 48109, USA Dr. Hermann Riedel, Fraunhofer-Institut für Werkstoffmechanik,Wöhlerstr. 11,79108 Freiburg, Germany Professor Asher A. Rubinstein, Department of Mechanical Engineering, Tulane University, New Orleans, LA 70118, U. S. A. Professor J. W .Rudnicki, Department of Civil Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3109 Dr. I. Sevostianov, Department of Mechanical Engineering, Tufts University, Medford, MA 02155 Dr. Y. Shim, Center for Simulational Physics, University of Georgia, Athens, GA 30602 Professor Z. Suo, Mechanical and Aerospace Engineering Department, and Materials Institute, Princeton University, Princeton, NJ 08544 Dr. S. C. Tang, Ford Research Lab., P.O. Box 2053, MD3135/SRL, Dearborn, MI 48121, U.S.A. Mr. Zhibo Tang, Division of Engineering, Brown University, Providence, RI 02912 Dr. Robb M. Thomson, Maaterials Science and Engineering Lab., National Institute of Standards and Technology, Gaithersburg, MD 20899 Dr. Jian-Sheng Wang, Northwestern University, Evanston, IL 60201, USA Dr. P. D. Wu, Alcan International Ltd., P.O. Box 8400, Kinston, Ontario, K7L 5L9, Canada Dr. Z. C. Xia, Ford Research Lab., P.O. Box 2053, MD3135/SRL, Dearborn, MI 48121

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Dr. Xiao J. Xin, Department of Mechanical and Nuclear Engineering, Kansas State University, 338 Rathbone Hall, Manhattan, KS 66506-5205 U. S. A. Professor Jin Yu, Department of Materials Science and Engineering,Korea Advanced Institute of Science and Technology, P.O. Box 201, Chongryang. Seoul, Korea

APPROXIMATE YIELD CRITERION FOR ANISOTROPIC POROUS SHEET METALS AND ITS APPLICATIONS TO FAILURE PREDICTION OF SHEET METALS UNDER FORMING PROCESSES W. Y. CHIEN, H.-M. HUANG AND J. PAN Department of Mechanical Engineering and Applied Mechanics The University of Michigan, Ann Arbor, MI 48109, USA AND S. C. TANG Ford Research Laboratory Ford Motor Company, Dearborn, MI 48121, USA

Abstract: An approximate anisotropic yield criterion for anisotropic sheet metals containing spherical voids is validated using a three-dimensional finite element analysis. An aggregate of periodically arranged cubes containing spherical voids is modeled using a unit cell method. Hill’s quadratic anisotropic yield criterion is used to describe the normal anisotropy and planar isotropy of the matrix. The metal matrix is first assumed to be elastic perfectly plastic and incompressible. The results of the finite element analysis can be in good agreement with those based on the proposed yield criterion by introducing three fitting parameters in the yield criterion. This modified yield criterion is adopted in a failure prediction methodology that can be used to determine the failure of sheet metals under forming operations. The material imperfection approach is employed to predict failure/plastic localization by assuming a slightly higher void volume fraction inside randomly oriented imperfection bands. Finally, the failure prediction methodology is applied to predict the failure of a mild steel sheet metal in a fender forming process.

1. Introduction Structural metals usually contain some form of second-phase particles such as maganese sulfides and carbides in steels. These particles usually provide strain concentration sites for void nucleation, growth and coalescence that lead to ductile fracture. In order to model the plastic flow and fracture of these ductile structural metals, Gurson (1977) conducted an upper bound analysis of simplified models containing voids and proposed an approximate yield criterion for porous materials where the matrices obey the von Mises yield criterion. 1 T.-J. Chuang and J. W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 1–15. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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Based on Gurson’s yield criterion (1977), Yamamoto (1978) investigated plastic flow localization with the assumption of a slightly higher initial void concentration as the initial material imperfection inside thin planar bands in a material element of interest. Needleman and Triantafyllidis (1978) further examined the effects of void growth based on Gurson’s yield criterion (1977) on localized necking in biaxially stretched sheets and compared the results with those from various types of initial imperfections. It was shown in Yamamoto (1978), Needleman and Triantafyllidis (1978), Saje et al. (1982) and Tvergaard (1981) that the porous material model based on Gurson’s yield criterion predicts failure qualitatively in accord with experimental results. In order to predict sheet metal failure quantitatively in accord with the experimental results under biaxial stretching conditions, Mear and Hutchinson (1985) introduced a family of constitutive laws to address the sensitivity of failure prediction to the yield surface curvature. They considered that the evolution of yield surface follows a simple rule of a combination of isotropic expansion and kinematic translation. Tvergaard (1981, 1982) introduced three additional fitting parameters in Gurson’s yield criterion by comparing the results of shear band instability in square arrays of cylindrical holes and axisymmetric spherical holes based on finite element models with those based on Gurson’s yield criterion. Saje et al. (1982) studied the void nucleation effects on shear localization in rate insensitive porous plastic solids using the modified yield criterion. A parallel work was carried out by Pan et al. (1983) with consideration of material rate sensitivity. The modified yield criterion was also used in the analysis of the cup-cone fracture in a round tensile bar by Tvergaard and Needleman (1984). The matrix material in the original Gurson model was assumed to be isotropic. However, sheet metals for stamping applications usually display certain extent of plastic anisotropy due to cold or hot rolling processes. In general, an average value of the anisotropy parameter R, defined as the ratio of the transverse plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial loading conditions, is used to characterize the sheet anisotropic plastic behavior. Graf and Hosford (1990) investigated the effects of R on forming limit using different yield criteria. They found that when Hill’s quadratic anisotropic yield criterion (1948) is employed, R has significant effects on forming limit. When the higher order yield criterion of Hosford (1979) is employed, R has virtually no effects on forming limit. This indicates that further research on the effects of plastic anisotropy or the effects of constitutive laws/plastic hardening on forming limit is needed. In view of possible significant effects of plastic anisotropy on the forming limit in sheet forming processes, a Gurson type of approximate yield criterion, which can be used to account for the matrix plastic anisotropy, is needed to investigate forming limit and to characterize ductile fracture processes in sheet metal forming applications (Liao et al., 1997). For simplicity, sheet metals were assumed to have normal anisotropy and planar isotropy. In order to develop an approximate macroscopic yield criterion for voided sheet metals with normal anisotropy and planar isotropy, a simplified sheet model containing a through-thickness hole under plane stress conditions was considered in Liao et al. (1997). In their investigation, the matrix material was characterized by Hill’s quadratic anisotropic yield criterion (1948) and Hill’s non-quadratic anisotropic yield criterion (1979). Upper bound analyses were carried out and the numerical results based on both Hill’s quadratic and non-quadratic anisotropic yield criteria were fitted well by a

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closed-form macroscopic yield criterion. An anisotropic Gurson yield criterion for sheet metals with spherical voids was also proposed. In order to validate the accuracy of the proposed anisotropic Gurson yield criterion by Liao et al. (1997), a three-dimensional finite element analysis is employed for a cube model containing a spherical void (Chien et al., 2000). The analysis is performed for various void volume fractions as well as different R values for elastic perfectly plastic materials. As in Tvergaard (1981, 1982), the anisotropic Gurson yield criterion is modified by adding three fitting parameters to fit the results based on the modified yield criterion with the finite element computational results. Finite element computations with consideration of material strain hardening under multiaxial proportional straining conditions are also performed. The results of finite element simulations are compared with those based on the modified anisotropic Gurson yield criterion. Finally, the modified anisotropic Gurson yield criterion is adopted here in a failure prediction methodology that can be used to predict failure in anisotropic, rate-sensitive sheet metals under forming processes. The material imperfection approach is used to predict sheet metal failure by assuming a slightly higher void volume fraction inside randomly oriented imperfection bands in the critical sheet element of interest. The failure of sheet metals under forming processes is defined where the failure or plastic flow localization in an imperfection band of the critical element is first met. Finally, conclusions are given.

2. Anisotropic Gurson Yield Criterion Liao, Pan and Tang (1997) considered a material element of sheet metals with arbitrary shaped voids as shown in Figure 1(a). The void volume fraction was assumed to be small. The sheet element was assumed to be subject to in-plane loading conditions for sheet metal forming applications. The matrix surrounding the voids was assumed to be rigid perfectly-plastic, incompressible and rate-insensitive to take advantage of the upper bound analysis. Liao et al. (1997) considered a simplified sheet model as shown in Figure 1(b) where a sheet contains a periodic array of circular through-thickness holes. A sheet cell model as shown in Figure 1(c) was then considered for upper bound analyses. Under axisymmetric loading, a closed-form upper-bound macroscopic yield criterion Φ CR can be derived as

(1) w h e r e Σe represents the macroscopic effective stress based on Hill’s quadratic anisotropic yield criterion, σo is the matrix yield stress under in-plane uniaxial loading conditions, ƒ is the void volume fraction, R is the anisotropic parameter, and Σ m ( = Σ kk /3 ) is the macroscopic mean stress. Figure 2 shows the upper bound solutions as open symbols for R = 1.8, which represents the typical value of R for low carbon steels. The results based on the closedform macroscopic yield criterion in Equation (1) are also plotted as various types of

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curves. Note that under macroscopic pure shear loading ( Σ m = 0), the solutions are located along the Σ e /σo axis. For a given ƒ, the solution with the largest macroscopic mean stress as shown in the figure represents the upper bound solution for equal biaxial tension. The solutions between the pure shear and the equal biaxial tension are for the other possible plane stress loading conditions. A reasonable match of the upper bound solutions with the macroscopic yield criterion for various plane stress deformation modes can be seen in Figure 2.

Figure 1(a). A sheet element with arbitrarily shaped voids.

Figure 1(b). A simplified sheet model with a periodic array of circular through-thickness holes.

Figure 1(c). A sheet cell for the simplified sheet model.

In order to understand the accuracy of the upper bound solutions and the closed-form solutions, the macroscopic in-plane mean stress is also obtained for the sheet cell model subject to a uniform radial traction under fully yielded conditions. The exact limit

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solutions for various values of ƒ are shown as solid symbols in Figure 2. As shown in Figure 2, the upper bound macroscopic effective stress and the mean stress are slightly larger than those of the exact limit solution for equal biaxial tension. The good agreement of the exact limit solution and the approximate yield criterion under equal biaxial tension indicates the accuracy of the closed-form upper-bound solutions and the validity of the approximate yield criterion near equal biaxial tension.

Figure 2. Comparison of the upper bound solutions based on the sheet cell model and the yield contours of the closed-form anisotropic Gurson yield criterion for various void volume fractions ƒ for R=1.8. The exact solutions of the macroscopic in-plane mean stress under fully yielded conditions are also shown as bullets.

In the original Gurson’s work (1977) for isotropic materials, the basic of the yield criterion for the cylindrical void model is

(2)

and the basic form of the yield criterion for the spherical void model is

(3)

It can be seen that both the yield criteria have the same form except a factor of in the cosh function. Therefore, the closed-form solution obtained from the sheet cell model

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with inclusion of a factor of 1/ can be used as a first-order approximate solution for the spherical void model with the matrix having mild normal anisotropy. An approximate anisotropic Gurson yield criterion for the spherical void model can be written as

(4)

3. Modified Anisotropic Gurson Yield Criterion Since Equation (4) is again approximate in nature, it is necessary to validate the accuracy of this yield criterion. Therefore, a three-dimensional finite element model is considered. As in Tvergaard (1981, 1982), a modified yield criterion is proposed in Liao et al. (1997) as (5)

where q 1, q2 , and q 3 are parameters to fit the finite element computational results. The finite element model considered here is a cube containing a spherical void, as shown in Figure 3. Due to the symmetry of the cube and the applied loads, only oneeighth of the cube is analyzed here, as shown in Figure 4.

Figure 3. A voided cube model.

Figure 4. One-eighth of the voided cube model.

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The boundary conditions of the model are set as follows. The bottom, front, and left surfaces are constrained to have zero normal displacements to satisfy the symmetry conditions. The void surface is specified to have zero traction. Uniform normal displacements are applied on the top, back, and right faces. Pure shear, uniaxial tension and equal biaxial tension are considered for plane stress conditions. In order to use the one-eighth voided cube model to investigate the plastic behavior under pure in-plane shear, a uniform normal displacement is applied to the right face in the x1 direction and the same amount of normal displacement is applied uniformly to the back face in the negative x2 direction to result in pure in-plane shear. A triaxial loading with high mean stress and a pure hydrostatic tension loading are also applied to complete the analysis. The matrix material is assumed to be elastic perfectly plastic. Poisson’s ratio v of the matrix material is assumed to be 0.33 and the ratio of the matrix yield stress σo to Young’s modulus E is set at 2 × 10-7 . With the small value of σ o /E, the matrix material can be considered as nearly rigid-perfectly plastic. Several initial void volume fractions (ƒ = 0.01, 0.04, 0.09 and 0.12) and R = 1.6 are used here to validate the proposed yield criterion in Equation (5). The commercial finite element program ABAQUS (Hibbitt et al., 1998) is used to perform the computations. Twenty node elements with a reduced integration scheme are used here such that the model is free from overconstraint. The macroscopic yield point is defined as the limited stress state where massive plastic deformation occurs. The corresponding macroscopic effective stress and macroscopic mean stress are then calculated and compared with those based on the anisotropic Gurson yield criterion in Equation (5).

Figure 5. Comparison of the modified anisotropic Gurson yield criterion (curves) and the finite element results (open symbols) for R = 1.6.

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In addition to the elastic perfectly plastic material model employed to calculate the fully plastic limits, the macroscopic plastic flow characteristics due to the material matrix hardening are also investigated for several proportional loading conditions (Chien et al., 2000). The in-plane shear stress and plastic shear strain of the matrix is assumed to follow a power-law relation. Then the matrix effective in-plane tensile stress σM as a P can be written as function of the effective in-plane tensile strain ε M

(6) where G represents the shear modulus under in-plane shear loading and N represents the hardening exponent. Here N = 0.1 is used.

Figure 6. The stress-strain behavior of the finite element results and the continuum models with ƒ = 0.09 for R = 1.6.

The finite element computational results are used to evaluate the accuracy of the anisotropic Gurson yield criterion. Figure 5 shows the finite element computational results (represented by open symbols) compared with those of the modified anisotropic Gurson yield criterion (represented by various types of curves). The values of the fitting parameters are selected as q1 = 1.45, q 2 = 0.9 and q 3 = 1.6 for R = 1.6. With the fitting parameters, it can be seen that the finite element computational results are in good agreement with those of the modified anisotropic Gurson yield criterion. It should be noted that when the macroscopic mean stress equals zero, the macroscopic effective stresses from our finite element computations are slightly lower than those predicted by the modified yield criterion. The reason for the earlier yielding from our finite element computations can be attributed to the shear localization in the matrix material. In order to investigate further the accuracy of the modified anisotropic Gurson yield criterion, the

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macroscopic hardening behavior from finite element computations is compared with that based on the modified anisotropic Gurson yield criterion. Figure 6 shows the normalized macroscopic tensile stress Σ 11 / σo as a function of the macroscopic tensile strain E11 for ƒ = 0.09 and R = 1.6 under uniaxial tensile conditions. As shown in the figure, the finite element computational results agree with those of the modified anisotropic Gurson yield criterion at low strains, and approach to those of the unmodified anisotropic Gurson yield criterion at large strains. Similar trends are also seen in Hom and McMeeking (1989) for isotropic materials. More computational results for various values of R under different multiaxial proportional loading conditions can be found in Chien et al. (2000).

4. Evolution of Void Volume Fraction Tvergaard and Needleman (1984) introduced the void volume fraction parameter ƒ° , which is a function of the void volume fraction ƒ, into the Gurson model in order to account for the gradual loss of stress carrying capacity due to void coalescence. The function ƒ°( ƒ ) in Tvergaard and Needleman (1984) is

(7) In Equation (7), the quantity ƒu° is defined as the limiting value of ƒ°( ƒ ) as the stress carrying capacity goes to zero. ƒc represents the critical void volume fraction, and ƒƒ represents the void volume fraction at final failure. Based on the experimental studies of Brown and Embury (1973) and Goods and Brown (1979) and the numerical analysis of Andersson (1977), the values of ƒ c and ƒƒ were chosen as 0.15 and 0.25, respectively (Tvergaard, 1982). Here, we adopt the function ƒ°( ƒ ) into the modified anisotropic Gurson yield criterion as

(8)

In addition to the effect of void coalescence that has been addressed by the function ƒ ° ( ƒ ) , the evolution of void volume fraction can be related to the nucleation of voids and growth/collapse of the existing voids. The increase/decrease of void volume fraction due to growth/collapse can be obtained with the assumption of plastic incompressibility of the matrix material based on the original work of Gurson (1977). For the evolution of void volume fraction due to nucleation, two models can be considered: one is the plastic strain controlled nucleation model suggested by Gurson (1977) based on the experimental data of Gurland (1972). The other is the stress controlled nucleation model in which void nucleation depends on the maximum stress transmitted across the particle-

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matrix interface as discussed in Argon and Im (1975). Thus, the rate of void volume fraction can be expressed as [Chu and Needleman (1980) and Saje et al. (1982)]: (9) Here tr

represents the macroscopic dilatational plastic strain rate,

represents the

represents the matrix average stress rate, and matrix average plastic strain rate, represents the Jaumann rate of the macroscopic mean stress. Here A and B are void nucleation parameters for the strain-controlled and stress-controlled nucleation models, respectively.

5. Yield Surface Curvature Effects Based on Gurson’s yield criterion for isotropic porous materials, Mear and Hutchinson (1985) considered a family of yield surfaces with different yield surface curvatures in order to fit well with the experimental results on necking instability. Mear and Hutchinson (1985) specified the size of the yield surface, σ F , which is a linear combination of the matrix initial yield stress σ y and the matrix flow stress σ M . Here, this yield (or potential) surface is regarded as the curvature surface. The details of including the curvature surface into our plasticity model can be found in Huang et al. (2000a). The tangent modulus procedure of Peirce et al. (1984) is employed to obtain the evolution of the rate-sensitive constitutive relations. The derivation of the constitutive relations including anisotropic hardening with consideration of void growth/collapse is detailed in Huang et al. (2000b).

6. Failure Localization Analysis We employ a Lagrangian formulation and take the initial undeformed configuration as the reference. The coordinates of a material point relative to a fixed Cartesian frame in the undeformed configuration, xi , are taken as the convected coordinates. In the current deformed state, the coordinates of a material point, referred to the reference Cartesian base vectors, are denoted by Latin indices range from 1 to 3 and summation convention is adopted for repeated indices. In the present analysis, many infinitely thin imperfection bands with different orientations are assumed to exist in a material element of interest. Figure 7 shows a material element having an imperfection band under plane stress conditions. In the figure, N represents the normal vector to the band in the undeformed state. The band angle θ , which is used to represent the orientation of the imperfection band, is the angle between the direction of N and a loading direction of interest, that is taken as the x2 direction as shown in Figure 7.

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Figure 7. A material element having an imperfection band under plane stress conditions

Homogeneous deformations inside and outside the band are assumed to occur throughout a deformation history. Here, a superscript or a subscript “b” represents a quantity inside the band, while a superscript or a subscript “o” represents a quantity outside the band. As the deformation proceeds, compatibility requires [Hill (1962) and Rice (1976)]: (10) where C is a vector denoting the discontinuity across the band. Also, equilibrium requires that the normal tractions are continuous over the band interface. Therefore, the equilibrium equation can be given in terms of the first Piola-Kirchoff stress S and the normal vector N as with

(11)

where T ik represent the contravariant components of the Kirchoff stress tensor T. Combining the rate form of the compatibility equations in Equation (10) and the rate form of the equilibrium equations in Equation (11) with the constitutive relations gives a set of equations for Given a prescribed deformation history outside the band,

, and the initial conditions,

the set of equations for can be solved incrementally to determine the deformation history inside the band. The condition of failure/plastic localization is reached when the ratio of the normal component of the deformation gradient rate in the direction of N inside the band to that outside the band becomes infinity. A sheet metal under a fender forming process, which is the benchmark problem of the 1993 NUMISHEET conference as shown in Figure 8, has been considered in Huang et

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al. (2000a). The deformation history including the relative rotation of principal stretch directions for the critical element as shown in Figure 8 has been identified. The failure of

Figure 8. The critical element and the initial major principal stretch directions of the element in a sheet metal under a fender forming operation.

Figure 9. The principal strain history of the critical element based on the FEM fender forming simulation and the corresponding predicted failure strains with rotating and fixed principal stretch directions. The calculated forming limit diagram (FLD) under proportional loading conditions for the mild steel is also presented as a solid curve for comparison.

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13

the critical element of the mild steel sheet is determined based on the growth of microvoids in imperfection bands under the non-proportional deformation history. The values of the initial imperfection in terms of the void volume fraction ƒ and the curvature parameter b are selected to fit the forming limit diagram (FLD) for the mild steel under proportional straining conditions. The values of ƒ and b are 0.000025 and 0.25, respectively. The value of b remains the same as that in Huang et al. (2000a). The value of f is smaller than that in Huang et al. (2000a) due to the selection of q 1 = 1.45, q 2 = 0.9 and q3 =1.6 in this investigation whereas q1 , q 2 and q 3 were set at unity in Huang et al. (2000a). Figure 9 shows the principal strain history for the critical element in the sheet metal based on the FEM fender forming simulation and the corresponding predicted failure strains with rotating and fixed principal stretch directions. The calculated FLD under proportional loading conditions for the mild steel is also presented as a solid curve in the same figure for comparison. Here, the predicted major principal failure strain with consideration of rotating principal stretch directions is 13.6 percent lower than that without consideration of rotating principal stretch directions under this specific deformation history. Note that in Huang et al. (2000a), where q1, q2 and q 3 were set at unity, the predicted major principal failure strain with consideration of rotating principal stretch directions is 8.2 percent lower than that without consideration of rotating principal stretch directions under this specific deformation history. This indicates that the selection of q1 , q 2 and q 3 different from unity does affect the prediction of failure. The effects of rotating principal stretch directions could be more difficult to predict when the non-proportional straining path encompasses the negative minor principal strain range. The detailed FEM forming simulation and failure prediction results are presented in Huang et al. (2000a).

7. Conclusions A simplified sheet cell model was first adopted in Liao et al. (1997) to obtain upper bound solutions for anisotropic voided sheets under plane stress conditions. In their work, the matrix of the voided sheet was assumed to be rigid perfectly plastic, incompressible and rate-insensitive. Hill’s quadratic and non-quadratic anisotropic yield criteria were used to describe the matrix normal anisotropy. A closed-form macroscopic yield criterion for the sheet model with a through-thickness hole was derived based on Hill’s quadratic anisotropic yield criterion under axisymmetric loading conditions. The upper bound solutions were fitted well by the closed-form macroscopic yield criterion under plane stress loading conditions. Based on the original Gurson models for isotropic materials, the inclusion of a factor of in the cosh function of the yield criterion was suggested to obtain a first-order approximation for the sheet material containing spherical voids (Liao et al., 1997). Here, a three-dimensional finite element analysis is employed to validate the anisotropic Gurson yield criterion. The results of the finite element calculations indicate that three fitting parameters are needed to improve the applicability of the anisotropic Gurson yield criterion. With appropriate selection of fitting parameters, it is found that the finite element results can be fitted by those based on the modified anisotropic Gurson yield

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W.Y.CHIEN, ET AL.

criterion for the range of strains and loading conditions investigated in Chien et al. (2000). The modified anisotropic Gurson yield criterion is adopted to predict the failure of a sheet metal under a fender forming operation. Note that a general deformation history for a critical element including the relative rotation of principal stretch directions was identified in a benchmark fender forming problem (Huang et al., 2000a). The failure of the critical sheet element is predicted based on the growth of voids in various oriented imperfection bands under this non-proportional deformation history. The current band orientation at failure implies the possible splitting failure direction. Here, the predicted major principal failure strain with consideration of rotating principal stretch directions is 13.6 percent lower than that without consideration of rotating principal stretch directions under this specific deformation history. As indicated in Huang et al. (2000b), elastic unloading/reloading can also have significant effects on failure in forming processes. Therefore, a more general constitutive relation that can be used to describe the material behavior under loading/unloading conditions with consideration of damage evolution was formulated (Huang et al., 2000b). This type of constitutive relation with consideration of void nucleation, growth/collapse and final coalescence should be further validated and modified by experiments. Then the constitutive relation can be implemented into computer codes to predict the failure of sheet metals in conjunction with the FEM simulations of forming processes.

Acknowledgement The support of this work by Ford Motor Company is greatly appreciated.

References Argon, A. S. and Im, J. (1975), Separation of second phase particles in spheridized 1045 steel, Cu-0.6pct Cr alloy, and maraging steel in plastic straining, Metall. Trans., 6A, 839. Andersson, H. (1977), Analysis of a model for void growth and coalescence ahead of a moving crack tip, J. Mech. Phys. Solids, 25, 217. Brown, L. M. and Emgury, J. D. (1973), Proc. 3rd Int.Conf. on Strength of Metals and Alloys, pp. 164-169, Inst. Metalls, London. Chien, W. Y., Pan, J and Tang, S. C. (2000), to be submitted for publication. Chu. C.-C., Needleman, A. (1980), Void nucleation effects in biaxially stretched sheets, J. Eng. Mater. Tech., 102, 249. Goods, S. H. and Brown, L. M. (1979), Nucleation of cavities by plastic deformation, Acta. Metall., 27, 1. Graf, A., Hosford, W. F. (1990), Calculations of forming limit diagrams, Metall. Trans., 21A, 87. Gurland, J. (1972), Observations on fracture of cementite particles in a spheroidized 1.05% C steel deformed at room temperature, Acta. Metall., 20, 735. Gurson, A. L. (1977), Continuum theory of ductile rupture by void growth: part I – yield criteria and flow rules for porous ductile media, J. Eng. Mater. Tech., 99, 2. Hibbitt, H. D., Karlsson, B. I. and Sorensen, E. P. (1998), ABAQUS user manual, Version 5-8. Hill, R. (1948), A theory of the yielding and plastic flow of anisotropic metals, Roy. Soc. London Proc., 193A, 281. Hill, R. (1962), Acceleration waves in solids, J. Mech. Phys. Solids, 10, 1. Hill, R. (1979), Theoretical plasticity of textured aggregates, Math. Proc. Camb. Philos. Soc., 85, 179. Hom, C. L. and McMeeking R. M. (1989), Void growth in elastic-plastic materials, J. Appl. Mech., 56, 309.

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Hosford, W. F. (1979), On yield loci of anisotropic cubic metals, Proc. 7 th North Am. Metalworking Res. Conf., SME, Dearborn, MI, p. 191. Huang, H.-M., Pan, J. and Tang, S. C. (2000a), Failure prediction in anisotropic sheet metals under forming operations with consideration of rotating principal stretch directions, Int. J. Plast., 16, 611. Huang, H.-M., Pan, J. and Tang, S. C. (2000b), Failure prediction in anisotropic sheet metals containing voids under biaxial straining conditions with prebending/unbending, to appear in Int. J. Plast.. Liao, K.-C., Pan, J. and Tang, S. C. (1997), Approximate yield criteria for anisotropic porous ductile sheet metals, Mech. Mater., 26, 213. Mear, M. E. and Hutchinson, J. W. (1985), Influence of yield surface curvature on flow localization in dilatant plasticity, Mech. Mater., 4, 395. Needleman, A. and Triantafyllidis, N. (1978), Void growth and local necking in biaxial stretched sheets, J. Eng. Mater. Tech., 100, 164. Marciniak, Z. and Kuczynski, K. (1967), Limit strains in the processes of stretch forming sheet metal, Int. J. Mech. Sci., 9, 609. Pan, J., Saje, M. and Needleman, A. (1983), Localization of deformation in rate sensitive porous plastic solids, Int. J. Fract., 21, 261. Peirce, D., Shih, C. F. and Needleman, A. (1984), A tangent modulus method for rate dependent solids, Comp. & Struct., 18, 875. Rice, J. R. (1976), Proc. 14t h Int. Cong. on Theoretical and Applied Mechanics, Ed. Koiter , W. T., 1, pp. 207220, Delft, North-Holland. Saje, M., Pan, J. and Needleman, A. (1982), Void nucleation effects on shear localization in porous plastic solids, Int. J. Fract., 19, 163. Tvergaard, V. (1981), Influence of voids on shear band instabilities under plane strain conditions, Int. J. Fract., 17, 389. Tvergaard, V. (1982), On localization in ductile materials containing spherical voids, Int. J. Fract., 18, 237. Tvergaard, V. and Needleman, A. (1984), Analysis of the cup-cone fracture in a round tensile bar, Acta Metal., 32, 157. Yamamoto, H. (1978), Conditions for shear localization in the ductile fracture of void-containing materials, Int. J. Fract., 11, 347.

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A DILATATIONAL PLASTICITY THEORY FOR ALUMINUM SHEETS

B. CHEN Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, IL 61801, U.S.A P. D. WU Alcan International Ltd., P.O. Box 8400, Kingston, Ontario K7L 5L9, Canada Z. C. XIA Ford Research Lab, P. O. Box 2053, MD 3135/SRL, Dearborn, MI 48121, U.S.A S. R. MAC EWEN Alcan International Ltd., P.O. Box 8400, Kingston, Ontario K7L 5L9, Canada S. C. TANG Ford Research Lab, P.O. Box 2053, MD 3135/SRL, Dearborn, MI 48121, U.S.A AND Y. HUANG Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, IL 61801, U.S.A

Abstract. The nucleation, growth and coalescence of micro-voids are important failure mechanisms in ductile materials. Gurson (1977) has developed a dilatation plasticity theory to quantitatively characterize the state of deformation and damage associated with micro-voids in isotropic materials. This theory, however, is not applicable to aluminum sheets because they are highly anisotropic. A dilatational plasticity theory for anisotropic ductile materials is developed in this study. The constitutive law is established for aluminum sheets that contain micro-voids, where the matrix material of aluminum is characterized by an anisotropic constitutive model developed by Barlat et al. (1991). Based on the numerical analysis, an approximate yield function is given in the closed form for anisotropic sheets. 17 T.-J. Chuang and J. W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 17–30. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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It shows that the mean hydrostatic stress plays an important role in the plastic behavior of anisotropic micro-voided aluminum sheets.

1. Introduction The ductile failure mechanism in most structural metals is the nucleation, growth and coalescence of microvoids that result from debonding and cracking of second-phase particles. It is well established that the growth of microvoids is governed by the hydrostatic stress field around the voids (e.g., Rice and Tracey, 1969; Huang, 1991). Gurson (1977) has incorporated this effect of void growth in the isotropic J2 continuum plasticity theories, and has developed a constitutive law for voided, dilating ductile materials. Unlike the classical plasticity theories, the hydrostatic stress component, σ kk , as well as the void volume fraction, ƒ, has appeared in Gurson’s (1977) dilatational plasticity theory. The Gurson’s theory has been widely used in the study of ductile failure of solids due to void growth or plastic flow localization (Needleman and Rice, 1978). The recent development on this subject can be found in the review article by Tvergaard (1990). There is an increasing need in recent years to use more aluminum alloys in industry because of their high strength/weight ratio. Similar to other ductile materials, void nucleation, growth and coalescence also govern the ductile failure of aluminum alloys. The constitutive behavior of aluminum, however, is highly anisotropic and cannot be characterized by Hill’s quadratic (1950) or non-quadratic (1979) yield functions (e.g., Mellor and Parmar, 1978; Mellor, 1981; Barlat et al., 1997). For example, the theoretical forming limit strains predicted by isotropic yield functions are unrealistically too high or too low. For this reason, Hosford (1979), Barlat and co-workers (1989, 1991, 1997), Karafillis and Boyce (1993) have made a series effort to develop the constitutive laws that are much more suitable for aluminum alloys. Gurson’s (1977) dilatational plasticity theory is not applicable to aluminum alloys because it is based on the isotropic J2 plasticity theories. There are very limited studies on the Gurson-type dilatational plasticity theory for aluminum because its anisotropic constitutive models are usually too complicated to provide analytical solutions for void growth. Liao et al. (1997) have recently developed the approximate yield criteria for anisotropic porous ductile sheet metals based on Hill’s quadratic (1950) as well as non-quadratic yield functions (1979). The yield criteria are similar to the Gurson’s (1977), except that the coefficient scaling the hydrostatic stress is different to reflect the anisotropy.

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In order to determine the state of deformation and damage in aluminum alloys, we develop a dilatational plasticity theory in this paper based on Barlat and co-workers anisotropic plasticity theory. We begin with a summary of Barlat and co-workers’ plasticity theory in Section 2, followed by the development of plane-stress anisotropic dilatational plasticity theory for aluminum sheets in subsequent sections. The proposed dilatational plasticity theory provides a means to quantitatively characterize the effect of microvoid damage in aluminum alloys. 2. Anisotropic Plasticity Theories for Aluminum Sheets Barlat and co-workers have developed the yield functions for anisotropic aluminum alloys (Barlat and Lian, 1989; Barlat et al., 1991; 1997). For example, Barlat and Lian (1989) and Barlat et al. (1991) have introduced yield functions that account for the differences of yield strengths in the rolling, transverse and normal directions. Some additional flexibility to the yield function of Karafillis and Boyce (1993) was recently introduced in the Barlat et al. (1997) model in order to account for the differences in the yield stress under pure shear for two types of Al-Mg alloys that have the same uniaxial yield stresses as well as the same equibiaxial yield stresses. As an initial attempt to develop a Gurson-type dilatational plasticity theory characterizing the effect of porosity on plastic yielding for anisotropic aluminum alloys, we use the 6-component anisotropic yield function proposed by Barlat et al. (1991). Since the thickness of aluminum sheets is typically much smaller than the in-plane dimension, we adopt the same assumption as Liao et al. (1997) that the aluminum sheets are under the plane-stress deformation. For simplicity, the plastic anisotropy within the sheet plane is neglected, i.e., the aluminum sheets are modeled as transversely isotropic. Let σ 11 , σ 22 , and σ12 = σ 21 represent a plane-stress state (σ 33 = 0) in the sheet plane and x 3 be the out-of plane direction. In order to account for anisotropy in plastic yielding, Barlat et al. (1991) have introduced a transformed stress state as

(1)

where c 1, c 3 and c 6 are non-dimensional material constants that reflect the At anisotropy of the solid, and the in-plane isotropy requires

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the limit of c1 = 1 and c 3 = 1, the transformed stress state s αβ degenerates to the deviatoric stress and the material degenerates to an isotropic solid. For aluminum sheets, the coefficient c3 can be more than 20% higher than c 1 (Barlat et al. 1997). The yield function is given in terms of the transformed stress state by (Barlat et al., 1991) (2) where s1 , s2 and s3 are the principal values of the transformed state (s11 , s22, s 33 , s12 ), i.e.,

(3) σ Y is the uniaxial yield stress in the sheet plane (e.g., rolling direction), and ρ depends on the anisotropy constants c1 and c 3 and is given by

(4) The elastic deformation is neglected such that the plastic strain rate becomes the same as the total strain rate. The plastic strain rate is normal to yield surface, and is given by (5) where α , β = 1, 2, and the proportionality coefficient represents the amplitude of plastic flow. The yield function (2) then gives the in-plane plastic strain rate as

(6)

PLASTICITY THEORY FOR ALUMINUM SHEETS where

12 =

2

12

21

is the engineering shear strain rate, and (7)

= σαβ αβ can be evaluated via (6) to give The plastic work dissipation the following simple expression in terms of the non-dimentional pareameter ρ in(4), uniaxial yield stress σY in the sheet plane, and the proportionality coefficient (8) The ℜ-value, defined as the ratio of transverse plastic strain-rate to the through-thickness plastic strain-rate, is obtained in terms of anisotropy constants c 1 and c 3 as

(9) The theoretical limit strains predicted by the above constitution model are in reasonable agreement with the experimental data for aluminum (Barlat et al., 1991). Accordingly, the constitute law described above is used to characterize the matrix material that contains microvoids in order to establish the Gurson-type dilatational plasticity theory for aluminum. 3. The Gurson-type Dilatational Plasticity for Aluminum Sheets We extend Gurson’s (1977) approach to establish the approximate yield function for an anisotropic aluminum sheet that contains microvoids. The sheet is under the plane-stress deformation. A finite circular sheet containing a single through-thickness hole is subjected to a general macroscopic strain rate α β on its outer boundary. The radius of the void is a, while the outer radius of the matrix is b. Their ratio, a/b, is related to the void volume fraction ƒ by (10) The microscopic strain rate is referred to the strain rate in the matrix and is nonuniform due to the existence of the void. The matrix material is characterized by Barlat et al.’s (1991) anisotropic plasticity model. An approximate, upper bound solution is obtained for the microscopic plastic work dissipation Its integration over the matrix material gives the macroscopic work dissipation whose derivative with respect to α β yields the stress state ∑ αβ at the macoroscale.

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The macroscopic strain rate α β can be decomposed to the volumetric part and deviatoric part for plane-stress deformation. The microscopic strain rate αβ , which are nonuniform, can be similarly decomposed to the volume-changing (constant shape) part and the shape-changing (constant volume) part ( 1 1) such that where Following Gurson (1977) and Liao et al. (1997), we approximate the at the microscale by the uniform field of macroshape-changing part scopic deviatoric strain rate (12) Also following Gurson (1977) and Liao et al. (1997), the volume-changing part at the microscale corresponds to an axisymmetric velocity field, (13) which gives the strain rates (14) where r is the polar radius measured from the center of the void, A1 a n d A 2 are coefficients to be determined in the following. Matching of the imposed deformation on the outer surface r = b of the matrix gives (15) where the right hand side represents the velocity associated with the macroscopic volumetric strain field . The other relation between A 1 and A 2 is derived from the traction-free condition on the void surface. In fact, the only non-zero stress component on the void surface is σθθ  r =a , which gives the transformed stress state in (1) as ( s rr,

s θθ ,

s33 ,

s rθ )

=

The constitutive relation (6) then gives the on the void surface. Their ratio yields the strain rates and second relation between A 1 a nd A 2 as (16) where the strain field in (14) has been used, and

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23

(17) is a non-dimensional parameter depending on the anisotropy constants c1 and c3 . The parameters A1 and A 2 are determined from (15) and (16) as

(18) where is the void volume fraction. The microscopic strain rates, accounting for both the volume-changing part and the shape-changing part, are found in terms of the macroscopic strain rates as

(19) The determination of microscopic plastic work dissipation is equivalent to finding the proportionality coefficient , according to (7). It is observed that, once αβ are obtained as in (19), there remain four unknowns, namely , s 11 , s 22 and s 12 , which are to be determined from three constitutive relations in (6) and the yield function in (2). The eliminination of s 11, s 22 a n d s 12 yields the following solution for (20) where

(21) and g is an implicit function of strain rates αβ , position (r, θ ), and the void volume fraction ƒ, and is governed by the following nonlinear equation, (22)

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and

(23)

The macroscopic plastic work dissipation scopic counterpart over the entire cell, i.e.,

is the average of its micro-

(24) where is given in (20). The upper bound analyses of Gurson (1977) and Liao et al. (1997) are

used to calculate the macroscopic stress, ∑ αβ , i.e.,

(25) The integration with respect to r and θ in (25) must be evaluated numerically. For any given macroscopic strain rates α β , (25) gives the corresponding macroscopic stress state. The macroscopic effective stress ∑ e is defined in the same way as the microscale yield function in (2), i.e. (26) where ρ is given in (4) and S1 , S 2 and S 3 are the principal values of the transformed stresses S 11 , S 22 , S 33 a n d S 12 at the macroscale, which are related to the macroscopic stresses ∑ αβ via the linear transformation (1),

(27)

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25

The substitution of (27) into (26) gives the effective stress ∑e in terms of the stress components ∑ αβ , (28) For the where and sheet subjected to the uniaxial tension ∑11 in the sheet plane, the effective stress in (28) degenerates to the uniaxial stress ∑ 11 . It should be pointed out that the effective stress depends on the anisotropy constants c1 and c 3 only through their ratio, For each given macroscopic strain rate tensor αβ , the macroscopic effective stress ∑ e and the in-plane mean stress can only be obtained numerically, and therefore yields an implicit relation between the two. This relation reflects the dependence of the macroscopic yield function on the mean stress and the void volume fraction for an anisotropic material and is presented in the next section, along with an approximate but analytic yield function. 4. An Approximate Yield Function and the Numerical Results The relation between the macroscopic effective stress ∑ e a n d t h e m e a n stress established in the previous section is fully implicit. It is desirable to derive an approximate but analytical expression for the yield function. Following Gurson (1977) and Liao et al. (1997), we derive an apin this section and compare it proximate relation between ∑ e and with the numerical results from the analysis in Section 3. Consider the same geometrical model of a sheet containing a throughthickness hole of the radius a as in Section 3. The outer radius of the sheet is denoted by b. An axisymmetric velocity field, corresponding to the macroscopic strain rate 11 = 22, is imposed on the outer boundary r = b. The sheet is also subjected to a uniform strain rate 33 in the out-ofplane direction. It should be pointed out the deformation is not plane-stress anymore because of the imposed 33 . The microscopic strain field in the matrix is then given by (29) where the coefficient A'1 is determined by the incompressibility of the plastic deformation, (30)

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The other coefficient, A' 2, is determined by the imposed velocity field on the outer boundary r = b, (31) which is similar to (15). The rest of the analysis is almost identical to that in Section 3, except that it is not plane stress (σ 33 ≠ 0) but axisymmetric. The matrix material is characterized by the Barlat et al. (1991) model, and the non-vanishing stresses on the microscale are σ rr, σ θθ and σ 33 . The corresponding nonvanishsing macroscopic stresses are ∑ 11 = ∑ 22 and ∑ 33. Therefore, the in-plane mean stress is while the effective stress defined by (26) becomes (32) Unlike Gurson (1977) and Liao et al’s (1997) analyses which yield the closed-form solutions for the approximate yield functions, the present axisymmetric analysis still does not warrant analytical solutions because the yield functions in (2) are highly nonlinear. However, the analytical solutions can be found for the following two limits: (1) uniaxial tension, It is straightforward to show A' 2 = 0 from (30) and (31) such that the microscale is also subjected to This makes sense because the existence uniaxial tension, of the through-thickness hole does not violate the condition σrr = σθθ = 0 in uniaxial tension along the thickness direction. The macroscopic in-plane stresses, ∑ 11 and ∑ 22, are the average of their microscale counterparts, σ 11 and σ 22 , within the unit cell (Gurson, 1977), and therefore also vanish. The closed-form solution gives the macroscopic effective stress as ∑e = (1–ƒ) σ, Y which also makes sense since the factor 1 – ƒ accounts for the reduced area to sustain the load. At this limit, the in-plane mean stress and the effective stress ∑e = (1 – ƒ) σY . (2) plane-strain deformation, 33 = 0; It is evident from (29) the microscale deformation is also under the plane-strain condition, 33 = 0. In conjuction with the normality of plastic flow, 33 = 0 gives that the microscale stress in the thickness direction is exactly the same as in-plane mean stress, i.e., By averaging over the unit cell, this condition which gives a vanishing effective stress, ∑ e = 0 becomes from (32). The closed-form solution also gives the in-plane mean stress as where ρ is given in (4) in terms of the anisotropy constants c1 a n d c3.

PLASTICITY THEORY FOR ALUMINUM SHEETS

27

These two limits lead to a very natural construction of the approximate yield criteria for the entire range of in-plane mean stress and effective stress ∑ e. It is observed that the first limit is equivalent to The second limit, after some manipulations, can and be written as and The approximate yield criterion that satisfies these tow limits and bears similarity with Gurson’s (1977) and Liao et al’s (1997) yield criteria for dilatational plasticity is simply the combination of the above two limits, (33) In particular, this relation is exact at the above two limits, i.e., ∑e = and In fact, (33) is a rather accurate representation of the numerical solutions for all combinations of strain rates. The yield function in (33) is extremely similar to those established by Gurson (1977) and by Liao et al. (1997), though the effective stress is defined by the Barlat et al. model (1991) via (28). The to scale the anisotropy comes into play through the coefficient in-plane mean stress as well as through the effective stress in (28). It is also observed that the yield function in (33) depends on the ratio of anisotropy constants c 1 a n d c 3 . It is recalled that the aim of the present study is to establish the yield function for plane-stress aluminum sheets containing through-thickness holes. Therefore, it is important to compare the above approximate yield function established from the axisymmetric analysis with the numerical solutions for plane-stress aluminum sheets. The relation between the effective from the previous section for stress ∑e and the in-plane mean stress the plane-stress is shown in Fig. 1. The approximate yield function in (33) is also presented in Fig. 1 for comparison. Both the effective stress and the in-plane mean stress are normalized by the uniaxial yield stress σ Y i n t h e sheet plane. The void volume fraction ƒ has a rather large variation, from 0.02 to 0.2, while the anisotropy coefficients are c 1 = 1 and c 3 = 1.2, which gives a very large ℜ -value ℜ = 1.80 from (9). It is clearly observed that, without any parameter fitting, the numerical results for plane-stress sheets agree very well with the approximate yield function in (33) for a large range of void volume fraction ƒ . We have also examined other combinations of the anisotropy constants, The numerical results all confirm that the yield function is well approximated by (33). Therefore, (33) gives a rather accurate measure of the yield function for anisotropic aluminum sheets.

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Figure 1. The yield function for anisotropic aluminum sheets containing microvoids; the matrix material is characterized by Barlat et al.’s (1991) anisotropic constitutive model; ∑e a n d are the effective stress and in-plane mean stress, respectively; σY is the uniaxial yield stress in the sheet plane; ƒ is the void volume fraction; the anisotropy c o n s t a n t s a r e c1 = 1 and c 3 = 1.2, p is given in (4), and the ℜ –value is 1.80 from (9).

It should be pointed out that the numerical results presented in Fig. 1 for plane-stress sheets have covered all possible combinations of macroscopic with strain rates, ranging from 12 = 0. In fact the left

while the right limit corresponds to limit corresponds to the most right point on each curve. The yield function (33) for anisotropic sheets has a remarkable resemblance to that of Gurson (1977) for isotropic solids, even though the effective stress is defined by the Barlat et al. (1991) model via (28) instead of the Von Mises effective stress. In fact, its interception with the vertical ( ∑ e) axis, 1 – ƒ, is identical to that in the Gurson model. The interception with the horizontal axis also has the same dependence on the void volume fraction, In ƒ– 1 ; though the associated coefficient is slightly different, reflecting the effect of the plastic anisotropy. The curves between these two limits are both characterized by the cosh function in (33) and in the Gurson

PLASTICITY THEORY FOR ALUMINUM SHEETS

29

model. Similar conclusions have been established by Liao et al. (1997) for Hill’s quadratic (1950) and non-quadratic (1979) anisotropic yield criteria. 5.

Conclusion

We have generalized Gurson’s (1977) isotropic dilatational plasticity theory to the anisotropic aluminum sheets in this paper. Following the same approach of Gurson (1977) and Liao et al. (1997), we have analyzed a matrix containing a single void subjected to an imposed deformation on the outer boundary. The anisotropic constitutive model of Barlat et al. (1991) is used to characterize the plastic behavior of aluminum matrix. The relation beis fully tween the effective stress ∑ e and the in-plane mean stress implicit and must be obtained numerically. However, we have obtained an approximate yield function that is rather accurate for the entire range of ∑e and This approximate yield function is similar to that in Gurson’s model, but the effect of plastic anisotropy has been accounted for. Acknowledgements Y.H. gratefully acknowledges the research grants from Ford Foundation and from Alcan Int. Ltd. References Barlat, F. and Lian, J. (1989) Plastic behavior and stretchability of sheet metals. Part I: A yield function for orthotropic sheets under plane stress conditions, Int. J. Plasticity 5, 51-66. Barlat, F., Lege, D.J., and Brem, J.C. (1991) A six-component yield function for anisotropic materials , Int. J. Plast 7 , 693-712. Barlat, F., Maeda, Y., Chung, K., Yanagawa, M., Brem, J. C., Hayashida, Y., Lege, D.J., Matsui, K., Murtha, S. J., Hattori, S., Becker, R.C. and Makosey, S. (1997) Yield function development for aluminum alloy sheets, J. Mech. Phys. Solids 4 5 , No. 11/12, 1727-1763. 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. Mater. Tech 99, 2-15. Hill, R. (1950) The mathematical theory of plasticity, Oxford University Press , L o n d o n . Hill, R. (1979) Theoretical plasticity of textured aggregates, Math. Proc. Camb. Philos. Soc. 8 5, pp. 179. Hosford, W.F. (1979) On yield loci of anisotropic cubic metals, Proc. 7th North American Metalworking Conf., SME, Dearborn, MI, pp. 191-197. Huang, Y. (1991) Accurate dilatation rate for spherical voids in triaxial stress fields, J . Appl. Mech. 5 8 , 1084-1086. Karafillis, A. P. and Boyce, M. C. (1993) A general anisotropic yield criterion using bounds and a transformation weighting tensor, J. Mech. Phys. Solids 4 1, 1859-1886. Liao, K.-C., Pan, J., Tang, S.C. (1997) Approximate yield criteria for anisotropic porous ductile sheet metals, Mechanics of Materials 2 6 , 213-226. Mellor, P.B. (1981) Sheet metal forming, Int. Metals Rev. 2 6 , 1-20.

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Mellor, P. B. and Parmar, A. (1978) Plasticity of sheet metal forming, in D. P. Koistinen and N. M. Wang (eds.), Mechanics of Sheet Metal Forming , Plenum Press, New York, pp. 53-74. Needleman, A. and Rice, J. R. (1978) Limits on ductility set by plastic flow localization, in D. P. Koistinen and N.-M. Wang (eds.), Mechanics and Sheet Metal Forming 17 , Plenum, New York, pp. 237. Rice, J.R. and Tracey, D.M. (1969) On the ductile enlargement of holes in triaxial stress fields, J. Mech. Phys. Solids 1 7, 201-217. Tvergaard, V. (1990) Material failure by void growth to coalescence, In: J. W. Hutchinson and T.Y. Wu (eds.), Advances in Applied Mechanics 2 7, pp. 83.

INTERNAL HYDROGEN-INDUCED EMBRITTLEMENT IN IRON SINGLE CRYSTALS JIAN-SHENG WANG Northwestern University Evanston, IL 60201, USA

Abstract: A thermodynamic model for internal hydrogen-induced embrittlement (HIE) in single crystals is proposed. The model is based on the assumption that the ductile versus brittle transition is controlled by the competition between dislocation emission from the crack tip and cleavage decohesion of the lattice. Embrittlement in single crystals is induced by segregation of hydrogen in solid solution to the crack tip and/or the fracture surfaces during separation, which reduces the cohesive energy of the lattice. This process will occur when the mobility of hydrogen atoms is high so that a surface excess of hydrogen can be built up during separation. The model predictions for hydrogen induced cleavage in iron single crystals are presented. 1. Introduction Hydrogen-induced embrittlement (HIE) has been observed not only in polycrystalline metals and alloys, but also in single crystals, e.g., in single crystals of Ni [1, 2], Nibased alloys [3, 4], Fe and FeSi alloys [1, 5-9], stainless steels [10], and intermetallics [11]. For a system where the formation of a hydride is thermodynamically unattainable or kinetically impractical, solution hydrogen-induced embrittlement may occur due to precipitation of gaseous hydrogen or methane; localized plastic deformation prompted by the interaction between hydrogen atoms and dislocations at the crack tip; or due to the reduction of the lattice or grain boundary cohesion. The hydrogen-enhanced localized plasticity theory suggests that hydrogen in a solid solution reduces the barrier to dislocation motion through an elastic shielding effect [12-14], thereby increasing the amount of plastic deformation that occurs in a localized region adjacent to the fracture surface, causing embrittlement. In contrast, the cohesion-reduction theory postulates that segregation or adsorption of hydrogen decreases the cohesive energy inducing embrittlement [e.g. 15-18]. Vehoff pointed out, in a recent review, that for HIE to occur, hydrogen has to enter the fracture processing zone (FPZ) to reduce the local atomic bonding strength at the crack tip [18]. In the presence of solute hydrogen in polycrystals, this can occur due to segregation of hydrogen to grain boundaries [15, 16]. In the presence of external gaseous hydrogen, this can occur due to adsorption of monatomic hydrogen at the newly formed fracture surfaces and the FPZ [2]. While the cohesion-reduction theory can explain hydrogen induced cleavage-like fracture in single crystals in the presence of external hydrogen gas, a rigorous thermodynamic analysis of the cohesion-reduction 31 T.-J. Chuang and J. W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 31–47.  2000 Kluwer Academic Publishers. Printed in the Netherlands.

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theory for HIE in single crystals in the absence of external hydrogen gas is not available. The present work suggests a thermodynamic model for HIE in single crystals when hydrogen is present in the solid solution and any damage induced by hydrogen charging, or any softening-hardening effect is negligible. This model is a natural extension of the thermodynamic analysis of segregation-induced interfacial embrittlement of Rice, to whom this book is dedicated, and his colleagues [15, 19-22]. To understand how a single crystal can be embrittled by hydrogen in solid solution, it might be worthwhile to review how Rice resolved the dilemma of segregation-induced interfacial embrittlement. 2. Thermodynamics of Segregation-Induced Interfacial Embrittlement How grain boundary segregation could induce embrittlement was a puzzle less than three decades ago. Thermodynamics asserts that a lower energy state is more stable. For example, special grain boundaries, such as ∑ 3, ∑ 11 coincide lattice site (CLS) grain boundaries, are more stable than random grain boundaries because of their low grain boundary energies. Segregation reduces the grain boundary energy; it would, intuitively, stabilize the grain boundary, but why, instead, does it promote intergranular brittle fracture? The cohesive energy of a grain boundary, or equivalently, the reversible work of intergranular separation, is conventionally (but, as Rice pointed out, not completely) defined as (1) γc = 2 γs – γ b where γ s and γ b are the surface and grain boundary energies, respectively. For a low temperature or fast interfacial separation process, when redistribution of the segregant at the newly created fracture surfaces is unattainable because of its low mobility, γs would remain unchanged. In this case, a reduction in γ b by segregation would increase γ c. Based on an incomplete thermodynamic analysis, it has been claimed that “low temperature work of grain boundary fracture is independent of segregation” [23], contradictory to experimental observations. A rigorous thermodynamic analysis of interfacial cohesion in the presence of solute atoms was provided by Rice and through this, the problem of segregation induced intergranular fracture was solved [19]. In his brilliant treatment of the interfacial cohesion problem, Rice introduced two new thermodynamic variables: the stress acting at the interface and the separation distance of the creating surfaces. The force per unit area acting on the separating atoms at the interface, σ, is a function of the separation distance δ. T h e cohesive energy of the interface, γc , is defined by the reversible work of separation, i.e., the area under the σ – δ curve: Fig. 1. Schematic of the σ – δ relation at a crack-tip.

(2)

where δ0 is the initial separation of the unstressed interface and δ is the separation

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excess under the stress (Fig. 1). Rice pointed out that the most important parameters associated with crack initiation were the cohesive strength of the interface σ c , i.e., the maximum of the σ – δ curve, and the cohesive energy of the interface, γc . Since the σ−δ relation is affected by segregation, both σc and γc may alter in the presence of segregants. Interface cohesion is not a state function in general, but depends on the thermodynamic path followed by separation. There are two limiting cases in interfacial separation processes in the presence of segregants: separation at constant interface concentration and separation at constant chemical potential. These two limiting cases identify two different thermodynamic paths. The first path is a “fast” separation on a time scale which does not allow further matter transport to the interface (the “immobile” case). The second is a “slow” separation on a time scale which allows full composition equilibrium between the interface and a matter source at a constant potential (the “mobile” case). By introducing two new valuables, σ and δ, Rice derived that

(3) for separation at constant Γ, where µ 0 (Γ ) is the potential corresponding to excess concentration Γ on the unstressed interface and µ ∞ ( Γ) the potential corresponding to the net excess concentration Γ on the two completely separated surfaces, and

(4)

for separation at constant µ, where Γ (µ) is the initial segregant excess on the unstressed interface and Γ∞ (µ) is the equilibrium excess on the two completely separated free surfaces. Rice’s analytic results were later given in terms of a reversible work cycle in chemical potential-composition space by Hirth and Rice [20] as shown in Fig. 2. The rigorous thermodynamic analysis of Hirth and Rice demonstrated that for a “fast” separation at constant Γ, the change in the cohesive energy is (5) corresponding to area OAYO along the µ = µ b( Γ ) curve in Fig. 2. For a “slow” separation at fixed µ the change in the cohesive energy is (6) corresponding to area OBYO along the µ = µs (Γ/2) curve in Fig. 2. Here Γs (µ) gives the segregant excess on a single free surface, and Γ b (µ) on a grain boundary, at equilibrating potential µ.

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Equations (5) and (6) link γ c , for separation at fixed composition or at fixed chemical potential, to quantities, which can, in principle, be estimated from solute segregation studies. Normally the potential necessary to equilibrate Γ on a grain boundary will be larger than the potential to equilibrate the same amount on a pair of free surfaces (at Γ /2 on each), i.e., µ b ( Γ ) > µ s ( Γ /2), and also 2Γs (µ) > Γ b (µ). With this normal type of segregation behavior ∆γ c > 0 for both paths, the segregation reduces γc , and thus is expected to promote embrittlement. The analysis of Hirth and Rice shows that for an interface with solute segregation, the definition for grain boundary cohesion given in (1) is valid only for the “mobile” case of fully equilibrated separation at constant chemical potential. Here, γs and γ b corresponds to the free energy of the completely separated surfaces and the unstressed interface, respectively, each of which is in equilibrium with the potential source, i.e., (7) Since a normal segregant reduces the free energy of the surface more than reducing the free energy of the grain boundary, the cohesive energy of the interface is thus reduced. Equation (1) is, however, invalid for “fast” separation when further matter transport is not allowed during separation. In this “immobile” and non-equilibrium case, the cohesive energy of the interface is given by (8) where γ b ( Γ0 ) is the free energy of the interface in which solute of concentration Γ0 equilibrates with the bulk phase, γ s ( Γ 0 /2) is the free energy of the surface in which solute of concentration Γ0 /2 equilibrates with the surface but not the bulk phase, and µ s and µ b are corresponding chemical potentials. Since γ s (Γ0 /2) < γ s0 , where γ s0 is the surface free energy without segregation, and for a normal segregant, the last term in the right hand side of (8) is negative, interface cohesion is reduced. The early dilemma in understanding segregation induced intergranular fracture is thus resolved.

Fig. 2. Schematic of the potential-excess spaces for a grain boundary and free surfaces. The original state is O( Γ b , µ 0 ). Two limiting cases and a transient case for grain boundary separation are shown.

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The more general case is that both the grain boundary excess and the chemical potential vary during separation following a transient path. Referring to Fig. 2, the separation process is a transition starting from the initial state O( Γ b , µ 0 ) to a non-equilibrium state T( Γt , µ t ). The change in the cohesive energy of the interface corresponds to area OTYO, where µ t (Γ t ) is the transient chemical potential of the segregant at a non-equilibrium surface excess of Γ t . Based on segregation kinetics, a model has been developed to evaluate the embrittlement propensity under transient conditions [22].

3. Thermodynamics of Segregation-Induced Embrittlement in Single Crystals Considering now a single crystal with a segregant in solid solution, a crack is initiated and propagating under stress leading to fracture, the temperature is high or the separation is slow under the “mobile” condition so that chemical potential is constant during matter transport. Following the same procedure of Hirth and Rice, the thermodynamics for segregation-induced embrittlement in single crystals is described as following. Thermodynamics states that for reversible alteration of state of a system dU = TdS + dw rev

(9)

where dw rev is the reversible work with the sign opposite to the conventional chemical thermodynamics usage. In an isothermal system consisting of a single crystal m, and a segregant of chemical potential µ, and capable of changing the surface area, dAs , under the surface tension, γs , in the absence of any external device work

where V is the volume of the system, P is the uniform pressure, dn and dn m are the exchanges of matter in moles. The surface tension is identical to the surface free energy if any change in the surface area by elastic stretching bonds can be ignored and, henceforth, we call γs the surface free energy. Equation (9) becomes

and in terms of the Helmholtz free energy, F = U–TS

The extensive quantities F, V and n can be divided into bulk quantities and surface excess quantities normalized to unit area of the surface, f, v, and Γ and Γm , thus (10) A simple argument of Hirth and Rice demonstrated that integration of (10) at constant T yields (11)

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Differentiation of (11) and by making use of the Gibbs-Duhen relation, it is easy to show that at constant T (12) Considering that the dividing surface can be chosen so that v = 0 (in the Gibbs sense) or the term vdP can be disregarded (in the Guggenheim sense), and that the phase law does not allow dµ m independent of dµ, (12) reduces to (13) Integration of (13) leads to the surface free energy (14) where γ s0 is the surface free energy of the pure single crystal and µ 0 ≅ RT lnc 0 is the potential of the solute of concentration c 0 in the bulk. The validity of this relation to fracture of a single crystal relies on the “mobile” condition that the chemical potential µ remains constant. Under this condition, the change in the cohesive energy of the lattice is (15) where Γ = Γ s (µ) is the excess on a single surface. The separation processes start from the origin O at constant µ = µ0 and end at B with surface excess Γs = Γ. The change in cohesive energy is represented by area OBYO along the µ = µ s (Γ ) curve in the chemical potential-excess space (Fig. 3). The original analysis of Hirth and Rice for a system with an initial interface (a grain

Fig. 3. Schematic of the potential-excess space for a free surface. Cleavage fracture at constant chemical potential or under transient conditions is shown.

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boundary) is valid for a system without a grain boundary, because thermodynamics does not distinguish the physical nature of the interface: no matter if it is a grain boundary or a surface. For a system without an initial grain boundary, (15) can be derived simply by setting the Γb term in (6) to zero. The same is true for Rice’s treatment with δ and σ as variables. Similarly, by setting Γ = 0 in (3) and (4), the cohesion of a single crystal under “immobile” and “mobile” conditions can be derived. Under “immobile” conditions, dγc /d Γ in (3) is identically zero, the cohesive energy of the crystal is independent of segregation if separation is “fast”. It ought to be understood that this statement is true if any changes in dislocation behavior, the bonding nature and the lattice distortion et cetera, induced by solute atoms can be ignored. 4. Micromechanics of HIE in Fe Single Crystals A micromechanical description of the ductile versus brittle fracture of a crystal concerns dislocation emission from the crack tip. It is generally believed that the tip response of a stressed crack is governed by the competition between dislocation nucleation from the tip and cleavage decohesion. This concept was modeled by Rice and Thomson [24] and advanced by Schoeck [25] and Rice [26]. In general, the critical energy release rate for dislocation nucleation from the crack tip, G disl , and the critical energy release rate for Griffith cleavage decohesion of the crystal, Gc l e a v , is compared. If Gdisl < G c l e a v , dislocation emission is predicted to occur first as the crack tip loading is increased, thereby blunting the crack tip and reducing the tip stress field required for cleavage. In this case, the crystal is interpreted as intrinsically ductile¹ and the crack tip is a dislocation emitting tip. Alternately, if G disl > G cleav , atomic decohesion occurs first, producing cleavage, or in essence, cleavage is potentially conceivable. In this case, the crystal is esteemed as intrinsically brittle and the crack tip is called a non-emitting tip. Understanding the HIE is thus reduced to evaluating how Gdisl and G c l e a v are affected by segregation of hydrogen to the crack tip and the fracture surfaces. 4.1. THE SEGREGATION INDUCED REDUCTION OF G c l e a v It has been derived from the cohesive zone model that the energy release rate for cleavage decohesion is equivalent to the cohesive energy given by (2), providing that self atom trapping at the crack tip is negligible, i.e., G cleav = γc

(16)

Because of the reduction of γc , ductile-brittle transition may occur if the inequality of G disl < Gcleav for ductile behavior is reversed to Gdisl > Gc l e a v for brittle behavior. When the surface excess Γ is less than values corresponding to full occupancy of a set of segregation sites, idealized as all having the same low energy relative to solute sites in the bulk, the simple Langmuir-McLean model [27, 28] may apply,

¹ An intrinsically ductile solid is not necessarily to fracture in a ductile manner. If the dislocation mobility is low, the newly formed dislocation may not be able to move away from the tip, exerting an image force to the tip and preventing further nucleation of dislocations from the tip, resulting in a brittle behavior.

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(17) where Γ is the surface saturation excess and the inherently negative segregation free energy ∆ g s is referenced to a bulk phase at the same T, i.e., is based on the expression µ = RT ln [x/( 1–x)] ≈ RT ln (x) for the equilibrating potential when a fraction x of available solute sites is occupied in the bulk and x<<1. Integration of (15) yields 0

(18) for an equilibrium separation, where θ s ≡ Γ /Γ 0 is the surface coverage: (19) It has been derived [22] that for a grain boundary separating at a constant chemical potential µ = µb ( Γb ) (20) where

is the equilibrium coverage in the grain boundary. Noting that Γ0 =

by setting θ b = 0 equation (20) for an interface reduces to (18) for a single crystal lattice where no initial interface coverage exists. Equation (18) gives the maximum reduction in cohesion when equilibrium segregation occurs during separation. This may not be attainable when separation is fast or the temperature is low. In reality, the more general case is that during separation the chemical potential varies following a transient path OT in Fig. 3. A non-equilibrium surface excess, Γt , is attained, which is controlled by segregation kinetics. Referring to Fig. 3, the separation is a transition starting from the initial state O(c 0 , µ 0 ) along a nonequilibrium path to a transient state T( Γt , µ t ). The change in the cohesive energy of the lattice corresponds to area OTYO. To determine the reduction in cohesion under transient conditions, ∆γ t, a knowledge of the actual details of the trajectory OT, i.e., the function µ = µ t ( Γ ), is needed, which are governed by the diffusional transport of the segregant from solid solution to the separating surfaces. In the case where no second phase is involved during separation, the function µt ( Γ ) must be continuous, single valued. Assuming a linear relation as a first order approximation, one may derive the reduction of cohesion under transient conditions:

(21) where θ t ≡ Γt /Γ 0 is the transient coverage and θ s is the maximum attainable surface coverage given by (19). The transient coverage might be derived from McLean segregation kinetics: (22)

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with

where D is the diffusion coefficient of the segregant, τ the fracture process time, h the thickness of the surface region and η ≡ θs /c 0 is the maximum surface enrichment factor. The fracture process time, τ, is related to the crack growth rate, , by τ = ω , where ω is the size of the cohesive zone at the crack tip. For an atomically sharp cleavage crack, ω ≅ 3~4 lattice spacings [21]. 4.2. EFFECTS OF SEGREGATION OF HYDROGEN ON G cleav IN IRON SINGLE CRYSTALS Hydrogen tends to occupy high symmetry adsorption sites: three-fold coordinated sites on (110) surfaces and bridge sites on (100) surfaces. The hydrogen covered (100) surface exhibits a c(2x2) reconstruction with a saturation occupancy of 1:1 atomic ratio [29]. Suppose hydrogen is induced into iron single crystals by cathodic charging at 298 K and fugacity f = 300 bar, the potential-excess space for cleaving along {100} plans is shown in Fig. 4. The maximum reduction in cohesion for the cleavage plane, calculated by (19) and represented by the shadowed area in Fig. 4, is 1.37 J/m². For comparison, the µ b -Γ space for a random grain boundary is also shown in the figure. The maximum Fig. 4. The potential-excess space for reduction in cohesion for the grain boundary, Fe(100) plane. Hydrogen pre-charging: calculated by (20), is 0.804 J/m². Under the T= 298 K and fugacity f =300 bar. same conditions, the propensity for HIE in single crystals might be greater than that in polycrystalline specimens. Under transient conditions, the reduction in cohesion is governed by the kinetics of segregation. Therefore, it is a function of the crack propagation rate and temperature. Figure 5 shows the reduction in cohesion of iron crystals versus the cracking rate at 298 and 373 K, where the cathodic charging is conducted during separation. Suppose a 5% reduction in cohesion is necessary for ductile-to-brittle transition, corresponding to ∆γ c = 0.1826 J/m², then HIE in iron single crystals occurs at room temperature when the –4 crack propagation rate is quite low: ≅ 4×10 µ/sec, showing a time delayed cracking behavior. Figure 6 shows the temperature dependency of the reduction in cohesion at a fixed cracking rate = 0.1 µ/sec for crystals cathodically charged at 343 K. The embrittlement intensity increases with increasing temperature because of the increase in hydrogen diffusivity. It decreases with further increasing temperature because of reduced segregation. The model predictions are consistent with generally observed

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experimental results. For comparison, the temperature dependency for polycrystals is also plotted in the figure, where the specimen is cathodically charged under the same conditions and equilibrium segregation of hydrogen in grain boundaries is assumed. For the transient case, the embrittlement intensity for polycrystalline specimens is greater than that for single crystals. 4.3. EFFECT OF SEGREGATION OF HYDROGEN ON G disl The critical energy release rate for dislocation nucleation from a crack tip, Gdisl , could be evaluated by the Rice-Thomson model. The original approach of Rice and Thomson takes into account the balance of the work done by the applied stress and the energy of a dislocation loop emanating from the crack tip. This procedure evokes a dislocation core cutoff, a poorly defined parameter in the continuum elastic dislocation theory and, in essence, it deals only with emission of dislocations from the crack tip, not dislocation nucleation. The newly developed Peierls-Nabarro type approach [25, 26, 30-34] realizes that a full dislocation is likely to emerge unstably from an incomplete, incipient

Fig. 5. The cracking rate dependency of HIE for Fe single crystals.

Fig. 6. The temperature dependency of HIE in Fe single crystals and polycrystalline specimens.

dislocation at the tip, and the barrier for nucleation of this incipient dislocation is proportional to Peierls barrier for dislocation motion. The analytic and numerical treatments of the model [26, 30] solve the elasticity problem of a traction free crack with a Peierls stress versus displacement relation being satisfied as a boundary condition along a slip plane ahead of a crack tip. Once this problem is solved for a suitable constitutive relation for material sliding and perhaps opening along a slip plane, there is no need for the core cut-off parameter. The Peierls-Nabarro type approach describes that the critical energy release rate for dislocation initiation from the tip, Gdisl , is scaled by the unstable stacking energy, γ us , a newly introduced solid state physics parameter [26]. Considering the softening effect of the mode I loading, atomistic simulations by the Embedded Atom Method (EAM) for Fe derived the fully relaxed unstable stacking energy γus = 0.44 J/m² [35]. Depending on the orientation of the active slip systems (the inclination angle of the slip plane against the crack plane and the angle between the Burgers vector and

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the crack front normal), Gdisl as a function of γ us can be obtained numerically [26, 31, 32, 35]. Suppose an atomically sharp crack at a (001) plane in an α-Fe crystal with the – crack front parallel to the [110] direction and the crack propagates in the [110] direction, the critical energy release rate for dislocation nucleation from the crack tip at 0 K was solved numerically: G disl = 3.55 J/m 2 [35]. Comparing with the cohesive energy of the cleavage plane, γ c = 3.652 J/m 2 [36], pure iron is predicted to be intrinsically ductile. The lowest ductile versus brittle transition temperature (DBTT) ever measured for α-Fe is 163 K [37]. It has been recognized that the DBTT of iron is strongly influenced by trace impurities, such as carbon, nitrogen, oxygen, sulfur, phosphorus, boron, and hydrogen. When iron is sufficiently purified, the physical and chemical properties are considerably different from accepted values. The experimentally observed low temperature brittleness of iron is related to the inevitable impurities and the low mobility of dislocations. Assuming a constant value of γ us ,for an ideally pure iron single crystal at 0 K, the ductile versus brittle transition might occur if the cohesive energy is reduced about 2.7%. At non-zero temperatures, Gdisl is lowered by thermal activation, a further reduction in cohesion is needed for the transition. This behavior is altered due to the presence of inevitable trace elements. Impurities affect not only the cohesion, but also Gdisl though the influence of solute atoms on γus and the dislocation line energy. Some of the trace impurities may increase γ us and some of them may decrease it. In addition, the pinning effect of large-atomic-size impurities on dislocations may impede dislocation nucleation from the crack tip [38]. Quantitative evaluation of the net effect of trace elements on the ductile versus brittle transition in Fe single crystals requires comprehensive atomistic simulations, which have not been available and is beyond the scope of the present work. The effects of interstitial atoms on dislocation emission from a crack tip was revealed by an atomic model in the spirit of the Peierls approach [39]. The model is based on an EAM-type potential for nickel. The energetics of dislocation emission from a crack-tip in nickel containing hydrogen is analyzed. The results show a substantial effect on the unstable stacking energy as the dislocation passes an interstitial on the slip plane, but the effect of an interstitial on the resistance to dislocation emission expressed in terms of the maximum lattice resistance, σr , is small and then only if it is confined to a region very near the crack tip. Similar simulations for iron have not been available. It is believed that the Peierls potential for screw dislocations in Fe single crystal is reduced by hydrogen [40], indicating a reduction in γ us . A thermodynamic model was proposed to evaluate the change in the dislocation line energy due to segregation of solute, in particularly, segregation of hydrogen [15]. Analogous to the surface or interface segregation, the reversible work of forming a unit length of dislocation in the presence of a solute is given (23) with e0 equal to the work of formation in the absence of solute, Γd is the solute excess at the dislocation and µ is the equilibrium chemical potential. Equation (23) is evaluated for conditions where the dislocation is formed at chemical equilibrium between the solute and the bulk and dislocation core sites. The McLean isotherm is used to define the equilibrium excess Γd at the dislocation in terms of the lattice concentration, the

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segregation free energy ∆ g d to the dislocation sites, and the saturation excess level Γ d0 at the dislocation. The reduction in the line energy is evaluated as (24) where θ d = Γd / Γ d , is the dislocation line coverage of the solute atoms. 0

For conditions where the dislocation core acts as segregation sites (i.e., ∆g d, < 0), the effect of adding an impurity to the bulk material is seen to reduce Gdisl . This effect is intended to complement impurity effects on atomic decohesion discussed earlier, and on diffuse plastic flow from external, non-crack-tip dislocations. A quantitative estimate of the effect of H on dislocation emission in Fe is difficult, since 0 Γ d is poorly known. For a reference, the binding energy of H to a non-screw dislocation in Fe is estimated as –58.6 kJ/mol [41], which is lower than the estimate for grain boundaries but considerably higher than the estimate for free surfaces [22]. One could expect that the hydrogen-induced reduction in Gdisl is less severe than in G cleav. The net effect of hydrogen in iron single crystals is to reverse the inequality of Gdisl < G cleav for an otherwise pure iron crystal to the inequality of G disl > Gcleav , or is to increase the ductile versus brittle transition temperature for an impurity-containing iron crystal. 5. Discussion The present thermodynamic and micromechanical model for HIE in single crystals states that because of the segregation-induced reduction of cohesion, the condition for an intrinsically emitting crack, Gdisl < Gcleav is reversed to G disl > G cleav for a nonemitting crack, leading to brittle fracture or an increase in the DBTT. A tactic assumption behind this statement is that while brittle fracture occurs, the crack tip is elastic, despite the fact that pre-existing dislocations in the near-tip region may induce a large amount of plastic deformation and hence, the brittle fracture energy of a metal is usually orders of magnitude higher than the cohesive energy. It has been realized that for a non-emitting crack, due to the microscopic discreteness of plastic flow at a length scale too small for continuum plasticity, there may exist an elastic enclave free of dislocations around the tip [42-45]. Because of the shielding effect of pre-existing dislocations, when the local tensile stress at the tip is great enough to meet conditions for Griffith cleavage, the corresponding concentrated stress field near the tip contains large enough shear stresses to move pre-existing dislocations. In other words, while conditions for cleavage are satisfied at the tip, a plastic zone develops near the tip. The motion of those pre-existing dislocations induces a strong shielding effect of the crack tip from the full effect of the externally applied load. Therefore, the external load level for cleavage is greatly increased, i.e. Kapp > > Ktip or equivalently, G far >> G tip . The shielding ratio, Gfar /G tip , is a strong function of γc . A small change in γ c causes a large change in the shielding ratio, i.e., the cohesive energy serves as a control “valve” for fracture. The self-consistent elastic enclave model provides a qualitatively sensible description of cleavage in intrinsically cleavable materials in the presence of a large amount of plastic deformation and insures the tip response upon stresses being governed by the competition between dislocation emission and cleavage decohesion, provided that the mobility of dislocations is not the limiting step. For an emitting crack tip, where

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G disl < G cleav , such an elastic core does not exist 2 ; the conditions for Griffith cleavage cannot be satisfied, resulting in a ductile fracture. Interaction between hydrogen atoms and the near tip stress field has been a great concern in the study of hydrogen assisted cracking. The near tip stress field may increase the solubility of hydrogen in solid solution; enhance the local concentration of hydrogen; accelerate hydrogen diffusion; promote segregation and therefore intensify HIE. While a general view of stress-enhanced diffusion appears to be adverse [46-48], it has been confirmed that the concentrated near tip stress enlarges the population of hydrogen in the tip area [e.g. 49-55]. Finite element analysis within the framework of continuum elasto-plastic theory shows that for an emitting crack with large plastic deformation around the crack tip, the accumulation of hydrogen is related to the population of defects associated with plastic deformation [49, 52]. Most of the hydrogen is trapped close to the blunted tip, the total population of hydrogen is dominated by the density of the deep traps, which can rise two orders of magnitude as the strain near the tip increases from zero to 0.8 and then saturates [49]. In this case, the solubility of hydrogen is increased because of the increase in the deep trapping sites [56]. When limited plastic strain precedes the fracture event, most of the hydrogen resides at the normal interstitial lattice sites at the hydrostatic stress peak located away from the tip [54, 55]. The concentration of hydrogen is increased within the solubility limit. The peak concentration of hydrogen is higher than the far field value only by a factor of 2~3, instead of the several orders of magnitude predicted by elastic models. This continuum elasto-plastic model may not be valid for a non-emitting crack. In the presence of the elastic enclave at a non-emitting crack tip, the maximum tensile stress near the tip is not 3 to 5 times of the yield strength located a distance away from the tip, as predicted by continuum elasto-plastic mechanics. When Griffith cleavage occurs, the normal stress within the cohesive zone reaches the theoretical bond strength, resulting in an enrichment of hydrogen population of several orders of magnitude [51]. In this case, predictions of elastic interaction models may apply and the chemical potential, µ, in (15) should be replaced by µ + ε , where ε is the interaction energy between hydrogen atoms and the tip stress field. Within the framework of linear fracture mechanics the elastic interaction energy has the form with

under plane strain conditions in a polar coordinate system (r, φ ) with the origin at the crack tip. Here KI i s the mode I stress intensity factor at the crack tip, and ∆Ω is the elastic relaxation volume of the solute atom. For a cleavage crack on Fe(100) plane within the cohesive zone, ε = –12.4 kJ/mol, which is about one third of the chemical potential of hydrogen in solid solution at fugacity f =300 bar. The enhancement of the tip stress field is significant.

2

This by no means excludes the existence of a dislocation free zone (DFZ). Deferring from the elastic core around a non-emitting tip, DFZ around an emitting crack tip is a zone through which dislocations nucleated from the tip can pass but otherwise is dislocation free.

J.-S. WANG

44 6. Summary

The present thermodynamic and kinetic analyses show that in a hydrogen-containing single crystal, segregation of hydrogen from solid solution to the crack tip and the fracture surfaces reduces the cohesive energy of the lattice, γ c . Hydrogen in solute solution may also reduce the critical energy release rate for dislocation nucleation from the crack tip, Gdisl . When the reduction in γ c is greater than the reduction in Gd i s l , hydrogen induces embrittlement. In the case where the effect of hydrogen on G disl could be neglected the maximum propensity for embrittlement is determined by segregation thermodynamics. The embrittlement intensity during separation is governed by the kinetics of segregation, which is controlled by diffusion of hydrogen from the interior of the crystal to the cohesive zone at the crack tip. The model predictions for the trends of the temperature and cracking rate dependence of HIE in iron single crystals are consistent with experimental observations. The model predicts that the maximum embrittlement propensity for a single crystal is greater than that for polycrystalline specimens. Under the same conditions the embrittlement intensity for a single crystal is less than that for polycrystalline specimens because of the kinetics. Acknowledgement This work was finished while the author was supported by the Industrial Consortium of Coating-by-Design at Northwestern University and the IHPTET Fiber Development Consortium of DARPA. The author would like to thank Dr. Richard Hoffman for his valuable suggestions and help in preparing the manuscript. References [1]. [2]. [3]. [4]. [5]. [6].

[7]. [8].

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HYDROGEN EMBRITTLEMENT IN Fe SINGLE CRYSTALS [9]. [10]. [11]. [12]. [13]. [14]. [15]. [16]. [17]. [18]. [19]. [20]. [21]. [22]. [23]. [24]. [25]. [26]. [27]. [28].

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Hu Z; Fukuyama S; Yokogawa K; Okamoto S., Hydrogen embrittlement of a single crystal of iron on a nanometre scale at a crack tip by molecular dynamics, Model Simul Mater Sci Eng 1999, 7, 541-551. Magnin T; Chambreuil A; Bayle B., The corrosion-enhanced plasticity model for stress corrosion cracking in ductile fcc alloys, Acta mater, 1996, 44, 14571470. Takasugi T; Hanada S., The influence of residual hydrogen and moisturereleased hydrogen on the embrittlement of Ni3 (Al, Ti) single-crystals, Acta metall mater 1994, 42, 3527-3534. Birnbaum H. K. and Sofronis P., Hydrogen-enhanced localized plasticity - a mechanism for hydrogen-related fracture, Mater Sci & Eng, 1994, A176, 191202. Sofronis P., and Birnbaum H. K., Mechanics of the hydrogen-dislocationimpurity interactions .1. increasing shear modulus, J. Mech Phys Solids, 1995, 43, 49-90. Ferreira P. J., Robertson I. M., and Birnbaum H. K., Hydrogen effects on interaction between dislocations, Acta mater. 1998, 46, 1749-1757. Anderson P. M., Wang J.-S. and Rice J. R., Thermodynamic and mechanical models of interfacial embrittlement, in Innovation in Ultrahigh Strength Steel Technology, ed. G. B. Olson, M. Azrin, and E. S. Wright, 1990, pp. 619-649. McMahon C. J., Hydrogen embrittlement of high-strength steels, in Innovation in Ultrahigh Strength Steel Technology, ed. G. B. Olson, M. Azrin, and E. S. Wright, 1990, pp. 597-618. Tromans D., On surface energy and the hydrogen embrittlement of iron and steels, Acta metall mater., 1994, 42, 2043-49. Vehoff H., Hydrogen related material problems, in Topics in Applied Physics, Vol. 73, Hydrogen in metals III, 1997, pp. 215-278. Rice J. R., Hydrogen and interfacial cohesion, in Effect of Hydrogen on Behavior of Metals, ed. A. M. Thompson and I. M. Bernstein, TMS-AIME, New York, 1976, pp.455-466. Hirth J. P., and Rice J. R., On the thermodynamics of adsorption at interface as it influences decohasion, Met. Trans. 11A, 1502, 1980. Rice J. R. and Wang J.-S., Embrittlement of interfaces by solute segregation, Mat Sci Eng, 1989, A107, 23-40. Wang J.-S., Hydrogen induced embrittlement and the effect of the mobility of hydrogen atoms, in Hydrogen Effects in Materials, ed. A. W. Thompson, and N. R. Moody TMS, Warrendale, 1996, pp. 61-75. Seah M. P., Segregation and the strength of grain boundaries, Proc. R. Soc. Lond. 1976, A345, 535-554. Rice J. R. and Thomson R., Ductile versus brittle behavior of crystals, Phil. Mag., 1974, 29, 73-97. Schoeck G., The formation of dislocation loops at crack tip in three dimensions, Phil. Mag, 1991, A 63, 111-120. Rice J. R., Dislocation nucleation from a crack tip - an analysis based on the peierls concept . J. Mech Phys Solids, 1992, 40, 239-271. McLean D., Grain Boundaries in Metals, Oxford Univ. Press, Oxford, 1957. Hondros E. D. and Seah M. P., Grain boundary segregation, Proc. R. Soc. Rond., 1973, A335, 191-212.

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[32]. [33].

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[39]. [40]. [41]. [42]. [43]. [44]. [45]. [46] [47] [48].

J.-S. WANG Heinz K and Hammer L., Hydrogen on metals: adsorption sites and substrate reconstruction, Phys. stat. sol. (a), 1997, 159, 225-233. Rice J. R., Beltz G. B. and Sun Y., Peierls Framework for dislocation nucleation from a crack tip, in Topics in Fracture and Fatigue, ed. A. S. Argon, Springer-Verlag, 1992, pp. 1-58. Beltz G. E. and Rice J. R., Dislocation nucleation versus cleavage decohesion, in Modeling the Deformation of Crystalline Solids: Physical Theory, Application, and Experimental Comparisons, ed. T. C. Lowe, A. D. Rollett, P. S. Follansbee and G. S. Daehn, TMS, 1991, pp. 457-489. Beltz G. E. and Rice J. R., Dislocation nucleation at metal ceramic interfaces, Acta metall mater, 40, S321-331, 1992. Sun Y ., Y ., Rice J. R., and Truskinovsky L., Dislocation nucleation versus cleavage in Ni3 Al and Ni, in High-Temperature Ordered Intermetallic Alloys, ed. L. A. Johnson, D. T. Pope and J. O. Stiegler, Proc. MRS, Vol. 213, 1991, pp.243-248. Sun Y., Beltz G. E. and Rice J. R., Estimates from atomic models of tension shear coupling in dislocation nucleation from a crack-tip, Mat Sci & Eng, 1993, A170, 67-85. Sun Y., Atomistic aspects of dislocation/crack tip interaction, Ph.D. Dissertation, Harvard University, Cambridge, MA, 1993. Cheung K. S., Atomistic study of dislocation nucleation at a crack tip, Ph.D. Dissertation, MIT, Cambridge, MA, 1990. Abiko K., The evolution of iron, Phys stat sol, (a) 1997, 160, 285-196. Yu J. and Rice J. R., Dislocation pinning effect of grain boundary segregated solute atoms at a crack tip, in Interface structure, Properties and Design, eds. M.H. Yoo, W.A.T. Clark and C.L. Briant, MRS Symposium Proc. Vol. 122, MRS, 1988, pp.361-366. Hoagland R. G., On the energetics of dislocation emission from a crack-tip in nickel-containing hydrogen, J. Mater Res, 1994, 9, 1805-1819. Kimura, H.,and H. Matsui, Scr metall., 1987, 21, 319. Hirth J. P., Effects of hydrogen on the properties of iron and steels, Met. Trans., 1980, 11A, 861-890. Suo Z., Shih C. F., and Varias A. G., A theory for cleavage cracking in the presence of plastic flow, Acta metall mater, 1993, 41, 1551-1557. Beltz G. E., Rice J. R., Shih C. F., and Xia L., A self-consistent model for cleavage in the presence of plastic flow, Acta mater, 1996, 44, 3943-3954. Lipkin D. M., and Beltz G. E., A simple elastic cell model of cleavage fracture in the presence of dislocation plascity, Acta mater, 1996, 44, 1287-1292. Lipkin D. M., Clarke D. R., and Beltz G. E., A strain-gradient model of cleavage fracture in plastically deforming materials, Acta mater, 1996, 44, 4051-4058. Fukai Y., and Sugimoto H., Hydrogen diffusion in metals-unsolved problems, Defect and Diffusion Forum, 1992, 83, 87-110. Suzuki T., Namazue H., Koike S. and Haykawa H., Phys Rev Lett, 1983, 51, 798-800. Beck W., Bockris J. O’M., McBreen J. and Nanis L., Hydrogen permeation in metals as a function of stress, temperature and dissolved concentration, Proc. Roy. Soc. London, 1966, A290, 220-235.

HYDROGEN EMBRITTLEMENT IN Fe SINGLE CRYSTALS [49]. [50]. [51]. [52].

[53]. [54]. [55]. [56].

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Sofronis P. and McMeeking R. M., Numerical-analysis of hydrogen transport near a blunting crack tip, J. Mech. phys. solids, 1989, 37, 317-350. Chen, X.F., Foecke T., Lii M., Katz Y. and Gerberich W.W., The role of stress state on H cracking in Fe-3%Si [001] single crystals, Eng Frac Mech, 1990, 35, 997-1017. Zhang T.-Y., Shen H. and Hack J. E., The influence of cohesive force on the equilibrium concentration of hydrogen atoms ahead of a crack tip in single crystal iron, Scrpta metall mater, 1992, 27, 1605-1610. Lufrano J; Sofronis P, Numerical analysis of the interaction of solute hydrogen atoms with the stress field of a crack, Int. J. Solids Struct., 1996, 33, 17091723. Toribio J., The role of crack tip strain rate in hydrogen assisted cracking, Corr. Sci., 1997, 39, 1687-1697. Lufrano J; Sofronis P, Enhanced hydrogen concentrations ahead of rounded notches and cracks-competition between plastic strain and hydrostatic stress, Acta mater, 1998, 46, 1519-1526. Sofronis P; Lufrano J., Interaction of local elastoplasticity with hydrogen: embrittlement effects, Mater Sci Eng A, 1999, 260, 41-47. Kiuchi K. and McLellan R. B., The solubility and diffusivity of hydrogen in well-annealed and deformed iron, Acta metall, 1983, 31, 961-984.

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A COMPREHENSIVE MODEL FOR SOLID STATE SINTERING AND ITS APPLICATION TO SILICON CARBIDE H. RIEDEL AND B. BLUG Fraunhofer-Institut für Werkstoffmechanik Wöhlerstr. 11 79108 Freiburg, Germany

Abstract: Previous models for partial aspects of solid state sintering and grain coarsening are combined to give a comprehensive model consisting of a set of equations. A series of sinter forging tests with a SiC powder is carried out, and the model is successfully adjusted to the experimental results. The resulting activation energy for bulk diffusion is substantially higher than activation energies reported in the literature.

1. Introduction There are various possible reasons for applying powder metallurgical techniques to the processing of materials. First, some materials such as ceramics, hard metals, refractory metals and even certain polymers are difficult or impossible to melt and cast, so that there is no practical alternative to the powder route. Second, steel parts are preferentially made from powders, if the part geometry is complex, if high dimensional accuracy is required, if large series are produced and if final machining must be avoided for economical reasons. Accordingly, most sintered steel parts are made for the automotive industry. Finally, powder metallurgy is applied, if a fine and homogeneous microstructure is needed, e.g. in critical and expensive parts for the aerospace industry. Various techniques are used to form a powder compact (the 'green body') with the desired shape. The most frequent shaping technique is probably die compaction, but cold isostatic pressing is also often applied, e.g. for spark plugs or lambda probes. Whiteware articles are shaped in large numbers by slip casting. In nearly all cases, the shaping process, which is carried out at or near room temperature, is followed by sintering at high temperature. In this step the fragile green body is transformed into a strong solid by the formation of necks between the particles. In many materials, such as engineering ceramics and hard metals, the density increases during sintering from the green density, which is typically 55% of the theoretical bulk density in these materials, to 95 to 100% density. Sintered steels, on the other hand, are usually sintered with less than 1% shrinkage, since the density achievable for iron powders by die compaction is already high enough (85 to 95%) to give good mechanical properties. Although strength and ductility could be improved substantially by further densification during sintering, one 49 T.-J. Chuang and J. W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 49–70. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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usually prefers to have no or little shrinkage, since dimensional accuracy of the parts is considered to be more important than optimum strength and ductility. Whenever the material undergoes an appreciable shrinkage during sintering, distortions of the parts can be a serious problem. The warpage results from different causes. Die compaction usually gives an inhomogeneous green density distribution, which leads to differential shrinkage of different volume elements. Intended property variations in gradient materials or in layered electronic circuits usually result in strong warpage, unless the geometrical features and the sintering characteristics are carefully balanced. Large and thin-walled parts undergo distortions due to gravity and to friction on the support plate. Temperature gradients also play a role, expecially in connection with microwave heating, since many materials absorb electromagnetic energy more effectively at higher temperatures, which tends to enhance temperature gradients and hence warpage. Like a few others, e.g. [1-8], the group of the present authors has applied the finite element method to predict the sinter distortions with the aim to minimize them by appropriate process control [9-13]. This paper describes a comprehensive constitutive model for solid state sintering, which summarizes various aspects published previously [14-21]. The model to be described has already been implemented in the FE Code ABAQUS and has been applied to the sintering of molybdenum cylinders [12]. A similar model is used by Kanters et al. [22]. A corresponding liquid phase sintering model and its implementation was published in [11,23]. The model in its present form is based on concepts developed, for example, by Ashby [24,25] and Arzt [26], as far as sintering mechanisms are concerned, and on work of Scherer [27], Abouaf et al. [1], Jagota and Dawson [28], and McMeeking and Kuhn [29], as far as mechanical aspects are concerned. It combines results on second and third stage sintering with models for grain growth in porous solids. The second and third sintering stages are characterized by open and closed porosity, respectively. In both stages, the pore surfaces are equilibrium surfaces, i.e. surfaces with minimum energy or uniform mean curvature [15,16]. In the first stage, the surface of the pore space is not yet in equilibrium, since it is still influenced by the initial shape of the powder particles. Although appropriate models for neck growth in the first stage are available [14], they are not embodied in the present model. Rather a purely phenomenological factor is used to describe particle rearrangement, which is another process in the first sintering stage. Since rearrangement and nonequilibrium neck growth have similar consequences for the constitutive response and since a detailed description of nonequilibrium neck growth would increase the conceptual complexity of the whole model, both processes are jointly described by one phenomenological factor. Grain boundary diffusion is assumed to be the dominant transport mechanism, but bulk diffusion through the grains is also taken into account as a parallel transport path. Surface diffusion acts like a process in series to grain boundary diffusion.

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2. The Model Equations 2.1. THE GENERAL FORM OF THE CONSTITUTIVE EQUATION Since the rates of stress-directed diffusion depend linearly on stress, the macroscopic strain rate tensor must be a linear function of the stress tensor: (1) where the prime denotes the deviator, σ m is the mean (or hydrostatic) stress, ∆p is a gas overpressure which may develop in closed pores, δ ij is the Kronecker symbol, G and K are shear and bulk viscosity, respectively, and σ s is the sintering stress, which arises from the surface tension forces of the pores. The densification rate is given by the trace of the strain rate tensor: (2) where ρ is the relative density. The dependences of G, K and σ s on temperature, density, grain size and possibly on other internal variables will be specified by the detailed model to follow. It should be mentioned that the relation between strain rate and stress may be nonlinear if the pore space does not have an equilibrium surface, i.e. if surface diffusion plays a role. Chuang 3/2 et al. [30,31], for example, obtain σ3 and σ dependences of the cavity growth rate under tensile stresses, when the pore shape deviates significantly from an equilibrium shape. Analogous solutions exist for neck growth during sintering, but for the present purposes these nonlinear relations are not relevant, since they are valid only for stresses several times greater than the sintering stress, while the stresses during sintering are usually smaller than the sintering stress. In this range, the linear dependence, eq. (1), is valid. 2.2. THE GENERAL FORM OF THE VISCOSITIES The viscosities are written in the following form:

(3) (4)

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with k = Boltzmann constant, T = absolute temperature, Ω = atomic (or molecular) volume, R = grain radius, δD b = grain boundary diffusion coefficient times grain boundary thickness; δD b exhibits the usual Arrhenius-type temperature dependence with activation energy Q b and pre-exponential factor δ D b0. Further, k1 and g l are normalized bulk and shear viscosities for open porosity, k2 and g 2 are normalized viscosities for closed porosity, θ gives a smooth transition from open to closed porosity, and U is a factor to describe the effect of grain rearrangement. Expressions for the normalized viscosities, for θ and for U are given in Sections 2.3 to 2.6. The viscosities and the sintering stress are calculated for the equilibrium pore surfaces given in [15]. Figure 1 shows examples of equilibrium surfaces for open porosity. Grain boundary diffusion in the approximately circular grain contact areas is the dominant densification mechanism. At a certain density, neighboring contact areas touch one another pinching off pore channels leading to isolated pores. The relative density at which this transition from open to closed porosity occurs is given by the following relation which approximates the numerical results of [15]: ρc l = 1.05 – 0.115 ψ

(5)

with the dihedral angle defined by

(6) where γ b and γ s are the specific energies of the grain boundary and the surface, respectively.

Figure 1. Equilibrium grain surfaces for open porosity (ψ = dihedral angle, ƒ = porosity)

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2.3. THE CONTRIBUTIONS OF GRAIN BOUNDARY, BULK AND SURFACE DIFFUSION TO THE BULK VISCOSITY Grain boundary diffusion is considered to be the primary transport mechanism for densification. Surface diffusion is needed to spread the material that flows out of a grain boundary over the pore surface. In principle, the assumption of equilibrium pore shapes implies that surface diffussion is infinitely fast. Finite surface diffusivities lead to nonequilibrium pore surfaces. However, the influence of finite, rather than infinite, surface diffusivities can be treated by an approximate method which is based on the assumption of equilibrium pore shapes [16-19]. One calculates the grain boundary and surface diffusion fluxes corresponding to a sequence of equilibrium configurations. Then one equals the dissipation rate associated with these fluxes to the negative rate of Gibbs free energy. In the resulting densification rates grain boundary and surface diffusion act like electric resistors in series. This approximation has been shown to give very accurate results compared to numerical solutions for pore skrinkage in a two-dimensional configuration [19], and there is no reason to assume that it is not applicable to 3D configurations. Bulk diffusion is generally understood as a parallel path to the grain boundary/surface diffusion path [24,25]. According to this understanding of the interaction between grain boundary, surface and bulk diffusion, the normalized bulk viscosities are written in the form (7)

The subscript i denotes open (i = 1) and closed (i = 2) porosity. The subscripts b, s and v denote grain boundary, surface and volume (or bulk) diffusion. The expressions for kib and ki s are taken from [16-18], and the contribution of volume diffusion is treated approximately as proposed by Ashby [24,25].

(8)

(9) (10) with the relative density ρ , the porosity f = 1 – ρ , the abbreviation Φ = 2(A3 + A 4 f) 2, the surface diffusion coefficient, δDs = δ Ds0 exp(-Q s /Rg T) and the bulk diffusion coefficient D v = D v0 exp(-Q v /R gT), where Rg is the gas constant; the functions of the dihedral angle, A0 to A10 , are given in the Appendix. Further

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(11)

(12) where (13) is the area fraction of grain boundaries covered by pores, and

(14)

whith ω = A8 ƒ 2/3. The distinction between ω and ω b is made since during grain growth, pores may detach from migrating grain boundaries (see Section 2.9). The volume fraction of pores that remain on grain boundaries, f b , is given by

(15)

Here β 0 describes the width of the range over which pore detachment occurs ( β 0 = 1.3 is chosen here), and f d is the porosity at which detachment occurs theoretically according to the condition derived in [20,21]: (16) where ωd = A 8 ƒd , and M b is the grain boundary mobility (see Section 2.9). 2/3

2.4. THE SHEAR VISCOSITY For open porosity, shear viscosities were calculated in [16,29] to be: g1 = β l k l

(17)

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An upper bound estimate for the ratio of shear to bulk viscosity is β1 = 0.6 [29] for freely sliding grain boundaries, while a self-consistent estimate is β 1 = 0.27 [16]. In the present paper β 1 is considered as an adjustable parameter,which is found to be 1.08. For closed porosity [17]: (18) with (19)

(20)

(21) The dimensionless factor β 2 should be β 2 = 1 according to the self-consistent estimate given in [17], but it is considered as an adjustable parameter in this paper.

2.5. INTERPOLATION BETWEEN OPEN AND CLOSED POROSITY The transition parameter θ is assumed to vary from 0 to 1 in a density range from ρ lo to ρ hi

(22)

(23) (24) with the relative density at pore closure ρ c l from eq. (5), and the arbitrarily chosen number 0.04 for the width of the transition range.

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2.6. PARTICLE REARRANGEMENT The phenomenological term for grain rearrangement is written in the form

(25)

where ρ 0 is the initial relative density, and the numbers 0.63 and 0.02 are chosen arbitrarily. The idea is that rearrangement can contribute to densification and deformation only in the initial sintering stages. Above a certain density (here 63%, the relative density of a random dense sphere packing) rearrangement can make no further contribution to densification. If the parameter α is zero, the rearrangement term has no influence. In the following a relatively small, fixed value, α = 0.2, is chosen, which means that the rearrangement term is considered to be not very important. 2.7. THE SINTERING STRESS Like the viscosities, the sintering stress is calculated by interpolating between the (numerical) results for open [15] and closed porosity [17] using the transition parameter θ: (26) with (27)

(28)

The functions of the dihedral angle, C 0 to C6, are given in the Appendix. 2.8. GAS PRESSURE After pore closure entrapped gas can no longer escape from the pore space. If the gas cannot diffuse through the solid and if ideal gas behavior is assumed, the overpressure in the pore is

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(29)

where the subscript cl denotes the values of density, temperature an external pressure, P ex, at the time of pore closure. In many cases the effect of gas pressure on the densification rate is negligibly small. In the experiments on SiC described in Section 3, the sintering stress is found to be of the order 5 MPa, whereas the gas overpressure is less than 0.5 MPa at relative densities up to 98%. 2.9. GRAIN COARSENING Sintering is usually accompanied by grain coarsening. The grain growth rate is described by the classical Hillert law with two modifications: (30)

The grain boundary mobility exhibits an Arrhenius-type temperature dependence Mb = M b0 exp (-Q b /RgT). The first modification of Hillert’s law is expressed by the factor F d. It is introduced to account for the fact that the powder usually does not have the steady-state grain size distribution, which is implicit in the Hillert solution. The following form is chosen for Fd : (31) where R 0 is the initial average grain radius and δ can lie between - ∞ and 1. In this paper δ = 0 is assumed, which corresponds to Hillert’s law without a correction for the size distribution. The second modification, expressed by the factor Fp in eq. (29), arises from the drag that pores exert on migrating grain boundaries. The specific form of Fp is taken from [20,21]. For open porosity (ρ < ρ c l ) : (32) For closed porosity (ρ > ρ c l ): (33)

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The D’s are functions of the dihedral angle (see Appendix). The term 1 – D3f 1/2 in eq. (32) represents the area fraction of grain boundary relative to the total grain surface area. Since the pore drag model was not designed to be accurate for large porosities, this term may erroneously become negative. If this happens, 1 – D 3f 1/2 is set to a small positive value. 2.10. SUMMARY OF THE MODEL Equations (1) to (33) define the solid state sintering model. Equations (1) and (30) are the most important ones, since they are the evolution equations for the strain rate and for the grain size. The rest of the equations explains the quantities that appear in the evolution equations. The whole set of equations has been implemented in the FE Code ABAQUS as a user supplied material routine (UMAT). An application to the sintering of Mo cylinders is described in [12]. Also a Fortran program for the solution of eqs. (1) to (33) for prescribed stresses was written. This is used to adjust the model parameters to the experiments on SiC described next.

3. Experimental 3.1. MATERIAL AND SPECIMENS An α-silicon-carbide powder of Elektroschmelzwerk Kempten, Ekasic D, was used for the tests. The powder has a grain size of 0.48 µ m and contains C, B and Al as sintering aids (<1% each). The theoretical density of the fully dense material is 3.19 g/cm3. For the subsequent sinter forging tests cylindrical specimens with 16 mm diameter and 20 mm height were pressed in a die to a relative density, ρ 0 = 52%, as measured after the removal of the organic binder. 3.2. SINTER FORGING TESTS The sinter forging test [32], which is also called load dilatometry, was applied to determine the constitutive response of the SiC powder compacts under sintering conditions. The principle is that during sintering an axial compressive load is applied to the cylindrical specimen and axial and radial strains are measured. A commercial hot press was modified for the tests. The specimens were heated in a CFC resistor furnace from room temperature to 2100°C at a rate of 15 K/min. Then the heating was switched off without a hold time leading to a cooling rate of 46 K/min. The temperature of the specimen was measured with a differential pyrometer, while the furnace temperature was controlled by a W-Re thermocouple placed slightly outside the hot zone to avoid excessive ageing. Load was applied by a hydraulic cylinder and transmitted by graphite punches coated with BN to minimize chemical reactions between the specimen and the punches. The furnace atmosphere was Argon with ambient pressure.

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The axial strain was measured at the load rod outside the furnace using an LVDT. Radial strains were transmitted by a graphite rod to a second LVDT outside the furnace. The dimensions of each specimen were measured after the test, and the LVDT signals were corrected accordingly if necessary. Further the strain signals were corrected for the thermal expansion of the experimental setup measured with a fully dense SiC dummy specimen. Depending on the axial load the specimens either shrink or expand in radial direction during the test. Friction between the specimen and the punches therefore causes the specimens to assume either hour-glass or barrel-type shapes. However, the BN coating apparently acts as a good lubricant, so that the deviations from cylindrical shapes remain small. Therefore stress and strain are evaluated, as if the specimens were cylindrical, but the changing diameter was taken into account. A PC program was written to control all relevant parameters of the tests (temperature, axial load, gas pressure). It also records these parameters and the strains. 3.3. MICROSCOPIC EVALUATION Sintered and partly sintered specimens were cut, lapped, polished and etched or fractured to study the evolution of the grain structure. A C++ program was written to evaluate the grain size distribution function according to the Standard Intercept Method ASTM/E112. Figure 2 shows a sequence of microstructures resulting from tests interrupted at various temperatures.

Figure 2. Microstructure after sintering to (a) 1911 °C (fracture surface), (b) 2100°C (polished and etched section). The grain size distributions obtained from these pictures are consistent with the steady-state distribution function predicted by Hillert [33]. Figure 3 shows an example for a specimen sintered at 2100°C. A log-normal distribution fits the data equally well. During grain growth, the distribution function should evolve in a self-similar manner according to the Hillert theory, and it actually does so to a high degree of accuracy.

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Figure 3. Observed grain size distribution function compared with Hillert’s distribution. Specimen sintered at 2100°C.

4. Experimental Results and Comparison with the Model Fifteen sinter forging tests were carried out under constant axial loads corresponding to initial stresses between 0 and 24 MPa. The sintering model outlined in Section 2 was adjusted to describe the evolution of axial and radial strains and of the grain size. It is recommended to adjust the grain coarsening kinetics first, since it is nearly independent of the densification process, whereas the densification rate depends strongly on the grain size. Figure 4 shows the fit with the final parameter set shown in Table 1 below. Coarsening occurs between 125 and 140 min, which corresponds to temperatures between 1875°C and the peak temperature 2100°C. The activation energy is found to be Q m = 900 kJ/mol, but 800 or 1000 kJ/mol would also give a reasonable fit. The final grain size at the end of the sintering cycle is nearly independent of the applied axial stress, both in the experiments and in the model.

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Figure 4. Evolution of the grain size – experiments and model. The peak temperature, 2100°C, is reached at 140 min. Next the grain boundary and bulk diffusion coefficients are adjusted to describe the densification rate at increasing temperature. It turned out that no reasonable fit of the experimental densification curves could be obtained with the activation energies reported in the literature, e.g. Ashby [25], who gives Q b = 557 kJ/mol and Q V = 696 kJ/mol. Figure 5 shows that with the diffusivities from [25] together with the observed coarsening kinetics the material would reach only 55% relative density as compared to 96% in the experiments. An increase of the pre-exponential factors would lead to a higher final density, but to a completely unsatisfactory time (or temperature) dependence. On the other hand, a very good fit is obtained, if higher activation energies are chosen. Figure 5 illustrates the improvement. The resulting activation energy for bulk diffusion is very much higher than values from the literature (Q V = 1300 kJ/mol vs. 696 to 912 kJ/mol [25,34-37]), but the fit of the data requires such a high value. The effects of surface diffusion and grain boundary diffusion on the densification and strain rates cannot easily be separated. Hence no attempt is made to determine the surface diffusion coefficient from the fit. Rather, the surface diffusion coefficient is set to a large value so that surface diffusion is effectively infinitely fast and therefore has no influence on the densification rate. An equally plausible assumption would be to set δ D s = δ D b which has been applied in other cases. To adjust the value of the dihedral angle one observes that the measured densification and strain rates exhibit a distinct drop at around 90% relative density, which is attributed to the transition from open to closed porosity. This feature is reproduced by the model, if the dihedral angle is set to ψ = 70° according to eq. (5). The value of the surface energy can, in principle, be derived from the stress dependence of the densification rate. As this dependence is small, however, the fit is relatively insensitive to the choice of the surface energies. A plausible value, γ S = 1 J/m² , leads to a stress dependence that is consistent with the experiments.

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Figure 5. Evolution of the density. Model predictions with activation energies from the literature [25] and with the adjusted activation energies 600 and 1300 kJ/mol.

An important parameter to fit the axial and radial strains measured in the sinter forging tests is the ratio of shear-to-bulk-viscosity, G/K, characterized by the parameters β 1 and β 2 . The G/K ratio is given by the ratio betwen volumetric and deviatoric strain rate. To fit the data it was necessary to use 4 times larger G/K ratios than predicted by the models that are based on freely sliding grain boundaries. Figures 6a-d show a comparison of measured and calculated strains. Apparently all sinter forging tests can be described rather accurately by the model with a single set of parameters. Table 1 shows the values of the parameters.

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Figure 6a-c. Radial and axial strains for various axial stresses. Solid lines: measured, dashed lines: predicted values. The peak temperature, 2100°C, is reached at 140 min.

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Figure 6d. Radial and axial strains for axial stress σz = -8.5 MPa. Solid lines: measured, dashed lines: predicted values. The peak temperature, 2100°C, is reached at 140 min. TABLE 1. Parameters of the model Unit Value 0.52 m 2.4 ·10-7 Pa 1.0 · 10 5 m³ 2.07·10- 2 9 m³/s 2.3·10 - 9 kJ/mol 600 Qb m³/s Surface diffusion δ D s0 4.0·10 -2 * kJ/mol 600 Qs m²/s Volume diffusion 1.2·10 15 D v0 kJ/mol 1300 Qv m²/s Grain boundary mobility 3.8·10 5 γ bM b0 /4 kJ/mol 900 Qm J/m² Surface energy 1.0 γs degree Dihedral angle 70 ψ α Exponent in rearrangement term 0.2 δ Initial grain size distribution 1 Pore detachment 1.3 β0 G/K for open porosity 1.08 β1 Multiplier for closed porosity 4 β2 *) The surface diffusion coefficient is not fitted, but set to a high value Meaning Green density Initial grain radius External gas pressure Atomic volume Grain boundary diffusion

Symbol ρ0 R0 pex Ω δ D b0

A MODEL FOR SOLID STATE SINTERING

5.

65

Discussion

In Section 4 it was demonstrated that the proposed sintering model describes the observed densification, deformation and grain coarsening kinetics in remarkable detail. The adjustment of the model to the measured data requires an activation energy for bulk diffusion that is much higher than values reported for SiC in the literature. With the diffusion data given by Ashby [25], for example, the material could never be sintered to a high density, since grain coarsening would prevent densification beyond some 55% in this case. Apparently the sintering aids in the present material (C, B, Al) promote a densification mechanism with a high activation energy. In general, the sintering aids have various effects on densification [38-40]. First they suppress grain coarsening mechanisms that are found to operate in pure SiC at moderate temperatures (e.g. 1200°C), which effectively prevents densification [40]. This kind of coarsening may be caused by the formation of elemental Si or of the volatile oxide Si0. This would allow coarsening either by rapid surface diffusion or by the evaporation/condensation mechanism. Apparently these low-temperature coarsening mechanisms are suppressed in the present material, since coarsening starts only at 1875°C, with the kinetics being consistent with a model based on grain boundary migration. Second the sintering aids in SiC are thought to enhance the diffusivities along grain boundaries and through the grains. According to the present analysis they do so especially at high temperature, since the activation energies required to fit the experimental data are much higher than those obtained from diffusion experiments with pure SiC. The conclusion that the activation energy for bulk diffusion should be high is probably independent of the specific sintering model proposed in this paper. Other models such as that of McMeeking and Kuhn [29] may differ in detail, but lead to similar trends. The activation energy for grain boundary diffusion resulting from the fit lies in the expected range, but the value is not certain, since it is sensitive to the choice of the rearrangement parameters, for which only assumed values could be used. Other choices of these parameters lead to somewhat higher activation energies for grain boundary diffusion.

6.

Conclusions

Previous models on various stages of sintering and grain coarsening were combined to form a comprehensive model of solid state sintering. This model consists of a set of equations which is cast in a Fortran program for the analysis of sinter and sinter forging tests. It is also implemented as a user defined material routine for the finite element code ABAQUS. A series of sinter forging tests (load dilatometry) was carried out with a SiC powder containing C, B and Al as sintering aids. It is possible to adjust the model parameters such that the model describes the densification, the deformation and the grain growth

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kinetics in all tests accurately. The model gives the general trend of the densification with time or temperature, it reflects the distinct drop of the densification rate at the transition from open to closed porosity, and it exhibits the observed stress dependence of the deviatoric strain rate. In this sense the model, especially in conjunction with the finite element method, is well suited to describe technological sintering processes. Admittedly, the model contains a number of adjustable parameters (primarily the diffusion coefficients) which are not measured independently. Hence it cannot be concluded definitively that the good agreement with the experiments supports all details of the model in a physical sense. The adjustment of the model to the experiments unequivocally requires an unusually high activation energy for bulk diffusion. The difference is tentatively explained by the effect of the sintering aids on the diffusion mechanisms.

References 1.

Abouaf, M., Chenot, J.L., Raisson, G., and Baudin, P.: Finite element simulation of hot isostatic pressing of metal powders, Int. J. Numer. Methods Engng. 25 (1988) 191-212. 2. Chenot, J.L., in: Numerical Methods in Industrial Forming Processes, Numiform 89, Thompson, E.G., Wood, R.D., Zienkiewicz, O.C., and Samuelson, H., A.A. Balkema, Rotterdam, 1989, p. 1. 3. Jagota, A., Mikeska, K.R., and Bordia, R.K.: Isotropic constitutive model for sintering particle packings, Am. Ceram. Soc. 73 (1990) 2266-2273. 4. Mori, K., Finite element simulation of nonuniform shrinkage in sintering of ceramic powder compact, in: Numerical Methods in Industrial Forming Processes, Numiform 92, Chenot, J.L., Wood, R.D., Zienkiewicz, O.C., A.A. Balkema, Rotterdam, 1992, pp. 69-78. 5. Cocks, A.C.F., and Du, Z.Z.: Pressureless sintering and hiping of inhomogeneous ceramic compacts, Acta metall. mater. 41 (1993) 2113-2126. 6. Gillia, O., Bouvard, D., Doremus, P., and Imbault D.: Numerical simulation of compaction and sintering of cemented carbide, in: European Conference on Advances in Hard Materials Production, 1996, pp. 61-68. 7. Shinagawa, K.: Finite element analysis of microscopic material behavior in sintering and prediction of macroscopic shape change in sintered bodies, in: Proceedings of rd 3 Asia-Pacific Symposium AEPA, 1996, pp. 439-444. 8. Olevsky, E.A.:Theory of sintering: From discrete to continuum, Mater. Sci. Engng. R23 (1998) 41-100. 9 . Riedel, H., and Sun, D.-Z.: Simulation of die pressing and sintering of powder metals, hard metals and ceramics, in: Numerical Methods in Industrial Forming Processes, Numiform 92, Chenot, J.L., Wood, R.D., and Zienkiewicz, O.C., Eds., A.A. Balkema, Rotterdam, 1992, pp. 883-886. 10. Sun, D.-Z., and Riedel, H.: Prediction of shape distortions of hard metal parts by numerical simulation of pressing and sintering, Simulation of Materials Processing: Theory, Methods and Applications, Numiform 95, Shen, S.-F. and Dawson, P.R., Balkema, Rotterdam, 1995, pp. 881-886.

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11. McHugh, P.E., and Riedel, H.: A liquid phase sintering model – application to Si3N4 and WC-Co, Acta metall. mater. 45 (1997) 2995-3003. 12. Plankensteiner, A.F., Parteder, E., Riedel, H., and Sun, D.-Z.: Micromechanism based finite element analysis of the sintering behavior of refractory metal parts using ABAQUS, in: ABAQUS User World Congress, Chester, 1999, pp. 643-657. 13. Kraft, T., Riedel, H., Stingl, P., and Wittig, F.: Finite element simulation of die pressing and sintering, Adv. Engng. Mater. 1(1999) 107-109. 14. Svoboda, J., and Riedel, H.: New solutions describing the formation of interparticle necks in solid-state sintering, Acta metall. mater. 43 (1995) 1-10. 15. Svoboda, J., Riedel, H., and Zipse H.: Equilibrium pore surfaces, sintering stresses and constitutive equations for the intermediate and late stages of sintering – Part I: Computation of equilibrium surfaces, Acta metall. mater. 42 (1994) 435-443. 16. Riedel, H., Zipse, H., and Svobada, J.: Equilibrium pore surfaces, sintering stresses and constitutive equations for the intermediate and late stages of sintering – Part II: Diffusional densification and creep, Acta metall. mater. 42 (1994) 445-452. 17. Riedel, H., Kozák V., and Svoboda, J.: Densification and creep in the final stage of sintering, Acta metall. mater. 42 (1994) 3093-3103. 18. Riedel, H., Svoboda, J., and Zipse, H.: Numerical simulation of die pressing and sintering – Development of constitutive equations, in: Powder Metallurgy World Congress PM94, D. Francois, Ed., Les Editions de Physique Les Ulis, Paris, 1994, pp. 663-671. 19. Svoboda, J., and Riedel, H.: Quasi-equilibrium sintering for coupled grain boundary and surface diffusion, Acta metall. mater. 43 (1995) 499-506. 20. Svoboda, J., and Riedel, H.: Pore-boundary interactions and evolution equations for the porosity and the grain size during sintering, Acta metall. mater. 40 (1992) 28292840. 21. Riedel, H., and Svoboda, J.: A theoretical study of grain coarsening in porous solids, Acta metall. mater. 41 (1993) 1929-1936. 22. Kanters, J., Eisele, U., and Rödel, J.: Scale dependent sintering trajectories, Acta mater. to be published (2000). 23. Svoboda, J., Riedel, H., and Gaebel, R.: A model for liquid phase sintering, Acta. metall. mater. 44 (1996) 3215-3226. 24. Ashby, M.F.: A first report on sintering diagrams, Acta metall. 22 (1974) 275-289. 25. Ashby, M.F.: HIP 6.0 Background reading, University of Cambridge (1990). 26. Arzt, E.: The influence of an increasing particle coordination on the densification of spherical powders, Acta metall. 30 (1982) 1883-1890. 27. Scherer, G.W.: Sintering inhomogeneous glasses: Application to optical waveguides, J. Non-Crystalline Solids 34 (1979) 239-256. 28. Jagota, A., and Dawson, P.R.: Micromechanical modeling of powder compacts – Unit problems for sintering and traction induced deformation, Acta metall. 36 (1988) 2551-2561 and 2563-2573. 29. McMeeking, R.M., and Kuhn, L.T.: A diffusional creep law for powder compacts, Acta metall. mater. 40 (1992) 961-969. 30. Chuang, T.-J., and Rice, J.R.: The shape of intergranular creep cracks growing by surface diffusion, Acta metall. 21 (1973) 1625-1628.

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31. Chuang, T.-J., Kagawa, K.I., Rice, J.R., and Sills, L.B.: Non-equilibrium models for diffusive cavitation of grain interfaces, Acta metall. 27 (1979) 265-284. 32. H. Zipse and H. Riedel, The mechanical behavior of sintering powder compacts, in: Ceramic Transactions, Vol. 51, Processing and Technology, H. Hausner, G.L. Messing and S.-i. Hirano, Eds., American Ceramic Society, Westerville, OH, 1995, pp. 489-493. 33. Hillert, M.: On the theory of normal and abnormal grain growth, Acta metall. 13 (1965) 227-237. 34. Hon, M.H., and Davis, R.F.: Self-diffusion of 14 C in polycrystalline β -SiC, J. Mater. Sci. 14 (1979) 2411-02421. 35. Hon, M.H., and Davis, R.F.: Self-diffusion of 30 Si in polycrystalline β -SiC, J. Mater. Sci. (1980) 2073-2080. 36. Hong, J.D., and Davis, R.F.: Self-diffusion of carbon-14 in high-purity and N-doped α-SiC single crystals, J. Am. Ceram. Soc. 63 (1980) 546-551. 37. Hong, J.D., and Davis, R.F.: Self-diffusion of silicon-30 in α -SiC single crystals, J. Mater. Sci. 16 (1981) 2485-2494. 38. Prochazka, S.: The role of B and C in the sintering of SiC, in: Special Ceramics, No. 6, P.Popper, Ed., British Ceramic Research Association, Manchester, 1975, pp. 171182. 39. van Rijswijk, W., and Shanefield, D.J.: Effects of carbon as a sintering aid in silicon carbide, J. Am. Ceram. Soc. 73 (1990) 148-149. 40. Greskovich, C., and Rosolowski, J.H.: Sintering of covalent solids, J. Am. Ceram. Soc. 59 (1976) 336-343.

A MODEL FOR SOLID STATE SINTERING

Appendix: Functions of the dihedral angle used in the sintering model In the following equations ψ is measured in radian. A 0 = 0.014573+0.0063822 ψ + 0.0009983 ψ ² A 1 = -0.092348 - 0.028098 ψ + 0.016495 ψ ² A 2 = 0.16242 - 0.0062352 ψ - 0.022826 ψ ² A 3 = 0.5998 + 0.00533 ψ A 4 = -1.271 + 0.4144 ψ

A 9 = A 0 + 0.32 A1 + 0.1024 A2 A 10 = (A l + 0.64 A2 )/A9 C 0 = - 4.069 + 6.557 ψ + 0.0253 ψ ² C1 = 26.75 - 42.58 ψ + 5.986 ψ ² C 2 = -51.01 + 82.12 ψ - 18.56 ψ ² C 3 = 3/2 (2 - 3 cos ψ + cos³ ψ )

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C4 = 3( ψ - π /6) - 2 √3 cos ψ sin ( ψ - π/6) C5 = C 0 + 0.32 C1 + 0.1024 C2 C6 = (C 1 + 0.64 C2) / C 5

MAPPING THE ELASTIC-PLASTIC CONTACT AND ADHESION

SINISA DJ. MESAROVIC Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22903 U. S. A.

Abstract: This paper is a review of current understanding of the behavior of elasticplastic spheres in contact – with or without adhesion. The emphasis is on recent computational advances. The results are presented in the form of behavior maps, which provide qualitative and quantitative description of the behavior, for a range of parameters. The gaps in understanding of contact and adhesion are identified.

1.

Introduction

The problems of elastic-plastic contact and adhesive pull-off between two smooth non-conforming bodies have a wide range of applications, which span several orders of magnitude of characteristic lengths. Indentation tests have been used from the th beginning of the 20 century to provide a quick measure of plastic properties of metals, but the current standards are based on empirical correlations. Friction and wear of solids depend critically on the details of the contacts between the asperities including the adhesive pull-off. Powder compaction processes rely upon the mutual plastic indentation of deformable particles. Response of an Atomic Force Microscope is intrinsically tied to the cohesive forces between the tip and substrate. Elastic-plastic contact alone is a complex process. Yet, such process is often only an element of a more complex processes, such as friction and wear or powder compaction. Therefore, there is a need for simple but comprehensive maps of contact behavior, which would provide guidance and criteria for application of contact models. The efforts in this direction until mid-eighties are summarized in the Johnson’s (1985) monograph. Recently, advances in analytical and computational modeling have been made. The progress in understanding the different regimes of elastic-plastic contact has been made possible by the advance of similarity solutions by Hill et al. (1989), Biwa and Storåkers (1995) and Storåkers et al. (1997). They showed that, under a restrictive set of assumptions, there exists a self-similar solution to the problem of contact between rigid-plastic solids with smooth non-conforming surfaces. This solution provides a second paradigm for the contact behavior (the first being the Hertz solution for elastic contact). This enables us to construct maps of contact behavior in 71 T.-J. Chuang and J. W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 71–85.  2000 Kluwer Academic Publishers. Printed in the Netherlands.

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the parameter space which consists of the regions dominated by the two analytical solutions and the regions where the analytical solution does not exist but a quantitative analysis can still be performed by interpolation of extrapolation. Using the finite element simulations Mesarovic and Fleck (1999, 2000) (hereafter referred to as I and II) defined the regime where the similarity solution is valid. They built upon the pioneering work of Johnson (1970) and refined and extended his master curves for correlation of indentation tests to the regimes where contact radius becomes large compared to the effective radius of curvature as well as to the contacts between dissimilar elastic-plastic spheres. Decohesion of a contact is a problem closely related to the one of crack growth. This has been first exploited by Johnson et al. (1971) who used the singular crack tip solution to model the decohesion process following elastic contact. Maugis (1992) observed that a relatively weak cohesion requires more detailed modeling with a cohesive zone model analogous to the Dugdale-Barenblatt crack tip model. Greenwood (1997), Barthel (1998) and Greenwood and Johnson (1998) extended Maugis' approach to more complex interface cohesion potentials. The advances in understanding the elastic-plastic and plastic fracture mechanisms (Rice, 1968, Hutchinson, 1979, Hutchinson, 1983) have not been exploited, in the context of contact behavior, until recently. Mesarovic and Johnson (2000) (hereafter referred to as III) the Maugis' cohesive zone and the results (I) and (II), to provide a model for decohesion after elastic-plastic indentation in the range of parameters where contact unloading/decohesion is predominantly elastic. Then, they used fracture mechanics concepts to generate a semi-phenomenological decohesion mechanism map. This paper represents an effort to provide a concise but comprehensive summary of the recent results (I, II and III) and the contact behavior maps that have emerged from these results, but it also provides some new insights and highlights the gaps in current understanding. Hereafter, contact between spheres is considered. This generalizes to contact between smooth, locally axisymmetric, non-conforming surfaces, such that, the variation in surface curvatures within a region of intensive deformation is sufficiently small.

2.

Analytic methods and scaling

2.1. INDENTATION The configuration considered is shown in Figure 1. Spheres of radii R l and R 2 are pressed together with the normal force F, so that the contact radius is a and the total overlap h. In the Hertz elastic solution for the frictionless contact (e.g., Johnson, 1985, Hills et al., 1993), only one out of four elastic constants of the two spheres, E , E , v and v , enters the solution. The equivalent modulus E* is given by l

2

1

2

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ELASTIC-PLASTIC CONTACT AND ADHESION (2.1) The equivalent radius R o suffices to describe a given geometry: 1 / Ro ≡ 1 / R 1 +1 / R2 .

(2.2) If the following assumptions hold: (i) The contact radius is sufficiently small compared to the radius of each sphere, so that each sphere can be treated as a semi-infinite half space, and, Strains and deformations are small, so that the spherical profile of the bodies (ii) in contact can be approximated by a paraboloid of revolution, Then, the overlap h, the contact radius a and the force F are related by (2.3) The solution for fully sticking elastic contact is given by Mossakovski (1963) and Spence (1968).

Fig. 1 Geometry of contact between two spheres with radii R1 and R 2 . Contact radius a corresponds to the overlap (of undeformed spheres) h.

The second "pinning point" for the contact maps is the similarity solution for the contact between two rigid-plastic spheres. The two spheres are composed of rigid – plastic, power law solids in accordance with J2 flow theory. In uniaxial tension, the stress σ is related to the strain ε according to σ = σ i ε l / m ; i=1,2 (2.4) where sphere 1 has a reference strength σ1 and sphere 2 has a reference strength σ 2 . Both solids have the same strain-hardening exponent (1 ≤ m ≤ ∞ ) . With the assumptions (i) and (ii), the indentation solution has the property of selfsimilarity, i.e., the geometry, stress and strain fields at any stage of indentation can be expressed in terms of an invariant solution (Storåkers et al., 1997). Moreover, the solution to the problem of contact between spherical bodies is a generalization of the solution for the contact between a rigid sphere and a semi-infinite solid, and is obtained

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from the latter by appropriate scaling. The scaling law is a generalization of the Hertzian scaling. An equivalent radius, and an equivalent strength σe describes the combined strengths of the two spheres, (2.5) From (2.4) and the elastic moduli, and depending on the details of the elasticplastic coupling (I), one can define the yield stresses (elastic limits) of the two solids in contact, σ y1 and σ y 2 , and the equivalent yield stress σo : (2.6) The average pressure is related to the contact radius a by the power law relation (2.7) and the contact area is proportional to the indentation depth, (2.8) where the constants c 2 ( m ) and k (m ) depend on m, but are independent of indentation depth, and of the diameters and strengths of the bodies in contact. They are tabulated Biwa and Storåkers (1995) for both frictionless and sticking contact. The relations (2.7) and (2.8) imply that the indentation force depends upon the indentation depth h according to (2.9) The similarity solution is expected to be accurate when the elastic deformation becomes negligible (compared to the plastic deformation) in the region around and under the contact. 2.2. ADHESIVE PULL-OFF Continuum mechanics analysis of adhesive contact between perfectly elastic spheres is well understood. The main results can be expressed in terms of two nondimensional parameters µ and (Johnson and Greenwood, 1997): (2.10) where E * and R o are defined in (2.1-2), w is the work of adhesion, and z o is the equilibrium spacing between the surfaces in the Lennard-Jones potential and P is the compressive contact force. The parameter µ is a measure of the elastic displacement due to adhesive forces, normalized by their range of action.

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Adhesion seems to add complexity to the already complex problem of elasticplastic contact. However, if one assumes that the unloading/decohesion process is predominantly elastic, an analytic solution becomes readily available. Given the contact pressure distribution from the indentation problem with the contact radius a0 , and a "fracture criterion", one can solve for the total contact pressure distribution for any contact radius a ≤ a 0 . The mathematical framework for such analysis is the rigid punch decomposition (Hill and Storåkers, 1990). Any axisymmetric pressure distribution p (r , a ) , over the circle of radius a, can be represented as superposition of pressures arising from elementary rigid punches ƒ ( ξ ) d ξ , 0 ≤ ξ ≤ a . Then, the pressure p (r, a) and the distribution of rigid punches ƒ ( ξ) are related by Abel’s integral equation and its solution: (2.11) Displacements are readily calculated from the rigid punch solution.

Fig. 2 (a) Geometry and pressure distribution of the adhesive contact. (b) Model for cohesion.

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The fracture criterion can be based on the work of adhesion only (for the singular crack tip model) or on both, work of adhesion and opening displacement (for the cohesive zone model, Figure 2). In the cohesive zone model the surfaces are assumed to separate slightly in the annulus (a < r
where

is the height of the crown arising from the elastically

unloaded contact.

3.

Elastic – plastic indentation

From the analytic solutions in Sec. 2.1, it would appear that, the frictionless contact problem in Fig. 1, depends on no more than two non-dimensional parameters: σ e / E * (or σ o / E * ) and m. The indentation force (or average pressure) and the indentation depth can then be computed as functions of the normalized contact radius a / R o . For elastic – ideally plastic solids ( m → ∞ ) only one non-dimensional parameter remains. Indeed, this is confirmed by numerical studies [(I) and (II)] for the range of parameters where the assumptions (i) and (ii) (Sec. 2.1) hold [but see (I) for the subtle effect of the Poisson’s ratii and the effects of the exact coupling between elastic and plastic deformation]. However, when the "small strain" assumption ceases to hold, i.e., when large sliding and rotations at the edge of the contact are present, at least two new independent parameters are needed. The choice is not obvious, but the simplest combinations seem to be to take the ratii of the two strengths and the two radii as additional parameters. Let σ y1 be the lower of the two yield strengths in (2.6). Tentatively, we define the parameters as: (3.1) (3.2)

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The numerical analysis has been reported in (I) and (II) for the combinations γ = 1 and 0, i.e. the spheres of equal strength and a deformable sphere against a rigid sphere, and β = 0, 1 and ∞. The cases β =0 and β = ∞ should be recognized as limiting cases of contact between a very large sphere and a very small one. 3.1. FRICTIONLESS CONTACT, ELASTIC – IDEALLY PLASTIC SOLIDS First, consider the results for the contact between a rigid and a deformable sphere ( γ = 0 ). The master curves of normalized average pressure

and the

geometric ratio a²/(2hR o) are shown in Figure 3. For small contact radii, both are functions of a single non-dimensional parameter

, where σ o = σ y1 . T h i s

part of the pressure master curve is the generalization of the Johnson's (1970, 1985) analysis of the Brinell indentation (β = ∞ ) . After the initial yield at

and a ²/2hR increase with increasing contact

size. The regime of constant average pressure begins at aE * / R σ o ≈ 50, but the full similarity solution (1) is not achieved until aE * / Rσ o ≈ 800 , when a ² / 2hR becomes constant (= c² ). For large contact radii, both and a² /2hR depend on a / R o and β (Fig. 3). In all of the cases shown, the onset of the finite deformation regime is given by the failure of the constant pressure condition (2.7). The finite deformation regime is characterized by The softening is caused by excessive rotations and sliding at the edge of the contact and is closely associated with the formation of the pile-up (Norbury and Samuel, 1928). This finite rotation/sliding effect is modulated by the effect of the relative size of the spheres in contact, so that the onset of the finite deformation regime depends on β . Consider now the contact between the spheres of equal strength ( γ = 1). Now the case β = 0 is equivalent to the case β = ∞ . The case of identical spheres ( γ = 1, β = 1) is identical boundary value problem to the one of deformable sphere against a rigid flat ( γ = 0, β = 0 ) considered earlier and th e master curve, such as the one in Fig. 3, is obtained by re-scaling the results for ( γ = 0, β = 0) to account for the definitions of R o (2.2) and h (Fig.1). The case of contact between a sphere and a half-space, both with the same properties ( γ = 0, β = 1 ), is of particular interest. This has been discussed earlier by Gampala et al. (1994). The numerical results (II) show that the smaller sphere takes almost all the deformation in the finite deformation regime (Figure 4). This will be discussed in Sec. 3.3.

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Fig. 3 Master curves for contact between rigid and elastic – ideally plastic spheres (γ = 0 ). The average pressure

(2.7) and the geometric ratio a² / 2hR o (2.8) are

functions of / R o σo for small a, and of a / R o and β for large a. Note that vertical scales for the two variables are different. aE *

Fig. 4 Initial and deformed configuration for the contact of a sphere and a half-space of identical material. (a) Elastic – ideally plastic solids, (b) Elastic-hardening solids, m=3. The above results can be represented in the form of contact maps. One such map for the case of the rigid ball indenting a half-space ( γ = 0, β = ∞ ) is shown in Figure 5. The map covers all the regimes mentioned above: the elastic regime, the elastic-

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plastic regime, the similarity regime and the finite deformation regime. It is emphasized that only the materials with a very low yield strain can experience the similarity regime. For most materials, a direct transition from the elastic-plastic to the finite deformation regime is expected. Contact maps for other cases, i.e. other values of β and γ , are given in (II). They differ from the one in Fig. 5 only by the positions of lines AC and CD.

Fig.5 Map of frictionless indentation of an elastic – ideally plastic solid, showing regimes of deformation mechanisms. Contours of average pressure (full lines) and the geometric ratio a ² /2hR o (dashed lines) are included. The map is based on the finite element results for E * / σ o = 3, 10, 30, 100, 250, 500, 1,000 and 10,000.

3.2. FRICTIONLESS CONTACT, ELASTIC - HARDENING SOLIDS To generalize this approach to include strain hardening (finite m ), define the reference stress

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(3.3) The similarity solution (2.7) now predicts that the reduced average pressure, (3.4) is constant. Indentation maps, similar to the one shown in Fig. 5, can now be constructed for strain hardening solids. As the hardening becomes stronger ( m smaller), the deformation pattern at the edge of the indenter transits from the pile-up to the sink-in mode. The pile-up vanishes around m =5. At about the same value of m, the onset of finite deformation regime becomes dependent on the variation of a² / 2hR with a / R

o

. For m =3, the reduced average pressure

o

remains constant up to a very

large values of a/ R o , while a ² / 2hR o begins to decrease at a /R o ≈ 0 . 3 . T h e indentation map for m=3, will then be similar to the one shown in Fig. 5, but with the line CD shifted to the right. The transition from the elastic-plastic to the similarity regime (line AB) will also have shifted to the right. This is expected since for a given contact radius, plastic deformation is more diffuse in a strain hardening materials then in the elastic – ideally plastic material, and the rigid-plastic similarity solution is applicable if elastic deformation is negligible in the region around and under the contact. The details are given in (I) and (II). 3.3. STRUCTURAL SOFTNESS, FRICTION, PRE-STRESS AND CONTACT STRESS DISTRIBUTIONS In the finite deformation regime, the smaller of the two contacting spheres of identical materials bears most of the deformation (Fig. 4). This observation is explained by the earlier (in terms of increasing a) decrease in the average pressure that the smaller sphere can sustain (cf. the pressure master curves in Fig.3 for different values of β ). While both spheres deform during the elastic, elastic-plastic and similarity regimes, the smaller sphere will enter the softening regime early and the larger sphere will then undergo elastic-plastic unloading. The effect is somewhat less pronounced if there is a strong strain hardening present (Fig. 4b). This structural softness is yet to be quantified for a full range of material parameters and relative sizes of the spheres in contact. The effects of contact friction have been discussed in (I) but only for the case of the rigid ball indenting a half-space (γ = 0 ,β = ∞ ). In this case, the average pressure remains constant up to a /R o = 0.7 (the end of simulation). This single result gives some insight into the qualitative effects of the contact friction, but much remains to be done. As an illustration of variability of the friction effect, consider the following observations. Fist, due to the symmetry, there is no friction effect for the contact between two identical spheres, whatever the friction law. Thus, the effects of friction will vary with β and γ . Second, Johnson’s (1968) experiment on copper spheres, indicate that only a part of the contact is sticking, while the rest is slipping, so that the fully sticking contact (I) may not be sufficiently accurate model for low friction.

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Pre-existing stresses in the solids affect contact behavior only in the elastic plastic regime (I). In the elastic – perfectly plastic case, the pressure is almost perfectly uniform throughout the contact area: (3.5) po = H , H ≤ 3 σ y , for a wide range of parameters, throughout both the similarity and finite deformation regimes (III). Experimental measurements by Johnson (1968) on compressed spheres of hard drawn copper confirm that the pressure distribution for a near – perfectly plastic material is very nearly uniform. The strain hardening material gives a pressure distribution, which rises towards the edge of the contact (III). This finding is supported by the experiments of Timothy et al., (1987) on mildly hardening lead spheres pressed onto a pressure sensitive film, as well as with the computational results of Ogbonna et al., (1995).

4.

Adhesive pull-off

Assuming that the unloading/decohesion after elastic-plastic contact is predominantly elastic, and using the rigid punch decomposition from Sec. 2.2, the problem of unloading/decohesion can be solved, provided that the initial (i.e., indentation) pressure distribution is known. So far only the case of uniform pressure distribution, resulting from the contact of elastic – ideally plastic solids, has been solved (III). As in the case of elastic-adhesive contact (Johnson et al. 1971), the main feature of the solution is the unstable pull-off; the pulling force (as a function of contact radius) has a maximum at a finite value of contact radius a . Two models adopted from the fracture mechanics analyses are used: the singular model, characterized by a critical stress intensity factor (or critical energy release rate), and the cohesive zone model, characterized by the energy of adhesion and the range of adhesive forces (see Fig. 2). The singular solution is a mathematical asymptote to the cohesive zone solution, when the cohesive zone (c – a ) becomes small compared to the contact size a. For range of relevant values of χ and S (2.13), the critical pull-off forces computed from the two models differ by 15% or less. The solutions given in (III) are valid only for the limited range of values of χ and S where the unloading/decohesion is predominantly elastic. Outside of this region, the pull-off behavior is governed by different physical mechanisms. These are analyzed in semi-phenomenological manner in (III). The results are summarized in Figure 6 where the parameter space ( χ, S ) is divided into regions where decohesion is governed by different physical processes. For S<1 any plastic flow on unloading will be small, contained, and unlikely to have any significant effect on the elastic unloading profile. If t o > p o ( S > 1), the yielding is widespread and the unloading/decohesion process must be analyzed as elastic-plastic (top center in Figure 6). The decohesion/fracture mechanism in this region cannot be determined a priori; both cleavage and void growth

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and coalescence are possible. However, if χ exceeds the limiting value of 2, the plastic zone will extend across the whole area of radius ao , such that separation takes place by bulk plastic flow, even to the extent of forming a neck. The pull-off force is then given 2 by Pc = π a H , i.e. the same as the force with which the spheres were pushed together. Johnson (1976) suggested a similar approximate condition for ductile separation. We note that low hardness and small contact size promote ductile separation. Bowden and Tabor (1950) reproduced this behavior on the macro scale by pressing a 3.2 mm diameter clean steel ball into a block of indium (H = 10 MPa). A ductile neck is generally observed in molecular dynamics calculations of plastic contact followed by separation, where the contact radius is of order 2 nm, e.g. Landman et al. (1992). Precise measurements of such behavior in an Atomic Force Microscope, with a gold tip on a gold surface, have been made by Agrait et al. (1996), where ao varied between 1 to 3 nm. We note that, for predominantly elastic decohesion, the work of adhesion w varies – both, in its magnitude and in its physical meaning – as one moves from the line χ = S ( S + 1) toward the upper left corner in Fig. 6. Along the line χ = S ( S + 1) the adhesion energy is approximately equal to the effective surface energy ∆γ , while in the domain of the singular solution the work of adhesion is taken to be the fracture toughness G IC for the given interface, whereby most of it is dissipated through the plastic work. To the right of the line χ = S ( S + 1) , the separation takes place with the whole initial contact subjected to traction t o until the gap at the edge of the initial contact reaches the value δ o , when the interface fails.

5 . Summary and suggestions for future research The frictionless elastic-plastic contact of smooth, locally axisymmetric, nonconforming solids has been modeled numerically for a wide range of parameters. The contact maps have been developed which determine the regions of parameter space where the contact deformation is governed by different deformation regimes: elastic, elastic-plastic, similarity, and finite deformation regimes. The finite deformation regime is of primary interest since it covers a largest part of the parameter space and since analytic solutions are available for the elastic and the similarity regimes. Four non-dimensional parameters, characterizing materials and geometry enter the solution: * σ e / E (or σ o / E ), m , β and γ. The results are quantitative, in that the relevant quantities such as force, pressure, contact radius can be obtained from the contact maps or master curves. The outstanding issue is the concept of structural softness, which, while qualitatively understood, requires quantification, particularly for strain hardening-materials. Contact friction is also understood only qualitatively. It diminishes the pile-up at the edge of the contact and it extends the constant average pressure regime. Thus, it may be conjectured that the friction effect diminishes with increasing strain-hardening *

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rate. The details, such as the extent of slipping versus sticking portions of the contact and tangential stress distributions are still unknown.

Fig. 6

Map of decohesion regimes in (χ , S ) space. v = 1/3, p o / σ y = 3 .

The problem of adhesive pull-off following elastic-plastic indentation has been solved only for the case of predominantly elastic unloading/decohesion and only for the elastic – ideally plastic solids. The extension to strain hardening solids is simple; the methodology is the same as in (III). While the phenomenological behavior map has been produced, the quantitative analysis of non-elastic unloading/decohesion has not been done. This will likely require extensive numerical effort.

S. MESAROVIC

84 6.

References

Agrait, N., Rubio, G. and Vieira, S. (1996) Plastic deformation in nanometre scale contacts. Langmuir 12, pp. 4505-4509. Barthel, E. (1998) On the description of the adhesive contact of spheres with arbitrary interaction potentials. J. Colloid Interface Sci. 200, pp. 7-18. Biwa, S. & Storåkers, B. 1995 An analysis of fully plastic Brinell indentation. J. Mech. Phys. Solids 43, pp. 1303-1334. Bowden, F. P. and Tabor, D. (1950) Friction and lubrication of solids, p.309. Clarendon. Gampala, R., Elzey, D.M. and Wadley, H.N.G. (1994) Plastic deformation of asperities during consolidation of plasma sprayed metal matrix composite monotape. Acta Metall. 42, pp. 3209-3221. Greenwood, J. A (1997) Adhesion of elastic spheres. Proc. Roy. Soc. Lond. A453, pp. 1277-1297. Greenwood, J. A. and Johnson, K. L. (1998) An alternative to the Maugis model of adhesion between elastic spheres. J. Phys. D: Appl. Phys. 31, pp. 3279-3290. Hill, R. and Storåkers, B. (1990) A concise treatment of axisymmetric indentation in elasticity. In Elasticity: Mathematical methods and applications, p. 199-210. Eason, G. and Ogden, R.W., eds. Ellis Horwood, Chichester, UK. Hill, R., Storåkers, B. and Zdunek, A. B. 1989 A theoretical study of the Brinell hardness test. Proc. Royal Soc. London A436, pp. 301-330. Hills, D. A., Nowell, D. and Sackfield, A. (1993) Mechanics of Elastic Contact. Butterworth-Heinemann, Oxford. Hutchinson, J. W. (1979) A course on nonlinear fracture mechanics. Technical University of Denmark. Department of Solid Mechanics. Hutchinson, J. W. (1983) Fundamentals of the phenomenological theory of nonlinear fracture mechanics. J. Appl. Mechanics 105, pp. 1042-1051 . Johnson, K. L. (1968) An experimental determination of the contact stresses between plastically deformed cylinders and spheres. In Engineering Plasticity, pp. 341 - 461. Heyman, J. and Leckie, F. A., eds. Cambridge University Press. Johnson, K.L. (1970) The correlation of indentation experiments. J. Mech. Phys. Solids 18, pp. 115-126. Johnson, K. L. (1976) Adhesion at the contact of solids. In Theoretical and Applied Mechanics, Proc. 4th IUTAM Congress, Ed. Koiter, p. 133. North Holland, Amsterdam. Johnson, K. L. (1985) Contact Mechanics. Cambridge University Press. Johnson, K. L. and Greenwood, J. A (1997) An adhesion map for the contact of elastic spheres.

J. Coll.

Interface Sci. 192, pp. 326-333. Johnson, K. L., Kendall, K. and Roberts, A. D. (1971) Surface energy and contact of elastic solids. Proc. Roy. Soc. London A 324, pp. 301-313. Landman, U., Luedtke, W. D. and Ringer, E. M. (1992) Molecular dynamics simulations of adhesive contact formation and friction. In Fundamentals of Friction, Eds. Singer and Pollock. Proc. NATO Ser. E, 220, Kluwer, pp.463-510. Maugis, D. (1992) Adhesion of spheres: the JKR-DMT transition using a Dugdale model. J. Colloid Interface Sci. 150, pp. 243-269. Mesarovic, S. Dj. and Fleck, N. A. (1999) Spherical indentation of elastic-plastic solids. Proc. Roy. Soc. Lond. A 455, pp. 2707-2728.

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Mesarovic, S. Dj. and Fleck, N. A. (2000) Frictionless indentation of dissimilar elastic-plastic spheres. Int. J. Solids Structures . In press. Mesarovic, S. Dj. and Johnson, K. L. (2000) Adhesive contact of elastic-plastic spheres.

J. Mech. Phys.

Solids . In press. Mossakovski, V.I. 1963 Compression of elastic bodies under conditions of adhesion (axisymmetric case). PMM 27 (3), pp. 418-427. Norbury, A. L. & Samuel, T. 1928 The recovery and sinking-in or piling-up of material in the Brinell test, and the effects of these factors on the correlation of the Brinell with certain other hardness tests. J. Iron Steel Institute 117(1), pp. 673-687. Ogbonna, N., Fleck, N.A. and Cocks, A. C. F. (1995) Transient creep analysis of ball indentation. Int. J. Mech. Sci. 37 (11), pp. 1179- 1202. Rice J. R. (1968) Mathematical analysis in the mechanics of fracture. In Fracture Vol. II, Ed. H. Liebowitz. Academic Press, New York. Spence, D. A. 1968 Self similar solutions to adhesive contact problems with incremental loading. Proc. Roy. Soc. A 305, pp. 55-80. Storåkers, B., Biwa, S., and Larsson, P.-L. (1997) Similarity analysis of inelastic contact.

Int. J. Solids

Structures, 34(24), 3061-3083. Timothy, S. P., Pearson, J. M., and Hutchings, I. M. (1987) The contact pressure distribution during plastic compression of lead spheres. Int. J. Mech. Sci. 29 (10/11), pp. 713-719.

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THE CRITICAL SHEAR STRESS TO TRANSMIT A PEIERLS SCREW DISLOCATION ACROSS A NON-SLIPPING INTERFACE

PETER M. ANDERSON Department of Materials Science and Engineering The Ohio State University 2041 College Road Columbus, OH 43210-1179 U. S. A. AND XIAOJ.XIN Department of Mechanical and Nuclear Engineering Kansas State University 338 Rathbone Hall Manhattan, KS 66506-5205 U. S. A.

Abstract: The critical resolved shear stress to transmit a screw dislocation through a non-slipping (welded) bimaterial interface is studied as a function of the elastic mismatch across the interface and the nonlinear shear stress-relative shear displacement relation across the incoming and outgoing slip planes. This study extends the work of Pacheco and Mura (1969), by using a numerical approach that incorporates a variety of slip plane relations and by adopting a formulation by Beltz and Rice (1991) that accounts for the finite interplanar spacing across a slip plane. The geometry is specialized to the case when slip planes are perpendicular to the interface and numerical results are obtained for values of mismatch, ∆µ, in elastic modulus equal to 20% of the average value. Numerical results in this regime confirm the Pacheco and Mura observation that the critical resolved shear stress is proportional to the mismatch in elastic shear modulus. A significant new result is that the critical resolved shear stress increases with the unstable stacking fault energy of the slip planes, but is relatively insensitive to the maximum shear strength of the slip planes. A simple model is constructed which adequately captures the dependence on stacking fault energy and elastic modulus mismatch. It is with pleasure and gratitude that this work is presented on the commemoration of the 60th birthday of Prof. James Rice, who as Ph.D. advisor to one of the authors (PMA), instilled a sense of enthusiasm and formalism to study dislocation-defect interactions of the type described herein. 87 T.-J. Chuang and J. W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 87–105.  2000 Kluwer Academic Publishers. Printed in the Netherlands.

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1. Introduction The critical resolved shear stress, τ*, to push a dislocation past an interface or grain boundary is a fundamental quantity that controls macroscopic yield in many materials. The connection between τ* and macroscopic yield can be understood through the familiar Hall-Petch analysis of a pile-up of screw dislocations against an obstacle. There, the leading dislocation in the pile-up is able to push past the obstacle if the macroscopic shear stress, resolved onto the slip plane in the slip direction, , reaches the critical value (Hall, 1951; Petch, 1953)

(1) Thus, macroscopic yield can be increased by either increasing the obstacle strength τ* or decreasing the spacing d between obstacles. Other parameters are the elastic shear modulus µ, magnitude b of the dislocation Burgers vector, and resistance τ o to dislocation motion in a material free of obstacles (d = ∞ ). Equation (1) implies that materials with stronger obstacles have larger values of kH-P , so that the yield strength or hardness of the material is more sensitive to a change in d. Multilayered systems consisting of alternating A/B metallic layers show distinct features about the strength of interfaces. First, Eqn. (1) can be fit to hardness measurements on a given A/B multilayer system, with a constant Hall-Petch slope (kH-P) over a layer thickness range of several microns to approximately 10nm (Shinn and Barnett, 1992). The constant slope implies that, for a given system, interfacial strength does not vary significantly over a large range of layer thickness. However, different metallic systems display a large range of slopes, from 7.0GPa nm 0.5 for epitaxial Ag/Cr multilayers to as large as 42GPa nm 0.5 for epitaxial Fe/Pt multilayers (Clemens et al., 1999). Thus, interfacial strength appears to vary significantly among metallic systems. At layer thickness less than 5nm, many metallic multilayered systems display a strong departure from Eqn. (1), in that experimental hardness values increase only modestly or even decrease with further reduction in layer thickness (Clemens et al., 1999). This sharp change in behavior is believed to occur since the structure of interfaces changes from a semicoherent one at larger layer thickness to a coherent one at smaller layer thickness. Semicoherent interfaces at larger layer thickness have misfit dislocations which appear to act as pinning sites for transmission of glide dislocations across interfaces, and these pinning sites disappear for coherent interfaces (Rao et al., 1995; Anderson et al., 1999a). Rao and Hazzledine (1999) estimate that misfit dislocations in a semicoherent Al/Ni interface increase the obstacle strength by µ /16, based on their Embedded Atom Method analysis of the transmission process. The same Embedded Atom Method approach applied to coherent interfaces predicts τ* = 0.015µ for transmission from Cu to Ni across either (111) or (100) epitaxial interfaces.

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Elastic continuum approaches to dislocation transmission generally assume that interfaces are welded, in the sense that interfaces or grain boundaries do not slip or open. Head (1953a) concluded that a Volterra dislocation with a tight, step-function distribution of slip is repelled from a welded interface if the dislocation lies in the lower modulus phase (µ 1 < µ 2 in Fig. 1) and it is attracted to the interface if it lies in the larger modulus phase. Deviations from this behavior can occur for edge dislocations, with along the x or ydirections in Fig. 1, if Poisson’s ratios ν1 ≠ ν 2 (Head, 1953b; Dundurs, 1969). For freely-slipping interfaces, Volterra dislocations of a screw type with along the z-direction or edge type with along the y-direction are always attracted to the interface. However, an edge type with along the x-direction is attracted to the interface only if it lies in the larger modulus phase. Recent models employing an Figure 1. The geometry of a interface with continuum frictional shear screw dislocation in material 1 at properties (Hurtado and Freund, 1998) or distance c from the interface. an interface having a linear relation Both Peierls and Volterra descriptions of a screw dislocation between shear traction and relative shear are shown. displacement (Shilkrot and Srolovitz, 1998) have been pursued as intermediate cases to the welded and freely-slipping models. However, all of these Volterra models predict a singular force (either attractive or repulsive) on the dislocation as it is moved toward the interface. As such, they are not able to furnish a finite value of τ*. The artificial singular force can be removed by using a Peierls description of the dislocation, with a smoothly varying slip distribution, or by having elastic properties that vary smoothly across a diffuse interface. Pacheco and Mura (1969) considered the first approach and concluded that the critical stress to push a Peierls-type screw dislocation across a welded interface from material 1 to 2 is τ1*/µ1 ≈ 0.2( µ 2 - µ 1)/(µ 2 + µ 1 ), 1. For the case of a Cu/Ni interface, the model provided that (µ 2 - µ 1 ) / ( µ 2 + µ1) predicts τ 1 */ µ1 = 0.054, based on µ Cu = 54.6 GPa and µ Ni = 94.7 GPa (Hirth and Lothe, 1982). More recent work has extended the Pacheco and Mura model to include a slipping interface with a sinusoidal shear traction-relative shear displacement relation (Anderson et al., 1999b). The results indicate that slipping interfaces can generate a significantly larger τ* than welded ones, since they allow for spreading of the dislocation core into the interface. Krznowski (1991) pursued the second approach of using a diffuse interface, with an elastic shear modulus that varies linearly from µ1 to µ 2 over a width w. In that case, the obstacle strength for a Volterra screw dislocation is τ* = ( µ2 - µ 1 )bln(w/2b)/4 πw. Thus,

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a Cu/Ni interface with w = 20b, for example, has τ 1 */ µ1 = 0.0067. In general, τ * decreases when interfaces become more diffuse. To summarize, continuum models suggest that interfaces are strong barriers to slip transmission when they are chemically sharp with large, abrupt changes in elastic moduli, when they are capable of slipping during the transmission process, and finally, when the slip profile of the transmitting dislocation has an abrupt, step-like shape. This manuscript extends the work of Pacheco and Mura (1969) on the critical stress to transmit a Peierls screw dislocation, with Burgers vector parallel to the z-direction, across a welded interface as shown in Fig. 1. One extension is that a numerical solution for the slip distribution of the dislocation is employed, as an alternative to a two-term Taylor series expansion used by Pacheco and Mura, in which the first term is the slip distribution in an infinite, homogeneous material. A second extension is to account for the finite interplanar spacing across a slip plane in the formulation of the shear constitutive relation for that plane, as proposed by Beltz and Rice (199 1). Finally, this analysis extends the Frenkel (1926) sinusoidal slip plane bonding relation used by Pacheco and Mura and the bonding relation used by Xu and Argon (1995) to a more complex description of the atomic shear traction-relative shear displacement relation across a slip plane. One motivation to do so is that the Frenkel sinusoidal relation is too stiff for many commonly encountered materials (Hirth and Lothe, 1982; Sun et al., 1993; Kaxiras and Duesbery, 1993; Xu and Argon, 1995). The other motivation is that the shear modulus, unstable stacking energy, and maximum material shear resistance can be varied independently on the incoming and outgoing slip planes. Thus, the effect of bonding properties across slip planes on τ * can be studied in detail, for the case of welded interfaces.

2. Model Development 2.1 THE ELASTIC SHEAR STRESS ALONG A SLIP PLANE Consider two semi-infinite, elastic phases with shear moduli µ1 and µ 2 that are bonded along the y-z plane as shown in Fig. 1. The phases are assumed to have the same lattice parameters and crystal structure so that the Burgers vector, of a screw dislocation in each phase is identical. The shear stress on the slip plane (y = 0) generated by a Volterra screw dislocation at x =x d is provided by Head (1953a). The screw dislocation is such that the portion of the slip plane x > xd has a relative shear displacement δ = u z(y = 0 +) u z(y = 0 -) = b across it. The portion of the slip plane x < x d has zero relative slip. In terms of the SF/RH sign convention employed by Hirth and Lothe (1982), the line sense ξ of the dislocation points along the +z-direction and the vector points along the -zdirection. If the dislocation is in material 1, at a position xd = c, then the yz-component of shear stress along the slip plane is

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STRESS TO TRANSMIT A SCREW DISLOCATION

(2)

and if the dislocation is in material 2, at a position xd = -c, then

(3)

Here, the mismatch in elastic shear modulus is measured by (4) The model of a Peierls dislocation assumes a continuous distribution dδ /dx of relative slip across a slip plane of infinitesimal thickness, so that in this case (5) The slip distribution is defined in Fig. 1 such that in the wake of the dislocation, δ (x = ∞ ) = b, and ahead of the dislocation, δ(x = −∞) = 0. The yz-component of elastic shear stress generated by this slip distribution is obtained by replacing b = (dδ /dx')dx' in Eqns. (2) and (3) and integrating over the domain x’ = −∞ to x’ = ∞,

(6)

Included in Eqn. (6) is the superposition of a yz component of remote shear stress τ∞, which serves to push the dislocation from material (1) toward material (2) in Fig. 1. Two types of normalization are used that have specific meanings, depending on the value of x. Both τ el and τ ∞ are normalized by µ, where µ = µ 1 if x > 0 and µ= µ 2 if x < 0. The normalized quantities τ∞ /µ and τ el /µ must be continuous across the interface since the shear stress γ yz must be continuous due to compatibility across a non-slipping interface. All length quantities in italics indicate normalization by b, so that x = x/b.

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2.2 THE SHEAR STRESS-SHEAR DISPLACEMENT RELATION FOR A SLIP PLANE The atomic prescription for shear stress on a slip plane is expressed most simply in terms of the relative slip ∆ between the two atomic planes on either side of the slip plane, as shown in Fig. 2. The yz-component of shear stress S is assumed to have a periodic dependence on ∆ and can be expressed in general by a Fourier series, (7)

This functional form was adopted by Xu and Argon (1995). The coefficients λn specify the detailed nature of the gradual breaking and remaking of bonds and, as such, they can take on a variety of values. Regardless, the functional form must satisfy the elastic shear relation for the material, so that in the limit of vanishing ∆, h ∂S/ ∂∆ = µ. Other important distinguishing features of the curves are γ us , the unstable stacking fault energy (Rice, 1992), and S max , the peak shear stress that occurs during shearing of the atomic plane. γ us is the maximum work that can be stored in a unit area of slip plane as it is sheared. As such, it corresponds to the maximum positive area under a S(∆) curve. A three-term version of Eqn. (7) is adopted so that µ, γ u s , and Sm a x can be varied independently. In particular, a set of parameters, (8)

Figure 2. Geometry of the relative shear between two adjacent atomic planes with interplanar spacing h. S denotes the local shear stress, τ yz , acting on the slip plane and relative interplanar shear.

∆ = u z ( y = + h / 2 ) - u z (y=-h/2) denotes the

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is specified for the incoming slip plane and a separate set for the outgoing slip plane. The first three coefficients in Eqn. (7) can be determined according to

(9)

where ∆m, the interplanar slip at which the maximum shear stress is reached, normalized by b, is given as the root of the condition,

(10)

Compared to earlier efforts, the three-term description in Eqn. (7) provides additional freedom to independently vary slip plane constitutive parameters. The Frenkel (1926) sinusoidal relation is reproduced by setting λ 1 = 1 and λ 2 = λ 3 = 0, so that γus = 1 and S max = 1. The result is shown as Curve HH in Fig. 3. As pointed out by Xu and Argon (1995), atomistic calculations using the Embedded Atom Method (Sun et al., 1993) and density functional theory (Kaxiras and Duesbery, 1993) show that the Frenkel relation often overestimates γ us and that, typically, γus < 1 and S m a x < 1 for most materials. The Xu and Argon (1995) two-term relation is reproduced by setting λ 3 = 0 and prescribing µ and γ us independently. In that case, the peak shear stress is specified in terms of γ u s . Table I shows sets of ( λl , λ 2 , λ 3) corresponding to a given pair (γ us , S max), over the range (1.0, 1.0) to (0.5, 0.5). In general, each pair ( γ us , S max) provides two values of ∆ m that satisfy Eqn. (10) and thus two sets of ( λ1 , λ 2 , λ 3) typically are shown in each entry. However, when γus < S max , the solutions to Eqn. (10) are not real and, in that case, an entry of (-) is listed. The range for γ u s is extended down to 0.5, although Xu and Argon (1995) show in a survey of slip systems in Si, Ni, Fe, and Al that γ us may be as small as 0.38. Five sets of ( λ1, λ2 , λ3) are highlighted in boldface and will be used to study the effect of slip plane properties on transmission of a screw dislocation across a welded interface. Set HH corresponds to ( γ us , S max ) = (1.0, 1.0) and yields the Frankel sinusoidal relation as shown in Fig. 3. The label HH denotes that the parameters ( γu s , S m a x ) are both high in value compared to other cases in Table I. Set HH2 corresponds to the second root for (γu s , S max ) = (1.0, 1.0). As such, the HH2 curve in Fig. 3 has the same maximum positive area and the maximum shear stress as curve HH. However, the

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P.M.ANDERSON AND X.J. XIN TABLE I. Values of (λ 1, λ 2 , λ 3 ) corresponding to values of γu s and Smax , to be used in the S(∆) relation shown in Eqn.(7).

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curve tends to rise more slowly to the maximum shear stress and, as a result, the maximum is positioned at ∆m = 0.33b, compared to ∆ m = 0.25b for set HH. Thus, comparison of τ∗ for sets HH and HH2 will assess the effect of bonding features other than ( γ u s , S max) on interfacial barrier strength. Set MH corresponds to ( γ u s , S max) = (0.8, 1.0), so that γ us takes on a moderate value in Table I but S m a x remains high as in sets HH and HH2. Set MM corresponds to a two-term series as used by Xu and Argon (1995), with ( γ us , S m a x) = (0.8, 0.8233) representing moderate values of each. Comparison of the MM and MH sets will assess the effect of S m a x on τ∗. Finally, set LL corresponds to the lowest combination of ( γu s , S max ) = (0.5, 0.5) considered in Table I. All sets follow the premise in Fig. 2 that ∆ = b/2 is a position of crystal symmetry for which S = 0 and for which the area under the S( ∆) curve reaches a positive maximum equal to γu s .

Figure 3. Candidate interplanar shear stress-relative shear displacement relations as summarized in Table I.

2.3 SOLUTION PROCEDURE The slip profile is determined by solving the equilibrium condition,

(11) It is understood that, in general, the atomic relation can change abruptly at the interface, since the set (λ1, λ2 , λ3 ) 1 corresponding to a given ( γ u s , S max ) 1 describes the incoming slip plane in material 1 and a second set ( λ l , λ 2, λ3) 2 corresponding to ( γu s , S max )2

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describes the outgoing slip plane in material 2. A simple assumption, ( λ 1, λ 2, λ 3) 2 = ω(λ1, λ 2, λ 3) 1, is proposed. The mismatch in unstable stacking energy between the materials can then be defined by (12) and a comparable expression defines the mismatch, κS , in S m a x of the two phases. The results will focus entirely on the simplest case of ω = 1 and κµ = 1, so that the ratio of An important distinction in Eqn. (11) is that the elastic shear stress is based on the distribution of slip, δ(x), across an infinitesimally thin plane while the atomic shear stress is based on the distribution of slip, ∆(x), across an interplanar thickness h as in Fig. 2. Beltz and Rice (1991) and Rice (1992) note this distinction between infinitesimal and interplanar slip in the treatment of dislocation emission from crack tips. We make this distinction also and note that the expression for elastic shear stress, Eqn. (6), can be changed to a functional dependence on ∆(x) using (13) Thus, the Beltz and Rice formalism used here is distinct from the Peierls analysis, for which δ (x) = ∆ (x) is assumed. For simplicity, h = b is used as a reasonable approximation to interplanar spacing. Pacheco and Mura (1969) obtained an approximate analytic solution to the Peierls formalism by approximating the slip distribution in terms of a departure from that for an isolated dislocation that is far from an interface. In the present work, the solution to Eqn. (11) is obtained numerically by discretizing the interplanar slip distribution into N+2 intervals so that (14) This slip profile corresponds to a step-wise distribution in which ∆ jumps by db = b/N+1 at positions x i, i = 1 to N + 1. The governing Eqn. (11) then becomes

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(15)

Thus, Eqn. (15) is a statement of N+1 equations to determine the equilibrium positions xi for N+1 model dislocations of Burgers vector magnitude db. This discretization addresses the Cauchy principal value of singular 1/(xi - x j) terms by noting that the net force of a dislocation on itself is zero. The discretization in displacement space helps to concentrate discretization points around the core of the dislocation, where dδ /dx is larger. Finally, the singular interaction of the nearest model dislocation with the interface is addressed with a singularity exclusion method discussed in the Appendix. There, Eqn. (A4) defines η i = 0 for all model dislocations i except the one nearest to the interface. If the applied shear stress is specified, then Eqn. (15) can furnish N+1 equations for the N+1 unknown values of xi. This corresponds to a stress-controlled transmission of the dislocation through the interface. A drawback to this approach is that it is difficult to solve numerically for the unstable portion of the transmission process, when the dislocation continues to transmit past the point of peak resistance provided by the interface. Rather, a more stable approach is to apply a form of displacement control, in which N is selected to be an odd number and the center of the dislocation is specified by prescribing the position x (N+1)/2 = c. In that case, Eqn. (15) still provides N equations for the remaining unknown xi . The equilibrium condition R(N+1)/2 = 0 for the center model dislocation is replaced by the statement of macroscopic equilibrium that ∑ Ri = 0. This statement furnishes the applied stress necessary to hold the entire screw dislocation in equilibrium,

(16)

An equivalent statement of macroscopic equilibrium in the continuum framework is obtained by multiplying both sides of Eqn. (11) by (d δ/dx)dx and integrating from x = - ∞ to +∞ ,

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To conclude, the procedure is to use the Peierls (1940) solution to generate an initial trial solution, (18) and then compute the initial R i(0) according to Eqn. (15) with applied stress given by Eqn. (16). The new iteration on the solution is determined by (19)

where K ij(0) is the inverse of the N x N Jacobian matrix, ∂ Ri /∂ xj at x j = x j(0) . This iteration process continues until the Euclidean norm of the residual ||R|| within a specified tolerance of 10-7.

3. Results 3.1 SENSITIVITY OF THE NUMERICAL SOLUTION TO NUMBER OF GRID POINTS In order to test the sensitivity of the numerical solution to N, a screw dislocation in an infinite, homogeneous material was modeled using the Peierls formalism, but with different N. The numerical version of the Peierls formalism is to use Eqns. (15) and (16), but with δ (x) = ∆ (x) so that the simplification, (1-S'/µ) = 1, is made in those equations. Table II shows %Error as the maximum value of |xi - x i(0) |/|x i(0) | for a solution with a specified N. The %Error ~ 1/N and, based on these results, N = 101 is used for subsequent calculations with an expected accuracy of approximately 1%. TABLE II. %Error in Peierls solution as a function of N

3.2 COMPARISON TO PRIOR WORK Figure 4 shows a comparison of the numerical results based on the slip plane relation HH to earlier work by Head (1953a) and Pacheco and Mura (1969). Two types of numerical solution are shown. The first is denoted as the Beltz and Rice formalism, which uses Eqns. (15) and (16) and second is the Peierls formalism which sets δ (x) = ∆ (x). In all cases, the mismatch in elastic shear modulus is κµ = 0.1, so that the applied stress drives

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the dislocation from a material with modulus µ 1 into a material with larger modulus µ 2 = 1.22µ 1. The unstable stacking fault energy and maximum shear resistance have the same relative values, so that γu s (2) = 1.22 γu s (1) and S max (2) = 1.22Smax (1). For reference, the applied stress to push a Volterra dislocation to a distance c from the interface in Fig. 1 is (Head, 1953a) (20)

while the corresponding stress from the Pacheco and Mura (1969) solution for a Peierls dislocation with atomic relation HH is

(21)

All solutions are nearly identical for c > 3b, but for c < b, the Head solution diverges noticeably due to the singular nature of the Volterra approach. The Pacheco and Mura analytic solution (solid line) and the numerical solution for the Peierls formalism (square symbols) essentially coincide over the entire range of positions, indicating that both solution techniques yield the same result. The numerical solution to the Beltz and Rice formalism has a noticeably smaller applied stress to drive the dislocation over the range 0.5b < c < 2b than the Peierls approach. However, both approaches predict barrier strengths at c = 0 that differ by less than 5%.

Figure 4. Comparison of the applied stress versus position of the dislocation center for different formulations.

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3.3 THE BARRIER STRENGTH FOR DIFFERENT SLIP PLANE RELATIONS Figure 5 shows the applied shear stress needed to push the center of the screw dislocation to a distance c from the interface, for the 5 slip plane constitutive relations highlighted in Table I. In all cases, the mismatch in elastic shear modulus is κµ = 0.1 and the Beltz and Rice formalism is used. The general trend for all cases is that the center of the dislocation is pushed closer to the interface as the remote stress is increased. The peak remote stress, or barrier strength τ∗, is reached when the center of the dislocation reaches the interface.

Figure 5. Comparison of the applied stress versus position of the dislocation center for the 5 slip plane constitutive relations summarized in Table 1. κµ = 0.1 and

Among the cases studied, the effect of unstable stacking energy is most important. In particular, Cases HH and MH differ only in that HH has stacking fault energies of µ 1 b/2 π2 and µ 2 b/2 π2 in materials 1 and 2, but MH has stacking fault energies of 0.8µ 1 b/2 π2 and 0.8µ 2b/2 π 2 in materials 1 and 2, respectively. This 20% reduction in stacking fault energies produces nearly a 25% reduction in τ∗. The rationale for this behavior will be understood when the slip profiles for these two cases are compared in the following section. The effect of maximum shear resistance in the slip plane constitutive relation is secondary to that of unstable stacking fault energy. In particular, Cases MH and MM differ only in that MH has S max (1) = µ1 /2 π and S max (2) =µ 2 /2 π, but MM has S max(1) = 0.8233 µ 1 /2π and S max (2) = 0.8233µ 2 /2 π. This 18% reduction in maximum shear resistance produces a modest increase (< 5%) in τ∗. A shorter range interaction also results, in that the rise to a peak occurs closer to the interface for Case MM.

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Changes in the shape of the slip plane constitutive relation appear to produce small changes in τ*, provided that stacking fault energies and maximum shear resistances are unchanged. Evidence for this stems from comparison of Cases HH and HH2, which both have the same stacking fault energies and maximum shear resistances. These cases display modest differences in τ ∞, even up to c = 0. In contrast, τ* for Case LL is nearly half that for HH and HH2. Although both the stacking fault energies and maximum shear resistances for LL are one half those for HH and HH2, the comparisons made earlier suggest that it is the reduction in stacking fault energy that contributes most significantly to the reduction in τ*. 3.4 THE CRITICAL SLIP PROFILES FOR DIFFERENT SLIP PLANE RELATIONS Figure 6 shows the critical dislocation slip profiles for the five cases discussed in the previous section. Again, the mismatch in elastic shear modulus is κ µ = 0.1. The critical configurations occur when the maximum remote stress τ∞ = τ * is applied and the dislocation center is pushed to the interface. It is clear that the most diffuse, spread out dislocation core occurs for Case LL, moderate spreading occurs for MH and MM, and the least spreading occurs for HH and HH2. The ordering of core spreading remains the same for c ≠ 0, when the dislocation is located away from the interface at τ ∞ < τ *. The results reflect the well-known feature of the Peierls model that the dislocation core width diminishes as stacking fault energy increases (Hirth and Lothe, 1982). The slip distribution for Cases MH and MM essentially overlap, as do those for Cases HH and HH2, so that changes in the maximum shear resistance of the slip plane or features other than unstable stacking energy appear to have little effect on the dislocation core width.

Figure 6. Comparison of the slip profiles for the 5 slip plane constitutive relations summarized in Table I, when the center of the dislocation is at the interface and the barrier strength, τ* ,is reached. κ µ = 0.1 and and

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The slip distribution profiles help provide an understanding of the relative magnitudes of τ* in Fig. 5. It appears that τ * increases as the dislocation core width diminishes or, equivalently, as the peak slope along the slip profile increases. This trend can be observed using a simple triangular approximation for the slope dδ/dx of the slip distribution as shown in Fig. 7. The maximum slope k at the center of the dislocation and the width 2w of the dislocation must satisfy the relation kw = b. If this profile is positioned so that the maximum slope is at the interface (i.e., c = 0 in Fig. 7), then Eqn. (17) yields (22) Thus, the barrier strength for this simple profile is directly proportional to the maximum slope in the profile. Figure 8 shows that there is reasonable agreement between the numerical results and the prediction from Eqn. (22), with only the first term contributing since the 5 cases considered do not have a mismatch in normalized unstable stacking fault energy. The barrier strength can also be correlated with the unstable stacking fault energy as shown in Fig. 9. A linear regression analysis of the 5 numerical cases yields (23) Comparison of Eqns. (22) and (23) suggests that the maximum slope is proportional to the normalized unstable stacking fault energy. However, the numerical results are limited, in that only cases with κ µ = 0.1 and κ γ = κ S = 0 have been considered. Larger values of κµ , for example, are expected to introduce a nonlinear dependence of τ* on κ µ , so that Eqn. (23) is not valid. The nonlinear dependence on κ µ is apparent in Eqn. (22), since dδ/dx| max there depends not only on γus , but also κ µ . A linear dependence on κµ would arise only if the slip profile

Figure 7. An idealized slip distribution used to develop Eqn. (22).

Figure 8. Barrier strength τ* as a function of maximum slope in the slip distribution, for the 5 cases highlighted in Table I. The line is the prediction of Eqn. (22).

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remained unchanged as the dislocation approached the interface, and this is expected to occur in the limit κ µ –> 0. Based on the numerical results, the correlation in Eqn. (23) appears to be reasonable for at least 0 ≤ κµ ≤ 0.1, and additional work is needed to determine the behavior at larger κ µ

Figure 9. Barrier strength τ* as a function of normalized unstable stacking fault energy, for the 5 cases highlighted in Table I. The line is Eqn. (23).

4.

Conclusions

The interaction of a distributed-core screw dislocation with a welded bimaterial interface is studied numerically, using a shear constitutive relation for the incoming and outgoing slip planes that permits the elastic shear modulus, unstable stacking fault energy, and maximum shear resistance to varied independently. The analysis adopts a modification to the Peierls approach as proposed by Beltz and Rice (1991), in which the finite thickness of the slip plane is accounted for in formulating the shear constitutive relation for the slip plane. The mismatch in elastic modulus across the interface generates an image force on the dislocation which tends to push the dislocation into the elastically softer medium. The numerical results here confirm a feature suggested by Pacheco and Mura (1969) that for small elastic mismatch, the critical stress τ * to push the dislocation through the interface into the elastically stiffer material is linearly proportional to the mismatch in elastic modulus. The results for small elastic mismatch also show that τ* is linearly proportional to the normalized unstable stacking fault energy, γus /µ b, employed in the constitutive relation for the slip planes, but it is weakly dependent on the maximum shear resistance of the slip planes.

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A simple model employing a quadratic description of the slip profile provides a useful interpretation of the numerical results. In particular, τ * is shown to be linearly proportional to product of elastic modulus mismatch and maximum slope in the slip profile. Thus, τ* is expected to increase for cases of larger unstable stacking fault energy, for which dislocations have small cores, and for cases with larger mismatch in elastic modulus. The numerical study suggests that the predictions of the simple model are reasonable, at least for small elastic mismatch in the range of (µ 2 - µ1)/(µ2 + µ1 ) < 0.1.

4.

Acknowledgements

PMA and XJX acknowledge support of the Air Force Office of Scientific Research, Grant F49620-96-1-0238 and the support of the Ohio Supercomputer Center.

5. References Anderson, P.M., Rao, S., Cheng, Y., and Hazzledine, P.M. (1999b) The Critical Stress for Transmission of a Dislocation Across an Interface: Results from Peierls and Embedded Atom Models, Mater. Res. Soc. Proc. 586. Anderson, P.M., T. Foecke, and P.M. Hazzledine (1999a) Dislocation-Based Deformation Mechanisms in Metallic Nanolaminates, MRS Bulletin 24(2), 27-33. Beltz, G.E., and Rice, J.R. (1991) Dislocation Nucleation Versus Cleavage Decohesion at Crack Tips, in T.C. Lowe, A.D. Rollett, P.S. Follansbee and G.S. Daehn (eds.), Modeling the Deformation of Crystalline Solids: Physical Theory, Application and Experimental Comparisons, TMS, Warrendale, PA, pp. 457-480. Clemens, B.M., H. Kung and S.A. Barnett (1999) Structure and Strength of Multilayers, MRS Bulletin 24(2), 20-26. Dundurs, J. (1969) Elastic Interaction of Dislocations with Inhomogeneities, in T. Mura (ed.) Math. Theory of Dislocations, ASME, NY, pp. 70- 115. Eshelby, J.D. (1949) Edge Dislocations in Anisotropic Materials, Philos. Mag. 40, 903-912. Frenkel, J. (1926) Zur Theorie der Elastizitatsgrenze und der Festigkeit Kristallinischer Korper, Z. Phys. 37, 572-609. Hall, E.O. (1951) The Deformation and Ageing of Mild Steel: III Discussion of Results, Proc. Roy. Soc. B64, 747-753. Head, A.K. (1953a) Edge Dislocations in Inhomogeneous Media, Proc. Phys. Soc. (London) B66, 793-801. Head, A.K. (1953b) The Interaction of Dislocations and Boundaries, Philos. Mag. 44, 92-94. Hirth, J.P. and Lothe, J. (1982) Theory of Dislocations, 2nd. ed., John Wiley and Sons, New York. Hurtado, J.A. and Freund, L.B. (1998) Force on a Dislocation Near a Weakly Bonded Interface, J. Elasticity 52(2), 167-180. Kaxiras, E., and Duesbery, M.S. (1993) Free Energies of Generalized Stacking Faults in Si and Implications for the Brittle-Ductile Transition, Phys. Rev. Lett. 70, 3752-3755. Krzanowski, J.E. (1991) Effect of Composition Profile Shape on the Strength of Metallic Multilayer Structures, Scripta Metall. Mater. 25(6), 1465-1470. Pacheco, E.S. and Mura, T. (1969) Interaction Between and Screw Dislocation and Bimetallic Interface, J. Mech. Phys. Sol. 17, 163-170. Peierls, R.E. (1940) The Size of a Dislocation, Proc. Phys. Soc. (London) 52, 34-37. Petch, N.J. (1953) Cleavage Strength of Polycrystals, J. Iron Steel Inst. 174, 25-28. Rao, S.I., P.M. Hazzledine, and D.M. Dimiduk (1995) Atomistic Simulations of Dislocation-Interface Interactions in Metallic Nanolayers, Mater. Res. Soc. Proc. 362, 67-72. Rao, S.I. and Hazzledine, P.M. (1999) (accepted for publication) Atomistic Simulations of DislocationInterface Interactions in the Cu-Ni System, Philos. Mag. A.

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Rice, J.R. (1992) Dislocation Nucleation from Crack Tips: An Analysis Based on the Peierls Concept, J. Mech. Phys. Sol. 40, 239-271. Shilkrot, L.E. and Srolovitz, D.J. (1998) Elastic Analysis of Finite Stiffness Bimaterial Interfaces: Application to Dislocation-Interface Interactions, Acta Mater. 46(9), 3063-3075. Shinn, M., Hultmann, L., and Barnett, S.A. (1992) Growth, Structure, and Microhardness of Epitaxial TiN/NbN, J. Mater. Res. 7, 901-911. Sun, Y., Beltz, G.E., and Rice, J.R. (1993) Estimates from Atomic Models of Tension-Shear Coupling in Dislocation Nucleation from a Crack Tip, Mater. Sci. Eng. A 170, 67-85. Xu, G., and Argon, A.S. (1995) Nucleation of Dislocations from Crack Tips Under Mixed Modes of Loading: Implications for Brittle Against Ductile Behavior of Crystals, Philos Mag. A 72, 415-451.

6. Appendix A modified exclusion method is used in the numerical method to avoid the singular interaction between the interface and the model dislocation of Burgers vector magnitude db that is nearest to it. Note that a Volterra screw dislocation generates a yz-component of shear stress along the x-z plane (Hirth and Lothe, 1982) (A1) For a Peierls screw dislocation, the yz component of shear stress along the x-z plane is (Eshelby, 1949) (A2) where η = b/2. The difference in Eqns. (A1) and (A2) is small when x > b, but Eqn. (A1) is singular at x = 0 and Eqn. (A2) is non-singular over the range of x. In addition, when η = 0, Eqns. (A1) and (A2) become identical. Thus, in order to eliminate the singularity associated with movement of the nearest model dislocation to the interface, the discretized form of Eqn. (11) is modified with the substitutions,

(A3) with ηi =

α db for dislocation i nearest to the interface

0 otherwise

(A4)

so that Eqn. (15) is produced. Final numerical results are very insensitive to the value of α in the range (0.1, 1); a value of 0.5 is adopted for the results presented.

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SELF-ORGANIZING NANOPHASES ON A SOLID SURFACE

Z. SUO AND W. LU Mechanical and Aerospace Engineering Department and Materials Institute Princeton University Princeton, NJ 08544

Abstract: A two-phase epilayer on a substrate may exhibit intriguing behaviors. The phases may select stable sizes, say on the order of 10 nm. The phases sometimes order into a periodic pattern, such as alternating stripes or a lattice of disks. The patterns may be stable on annealing. This paper describes an irreversible thermodynamic model that accounts for these behaviors. The phase boundary energy drives phase coarsening. The concentration-dependent surface stress drives phase refining. Their competition may stabilize nanoscopic phases and periodic patterns. 1. Introduction Rice and co-workers wrote a series of papers on polycrystalline materials subject to both stress and heat (Chuang and Rice, 1973; Chuang et al., 1979; Needleman and Rice, 1980; Rice and Chuang, 1981). The phenomenon concerned a cavity on a grain boundary. The cavity could change shape and size via mass transport processes (creep, diffusion on the cavity surface, and diffusion on the grain boundary), driven by thermodynamic forces (stress, surface energy, and grain boundary energy). Building on those of Herring (1951), Mullins (1957), and Hull and Rimmer (1959), Rice and co-workers developed a general approach to this complex phenomenon. Figure 1 outlines this approach. The basic ingredients are kinematics, energetics, and kinetics. To model an evolving structure, one first describes its configuration with kinematic quantities: the shape of the structure, the deformation field, the concentration field, etc. These kinematic quantities are thermodynamic coordinates. One then equips the structure with a free energy as a functional of the kinematic quantities. The variation of the free energy associated with the variation of the kinematic quantities defines the driving force. Finally one provides the kinetic relations between the rate of the kinematic quantities and the driving forces. These ingredients are combined into a variational statement. Depending on the type of the kinematic quantities, the variational statement ramifies into several routes to simulate the evolution of the structure. If the kinematics comprises fields, the variational statement leads to partial differential equations and boundary conditions. If the kinematics comprises discrete variables, the variational statement leads to ordinary differential equations. If the structure is divided into elements, the variational statement leads to a finite element model. 107 T.-J. Chuang and J. W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 107–122. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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Figure 1. A general approach to evolving structures.

In the world of dissipative, time-dependent phenomena, this approach has a long tradition, dating back to the treatment of damped vibration by Rayleigh (1894), irreversible thermodynamics by Prigogine (1967) and others, spinodal decomposition by Cahn and Hilliard (1958), and heat transfer by Biot (1970). The approach, in various forms, has been used to study diverse material structures (e.g., Khachaturyan, 1983; Srolovitz, 1989; McMeeking and Kuhn, 1992; Gao, 1994; Freund, 1995; Suo, 1997; Carter et al., 1997; Bower and Craft, 1998; Cocks et al., 1999). Recent applications include self-assembled quantum dots, electromigration voids, ferroelectric domains, and emerging crack tips; see reviews by Freund (2000) and Suo (2000). Like other fields to which Rice has made seminal contributions, this field has advanced solid mechanics by developing basic theories and methods to understand phenomena of practical significance. 2. Self-Organization by Competitive Coarsening and Refining This paper considers a particular phenomenon: self-organizing nanophases in binary epilayers. For a decade high-resolution imaging techniques, such as Scanning Tunneling Microscopy (STM) and Atomic Force Microscopy (AFM), have spurred intense studies of nanoscopic activities on solid surfaces. Kern et al. (1991) deposited a submonolayer of oxygen on a copper (110) surface. On annealing, the oxygen atoms arranged into stripes that alternate with bare copper stripes. The width of the stripes was on the order of 10 nm. The self-assembled nanostructure can be a template for making functional structures. Li et al. (1999) grew ferromagnetic iron films on the oxygen-striped copper substrate. The stripe structure was retained up to several monolayers of the iron films. Periodic patterns have also been observed in other material systems. Pohl et al. (1999) deposited a monolayer of silver on a ruthenium (0001) surface, and then exposed the silver-covered ruthenium to sulfur. The epilayer became a composite of sulfur disks in a silver matrix. The sulfur disks were of diameter about 3.4 nm, and formed a triangular lattice. The observations include nanoscopic phases, periodic patterns, and stability on annealing. These behaviors are intriguing because they are absent in bulk phase separation. The basic behaviors of bulk phase separation are well known. Below a critical temperature, a miscible solution becomes unstable and separates into two phases. One phase may form particles, and the other a continuous matrix. In the beginning, the particles are small and the total area of the phase boundaries is large. The atoms at the

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phase boundaries have excess free energy. To reduce the free energy, the total area of the phase boundaries must reduce. Consequently, atoms leave small particles, diffuse in the matrix, and join large particles. Over time the small particles disappear, and the large ones become larger. The process is known as phase coarsening. Time permitting, the phases will coarsen until only one big particle left in the matrix. The two phases usually have different atomic lattice constants. If the phase boundaries are coherent, the lattice constant misfit induces an elastic field. For simplicity, assume that both phases have cubic atomic lattices, with lattice constants a 1 and a 2 , respectively. The misfit strain is ε M = ( a1 – a 2 ) / a1 . Let E be an elastic modulus. The average elastic energy of the two-phase mixture scales as , invariant with the particle size. Consequently, bulk elastic misfit does not stop phase coarsening. In the preceding paragraph, we have excluded other size scales of the system. Imagine, for example, a thin film bonded to a substrate. The film undergoes phase separation, but the substrate does not. The film thickness provides a length scale. When the particle size approaches and exceeds the film thickness, the total elastic energy of the system increases with the particle size in the lateral direction (Roytburd, 1993; Pompe et al. 1993). Consequently, the elastic energy in the film-substrate composite causes phase refining. The two competing actions—refining due to elasticity and coarsening due to phase boundaries—can select an equilibrium phase size. Similar competing actions are well known in ferroelectric films and polycrystals; see Suo (1998) for review. In the latter, the grain size provides the needed length scale. Long range interactions other than elasticity, such as electrostatics, can also refine phases (Chen and Khachaturyan, 1993; Ng and Vanderbilt, 1995; Ball, 1999). Elasticity-mediated refining may account for composition modulation sometimes observed in multi-component semiconductor films, although we cannot be certain until a detailed model is developed and compared with experimental observations. The existing models (Glas, 1997; Guyer and Voorhees, 1998) do not include the film thickness effect, so they do not have the phase refining action. It would be significant to see if the model with both coarsening and refining actions can stabilize composition modulation. The instability of self-assembled quantum dots provides another case study; see Freund (2000) for review. Imagine an elemental semiconductor film on another elemental semiconductor substrate, such as germanium film on silicon substrate. The misfit lattice constants of the two crystals induce an elastic field, assuming that the interface is coherent. The film may break into islands to reduce the elastic energy of the system. The shape change is via atomic diffusion on the surface. It is sometimes observed that the islands have a narrow size distribution, and even order into periodic patterns. Nonetheless we note that in this case elasticity cannot stop island coarsening. Surface diffusion allows the islands to change both lateral size and height. For example, if the islands coarsen with a self-similar shape, the average elastic energy is invariant with the island size. Should stable, periodic islands ever be observed, something in addition to classical elasticity must be invoked to stop coarsening. 3. A Model of a Binary Eiplayer on a Substrate From the above discussion, it is clear that a model of self-organizing phases should contain the following ingredients: phase separation, phase coarsening, and phase refining. Each ingredient may be given alternative mathematical and physical representations. We next summarize a model proposed by Suo and Lu (2000).

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Imagine an epilayer of two atomic species A and B on a substrate of atomic species S. The epilayer is one atom thick, and the substrate occupies the half space x 3 < 0 , bounded by the x 1 -x 2 plane. The two species A and B can be both different from that of the substrate (such as sulfur-silver on ruthenium). Alternatively, only one species of the epilayer is different from that of the substrate (such as oxygen on copper). The epilayer is a substitutional alloy of A and B. Atomic diffusion is restricted within the epilayer. 3.1 KINEMATICS Two sets of kinematic quantities describe the configuration of the epilayer-substrate composite: one for elastic deformation, and the other for mass transport. Let u i be the displacements in the substrate. A Latin subscript runs from 1 to 3. We assume that the epilayer is coherent on the substrate. When the substrate deforms, the epilayer deforms by the same amount as the substrate surface. Consequently, the displacement field of the substrate completely specifies the deformation state of the epilayer-substrate composite. The misfit strains in the epilayer remain constant, and therefore are not represented as thermodynamic variables. Let C be the fraction of atomic sites on the surface occupied by species A. Imagine a curve on the surface. When some number of A-atoms cross the curve, to maintain a flat epilayer, an equal number of B-atoms must cross the curve in the opposite direction. Denote the unit vector lying in the surface normal to the curve by m. Define a vector field I in the surface (called the mass relocation), such that Iα m α is the number of Aatoms across a unit length of the curve. A Greek subscript runs from 1 to 2. A repeated index implies summation. Mass conservation requires that the variation in the concentration relate to the variation in the mass relocation as Λδ C = – δIα , α ,

(1)

where Λ is the number of atomic sites per unit area. Similarly define a vector field J (called the mass flux), such that Jα m α is the number of A-atoms across a unit length of the curve on the surface per unit time. The relation between I and J is analogous to that between displacement and velocity. The time rate of the concentration compensates the divergence of the flux vector, namely, Λ∂ C/ ∂ t = – J α,α .

(2)

3.2 ENERGETICS We next specify the free energy as a functional of the kinematic quantities ui and C. Let the reference state for the free energy be atoms in three unstrained, infinite, pure crystals of A-atoms, B-atoms and S-atoms. When atoms are taken from the reference state to form the epilayer-substrate composite, the free energy changes, due to the entropy of mixing, the misfits among the three kinds of atoms, and the presence of the free space. In addition, the misfits can induce an elastic field in the substrate. Let G be the free energy of the entire composite relative to the same number of atoms in the reference state. For an epilayer only one atom thick, we cannot attribute the free energy to individual kinds of misfit. Instead, we lump the epilayer and the adjacent monolayers of the substrate into a single superficial object, and specify its free energy. The free energy of the composite consists of two parts: the bulk and the surface, namely,

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(3) The first integral extends over the volume of the entire system, W being the elastic energy per unit volume. The second integral extends over the surface area, Γ being the surface energy per unit area. Both the volume and the surface are measured in the unstrained substrate. As a convention, we extend the value of the substrate elastic energy W all the way into the superficial object. Consequently, the surface energy Γ is the excess free energy in the superficial object in addition to the value of the substrate elastic energy. The convention follows the one that defines the surface energy for a onecomponent solid. The elastic energy per unit volume, W, takes the usual form, being quadratic in the displacement gradient tensor, u i , j . We assume that the substrate is isotropic, with Young’s modulus E and Poisson’s ratio v. The elastic energy density function is (4) The stresses σ ij are the differential coefficients, namely, δW = σ ij δu i,j . The surface energy per unit area, Γ , takes an unusual from. Assume that Γ is a function of the concentration C, the concentration gradient C,α , and the displacement gradient in the surface, u α,β . Expend the function Γ (C,C,α ,uα,β ) to the leading order terms in the concentration gradient C,α and the displacement gradient uα,β , namely, (5) where g, f and h are all functions of the concentration C. We have assumed isotropy in the plane of the surface; otherwise both f and h should be replaced by second rank tensors. The leading order term in the concentration gradient is quadratic because, by symmetry, the term linear in the concentration gradient does not affect the surface energy. We have neglected terms quadratic in the displacement gradient tensor, which relate to the excess in the elastic constants of the epilayer relative to the substrate. We next explain the physical content of (5) term by term. When the concentration field is uniform in the epilayer, the substrate is unstrained, and the function g(C) is the only remaining term in G, the excess free energy of the composite relative to the reference state. Consequently, g(C) is the surface energy per unit area of the composite of the uniform epilayer on the unstrained substrate. We assume that the epilayer is a regular solution, so that the function takes the form (6) Here g A and g B are the excess energy of the superficial object when the epilayer is pure A or pure B. In the special case that A, B and S atoms are all identical, gA and g B reduce to the surface energy of an unstrained one-component solid. Due to mass conservation, the average concentration is constant when atoms diffuse within the

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epilayer. Consequently, in (6) the terms involving g A and g B do not affect diffusion. Only the function in the bracket does. The first two terms in the bracket result from the entropy of mixing, and the third term from the energy of mixing. The dimensionless number Ω measures the exchange energy relative to the thermal energy kT. The g ( C) function is convex when Ω < 2, and nonconvex when Ω > 2. The function is mainly responsible for phase separation; it favors neither coarsening nor refining. We assume that h(C) is a positive constant, h(C)= h 0 . Any nonuniformity in the concentration field by itself increases the free energy Γ . Consequently, the second term in (5) is taken to represent the phase boundary energy; the term drives phase coarsening. The first two terms in (5) are analogous to those in the model of bulk phase separation of Cahn and Hilliard (1958). The model represents a phase boundary by a concentration gradient field. An alternative model would represent a phase boundary by a sharp discontinuity. The merits of the two models have been extensively discussed in the literature, and will not be repeated here. Now look at the last term in (5), where u 1,1 and u 2,2 are the strains in the surface. By definition, f is the change in the surface energy per unit strain. Consequently, f represents the residual stress in the superficial object. More precisely, it is the resultant force per unit length. The quantity f is known as the surface stress (Cahn 1980, Rice and Chuang 1981). The existing literature mainly concerns the surface stress for onecomponent solids (Cammarata, 1994; Cammarata and Sieradzki,1994; Freund, 1998; Gurtin and Murdoch, 1975; Willis and Bullough, 1969; Wu, 1996). In the present problem, when the concentration is nonuniform, the surface stress is also nonuniform, and induces an elastic field in the substrate. As stated in Section 2, such an elastic field will refine phases. For simplicity, we assume that the surface stress is a linear function of the concentration, f(C) = f 0 + f 1 C . Ibach (1997) has reviewed the experimental information on this function. Surface energy can also be a function of an order parameter. Alerhand et al. (1988) used the idea to model surface domain patterns. 3.3 KINETICS The composite evolves by making two kinds of changes: elastic deformation in the substrate, and mass relocation in the epilayer. Elastic deformation does not dissipate energy, but mass transport does. Define the driving force Fα as the reduction of the free energy of the composite when an atom relocates by unit distance. Following Cahn (1961), we specify a kinetic law by relating the atomic flux linearly to the driving force: J α = M Fα ,

(7)

where M is the mobility of atoms in the epilayer. Again we have assumed isotropy in the surface; otherwise M should be replaced by a second rank tensor. 4. Variational Statement and Partial Differential Equations We now mix the ingredients. Recall that the driving force is defined as the reduction of the free energy of the composite when an atom relocates by unit distance. Translating this definition into a mathematical description, we have

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113 (8)

The two vector fields, u and I, are basic kinematic variables; they vary independently, subject to no constraint. Mass transport dissipates energy, but elastic deformation does not. The variational statement (8) embodies these considerations. Calculate δG using the equations in Section 3, giving

(9)

where Now compare (8) and (9). The free energy variation with the mass relocation gives the expression for the driving force for diffusion: (10) Because elastic deformation does not dissipate energy, the free energy variation with the elastic displacement vanishes, leading to (11) in the bulk and (12) on the surface. Equation (11) recovers the equilibrium equation in the elasticity theory. Equation (12) has a straightforward interpretation. Recall that the surface stress is the resultant force (per unit length) of the residual stress in the surface. Force balance equates the gradient of the surface stress to the tangential traction. Equation (12) sets the boundary conditions of the elastic field in the substrate. Observe that the last term in (5) varies with both fields u and I, and thereby couples the two fields. The substrate displacements enter the diffusion driving force (10), and causes the concentration field to change over time. Once concentration field changes, the surface stress changes and, through the boundary conditions (12), alters the displacements in the substrate. The elastic field in a half space due to a tangential point force acting on the surface was solved by Cerruti (see p. 69 in Johnson, 1985). A linear superposition gives the field due to distributed traction on the surface. Only the expression u β,β enters the diffusion driving force, given by

(13)

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The integration extends over the entire surface. A combination of (2), (7) and (10) leads to a diffusion equation: (14) Equations (6), (13) and (14) define the evolution of the concentration field. Once the concentration field is given at t = 0, these equations update it for the subsequent time. Equation (14) looks similar to that of Cahn (1961) for spinodal decomposition. The main difference is how elasticity is introduced. Cahn considered misfit effect caused by composition nonuniformity in the bulk. As discussed in Section 2, such an elasticity effect does not refine phases. Consequently, no stable pattern is expected. Indeed, numerical simulations have shown that the phases coarsen indefinitely, limited only by the computational cell size or computer time; see review by Chen and Wang (1996). By contrast, the elasticity effect in our model comes from nonuniform surface stress, which is similar to the nonuniform residual stress in the thin film discussed in Section 2. This elasticity effect does refine phases. 5. Scales and Parameters A comparison of the first two terms in the parenthesis in Eqn. (14) sets a length: (15) This length scales the distance over which the concentration changes from the level of one phase to that of the other. Loosely speaking, one may call b the width of the phase boundary. The magnitude of h 0 is on the order of energy per atom at a phase boundary, and kT ~ 10 –20 J namely, h 0 ~1eV . Using magnitudes h 0 ~ 10 J, Λ ~ 10 m (corresponding to T = 700 K), we have b = 0.3 nm. The competition between coarsening and refining (i.e., between the last two terms in Eqn. 14) sets another length: –19

20

–2

(16) This length scales the equilibrium phase size. Young’s modulus of a bulk solid is about E ~ 10 11 N/m 2 . According to the compilation of Ibach (1997), the slope of the surface stress is on the order f1 ~ 1N/m . These magnitudes, together with h0 ~ 10 – l 9 J , give l ~ 1 0 n m . This estimate is consistent with the experimentally observed stable phase sizes. From (14), disregarding a dimensionless factor, we note that the diffusivity scales as D ~ MkT/Λ . To resolve events occurring over the length scale of the phase boundary 2 width, b, the time scale is τ = b / D , namely,

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115 (17)

To resolve events over the length scale of the phase size, l, the time scale is 2 2 l / D = (l / b) τ. Normalize the coordinates x α and ξ α by b, and the time t by τ . In terms of the dimensionless coordinates and time, Eqns. (13) and (14) are combined into

(18)

The system is nonlinear and nonconvex because of the function g( C). The first two terms in (6) disappear after the differentiation in (18). The expression for g in (18), normalized by Λ kT , is (19) The problem has two dimensionless parameters: l/b a n d Ω . The parameter l/b measures the ratio of the equilibrium phase size to the phase boundary width; a representative value is l/b ~ This ratio appears in front of the refining term in (18) as a small parameter. The parameter Ω measures the degree of the convexity of the function g(C), which is nonconvex when Ω > 2. Parameters describing the initial concentration field also enter the problem. In so far as the equilibrium pattern is concerned, only the average concentration, Cave , is important. Recall that C ave is timeinvariant because of mass conservation. 6. Linear Perturbation Analysis This section summarizes the results of a linear perturbation analysis (Lu and Suo 1999). As stated before, when the concentration field is uniform, the substrate is unstrained, and the composite is in an equilibrium state. To investigate the stability of this equilibrium state, we superpose to this uniform concentration field a perturbation of a small amplitude. The small perturbation can be represented by a superposition of many sinusoidal components. Consider one such component, which is a sinusoidal field in the x 1 direction. Let C 0 be the uniform concentration from which the system is perturbed, q 0 be the perturbation amplitude at time t = 0, and β be the perturbation wavenumber. The wavenumber relates to the wavelength λ as β = 2 π / λ . According to the linear perturbation analysis, at time t the concentration field becomes (20)

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Over time, the concentration field keeps the same wavenumber, but changes the amplitude exponentially. The characteristic number α is given by (21) with (22) If α > 0, the perturbation amplitude grows exponentially with the time, and a nonuniform epilayer is obtained. If α < 0, the perturbation amplitude decays exponentially with the time, and the uniform epilayer is stable. Figure 2 plots α as a function of the wavelength. We distinguish three cases: When η > 0.5 , α < 0 for all wavelengths, so that the uniform epilayer is stable against perturbation of all wavelengths. When η < 0, the curve intersects with the horizontal axis only at one point, so that • the uniform epilayer is stable for short wavelengths, but unstable for long wavelengths. • When 0 < η < 0.5 , the curve intersects with the horizontal axis at two points, so that the uniform epilayer is stable against perturbations of long and short wavelengths, but unstable against perturbations of an intermediate range of wavelengths. From (22) 0 < η < 0.5 means that g (C) is convex at C 0 , but is very shallow. Acting by itself, g (C) would stabilize the uniform epilayer. In the presence of concentrationdependent surface stress, however, the shallow convex g (C) is insufficient to stabilize the uniform epilayer. For a sufficiently small η , the curve reaches a peak at wavelength •

(23) This wavelength corresponds to the fastest growing perturbation mode. The linear perturbation analysis is valid so long as the perturbation amplitude q0 exp( αt) is small compared to C 0 . The results are useful to check numerical simulation. However, the linear perturbation analysis cannot predict the equilibrium pattern, where the concentration nonuniformity has large magnitudes. These considerations will become clear below.

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Figure 2 The characteristic number as a function of the wavelength.

7. Numerical Simulation As discussed in Section 5, this is a multiscale problem. In numerical simulation, the epilayer is divided into grids. To resolve a phase boundary, the grid size should be smaller than b, and the time step should be smaller than τ . Only a finite area of the epilayer is simulated; the infinite epilayer is represented by periodic boundary conditions. To reduce the effect of the boundary conditions on the phase pattern, the period simulated should be much larger than l. For the diffusion process to affect events at size scale l, the total time should be on the order of (l/b)² τ . We have also developed program in the reciprocal space (Chen and Shen, 1998). In addition, without using the diffusion equation, we can minimize the free energy to obtain equilibrium phase patterns. 7.1 A SMALL SINUSOIDAL PERTURBATION AS THE INITIAL CONDITION The parameters in this example are l/b = , Ω = 2.6 and C ave = 0.5. The initial -3 concentration perturbation is sinusoidal, with amplitude 10 and wavelength 2l. We assume that the concentration field varies with x1 but not with x2 . Consequently, the diffusion equation is one dimensional, and the substrate is in the state of plane strain deformation. Figure 3 shows the evolving concentration field. The evolution process appears to have three stages. In the first stage, the perturbation amplitude increases with time exponentially, but the wavelength remains constant, as anticipated by the linear perturbation analysis. In the second stage, stripes of a narrower width grow. This new wavelength is selected by the fastest growth mode predicted by the linear stability analysis. For the present parameters, Eqn. (23) gives λf /l = 0.35. In the third stage, the

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concentration field approaches an equilibrium pattern, with stripe width about l. Energy minimization gives the same equilibrium pattern.

Figure 3 Evolving concentration field. The initial condition is a small perturbation from a uniform field.

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7.2 A LARGE PERTURBATION AS THE INITIAL CONDITION In this example the initial concentration field has a large-amplitude island. All the other parameters are kept the same as before: and C ave = 0.5. Figure 4 shows the evolving concentration field. The evolution process differs conspicuously from that of the previous example, but ends with the same equilibrium pattern.

Figure 4 Evolving concentration field in one dimension. The initial condition is a concentration island.

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7.3 PRELIMINARY RESULTS OF 2D CONCENTRATION FIELDS When the concentration field is two dimensional, the nonuniform surface stress sets a three dimensional elastic field in the substrate. Numerical simulation is time consuming. At this writing, we have limited experience with this general situation. Figure 5 shows a time sequence of concentration field. The basic parameters are kept the same as before. The initial concentration field is a random perturbation of a small amplitude from the average concentration. The computation cell size is 6.4l. The small cell size may affect the phase pattern. It is premature to draw any conclusion from this simulation. Nonetheless, the simulation does produce intricate patterns often seen in experiments.

Figure 5 Evolving concentration field in two dimensions.

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8. Conclusion This paper considers a two-phase epilayer on an elastic substrate. The ingredients for ordering a stable, nanoscopic, periodic phase pattern are identified: (i) unstable solution for phase separation, (ii) phase boundaries for phase coarsening, and (iii) concentrationdependent surface stress for phase refining. We include these ingredients in a CahnHilliard type model. The concentration-dependent surface stress induces an elastic field in the substrate, which is determined by a linear superposition of the Cerruti solution. The elastic field enters the diffusion equation, which updates the concentration field. The results of this model available so far are surveyed, including the governing equations, length and time scales, linear perturbation analysis, and numerical simulation. When the concentration field is restricted within one dimension, our numerical simulations show that the same periodic phase pattern emerges from very different initial conditions. More simulations need be carried out for two dimensional concentration fields. The equations can also be used to study imperfections in a nearly ordered phase pattern, such as dislocations and domain boundaries.

Acknowledgements This work is supported by the Department of Energy through contract (DE-FG0299ER45787). References Alerhand, O.L., Vanderbilt, D., Meade, R.D., and Joannopoulos, J.D. (1988) Spontaneous formation of stress domains on crystal surfaces. Phys. Rev. Lett. 61,1973-1976. Ball, P. (1999) The Self-Made Tapestry, Oxford University Press, UK. Biot, M.A. (1970) Variational Principles in Heat Transfer, Oxford University Press, Oxford. Bower, A.F. and Craft, D. (1998) Analysis of failure mechanisms in the interconnect lines of microelectronic circuits. Fatigue Fracture Engineering Materials Structure 21, 611-630. Cahn, J.W. (1961) On spinodal decomposition. Acta Metall. 9,795-801. Cahn, J.W. (1980) Surface stress and the chemical equilibrium of small crystals—I. the case of the isotropic surface. Acta Metall. 28, 1333-1338. Cahn, J.W. and Hilliard, J.E. (1958) Free energy of a nonuniform system. I. interfacial free energy. J. Chem. Phys. 28,258-267. Carter, W.C., Taylor, J.E., and Cahn, J.W. (1997) Variational methods for microstructural-evolution theories. JOM, 49, No. 12, pp.30-36. Cammarata, R.C. (1994) Surface and interface stress effects in thin films. Prog. Surf. Sci. 46, 1-38. Cammarata, R.C. and Sieradzki K. (1994) Surface and interface stresses. Annu. Rev. Mater. Sci. 24,2 1 5 - 2 3 4 . Chen, L.-Q. and Khachaturyan A.G. (1993) Dynamics of simultaneous ordering and phase separation and effect of long-range coulomb interactions. Phys. Rev.Lett. 70, 1477-11480. Chen, L.-Q. and Shen J. (1998) Applications of semi-implicit Fourier-spectral method to phase field equations. Computer Physics Communications 108, 14-158. Chen, L.-Q. and Wang, Y. (1996) The continuum field approach to modeling microstructural evolution. JOM, Vol. 48, No. 12, pp.13-18. Chuang, T.-J., Kagawa, K-I., Rice, J.R., and Sills, L.B. (1979) Non-equilibrium models for diffusive cavitation of grain interfaces. Acta. Metall. 27, 265-284. Chuang, T.-J. and Rice, J.R. (1973) The shape of intergranular creep cracks growing by surface diffusion. Acta. Metall. 21, 1625-1628. Cocks, A.C.F., Gill, S.P.A., and Pan, J. (1999) Modeling microstructure evolution in engineering materials. Advances in Applied Mechanics, 36, 81-162.

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Freund, L.B. (1995) Evolution of waviness on the surface of a strained elastic solid due to stress-driven diffusion. Int. J. Solids Structures 32, 911-923. Freund, L.B. (1998) A surface chemical potential for elastic solids. J. Mech. Phys. Solids 46, 1835-1844. Freund, L.B. (2000) The mechanics of electronic materials. Int. J. Solids Structures. 37, 185-l96. Gao, H. (1994) Some general properties of stress-driven surface evolution in a heteroepitaxial thin film structure. J. Mech. Phys. Solids 42, 741-772. Glas, F. (1997) Thermodynamics of a stressed alloy with a free surface: coupling between the morphological and compositional instabilities. Phys. Rev. B 55, 11277-11286. Gurtin, M.E. and Murdoch, A.I. (1975) A continuum theory of elastic material surface. Arch. Rat. Mech. Anal. 57, 291-323. Guyer, J.E. and Voorhees, P.W. (1998) Morphological stability and compositional uniformity of alloy thin films. J. Crystal Growth 187, 150-165. Herring, C. (1951) Surface tension as a motivation for sintering. The Physics of Powder Metallurgy, McGraw-Hill, editor Kingston, W.E., New York pp. 143-179. Hull, D. and Rimmer, D.E. (1959) The growth of grain-boundary voids under stress. Phil. Mag., 4, 673-687. Ibach, H. (1997) The role of surface stress in reconstruction, epitaxial growth and stabilization of mesoscopic structures. Surf. Sci. Rep. 29, 193-263. Johnson, K.L. (1985) Contact Mechanics, Cambridge University Press, UK. Kern, K., Niebus, H., Schatz, A., Zeppenfeld, P., George, J., Comsa, G. (1991) Long-range spatial selforganization in the adsorbate-induced restructuring of surfaces: Cu{110}-(2x1) O. Phys. Rev. Lett. 67, 855-858. Khachaturyan, A.G. (1983) Theory of Structural Transformation in Solids, Wiley, New York. Li, D., Diercks, V., Pearson, J., Jiang, J.S., and Bader, S.D. (1999) Structural and magnetic studies of fcc Fe films with self-organized lateral modulation on striped Cu{110}-O(2x1) substrates. J. Appl. Phys. 85, 5285-5287. Lu, W. and Suo, Z. (1999) Coarsening, refining, and pattern emergence in binary epilayers, the Fred Lange Festschrift in the journal Zeitschrift fur Metallkunde. In Press. McMeeking, R.M. and Kuhn, L.T. (1992) A diffusional creep law for powder compacts. Acta Metall. Mater. 40, 961-969. Mullins, W.W. (1957) Theory of thermal grooving, J. Appl. Phys., 28, 333-339. Needleman, A and Rice, J.R. (1980) Plastic creep flow effects in the diffusive cavitation of grain boundaries. Acta Metall. 28, 1315-1332. Ng, K.-O. and Vanderbilt, D. (1995) Stability of periodic domain structures in a two dimensional dipolar model. Phys. Rev. B 52, 2177-2183. Pohl, K., Bartelt, M.C., de la Figuera, J., Bartelt, N.C., Hrbek, J., Hwang, R.Q. (1999) Identifying the forces responsible for self-organization of nanostructures at crystal surfaces. Nature 397, 238-241. Pompe, W., Gong, X., Suo, Z. and Speck, J.S. (1993) Elastic energy release due to domain formation in the strained epitaxy of ferroelectric and ferroelastic films. J. Appl. Phys. 74, 6012-6019. Prigogine, I. (1967) Introduction of Thermodynamics of Irreversible Processes, 3rd edition, Wiley, New York. Rayleigh, J.W.S. (1894) The Theory of Sound, Vol. 1, Art. 81. Reprinted by Dover, New York. Rice, J.R. and Chuang, T.-J. (1981) Energy variations in diffusive cavity growth. J. Am. Ceram. Soc. 64, 4653. Roytburd, A.L. (1993) Elastic domains and polydomain phases in solids. Phase Transitions, 45, 1-33. Srolovitz, D.J. (1989) On the stability of surfaces of stressed solids. Acta Metall. 37, 621-625. Suo, Z. (1997) Motions of microscopic surfaces in materials. Advances in Applied Mechanics. 33, 193-294. Suo, Z. (1998) Stress and strain in ferroelectrics. Current Opinion in Solid State & Materials Sicence, 3, 486489. Suo, Z. (2000) Evolving materials structures of small feature sizes. Int. J. Solids Structures. 37, 367-378. Suo, Z. and Lu, W. (2000) Composition modulation and nanophase separation in a binary epilayer, J. Mech. Phys. Solids. In press. Willis, J.R. and Bullough, R. (1969) The interaction of finite gas bubbles in a solid. J. Nuclear Mater. 32, 7687. Wu, C.H. (1996) The chemical potential for stress-driven surface diffusion. J. Mech. Phys. Solids 44, 20592077.

ELASTIC SPACE CONTAINING A RIGID ELLIPSOIDAL INCLUSION SUBJECTED TO TRANSLATION AND ROTATION

M. KACHANOV, E. KARAPETIAN AND I. SEVOSTIANOV Department of Mechanical Engineering, Tufts University Medford, MA 02155

1. Introduction The problem of a linear elastic space containing a rigid ellipsoidal inclusion subjected to translation and rotation is critically overviewed and further developed. Full elastic fields, as well as the “stiffness relations” that give forces and moments, that have to be applied to the inclusion in order to produce the given rotations and displacements, are given. This problem can be viewed as supplementary to Eshelby's problem for an ellipsoidal inhomogeneity (Eshelby, 1961) which does not cover the rigid body motion of the inhomogeneity. The problem is of a practical interest, for example, for the geotechnical applications (Selvadurai, 1976). In contrast with Eshelby’s problem, it has not been fully analyzed. The problem was first considered, probably, by Keer (1965) in the special case of a rigid circular disc subjected to translation in the disc plane. The case of a general ellipsoid was considered by Lur’e (1970). However, his solution is incomplete: it is not expressed in terms of any standard functions and results for the important case of a spheroid (that do not follow from the general case in a straightforward way) were not given; besides, his work contains misprints and minor errors. Kanwal and Sharma (1976) considered the case of a spheroid and derived tractions on the spheroid’s boundary and relations between the overall forces and moments applied to the spheroid and its displacements and rotations (“stiffness relations”). However, the full fields in the elastic space were not given. Selvadurai (1976) considered the spheroidal inclusion subjected to translation parallel to the spheroid’s axis and gave full fields in this special axisymmetric case. The analysis was extended to the case of the transversely isotropic space with a rigid spheroidal inclusion in several works. Selvadurai (1979, 1980) considered this problem in the case of a circular disk subjected to translation and to rotation about the axis of the symmetry. Zureick (1988, 1989) considered the case of a spheroid, but the solution is not given in the closed form. Rahman (2000) considered a rigid disc of the elliptical shape subjected to translation normal to the disc plane and gave the closed form solution. The present work focuses on the case of the isotropic space and further advances the existing results, by explicitly deriving the full set of elastic fields and “stiffness relations” 123

T.-J. Chuang and J. W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 123–143. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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in the closed form for all cases, including, in particular, the important case of a spheroid (requiring a non-trivial limiting transition). The derivation generally follows the approach of Lur’e (1970).

2. General Ellipsoid We consider a rigid ellipsoid, with the surface given by equation

(2.1)

embedded into an infinite elastic medium. It is subjected to small translation u0 and small rotation ù , both of arbitrary directions. Equation (2.1) can be rewritten in terms of (instead of a 1 , a 2 , a3)

parameters as follows:

(2.2)

where a ρ0 ,

are ellipsoid's semiaxes. The points of the boundary

( ρ = ρ0 ) undergo displacements given by u

ρ=ρ 0

= u0 + ù × R 0

(2.3)

where R 0 is the position vector of a point on the ellipsoid's surface.

We use orthogonal ellipsoidal coordinates (ρ, µ,v) that are related to cartesian coordinates ( x1 , x 2 , x 3 ) as follows:

(2.4)

where

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2.1. FIELDS PRODUCED BY TRANSLATION OF AN ELLIPSOID We utilize Papkovich-Neuber’s general representation of displacements in terms of harmonic vector B and harmonic scalar B 0 :

(2.5)

where we further express B 0 and components B m of B in terms of four potential functions F0 , F1 , F 2 , F3 as follows: B m = C m F0 , B

0 =D m Fm ,

m =

1,2,3

(2.6)

= u0. We seek such potential functions that they are harmonic everywhere except for the ellipsoid’s surface and tend to zero at ρ → ∞. Therefore, they are taken as potentials of a simple layer and are as follows. Inside the ellipsoid: and constants C m , Dm are to be determined from boundary conditions u

F0 =1,

Fm =x m ,

m = 1,2,3,

ρ ≤ ρ0

ρ =ρ0

(2.7)

Outside of the ellipsoid: (2.8) where ψ i (ρ ) are given by the elliptic integrals:

(2.9)

Utilizing (2.6-8), representation (2.5) can be written in terms of ψi (ρ) :

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(2.10)

Differentiating functions ψ i ( ρ) :

(2.11)

brings (2.10) to the form:

(2.12)

Boundary conditions uρ = ρ0 =u0 lead to the following system of six linear algebraic equations for constants C m , D m (that decouple into three separate systems of two equations each):

(2.13)

Utilizing the relationship for curvilinear coordinates Lame’s coefficients, yields:

, where H s a r e

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(2.14)

Substituting (2.14), along with expressions for constants C m ,D m obtained by solving (2.13), into (2.12) yields the displacement field outside of the ellipsoid ( ρ > ρ 0 ):

(2.15)

where the following notations are used:

(2.16)

(2.17)

and where ψ i ( ρ) entering (2.16), after evaluation of integrals in (2.9), take the form:

(2.18)

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Here

are the incomplete

elliptic integrals of the first and the second kinds, respectively, and ϕ = arcsin(1/ ρ). We now derive the expression for the stress vector t on the surface of the ellipsoid ρ = ρ 0 . We express functions B m in the alternative form, namely, as potentials of a simple layer, with t being the density of the layer: (2.19)

On the other hand, relations (2.6), (2.8) and (2.13) yield the following expressions for components of vector B:

(2.20)

The theorem on discontinuity of the normal derivative of the potential of a simple layer implies that

(2.21)

Therefore, based on (2.20), we have:

(2.22)

Further, utilizing (2.11), (2.14) and the third relationship of (2.17), the tractions on the ellipsoid’s surface ρ = ρ 0 are obtained as follows: (2.23)

0

Components of the resultant force T that is required to produce translation u (obtained by integration of (2.23) over the ellipsoid’s surface) are given by the following “stiffness relations”:

ELASTIC SPACE WITH AN INCLUSION

129 (2.24)

The resultant moment M = 0 . 2.2. FIELDS PRODUCED BY ROTATION OF AN ELLIPSOID The solution due to an arbitrary rotation ù of the ellipsoid is obtained by taking harmonic vector B and harmonic scalar B 0 in Papkovich-Neuber’s representation (2.5) in the form:

(2.25) Here, functions ψ 1 ( ρ),ψ ψ 6 (ρ ) are as follows:

2

( ρ) and ψ 3 (ρ) are given by (2.9) and ψ 4 ( ρ ),ψ 5 ( ρ ) and

(2.26)

Nine constants N m , D m , D' m condition u ρ =

ρ0

(m = 1,2,3) are to be determined from the boundary

= ù × R 0 , or, in components: (2.27)

Substitution of (2.25) into (2.5) and utilization of (2.11) yield expressions for the displacement components. Boundary conditions (2.27) lead to a cumbersome system of nine linear algebraic equations for the constants N m , D m , D ' m . However, solving this system can be simplified by utilizing the superposition: the problem for an arbitrary rotation vector ù is represented as system of the three sub-problems corresponding to components ω1 , ω 2 , ω3 . Each of these sub-problems gives rise to a system of only three equations for three constants. In the case of ω2 = ω3 = 0:

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M. KACHANOV, ET AL. (2.28)

In the case of ω1 = ω 3 = 0: (2.29)

In the case of ω1 = ω 2 = 0 : (2.30)

Finding the constants this way yields the displacement field due to an arbitrary rotation ù (outside of the ellipsoid, ρ > ρ 0 ) as follows:

(2.31)

where the following notations are used:

(2.32)

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131

and where ψ i ( ρ), after evaluation of integrals in (2.26), takes the form:

(2.33) (m = 1,2,3) are obtained from g m ( ρ) by changing the first term of g m (ρ) according to the following rule:

Functions

Note that Traction t on the ellipsoid’s surface is obtained by using the theorem on discontinuity of the normal derivative of the potential of a simple layer, see (2.21) and utilizing (2.11), (2.14) and the third relationship of (2.17). The expression for t is as follows:

(2.34)

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Components of the resultant moment M that is required to produce rotation ù (“stiffness relations”) are:

(2.35) where

(2.36)

The resultant force T = 0 .

3.

Oblate Spheroid

In this case, a 1 = a2 ( ≡ a ρ 0 ) . The solution in this case cannot be obtained from the one for the general ellipsoid by a straightforward substitution, but requires a non-trivial limiting procedure. This is related to the fact that the ellipsoidal coordinates have to be changed to the oblate spheroidal ones. In all the equations for the general ellipsoid, we impose the following condition: e → 0, v → 0 in such a way that the ratio v /e remains finite. We also set: (3.1) The relationships between cartesian coordinates (x1 ,x 2 , x3 ) and curvilinear coordinates ( s , q , φ ), obtained from (2.4) by the limiting transition, take the form:

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133 (3.2)

where 0 ≤ s < ∞ , –1 ≤ q ≤ 1. The boundary of the oblate spheroid corresponds to s = s0 and is given by the following equation in Cartesian coordinates:

(3.3)

3.1. FIELDS PRODUCED BY TRANSLATION OF AN OBLATE SPHEROID Utilizing (3.1-3), the expressions in (2.17,18) can be reduced to elementary functions, as follows:

(3.4) The displacement field outside of the spheroid, s > s0 , due to translation u0 takes the form:

(3.5)

where the following notations are used:

(3.6)

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Traction vector t on spheroid’s surface s = s 0 is:

(3.7)

Components of resultant force T , that is required to produce translation u 0 , are given by the following “stiffness relations”:

(3.8)

The resultant moment M = 0 . 3.2. FIELDS PRODUCED BY ROTATION OF AN OBLATE SPHEROID The expressions in (2.33), in the case of the oblate spheroid, can be reduced to the following elementary functions:

(3.9)

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The displacement field outside of the spheroid, s > s 0 , due to rotation ù , after some algebra, takes the form:

(3.10)

where the following notations are used:

(3.11)

Traction vector t on the oblate spheroid’s surface, s = s 0 , due to rotation ù , is:

(3.12)

Components of resultant moment M , that is required to produce rotation ù , are given by the following “stiffness relations”:

M. KACHANOV, ET AL.

136

(3.13)

where

The resultant force T = 0 .

4.

Prolate Spheroid

In this case, a 2 = a 3

Similarly to the case of the oblate spheroid, the

solution cannot be obtained from the one for the general ellipsoid by a straightforward substitution, but requires a non-trivial limiting procedure. This is related to the fact that the ellipsoidal coordinates have to be changed to the prolate spheroidal ones. In all the equations for the general ellipsoid, we impose the following conditions:

(4.1)

The relationships between cartesian coordinates (x 1 , x 2 , x 3 ) and curvilinear coordinates ( s, q, φ ), obtained from (2.4) by the limiting transition, take the form: (4.2) where 1 ≤ s < ∞, –1 ≤ q ≤ 1. The boundary of the prolate spheroid corresponds to s = s 0 and is given by the following equation in Cartesian coordinates: (4.3)

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4.1. FIELDS PRODUCED BY TRANSLATION OF A PROLATE SPHEROID Utilizing (4.1-3), the expressions in (2.17,18) can be reduced to elementary functions, as follows:

(4.4)

The displacement field outside of the spheroid, s > s 0 , due to translation u 0 , takes the form:

(4.5)

where the following notations are used:

(4.6)

Traction vector t on the spheroid's surface s = s 0 is:

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M. KACHANOV, ET AL. (4.7)

Components of resultant force T , that is required to produce translation u0 , are given by the following “stiffness relations”:

(4.8)

The resultant moment M = 0 . 4.2. FIELDS PRODUCED BY ROTATION OF A PROLATE SPHEROID The expressions in (2.33), in the case of the prolate spheroid, can be reduced to the following elementary functions:

(4.9)

The displacement field outside of the spheroid, s > s0 , due to rotation ù , after some algebra, takes the form:

(4.10)

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139

where the following notations are used:

(4.11)

Traction vector t on the prolate spheroid’s surface, s = s 0 , due to rotation ù , is: (4.12)

Components of resultant moment M , that is required to produce rotation ù , are given by the following “stiffness relations”:

(4.13)

M. KACHANOV, ET AL.

140 where

The resultant force T = 0 . 5.

Rigid Sphere

We now consider the simplest case of a rigid sphere of radius R 0 embedded into an infinite elastic medium. It is given small translation u 0 and small rotation ù , so that displacements of the points of the sphere’s boundary are: (5.1) where R 0 is the position vector of points on sphere's surface. To determine the displacement and stress fields outside of the sphere, we utilize Trefftz's general solution for displacements. The displacement field, outside of the sphere ( R > R 0 ), can be obtained in the following form:

(5.2)

(5.3) In the text to follow, we transform these expressions to a more convenient representation of displacements (and stresses) in spherical coordinates (R , θ , φ), with unit vectors e R , eθ , e φ . 5.1. FIELDS PRODUCED BY TRANSLATION OF A SPHERE In this case u 0 = u 0 i , where i is the unit vector along x - axis. Using the relationship

(5.4)

where H R =1, H θ = R , H φ = R sin θ are Lame’s coefficients, we have: (5.5)

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and, since R / R = e R , we also have (5.6)

After substitution of (5.5,6) into (5.2) and some algebra the displacement components in spherical coordinates take the form:

(5.7)

and the dilatation is: (5.8)

Stress components in spherical coordinates are obtained as follows:

(5.9)

On the surface of the sphere ( R = R 0 ), the stresses are:

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(5.10)

so that the traction vector on the sphere's surface is: (5.11)

The resultant force T, required to produce translation u 0 , is given by the following “stiffness relation”: (5.12)

The resultant moment M = 0. 5.2. FIELDS PRODUCED BY ROTATION OF A SPHERE We now consider the displacement field u due to rotation vector ù = ω 0 k. Since k = e R cos θ – e θ sin θ , substitution of ù into (5.3) gives the only non-zero displacement component in spherical coordinates as follows:

(5.13) The only non-zero stress component is:

(5.14) The traction vector on the surface of the sphere is: (5.15) The resultant moment M, required to produce rotation ù , is given by the following “stiffness relation”:

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143 (5.16)

The resultant force T = 0 .

6.

Conclusions

Closed form expressions are derived for the full set of elastic fields generated by a rigid ellipsoidal inclusion embedded into an infinite elastic space and subjected to (arbitrarily oriented) translations and rotations. “Stiffness relations” that interrelate the displacements and rotations of the inclusion to the forces and moments applied to it are also given.

Acknowledgement. The authors are grateful to Dr. Rahman for making a preprint of his work available. This work was supported by the National Science Foundation and U.S. Department of Energy through grants to Tufts University. References Eshelby, E.J. (1961) Elastic inclusions and inhomogeneities. Progress in Solid Mechanics, eds. Sneddon J.N., Hill R., North-Holland, Amsterdam, V. 2, pp. 89-140. Kanwal R.P and Sharma D.L. (1976) Singularity methods for elastostatics. J. Elasticity, 6, pp.405-418 Keer, L.M. (1965) A note on the solution for two asymmetric boundary value problems. Int. J. Solids Structures, 1, pp. 257-264. Lur’e, A.I. (1970) Theory of Elasticity. Nauka, Moscow (in Russian). Rahman, M. (2000) The normal shift of a rigid elliptical disk in a transversely isotropic solid. Int. J. Solids Structures (in press). Selvadurai A.P.S. (1976) The load-deflection characteristics of a deep rigid anchor in an elastic medium. Geotechnique, 26, pp. 603-612. Selvadurai A.P.S. (1979) On the displacement of a penny-shaped rigid inclusion embedded in a transversely isotropic elastic medium. SM Arch., 4, pp. 163-172. Selvadurai A.P.S. (1980) Asymmetric displacements of a rigid disc inclusion embedded in a transversely isotropic elastic medium of infinite extent. Int. J. Engng. Sci., 18, pp. 979-986. Zureick A.H. (1988) Transversely isotropic medium with a rigid spheroidal inclusion under an axial pull. ASME J. Appl. Mech., 55, pp. 495-497. Zureick A.H. (1989) The asymmetric displacement of a rigid spheroidal inclusion embedded in a transversely isotropic medium. Acta Mech., 77, pp 101-110.

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STRAIN PERCOLATION IN METAL DEFORMATION

R. M. THOMSON Emeritus, Materials Science Engineering Laboratory, NIST, Gaithersburg, MD 20899 L. E. LEVINE Materials Science Engineering Laboratory, NIST, Gaithersburg, MD 20899 AND Y. SHIM Center for Simulational Physics, University of Georgia, Athens, GA 30602

Abstract. In previous papers, we have introduced a percolation model for the transport of strain through a deforming metal. In this paper, we summarize the results from that model, and discuss how the model can be applied to the deformation problem. In particular, we outline the primary experimental features of deformation which the model must address, and discuss how the model is to be used in such a program. It is proposed that the discrete percolation events correspond to slip line formation in a deforming metal, and it is shown that the deforming solid is a self organizing system. It is recognized that deformation is localized in space and time, that deformation is fundamentally rate dependent, that hardening depends upon relaxation processes associated with discrete percolating events, and that secondary slip is an essential part of band growth and relaxation processes.

1. Preface It is a pleasure to participate in this celebratory volume to Jim Rice. Our collaboration on the problem of dislocation emission from cracks began in the Summers at the ARPA Materials Research Council, and eventually matured in our 1974 paper. I learned from that experience that Jim is one of those rare people who can work creatively in more than one field at once, 145 T.-J. Chuang and J. W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 145–157. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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because he so quickly grasps the essence of the main issues, even when the scientific language and usage is quite different from his own. In addition, I have been privileged to be a friend of an unusually kindly and generous person over the years. Jim, I wish you many more years of productive and distinguished accomplishment in science! 2. Introduction A successful theory of work hardening in metals would include both an analysis of the pattern formation of dislocations during metal deformation and the transport of mobile dislocations through the dislocation structures that are formed. A treatment of pattern formation would include the change from one type of pattern to another (e.g. carpets to cells) as well as the evolution of the size, distribution and shape of a basic pattern as deformation proceeds. We have not attempted such a complete theory, and suspect that a satisfactory attempt is still some distance in the future. Rather, we have noted that the pattern formation and transport parts of the problem are logically rather different, and have chosen to focus on just the transport half. In our work, then, the structure and its evolution is an input, although we find that certain features of the transport problem are useful in understanding the pattern size evolution. A useful dividend of our focus on the transport problem is that the constitutive laws and stress/strain relations are closely related to the transport problem, and should follow rather straight forwardly from an adequate understanding of it. Another important part of our approach and our motivation is the realization that the metal deformation problem is dominated by probability and statistics, and we have been guided by a hope that developments in modern statistical physics and critical phenomena might provide useful insights and concepts for understanding the deformation problem. We believe this work has been at least a partial success in implementing that hope. One of the most active areas in modern research on deformation has been the direct application of computer simulation to deformation, and the development of what is now called dislocation dynamics. We believe that the work here is complementary to dislocation dynamics in the sense that our model is viewed as a statistical theoretical framework, not a first principles theory, and contains a variety of internal variables and input parameters, which must be obtained from outside the model. These inputs are expected to be supplied by some combination of incisive experiment and dislocation dynamics addressing the question of how dislocation cells operate both as sources and sinks for mobile dislocations. This paper will first provide an overview of the model and its primary

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mathematical features and results, including a “universal” stress/strain relation. We then summarize the essential experimental findings to which the model must relate, discuss time dependent effects, and finally, list experimental and dislocation dynamics studies that are needed to provide essential inputs to the overall modeling effort. 3. The Model The model ((Thomson and Levine, 1998; Thomson et al., 2000)) is built on the premise that the system has been brought to a level of strain corresponding roughly to stage III, where a cellular structure has developed. (The model can probably be applied to more general situations, but these have not yet been explored.) The basic idea is that the walls of the cell structure provide both sources of mobile dislocations which move through the interior of the cell, and barriers to the fully free motion of those dislocations. When mobile dislocations are arrested by the other walls of the cell in which they have been formed, they exert concentrated stresses on the incipient sources in these arresting walls, and the process repeats. In detail: 1. The system is brought to a definite state of unidirectional strain characterized by the plastic strain, εp , and a flow stress, 2. The weakest cell in a band of slip undergoes a burst of strain which initiates a cluster of newly strained cells. The initiating strain is s0 . For convenience, we will simply take the strain variable for each cell as the number of mobile dislocations created in that cell during a specific percolation event. 3. The transmission of strain from a strained cell to a neighboring unstrained cell is given by s* = as, (1) where s is the number of dislocations in the strained cell, s* is the number induced in the previously unstrained neighboring cell, and a is the amplification factor, a stochastic function characterizing the wall between a strained and unstrained cell. 4. Because s corresponds to the discrete number of mobile dislocations in a cell, s* ≥ 1. (We do not take the trouble to discretize the variable, but it is necessary that it always be at least unity.) 5. Growth takes place only on the boundary of a strained 2-d cluster of cells, because any cell strained in a percolation event will be hardened against further straining during that event. 6. The stochastic amplification factor depends on the properties of the wall, and can be written a = a ( P1, P 2 ,…). where P 1 , P 2 , etc. are parameters. We visualize at least two separate mechanisms for transmit-

148

R. M. THOMSON ET AL. ting strain from cell to cell, a source mechanism and a lock breaking mechanism. We have investigated two possible cases. In case I, only the source mechanism operates. In case II, both mechanisms play a role. In the source mechanism, the wall possesses a distribution of dislocation sources of varying strength whose maximum strength is given by the parameter P1 . For case I, we shall write (2) where ζ is a random number. In the second mechanism for transmitting strain, we presume that the cell walls are localized in the lattice by locks of various sorts and strengths, such as Lomer-Cottrell locks, and that these locks can be broken, (unzipped) under the stress of the pile-up dislocations. When that happens, the region of the wall stabilized by the lock will break away as mobile dislocations into the next cell, with a large amplification factor. For case II, we then modify Eqn. (2) to give

(3) where P 2 is the amount of strain released in an unzipping event and k is a measure of the probability of such an event occurring. It will be assumed that P 2 >> 1 and k << 1 ,corresponding to the assumption that unzippable locks are relatively rare, but when they do occur, the resulting strain is large. 7. In a computer implementation, one begins a simulation at the cell at the origin, for which an initiating value of strain, s 0 , is assumed. Given the rules for strain transmission, the initiation value must be greater than a limiting critical value, s 0 > 1/P 1 , in order for the strain to grow out of the origin. (This critical value is only correct if, as in case I, the sites next to the origin have no walls with unzippable locks.) These rules constitute a well posed percolation problem, and a unique percolation threshold is observed at P1 = P1 c in case I. In case II, a percolation threshold also exists, but in this case the threshold lies on a critical surface, C ( P1 , P 2, k ) = 0. We have shown that the geometrical aspects of the percolating strain clusters conform to the universality class of ordinary percolation theory ((Stauffer and Aharony, 1992)). Results for case I have been published ((Thomson et al., 2000)). Fig. 1 shows a typical run for case II for a cluster which spans the system size chosen at the critical point.

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Spanning cluster at the critical point. Dark regions represent larger strain Figure 1. and lighter represent smaller strain in a cell. Here, s 0 = 2.2, P 2 = 40, and k 0 = 0.01 for L = 401 with P 1c = 0 . 6 4 5 2 . L is the linear dimension of the system.

Since the strain percolation problem contains the strain variable, s , which has no counterpart in standard percolation theory, the scaling laws for 〈 s 〉 must also be worked out. We have done so for case I ((Shim et al., 2000)), but case II is still incomplete. In case I, the scaling law for the strain is found to be (4) where T is the strain per cell site in the system, γ = 2.389 is one of the exponents of standard percolation theory, S is the average number of strained sites in the system, and χ is a new exponent whose value is determined to

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be χ = 0.75 ± 0.25 (one standard deviation), and c is a constant. The ratio Tc /Sc is the strain per strained site in the system at the critical point. For this ratio to be finite, it is necessary that T have the same fractal dimension with at the critical point as does the geometric cluster, d ƒ = 1.896. In the discussion immediately following, the behavior of the strain above the percolation threshold is found to be the important physical quantity. Equation (5) is valid both below and above the critical point, but above the critical point, the law takes the simpler form, (5) where β = 0.14 is one of the standard percolation parameters. Since the geometrical density of strained sites per system site grows as above the critical point in case I (with just one percolation critical parameter), the meaning of Eqn. (5) is that the strain growth is controlled and dominated by the geometry above the critical point. This super critical growth does not extend very far above the critical point, because as P1 – P c increases, the system reaches a second critical point where the average strain diverges in a strain avalanche. That is, immediately above the critical point, even though the geometry has exceeded its percolation threshold, and its fractal character has disappeared, the strain seems only subliminally aware that something significant has happened. That is, if we examine the strain per strained site, in the system, has a well defined fixed value for large system sizes below the critical point, and grows only slowly in nearly linear fashion above the critical point, increasing only about 40% in the super critical region below the avalanche. Since this contribution to the strain per cell is only slowly varying in the super critical regime, and the geometrical density of strained sites is increasing quite rapidly, the geometrical increase dominates. That is, when one plots the strain per cell in the super critical regime, the linear increase mentioned above gets lost in the strong increase which signals the approach of the avalanche. This change in ∈( P ) can therefore be realistically neglected when plotting T ( P ), below the avalanche, so that T can simply be regarded as exhibiting the geometrical increase of strained clusters above the percolation critical point. But it is interesting that the mathematical model exhibits two separate critical points slightly separated from one another: a percolation threshold and an avalanche point. Of course, any physical significance of the avalanche point must be considered in the light of the very severe limits one must put on the upper limit to the number of dislocations which a wall can produce.

STRAIN PERCOLATION IN METAL DEFORMATION 4 . A Universal

Constitutive

151

Law.

In the following, we are mainly interested in generic results, so we shall limit ourselves to case I with the single percolation parameter, P1 . We postulate that the model corresponds to actual strain percolation events in a real system, and that the strain in the deforming solid is the accumulated strain in the percolation events. It turns out that the strain at the percolation threshold is exactly zero in the “thermodynamic limit” of infinite system size, and thus finite strain only appears slightly above the threshold in the super critical regime. Thus, as the system moves along the stress strain curve, it remains always at the critical threshold, or actually just slightly above it. This behavior defines a self organizing critical (SOC) system. Since the percolation parameters describe the production of dislocations from the cell walls, and since this ability depends on both the flow stress, as well as the state of strain in this unidirectional straining system, the percolation parameters must depend on the strain, ε , and the stress, . Thus, in the simple case I, P1 = P 1 ( , ε ), and on the stress/strain curve, (6) (A somewhat more complicated, but equivalent, treatment follows for case II, where the weak walls are included.) If one remembers that the percolation events are discrete, then from one percolation event to the next, we can write the stress/strain law as a “universal” relation, (7) This equation is rigorous, and follows from the SOC of the deformation, which is the reason we call it a universal law. But it can take on physical meaning only when it is possible to relate the percolation parameter to the stress and strain. In . view of the way the model is constructed, and its fundamental relation to the way the cell walls create and obstruct mobile dislocations in the system, this functional dependence can only be determined by a detailed study of what we term dislocation “cell physics”. Hopefully, from this study, the two physical mechanisms by which we believe dislocations are produced (source action and unzipping) can be distinguished, and stochastic distributions determined, from which percolation variables can be deduced. It is in this sense that we claim that the strain percolation model provides a stochastic framework for a deformation theory, but the inputs to this framework must be determined by separate study of the dislocation cell phenomenology.

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In the following section, we will take some tentative steps in this direction by looking at what the existing experimental lore can tell us about the cell physics, and ultimately the constitutive relations. 5. Real Slip In this section, we review the major features of slip in a deforming metal to which the percolation model must conform. The model is already consonant with the existence of the cell structure, by its nature. But slip in metals is localized in space and time in characteristic ways which we now explore. In all of the standard stages of strain, the elementary slip events are observed as fine slip lines on the surface of the deforming metal. Up through Stage III, these lines are many micrometers in length (but they shorten with strain), and are up to several hundred Burgers vectors in height. We will identify the slip which occurs in a percolation event in our percolation model with this observed “universal” elementary unit of slip. A striking observation which begins in Stage II, and is fully developed in Stage III, is the spatial localization of the percolation events into bands of slip. This localized ordering of the percolation events into bands is critical to understanding the stress-strain law, because the percolation events in a band interact strongly with one another. Our principal focus is on aluminum, and the quantitative study of the bands in Al goes back to early electron microscope replica studies by H. and D. Wilsdorf ((Wilsdorf and Kuhlmann-Wilsdorf, 1952)) and by Noggle and Koehler ((Noggle and Koehler, 1957)). In addition to this early work, we are aware of a more modern measurement of Al band structure by W. Tong et al. ((Tong et al., 1997)) using atomic force microscopy (AFM). One of the authors, H. Weiland ((Weiland, )), reported some additional AFM results to us privately. These AFM measurements are consistent with the early and much lower resolution electron microscope replica work. All of the results are quoted for room temperature aluminum at strain levels where the cell structure should be well developed. Weiland differs from the other authors in that his aluminum samples were doped with Mg, while the others worked with high purity metal. A summary of the main conclusions of these studies is: 1. About 80% of the total strain is concentrated in the bands. But the minority strain taking place in the matrix means that the matrix is hardening along with the bands. 2. Both bands and slip lines are very long compared to a typical cell size. 3. The number of slip lines contained in a band varied from a few to the order of 100, with an average in the range of 20. This number increased slowly with total strain.

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4. The slip line height increases with strain, but in general, the authors find heights between about 10 and several hundred Burgers vectors, with an average in the neighborhood of 30-50 ((Noggle and Koehler, 1957; Weiland, )). 5. The distance between slip lines appears to have a minimum value of about 30 atomic distances, so there is a natural width to a percolation event. Data on the time localization of slip are available from experiments reported by Pond ((Pond Sr., 1972)) on high purity Al at room temperature, but total strain levels are not reported. From the author’s comment that measurements were made till the slip lines interfered with one another, it is presumed that these data are also representative of strain at a level where the cells were well developed. The data were obtained from optical cinematography on the metal surfaces. Pond finds that a slip band grows in discrete jumps, with individual jumps taking place in about 0.1 sec., and with several to many seconds between jumps. The amount of growth in a jump varies from 50 b to 500 b, with an average around 150 b to 200 b. If these data can be correlated with the slip line heights reported above, it appears that the band growth takes place with the production of a small number (around 3 to 7) of slip lines in an average growth event. In summary, the slip is localized in both time and space, probably with multiple percolation events energizing one another in the discrete growth jumps occurring in a band. The band grows by both filling in the allowable space between slip lines (percolation events) and by growth in width of the band. (In experiments in Cu ((Mader, 1957)), it is known that individual bands broaden with strain, and that new slip bands are nucleated out of the matrix, but corresponding experiments have not been done in Al.) The slip line heights are greater at larger strain, which may be because multiple percolation events can occur in the same location, or perhaps because the slip generated in a percolation event at the higher strain levels is greater. Clearly, the phenomenological picture needs much clarification, especially as regards the quantitative aspects of band growth. 6. Relaxation Modeling of Bands We have shown how a “universal” stress-strain law follows from a simple assumption that our discrete percolation model contributes all the strain in the deforming metal. If the percolation events of our model are identified with the elementary slip lines observed in all deforming metals, one can have high confidence that it is correct. But there is much physics hidden in the universal law, and we will now construct an additional level of detail

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based on the findings we have summarized in the previous section. In the current section, we focus on the growth of a well developed band containing a relatively large number of elementary slip lines. The first question that immediately asserts itself is why the bands exist in the first place? Equivalently, why is a local region where multiple slip lines have been produced softer than the matrix? There is not space here to enter into an adequate discussion of this question, but we believe the relevant mechanism is the rotation created by the, not necessarily local, interaction between the majority primary dislocations which have been created in the slip lines, and the secondary dislocations. In brief, the secondary dislocations which are produced in the slip band are all of such a character as to rotate the lattice in a slip band region in a softening direction relative to the matrix. That is, the rotation is such as to increase the resolved shear stress on the primaries. Since at the start of a percolation event, the flow stress in the band region must be the same as that in the matrix, because the matrix and band regions both deform, the rotation of the band constitutes an instability of the band relative to the matrix. This mechanism has its seat in the uncompensated dislocations which are produced in regions of strain gradients. We now consider in a very generic way the effect of a percolation event on the stress level in a band. In the following, we shall not distinguish between a single percolation event and several correlated simultaneous percolation events. If we assume the system is in a hard tensile machine with a prescribed strain rate, then the strain, ε , in the sample can be written as a linear function of time, t, as (8) where subscripts p and e refer to the plastic and elastic parts. We consider the time between one (the reference) percolation event and the next. At the moment of the event when t = 0, the strain in the system is increased suddenly by the strain, δ ε p , in the percolation event, and the stress in the system is lowered by the amount, – µ δε p , where µ is the elastic constant. After time zero, the stress increases linearly because of the way the system is loaded, and (t) = µ ( t – δ ε p ). When the actual stress, (t), reaches the flow stress, p (t), another event will occur, and the system cycles. By definition, the system at time t = 0 was at the flow stress for that state of strain, and if the flow stress does not change, then the time between events will be given by the time for the tensile machine to build the stress back to the initial level. But the flow stress is altered by the strain in the percolation event. The flow stress in the slip plane of the event(s) is immediately increased because of the local increase of immobilized dislocations in the slip line plane(s). In the time provided by the stress drop

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and subsequent build up, the immobilized dislocations will recover by cross slip, climb, etc., thereby altering the flow stress both within the immediate slip plane of the event as well as a region parallel to that plane which is reachable by short range cross slip and climb processes. In addition, secondary slip will be initiated by the percolating primary slip, and these non-percolating secondary dislocations will move on slip planes at a large angle to the primary plane. Presumably the secondaries have a mean free path of the order of a single cell size. Their effect is three fold. First, they will contribute forest dislocations to cell walls throughout the active band, and second, they will interact with incipient sources on other slip-line planes in the band. This second interaction is a mechanism for initiating multiple percolation events simultaneously, and it is also a possible mechanism for enhancing the source distribution throughout the band for a subsequent percolation event. Third, the additional secondaries, in their interaction with the primary dislocations of the band will rotate the band relative to the matrix, thereby softening it. This complex set of processes has the effect of increasing the flow stress of the band over the time between successive slip events, but its precise form is very difficult to surmise without more detailed knowledge about the cell physics. We write this unknown function as (9) because the stress relaxation function, R(t), must be linear in the strain increment of the percolation event. µ' is a constant. The next event will occur at the time when the new flow stress is equal to the stress building in the deforming sample, (10) which has a complicated implicit dependence on the time. However, since the slope of the stress strain curve in the plastic regime is much smaller than the elastic constant, µ, we can take the time between events as simply δ t = δε p / , and the stress-stain relation follows, (11) In writing this equation, we make the further assumption that R(t) is a slowly varying function in the neighborhood of t = δ t . This equation now fleshes out some of the underlying relaxation physics in the earlier universal relation, by focusing attention on the mechanisms by which the flow stress relaxes during band growth. It also emphasizes that deformation in metals in Stage III is at its most fundamental level, a rate process. Finally, it shows that deformation at this most fundamental level is a discrete process, and

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that the average strain in a discrete percolation event, δε p , is a fundamental quantity, which appears explicitly in the final stress-strain relations. The connection between the universal equation and the relaxation equation appears when one relates the percolation parameter, P 1 c to the relaxing flow stress, δ p . 7.

Conclusions

and Needed Experiments

The universal stress-strain law obtained from the SOC character of the deforming metal must be extended to include the consequences of the ordering of the percolation events into spatially localized slip bands. This ordering points to the importance of the relaxation of the flow stress between timelocalized clusters of percolation events in the band. We believe the identification of mechanisms for band growth and the relaxation of the flow stress between time-localized clusters of percolation events requires new experiments and modeling focused on these issues. Specifically: 1. Use AFM to obtain 3D plots of the slip-line and slip-band structures and explore the growth of the bands, hopefully as a function of orientation, alloy composition, strain and strain rate. 2. Correlate the AFM measurements with the underlying cell structure obtained by TEM and synchrotron X-ray measurements. 3. Do time localization studies of band growth bursts by acoustic emission, hopefully with correlation of the time bursts with particular bands. 4 . Model cell wall structure and source formation by dislocation dynamics. If computationally possible, also explore band formation and growth by dislocation dynamics. Results from such experiments and modeling will supply information on the parameters involved in the model, and also on the strain, δε , of an individual percolation event. This quantity is an output of our model, but we have not studied how case II behaves above the critical point sufficiently, and we believe that is the relevant physical case. Further, the observed values of δε in Pond’s experiments seem to be of the order 50, which is rather large compared to our expectations for wall source maximum outputs. But Pond’s experiments may involve more than one percolation event in a strain burst, as discussed in the text, so the connection between the experiments and the theory must still be considered uncertain. Thus, further study is required both of the model and experimental measurements of this critical parameter, before we can present a satisfactory physical picture. Finally, we note that the percolation model has implicit in it a mechanism for cell evolution through the nucleation of new cell walls by means

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of the capture of mobile dislocations in mid-cell by secondary dislocations. Incorporation of such cell evolution features is required for producing a complete stress-strain law. References Mader, S. (1957). Z. Phys., 149:73. Noggle, T. and Koehler, J. (1957). J. Appl. Phys., 28:53. Pond Sr., R. B. (1972). The inhomogeneity of plastic deformation. In Reed-Hill, R. E., editor, ASM Seminar Series, page 1, Metals Park, Oh. ASM. Shim, Y., Levine, L., and Thomson, R. (2000). Mater. Sci. Eng. A. in press. Stauffer, D. and Aharony, A. (1992). Introduction to Percolation Theory. Taylor and Francis. Thomson, R. and Levine, L. E. (1998). Theory of strain percolation in metals. Phys. Rev. Lettr, 81:3884–3887. Thomson, R., Levine, L. E., and Stauffer, D. (2000). In press. Tong, W., Hector, Jr., L. G., Weiland, H., and Wieserman, L. F. (1997). In-situ surface characterization of a binary aluminum alloy during tensile deformation. Scripta Mater., 36(11):1339–1344. Weiland, H. private communication. Wilsdorf, H. and Kuhlmann-Wilsdorf, D. (1952). Z. Ang. Phys, 4:23.

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DIFFUSIVE INSTABILITIES IN DILATING AND COMPACTING GEOMATERIALS J. W. RUDNICKI Department of Civil Engineering Northwestern University 2145 Sheridan Road Evanston, Illinois 60208-3109

This chapter reviews and extends analyses of diffusive instabilities in Abstract. inelastically deforming geomaterials. The onset of these instabilities is connected with the conditions for shear localization in the limiting cases of drained (constant pore pressure) and undrained (constant fluid mass) deformation and depends on whether inelastic volume change is dilation or compaction. Rice [1975] showed that homogeneous shear deformation of a layer was stiffer for undrained than for drained conditions but was unstable in the sense that the magnitude of infinitesimal spatial nonuniformities begins to grow exponentially in time when the condition for localization is met in terms of the underlying drained response. As the condition for localization in terms of the undrained response is passed, infinitesimal spatial perturbations experience infinitely rapid decay and then infinitely rapidly growth. For materials that dilate with inelastic shearing the condition for localization is met for the drained response before it is met for the undrained response. For materials that compact and for which the shear yield stress increases with normal stress, the undrained response is softer than the drained and conditions for localization are met for undrained response before drained. If the shear yield stress decreases with normal stress, as for materials modeled by a “cap” on the yield surface, results for compacting materials are identical to those for the dilating materials. Generalization of the layer results to arbitrary deformation states reveals the same relation for the onset of diffusive instability: spatial nonuniformities begin to grow exponentially when the condition for localization is met in terms of the underlying drained response. In contrast to the result for the layer, the growth rate of perturbations does not necessarily become unbounded when the condition for localization is met in terms of the undrained response. The difference is due to a lack of symmetry in the constitutive tensors that is typical of geomaterials. Explicit expressions are given for the undrained response in terms of the drained for an elastic-plastic relation with yield stress and flow potential depending on 159 T.-J. Chuang and J. W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 159–182. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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first and second stress invariants. For this relation and the limit of incompressible solid constituents, the lack of symmetry just-mentioned disappears. If the fluid constituent is also incompressible, the analysis confirms a result of Runesson et al. [1996] that the undrained response is independent of mean stress and the predicted direction of shear bands is 45° to the principal axes of stress. 1

Introduction

In contrast to metals, inelastic deformation of geomaterials typically involves volume change. In low porosity rocks, dilatancy (volume increase) can occur during inelastic shearing under compressive mean stress because of local tensile microcracking at the tips of sliding fissures or at local property mismatches, and from uplift in sliding over asperity contacts on fissure surfaces. In soils, dilatancy results from rearrangement of close-packed particles due to shearing. Compaction in high porosity rocks can result from the collapse of pore structures due to shearing or high mean stresses and, at very high mean stresses, from grain crushing. Compaction of low density soils occurs when shearing causes a closer packed arrangement of particles. When the geomaterial is fluid-saturated, inelastic volume changes tend to cause a change in pore fluid pressure. If the deformation is slow enough and drainage from the boundaries is possible, alterations in pore pressure will be equilibrated by fluid mass flow. In this drained limit, the pore pressure is constant. For volume changes that occur without allowing drainage from material elements, the pore pressure changes. This undrained limit can occur if volume changes occur too rapidly (though still slow enough so that inertia is not significant) to allow time for fluid mass flow or during homogeneous deformation if fluid flow from the boundaries of the body is prevented. Because the inelastic response of geomaterials is affected by the mean effective stress, that is, the total compressive mean stress minus the pore pressure, alterations in pore pressure will either inhibit or promote further inelastic straining. Consequently, the inelastic response differs in the drained and undrained limits and, for intermediate cases, is coupled to the diffusion of pore fluid. Conditions for failure and, in particular, for localization of deformation depend on this coupling. In a seminal analysis, Rice [1975] examined the coupling of pore fluid diffusion and inelastic response for combined shear and compression of a layer that exhibited

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dilatant volume changes. He showed that dilatancy during homogeneous shearing of the layer without allowing fluid drainage from the boundaries prevented caused a reduction of pore fluid pressure and, thus, an increase in effective compressive stress. This increase inhibited further inelastic deformation. But Rice [1975] proceeded to show that homogeneous dilatantly strengthened response becomes unstable when the condition for localization of deformation [Rudnicki and Rice, 1975] is met in terms of the underlying drained (constant pore pressure) response. For the dilatant behavior and layer model considered in Rice [1975], this condition occurs when the drained shear stress versus shear strain curve reaches a peak even though the undrained response curve is still rising. When this condition is met, spatial non-uniformities grow exponentially in time with the smallest wavelengths growing the fastest. Because spatially nonuniform deformation causes fluid flow in response to pore pressure gradients, homogeneous undrained response cannot be realized beyond this point. Dilatant strengthening has been widely observed in granular materials dating back to the experiments of Reynolds [1885]. More recently, it has been observed in laboratory tests on both rocks [Brace and Martin, 1968; Martin, 1980] and soils[Mokni and Desrues, 1999]. Vardoulakis [1985, 1986, 1996a, b] has adapted Rice’s analysis for the biaxial deformation of both dilating and compacting water saturated sand and used it to interpret laboratory observations on the development of localization of deformation. Recent theoretical analyses [Runesson et al., 1996; 1998] have examined conditions for localization in the limit of undrained deformation. In this article, I review Rice’s [1975] analysis and discuss more generally its implications, in particular, for compacting materials. Compaction can result not only from inelastic shearing but also from purely hydrostatic stress. The inelastic response of materials that compact under hydrostatic stress is often modeled by a “cap” on the yield surface enclosing the hydrostatic axis [Dimaggio and Sandler, 1971; Wong et al., 1997]. The presence of this cap implies that inelastic shearing is enhanced, rather than inhibited, by an increase in mean compressive stress. Recently, Issen and Rudnicki [2000] have shown that for compacting materials the conditions for localization admit solutions not only for shear bands, but also for compaction bands. Compaction bands, narrow planar zones of compacted material that form perpendicular to the largest compressive principal stress, have been observed both in the field [Mollema and Antonellini, 1996] and in laboratory experiments [Olsson, 1999]. For the deformation state considered in Rice [1975] the conditions for localization are met at the peak of the shear stress versus shear strain curve. For more

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general deformation states, Rudnicki and Rice [1975] have shown that the condition for localization may be met before or after peak stress. Here, I show the conclusions of Rice [1975] concerning the stability of undrained deformation can be generalized to arbitrary deformation states. A previous analysis of this type has been outlined in Rudnicki [1983] but simplifies the description of fluid flow. Rudnicki [1983] assumes the pore pressure is uniform in a planar band and in the surrounding material, and, in addition, that fluid mass exchange between the layer and the surrounding material is proportional to the difference in pore fluid pressures. These simplifications are avoided here by resolving the perturbations from the uniform fields in terms of Fourier components relative to the putative band. The implications of the results are examined for a general elastic plastic model and used to illuminate the role of pore fluid compressibility on conditions for localization in undrained deformation [Runesson et al., 1996]. The next section briefly summarizes Rice’s [1975] analysis. Succeeding sections develop the extensions to compacting materials and arbitrary deformation states. 2 2.1

Rice’s

Analysis

Formulation

The geometry of the problem considered by Rice [1975] is shown in Figure 1: plane strain deformation of a layer extending indefinitely in the x-direction. Displacements in the x and y directions are u (y, t) and v (y, t), respectively, where t is time. Only the normal strain ∈ (y, t ) = ∂ v /∂y (positive in extension) and the shear strain γ (y, t ) = ∂ u /∂ y are nonzero. Stresses work-conjugate to ∈ and γ are normal stress σ (positive in compression) and shear stress . Equilibrium (in the absence of body forces) requires that σ and be uniform. Hence, they are functions only of time. Other reaction stresses exist to maintain the constraints of zero strain in the x and out-of-plane directions Constitutive relations relate increments of ∈ and γ to increments of σ and . For constant pore pressure, Rice [1975] gives these as follows:

(1a) (1b) The first term in each expression is the elastic portion of the increment; G and M are elastic moduli. The second terms in (1) are the inelastic portions. For constant σ, the hardening modulus H is related to H tan , the slope of a curve of versus γ,

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Figure 1:

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Geometry of the layer problem analyzed by Rice [1975].

by (2) Thus, H ≈ H tan, for H << G The yield surface is the boundary of the stress states ( , σ) that cause only elastic response for a given state of inelastic deformation; µ, the local slope of this surface, is referred to as a friction coefficient (Figure 2). Thus, when H > 0, deformation increments tending to make d ≤ µd σ are purely elastic and the second terms in (1) are dropped. (When H < 0, elastic unloading corresponds to d ≥ µd σ.) Thus, increases in compressive normal stress inhibit further inelastic deformation. Inelastic increments of volume strain d p ∈ are related to inelastic increments of shear strain d p γ b y (3) where β is a dilatancy factor. When the pore pressure is not constant, the constitutive relations (1) are modified by replacing the increment of normal stress by an increment of the effective normal stress, a linear combination of d σ and dp. Experiments on failure of rocks (e.g., Paterson [1978]) suggest that σ – p is the appropriate form for the effective

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Figure 2: [1975].

J. W. RUDNICKI

Geometric interpretation of the constitutive parameters used by Rice

stress for inelastic response. In addition, Rice [1977] has argued on theoretical grounds that this is the appropriate form for inelasticity due to microcracking from the tips of sharp fissures and to frictional sliding on surfaces with small real contact areas. Generally, a different form is needed for elastic deformation [Nur and Byerlee, 1971; Rice and Cleary, 1976]. But, when the solid and fluid constituents are much less compressible than the porous matrix, as for most soils, σ – p is also the appropriate form for elastic straining. In this case, equations (1) become (4a) (4b) An additional constitutive relation is Darcy’s law which, in the absence of body forces, has the following form: (5)

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where q is fluid mass flow rate per unit area in the y direction and ρ is the fluid mass density. The coefficient κ is often expressed as the ratio of a permeability, with dimensions of length squared (frequently measured in darcies; 1 darcy = 10– 8 cm2 ) to the fluid viscosity. If the fluid phase is incompressible, the density is constant and the equation expressing conservation of fluid mass is (6) Substituting (5) into (6) yields an equation relating gradients in pore pressure to changes in volumetric strain: (7) 2.2

Undrained Homogeneous Deformation

If drainage from the boundaries of the layer is prevented and the deformation and properties are uniform, (5) yields q = 0 throughout the layer and, from (7), increments in volume strain are zero. Setting d ε = 0 in (4) yields the following expression for the change in effective normal stress: (8) Since M is an elastic modulus and positive, the pore pressure decreases for dilation ( β > 0) at fixed normal stress. Substitution of (8) into (4a) reveals that the slope of the shear stress versus shear strain curve (no longer at constant effective normal stress) is still given by (2) but with the hardening modulus H replaced by the augmented value: (9) 2.3 Instability Rice [1975] proceeds to show, however, that the homogeneous, undrained solution is unstable with respect to small spatial perturbations in the strain or pore pressure. In particular, linearization of the governing equations about the undrained homogeneous solution yields the homogeneous diffusion equation for the perturbations in pore pressure : (10) where the diffusivity c is given by (11)

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and the constitutive parameters are to be evaluated at homogeneous undrained deformation. Perturbations with a Fourier wavelength λ grow exponentially at a rate r = – 4 π2 c /λ2 (12) If both β and µ are positive, then, since M > 0, H u n d r a i n e d > H and t he homogeneous, undrained response is dilatantly hardened. But, because c ( 1 1 ) passes through zero from positive to negative when H = 0 , the magnitude of spatial perturbations, instead of decaying exponentially, grow exponentially (12). As noted by Rice [1975] this is analogous to running the heat equation backwards in time: non-uniformities become more localized rather than more diffuse with time. Thus, dilatantly hardened response becomes unstable when the underlying drained response passes through a peak. Since H is generally a decreasing function of inelastic deformation, H = 0, corresponding to a peak in the drained vs. γ curve (at constant σ ) will occur before a peak in the undrained response curve, H u n d r a i n e d = 0. The Appendix of Rice [1975] develops the analysis for arbitrarily compressible solid and fluid constituents. The effect is to replace the elastic modulus M in (8), (9) and (11) by a modified value (13) where φ is the apparent void volume fraction, Kƒ is the bulk modulus of the pore fluid and M s and N s are additional moduli associated with the solid constituents. When both the solid and fluid constituents are effectively incompressible, M' = M. If the solid constituents are incompressible, i.e., Ms , N s >> M , (14) If the fluid is very compressible Kƒ /φ << M and M' ≈ Kƒ / φ . Thus, the dilatant hardening effect vanishes in the limit that the pore fluid bulk modulus goes to zero. 2.4

Discussion

Rice [1975] remarks that if initial material non-uniformities or variation of constitutive parameters on the time scale of perturbation evolution are included, these introduce additional inhomogeneous terms into (10). In the linearized analysis, these terms are linear in the perturbation magnitudes and proportional to the stress or strain rate of the uniform solution. For infinitesimal wavelengths, λ → 0,

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the growth rate in (12) is unbounded and these additional terms do not affect the conditions for instability. In actuality, the perturbation wavelength is limited from below by the material grain size lg . Consequently, the growth rate r is bounded from above by Thus, Rice’s [1975] analysis corresponds to the case when the perturbation growth rate is much larger than the layer deformation rate, r max >> That is, perturbation growth occurs “instantaneously” on the time scale of the layer deformation and the onset of perturbation growth ( c = 0) will coincide with instability. For very low permeability rocks (or fast loading rates), the diffusion length scale l d = (– c / ) 1/2 may be comparable to the grain size, yielding rmax ~ In this case, the actual instability, defined as a certain increase of the perturbation magnitude over its initial value, will be delayed from the onset of perturbation growth. To estimate the delay of undrained instability, Garagash and Rudnicki [2000] have extended Rice’s analysis to include the coupling between the perturbation and the evolution of the constitutive parameters with uniform background deformation. Rudnicki [1984b] examined the effect of an initial non-uniformity by considering the shear of a weakened layer embedded in an infinite body. Both the layer and the surrounding material deform non-elastically but the peak stress in the layer is slightly less than that in the surrounding material. The pore pressure is assumed to be uniform in both the layer and the surrounding material and the fluid mass exchange is assumed to be proportional to the difference. The development of instability in time depends on the ratio of the rate of imposed farfield strain-rate ∞ to the rate of exchange of fluid mass between flow from the layer, d. In the limit ∞ /d → 0, the pore pressure in the layer is the same as in the surrounding material and instability occurs at the peak of the drained stress-strain curve. For finite ∞ /d, instability is delayed until the weakened layer reaches the peak of its dilatantly hardened stress-strain curve. For small ∞ /d, as appropriate for most applications, an asymptotic analysis predicts that the time delay is given by where λ is the half-width of the peak of the stressstrain curve, ∆ is the difference in the peak stresses of the weakened layer and the surrounding material divided by λ times the elastic shear modulus and α is a nondimensional measure of the strength of dilatant hardening. These delay times are less than a few hours for tectonic strain rates and less than a few tens of seconds for typical laboratory strain-rates. If dilatant hardening causes a sufficient reduction of the pore fluid pressure, exsolution of dissolved gases or cavitation of the fluid may occur and the pore fluid bulk modulus will be dramatically reduced. Equations (13) and (9) indicate

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that the dilatant hardening effect diminishes with reduction of the pore fluid bulk modulus and vanishes in the limit Kƒ → 0. Rudnicki and Chen [1988] have proposed that cavitation is a limit to strong dilatant hardening associated with slip on a weakening frictional surface and suggest that this effect is consistent with observations by Martin [1980] of pore fluid stabilization of rock failure. In addition, in undrained biaxial experiments on a quartz sand, Mokni and Desrues [1999] have observed that the formation of shear bands in dilatant specimens does not occur until cavitation of the pore fluid. The numerical simulations of Schrefler et al. [1996] also indicate the importance of cavitation in the formation of shear bands. 3

Application

to

Compacting

Materials

Although Rice [1975] considers only dilating materials ( β > 0), he notes that for loosely packed granular materials β < 0. This will also be the case for high porosity rocks that compact when sheared. For β < 0, (8) indicates that the effective compressive normal stress decreases and (9) that H u n d r a i n e d < H. T h i s phenomenon could be described as compaction softening. Consequently, the denominator of (11) will pass through zero before the numerator. The exponent r in (12) reveals that spatial perturbations are damped infinitely fast and then grow infinitely fast as H u n d r a i n e d , the denominator of (11), passes through zero. This singular jump in diffusivity suggests that variation of constitutive parameters with the uniform background deformation should be included in the perturbation solution. By including this variation, Garagash and Rudnicki [2000] show that undrained deformation of compacting material is stable (the perturbation magnitude vanishes) as H undrained passes through zero if the ratio of the diffusion length l d = (–c / )1/2 to the maximum perturbation wavelength (defined by the layer width 2h) is larger than a critical value defined by the dependence of the inelastic moduli H a n d H u n d r a i n e d on the deformation State: (15) This inequality indicates that the maximum perturbation wavelengths (h /2π) are most unstable for compaction softening in contrast to dilatant hardening for which the shortest wavelengths grow most rapidly (12). This suggests that failure will occur by a diffuse mode rather than a localized mode, as for dilatant hardening. The stability criterion (15) implies an interesting scale effect: compaction softening is stable near the peak shear stress (H u n d r a i n e d = 0) if the specimen size h is small enough. This result is consistent with the small scale laboratory results

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of Han and Vardoulakis [1991] and Finno et al. [1995]. If the stability criterion is not satisfied, the perturbation magnitude becomes algebraically singular at H u n d r a i n e d = 0. Vardoulakis [1996b] has applied Rice’s analysis to study the stability of biaxial deformation of water-saturated sand and noted that the instability at H undrained = 0 can also be mitigated by the introduction of rate-dependence in the material constitutive behavior. The discussion of compaction in the preceding paragraphs assumes µ > 0 and the physical interpretation of µ as a friction coefficient would seem to preclude the possibility that µ < 0. But, µ enters (1) as the local slope of the yield surface. As sketched in Figure 2, the yield surface is open on the normal stress axis. Consequently, purely normal stress σ does not cause inelastic deformation, as appropriate for low porosity rocks. For loose soils or highly porous rock, inelastic compaction (volume decrease) is due not only to inelastic shearing but may be caused by purely hydrostatic (or in the one dimensional layer model here, by purely normal) stress. In this case, the yield surface is closed on the normal stress axis, as depicted in Figure 3. So-called “cap” models were introduced by Dimaggio and Sandler [1971]. As sketched in Figure 3, µ < 0 is appropriate for this class of models and the magnitude of µ becomes large as the σ axis is approached. Because the stability condition contains only the product β µ, the case of a material that compacts under a stress state on the cap of the yield surface reduces to that analyzed by Rice [1975]. The form of equation (3) assumes that all inelastic volume strain is associated with inelastic shear strain. Consequently, no inelastic volume strain would occur for loading by purely normal stress. For high porosity rocks and loose soils, this will not be a complete description and a term contributing inelastic volume strain for purely (effective) normal stress can be added to (3) (16) where k is an inelastic modulus that is the slope of a drained σ versus ε p at constant shear stress (d = 0). The effect of is to modify the elastic contribution to the volumetric strain stability analysis to replace the elastic modulus M (or M ') by

( dp = 0) curve of including this term in (4b) and in the the effective value (17)

For typical cases, k ≥ 0 but k << M " . Decreasing k will reduce the difference between the drained and undrained hardening moduli. Thus, for small k, differences in drained versus undrained deformation may be difficult to observe unless

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Figure 3: Schematic illustration of a cap on the yield surface. µ = d /d σ < 0.

Note that

the stress state is near enough to the σ axis so that values of µ are large and negative. Many geomaterials exhibit both compaction and dilation depending on the initial confining stress, initial porosity and load path. Examples include simulated fault gouge [Marone et al., 1990], loose sand [Finno et al., 1995] and limestone [Baud et al., 2000]. When these materials are fluid-saturated and deformed without allowing fluid flow from the boundaries, the evolution of localized zones depends on the local rate of fluid flow, the imposed rate of straining, and the transition from contraction to dilation, A simple analysis [Rudnicki, 1996] shows that small variations in the evolution of porosity with shear strain can dramatically alter the undrained response. Rudnicki et al. [1996] have suggested that compaction softening followed by dilatant hardening may be an explanation for the evanescent shear band structures observed in some experiments of Finno et al. [1995]. If compaction softening causes the onset of localized deformation in a narrow zone but gives way to dilatant hardening before full development of the band, formation

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of a shear band in another orientation may occur when the dilatant hardening response becomes unstable. 4 Generalization to Arbitrary Deformation States In this section, I extend Rice’s [1975] analysis to arbitrary deformation states. The analysis follows the lines of Rice’s [1976] general treatment of conditions for localization of plastic deformation in rate-independent solids. That analysis is phrased in terms of measures of stress and strain appropriate for arbitrary deformation magnitudes and discusses localization both as a bifurcation from homogeneous deformation and as the growth of an initial non-uniformity. Here, for simplicity, I restrict attention to small strains and consider only small perturbations from completely uniform deformation. Runesson et al. [1998] have presented an analysis of localization for undrained conditions for finite strains. Generalizations of the present analysis to finite strains can be adapted from the treatments of Rudnicki [1983], Runesson et al. [1998] or Coussy [1995]. 4.1

Localization in Rate-independent Solids

Consider a homogeneous solid deforming in homogeneous fashion described by Perturbations from these uniform fields are a strain rate d 0 and stress-rate To investigate the possibility of localization in a planar denoted by ∆ d and band the perturbed fields are taken to vary with distance from a plane with normal n. The requirements of equilibrium and continuous velocities place the following restrictions on the form of the perturbed fields [Rice, 1976]): (18) (19) where g is a function of n · x. For nonlinear elastic or rate-independent elastic-plastic solids with a smooth yield surface and plastic potential, the strain-rate can be given by an incrementally linear relation: (20) where If K can be inverted, (20) can be written in the alternative form: (21) where K and M are mutual inverses with components satisfying (22)

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and δ ij is the Kronecker delta. The components of the incremental compliances K i j k l and the moduli M ijkl are symmetric with respect to interchange of the first two indices and the last two indices, i.e., M i j k l = M j i k l and M i j k l = M i j l k , but are not, in general, symmetric with respect to interchange of the first pair and the last pair, i.e., M i j k l ≠ M klij . The last is typical of geomaterials for which inelastic strain increments are not perpendicular to the yield surface or, in other words, the plastic flow rule is not associated with the yield function. Substitution of (18) into (21) and the result into (19) yields (23) where M 0 denotes the incremental moduli for the uniform fields. In the simplest case, the constitutive response of the perturbed field is identical with that of the uniform field and the right hand side of (23) vanishes. The first possibility for a non-trivial solution for g occurs when the determinant of the coefficient matrix vanishes: (24) w h e r e m jk = (n ⋅ M ⋅ n ) jk If the constitutive responses of the perturbed and homogeneous fields differ, the condition (24) is still limiting in the sense that g determined from (23) will be unbounded when (24) is met. 4.2

Inclusion of Pore Fluid

When the solid is saturated with a fluid that can be described in terms of a single scalar pore pressure (see Cleary [1978] for a discussion of other possibilities), a term involving the rate of pore pressure must be appended to (20): (25) Alternatively (25) can be written as (26) w h e r e α = M : A. The introduction of the pore pressure as an additional field variable requires an additional constitutive equation. This is conveniently taken to be a relation the rate of change of fluid mass content per unit volume of porous solid m = ρ υ where ρ is the density of homogeneous pore fluid and υ is apparent void volume fraction. For arbitrarily compressible constituents, can be expressed in terms of the rate of strain and pore pressure change: (27)

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or in terms of the stress-rate and pore pressure change by substituting (25) into (27): (28) In (25), (26) and (27), A , α and R are symmetric because of the symmetry of d and For drained response, the pore pressure is constant and (25) and (26) reduce to (20) and (21). The condition for localization is given by (24). For undrained response, the fluid mass in material elements is constant, the right hand side of (27) is zero, and the pore pressure is given by (29) or, equivalently, in terms of the stress from (28) (30) Substitution of (29) into (26) gives (31) where the undrained incremental moduli are given by (32) The condition for localization (24) evaluated in terms of the undrained moduli (32) is where m u = n ⋅ M u ⋅ n. The matrix m u is related to m (24) by

(33)

(34) where r = n ⋅ R and a = n⋅ α. The earlier discussion of the layer problem examined by Rice [1975] suggests that (33) may be met before or after (24) depending on the nature of the constitutive relation and, in particular, the inelastic volumetric deformation. If (24) is met before (33), Rice’s [1975] analysis also suggests that undrained homogeneous deformation is unstable, in the sense that small spatial perturbations will grow exponentially in time. As a result, the condition (33) may be irrelevant since instability may occur well before it is met.

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4.3 Diffusive Instability In order to analyze the stability of undrained homogeneous deformation, an equation expressing fluid mass conservation (in the absence of sources) must be added to the field equations and an additional constitutive equation relating the fluid mass flux to gradients in pore pressure must be specified. Fluid mass conservation (in the absence of sources) is given by (35) where q is the mass flow rate per unit area of porous solid. Fluid mass flow is assumed to be related to the pressure gradient by Darcy’s law: (36) where κ is the ratio of a symmetric permeability tensor to the (scalar) fluid viscosity and ρ 0 is the (constant) fluid mass density in the reference state. Substitution of (36) into (35) yields (37) For simplicity, we assume, as in Rice [1975], that perturbations from undrained homogeneous deformation are small enough so that incremental constitutive parameters are the same for the uniform and perturbed fields. If this is not the case, additional inhomogeneous terms would enter the right hand sides of the equations. As in (23), these terms involve the differences of the constitutive parameters multiplied by the strain-rate and pore pressure rate in the uniform field. Forming the difference of (26) and substituting into (19) yields (38) where is the difference between the pore pressure-rates in the uniform and perturbed fields and, again, a = n ⋅ α. Because the unperturbed fields are homogeneous and undrained, both sides of (37) are identically zero. The kinematic restriction on the perturbed strain-rate field (18) suggests that ∆ p depends on distance n ⋅ x perpendicular to the orientation of a planar band and, hence, can be decomposed into Fourier components of the form exp λ n ⋅ x). This type of deformation mode has also been discussed for sands by Vardoulakis [1996b]. For this decomposition, using (27) in (37). Since both sides of (37) are zero for the unperturbed (homogeneous, undrained) field, the difference fields also satisfy (37). Using the Fourier decomposition on the right hand side and assuming the permeability is uniform yields (39)

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where κ = λ² n ⋅ κ ⋅ n. Forming the difference of (27) and substituting into (39) yields (40) The vector g can be eliminated from (38) by writing the inverse of m as k / m where k is the matrix of co-factors and m = det( m i j ). Substituting the result into (40) yields (41) This is a linear ordinary differential equation for the time evolution of the perturbed pressure field. The solutions change from exponentially decaying to exponentially growing as m passes through zero from positive to negative values (at least if the term in square brackets is positive). Since the vanishing of m is the condition for localization in terms of the drained incremental moduli (24), this condition controls the growth rate of spatial perturbations as in the layer analysis of Rice [1975]. The similarity with Rice’s [1975] analysis suggests that the square bracket in (41) will vanish when the condition for localization is met in terms of the undrained moduli (33). Using m – 1 = k /m in (34), pre- and post-multiplying by a and dividing through by a ² = a ⋅ a yields (42) Using this expression to eliminate m η in (41) gives (43) If R = α (so that a = r ) or if k is symmetric, then the second term in brackets vanishes and the coefficient of is expressed in terms of the matrix entering the localization condition in terms of the undrained moduli. Even in this case, however, it is not clear that the sign of this term changes when (33) is met. For geomaterials, M is generally not symmetric with respect to interchange of the first and last pair of indices. Consequently, neither m nor k will be symmetric. In addition, for the particular form of a constitutive relation discussed in the next section, this lack of symmetry of M causes R and α to differ. 5 Application to an Elastic-Plastic Relation This section illustrates the analysis of the preceding section by specializing to a particular form of elastic plastic constitutive relation. The strain-rate for plasticloading is given by

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(44) where the first term is the elastic contribution, C is the tensor of elastic compliances, P is a tensor specifying the direction of inelastic strain increments in stress space, Q is the tensor giving the normal to the yield surface in stress space, and h is an inelastic hardening (softening) modulus. Deformation increments that tend to make Q : ≤ 0 for h > 0 cause elastic unloading and for these the second term is dropped. (If h < 0, increments tending to make Q : > 0 cause elastic unloading). Thus, the tensor K in (20) is given by (45) The inverse of K, M in (21) is given by (46) where L = C – 1 is the tensor of elastic moduli, p = L : P and q = L : Q . The constitutive relation used by Rudnicki and Rice [1975] in their study of localization has the form of (44) but the yield function and plastic potential depend only on the first and second stress invariants. In this case, P and Q are given by

(47a) (47b) 1/2

where N = s/2 , s = σ – (1/3) tr σ I is the deviatoric stress tensor, = (s : s/2) is the Mises equivalent shear stress, I is the identity tensor and µ and β h a v e interpretations similar to those in (1). If the elasticity is assumed to be isotropic with shear and bulk moduli, G and K, respectively, and P and Q are given by (50) (48a) (48b) (48c) Rudnicki [1984a, 1985] has extended the formulation to include pore fluid effects in the form of (44) used by Rudnicki and Rice [1975]. He assumes isotropic

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poroelasticity for the elastic strain increments and inelastic strain increments are given by the second term of (44) with the stress replaced by the effective stress σ + p I . As already noted, much experimental evidence is consistent with this form and Rice [1977] has argued that it is appropriate for inelasticity arising from microcracking and frictional slip on surfaces with small real areas of contact. For isotropic poroelasticity, Nur and Byerlee [1971] have shown that the proper form of the effective stress is σ + p ζI. The Biot porous media parameter ζ (often denoted α ) is equal to 1 – where K is the elastic bulk modulus of the porous solid and is an additional modulus related to the bulk modulus of the solid constituents [Rice and Cleary, 1976]. Thus, if the solid constituents are incompressible, and ζ = 1 This is a suitable approximation for soils, in which the compressibility of the grains is much less than that of the porous matrix. Under the same conditions for which σ + p I is the form of the effective stress for inelastic deformation, Rice [1977] has also argued that the inelastic increment of the apparent void volume fraction where υ = m /ρ ) is equal to the inelastic volume strain increment. Under these circumstances, the tensors A in (25) and α in (26) are given by (49a) (49b) The tensor R and the coefficient η entering (27) are given by (50a) (50b) where B is Skempton’s coefficient, the negative of the ratio of pore pressure to mean normal stress during undrained elastic deformation [Rice and Cleary, 1976]. The rate of change of pore pressure during undrained deformation (30) is given by (51) For isotropic elasticity, C : I = I : C =(1/3 K)I and Q : L : P = G +tr(Q ) tr(P)K . The compliance tensor for undrained deformation can be formed from these results (52)

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where hu n d = h + BK tr P tr Q /ζ and the primes on P and Q denote the deviatoric parts. For isotropic elasticity, the first two terms correspond to replacing K , the elastic bulk modulus for drained deformation by K u the value for undrained deformation, where (53) The contribution to the inelastic strain-rate, the last term in (52), is identical in form to that for drained deformation (44), with h replaced by h und , tr P replaced by (1 – B ) tr P and tr Q by (1 – B ) tr Q. If the expression for the compliance is specialized to pure shear, the result is exactly analogous to that of Rice [1975]: The undrained response is identical in form to that for drained but with h replaced by h und . Since B, K, and ζ –1 are positive, h und is greater or less than h depending on the sign of tr P tr Q, in a manner identical to the dependence on the sign of β µ in Rice [1975]. These expressions can also be used to examine the role of the compressibilities of the individual solid and fluid constituents. If the solid constituents are incompressible, ζ = 1, and α = R = I. Consequently, the results simplify considerably. In particular a = r in equation (34) and the term r ⋅ k ⋅ a – a ⋅ k ⋅ r vanishes in (43). The compressibility of the pore fluid enters only through the Skempton coefficient B . For isotropic poroelasticity, B can be expressed as (54) where υ 0 is the apparent void volume fraction in the reference state, Kƒ is the pore fluid bulk modulus, and is another modulus related to the bulk modulus of the solid constituents [Rice and Cleary, 1976]. If the solid constituents are If the fluid is also incompressible, incompressible, t h e n B = 1. In this limit, the bulk modulus is eliminated from (52) and so is tr P. Consequently, both the elastic and inelastic components are incompressible, consistent with the limit of incompressible constituents. Furthermore, tr Q is eliminated from (52). Therefore, any dependence of the yield function on mean stress for drained response is exactly compensated by changes in pore pressure during undrained response. Thus, the undrained response is incompressible and does not depend on the mean stress. For such a solid, shear bands are predicted to occur at 45° from the principal axes, at least if P' = Q ' . Runesson et al. [1996] obtain this result by direct calculation for undrained deformation of a porous elastic plastic solid with an incompressible pore fluid. The result pertains, however, only when the solid constituents are also incompressible. Although Runesson et al. [1996]

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do not explicitly assume that the solid constituents are incompressible, they use σ + p I as the effective stress for both inelastic and elastic deformation. As discussed earlier, this form is consistent with linear poroelasticity only when the solid constituents are incompressible. 6 Concluding Discussion Application of Rice’s [1975] results to materials that compact with inelastic shearing suggest that the accompanying pore pressure changes will be destabilizing. That is, the effective hardening modulus for undrained response will be less than that for drained response and conditions for localization will be met for undrained constitutive response before they are met in terms of the drained response. But this conclusion appears to apply only when the yield stress in pure shear increases with increasing mean stress. For highly porous materials that compact with mean stress, the yield surface typically has a cap and, as a result, the yield stress in pure shear decreases with increasing compressive mean stress. In this case, the conditions for linearized stability of undrained deformation revert to those obtained by Rice [1975]: undrained deformation is stiffer than the drained response, but homogeneous undrained response becomes unstable when conditions for localization are met in terms of the drained response. The conclusions of Rice [1975] based on analysis of a layer subject to simple shear and compression are shown to apply to arbitrary deformations, although conditions for localization are not, in general, met at the peak of the stress versus strain curve as they are for the layer. Spatial perturbations from the undrained solution grow exponentially when the condition for localization is met in terms of the drained response. In the layer problem, the growth rate is unbounded immediately after the condition for localization is met in terms of the undrained response. This is not necessarily the case for arbitrary three dimensional deformations because of the lack of symmetry in the constitutive tensors that is typical of geomaterials. For elastic plastic solids, the correspondence with the simple layer analysis is direct. The effective hardening modulus governing shear for undrained conditions is the sum of the value for drained response and a term that is the product of an elastic bulk modulus and the mean parts of tensors giving the plastic flow direction and the normal to the yield surface. The incremental compliance for undrained deformation of an elastic plastic solid gives some insight into the surprising result of Runesson et al. [1996]: if the pore fluid is completely incompressible, zones of localization are predicted to occur at 45° to the principal axes regardless of the form of the yield function and plastic potential (at least, if the deviatoric portions of the tensors giving the plastic flow direction and the normal to yield surface

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coincide). This results because, if the solid constituents are also incompressible, undrained response is completely incompressible and the induced changes in pore pressure exactly compensate for any dependence of the yield function on the mean stress. I am grateful to Dmitry Garagash for many helpful Acknowledgement. discussions. I also wish to acknowledge interesting correspondence of several years ago with Stein Sture and Kenneth Runesson on conditions for localization for undrained deformation of fluid-saturated porous solids. Partial financial support for this work was provided by the U. S. Dept. of Energy, Office of Basic Energy Sciences, Geosciences Research Program through Grant No. DE-FG02-93ER14344/09 to Northwestern University.

References Baud, P. Alexandre Schubnel and T. -F. Wang, Dilatancy, compaction and failure model in Solnhofen limestone, J. Geophys. Res., in press, 2000. Brace, W. F. and R. J. Martin, III, A test of the law of effective stress for crystalline rocks of low porosity, Int. J. Rock Mech. Mining Sci., Vol. 5, 415-436, 1968. Cleary, M. P. Elastic and dynamic response regimes of fluid-impregnated solids with diverse microstructures, Int. J. Solids Structures, 795-819, 1978. Coussy, O. Mechanics of Porous Media, John Wiley and Sons, Ltd., Chichester, 1995. Dimaggio, F. L. and I. S. Sandler, Material model for granular soils, J. Eng. Mech. Div. ASCE, Vol. 97, 935-950, 1971. Finno, R. J., W. W. Harris, M. A. Mooney, and G. Viggiani, Shear bands in plane strain compression of loose sand, Géotechnique, Vol. 47, 149-165, 1997. Garagash, D. and J. W. Rudnicki, Stability of undrained deformation of fluid-saturated dilating/compacting solids (Abstract), 20th Int. Cong. of Theor. and Appl. Mech., Chicago, II, Aug. 27 - Sept. 2, 2000. Han, C. and I. G. Vardoulakis, Plane strain compression experiments on water-saturated fine-grained sand, Géotechnique, Vol. 41, 49-78, 1991. Issen, K. A. and J. W. Rudnicki, Conditions for compaction bands in porous rock, J. Geophys. Res., in press, 2000. Marone, C., C. B. Raleigh, and C. H. Scholz, Frictional behavior and constitutive modeling of simulated fault gouge, J. Geophys. Res., Vol. 95, 7007-7025, 1990. Martin, R. J. III, Pore pressure stabilization of failure in Westerly granite, Geophys. Res. Letters, Vol. 7, 404-406, 1980. Mokni, M. and J. Desrues, Shear localization measurements in undrained plane-strain biaxial tests on Hostun RF sand, Mech. of Cohesive-Frictional Materials, Vol. 4,

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419-441, 1999. Mollema, P. N. and M. A. Antonellini, Compaction bands: a structural analog for antimode I cracks in aeolian sandstone, Tectonophysics, Vol. 267, 209-228, 1996. Nur, A. and J. D. Byerlee, An exact effective stress law for elastic deformation of rock with fluids. J. Geophys. Res., Vol. 76, 6414-6419, 1971. Olsson, W. A., Theoretical and experimental investigation of compaction bands, J. Geophys. Res., Vol. 104, 7219-7228, 1999. Paterson, M. S. Experimental Rock Deformation: The Brittle Field, New York: SpringerVerlag, 1978. Reynolds, O., On the dilatancy of media composed of rigid particles in contact, with experimental illustrations, Phil. Mag. (reprinted in Papers on Mechanical and Physical Subjects by O. Reynolds, Cambridge University Press, New, York, 1901, Vol. 2, pp. 203-216), 1885. Rice, J. R. On the stability of dilatant hardening for saturated rock masses. J. Geophys. Res., Vol. 80, 1531-1536, 1975. Rice, J. R. The localization of plastic deformation, in Proceedings of the 14th Int. Union Theor. and Applied Mech. Congress, ed. W. T. Koiter, pp. 207-220, North Holland, Amsterdam, 1976. Rice, J. R. Pore pressure effects in inelastic constitutive formulations for fissured rock masses. In Advances in Civil Engineering through Engineering Mechanics, pp. 360363. New York: American Society of Civil Engineers, 1977. Rice, J. R. and M. P. Cleary, Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents, Rev. Geophys. Space Phys., Vol. 14, pp. 227-241, 1976. Rudnicki, J. W. and J. R. Rice, Conditions for the localization of deformation in pressuresensitive dilatant materials. J. Mech. Phys. Solids, Vol. 23, 371-394, 1975. Rudnicki, J. W. A formulation for studying coupled deformation - pore fluid diffusion effects on localization. In Geomechanics, Proceedings of the Symposium on the Mechanics of Rocks, Soils and Ice, Applied Mechanics Division, Vol. 57 (edited by S. Nemat-Nasser), pp. 35-44, American Society of Mechanics Engineers, New York, 1983. Rudnicki, J. W. A class of elastic-plastic constitutive laws for brittle rock, J. of Rheology, Vol. 28, 759-778, 1984a. Rudnicki, J. W. Effects of dilatant hardening on the development of concentrated shear deformation in fissured rock masses, J. Geophys. Res., Vol. 89, 9259-9270, 1984b. Rudnicki, J. W. Effect of pore fluid diffusion on deformation and failure of rock, in Mechanics of Geomaterials (edited by Z. P. Ba ant), pp. 315-347, John Wiley & Sons, Ltd., New York, 1985. Rudnicki, J. W. and C.-H. Chen, Stabilization of rapid frictional slip on a weakening

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fault by dilatant hardening, J. Geophys. Res., Vol. 93, 4745-4757, 1988. Rudnicki, J. W., R. J. Finno, M. A. Alarcon G. Viggiani, and M. A. Mooney, Coupled deformation-pore fluid diffusion effects on the development of localized deformation in fault gouge, in Predictions and Perfomance in Rock Mechanics and Rock Engineering, EUROCK’96, edited by G. Barla, pp.1261-1268, Balkema, 1996. Runesson, K., D. Peri and S. Sture, Effect of pore fluid compressibility on localization in elastic-plastic porous solids under undrained conditions, Int. J. Solids Structures, Vol. 33, 1501-1518, 1996. Runesson, K. R. Larsson, and S. Sture, Localization in hyperelasto-plastic porous solids subjected to undrained conditions, Int. J. Solids Structures, Vol. 35, 4239-4255, 1998. Schrefler, B. A., L. Sanavia and C. E. Majorana, A multiphase medium model for localisation and post localization simulation in geomaterials, Mech. of Cohesive-Frictional Materials, Vol. 1, 95-114, 1996. Vardoulakis, I. Stability and bifurcation of undrained, plane rectilinear deformations on water-saturated granular soils, Int. J. Num. and Anal. Meth. Geomech., Vol. 9, 399414, 1985. Vardoulakis, I. Dynamic stability analysis of undrained simple shear on water-saturated granular soils, Int. J. Num. and Anal. Meth. Geomech., Vol. 10, 177-190, 1986. Vardoulakis, I. Deformation of water-saturated sand: I. uniform undrained deformation and shear banding, Géotechnique, Vol. 46, 441-456, 1996a. Vardoulakis, I. Deformation of water-saturated sand: II. effect of pore water flow and shear banding, Géotechnique, Vol. 46, 457-472, 1996b. Wang, T.-F., C. David, and W. Zhu, The transition from brittle faulting to cataclastic flow in porous sandstones: Mechanical deformation, J. Geophys. Res., Vol. 102, 30093025, 1997.

FRACTURE MECHANICS OF AN INTERFACE CRACK BETWEEN A SPECIAL PAIR OF TRANSVERSELY ISOTROPIC MATERIALS

LESLIE BANKS-SILLS AND VINODKUMAR BONIFACE The Dreszer Fracture Mechanics Laboratory Department of Solid Mechanics, Materials and Structures The Fleischman Faculty of Engineering Tel Aviv University 69978 Ramat Aviv, Israel

Abstract. In this investigation, the Stroh formulation is employed to develop the stress and displacement fields in the vicinity of an interface crack between two specially oriented transversely isotropic materials. The lower material is mathematically degenerate. In addition, a conservative integral is employed in conjunction with the finite element method to calculate stress intensity factors. The derived stress and displacement fields are used as auxiliary fields in the M-integral for extraction of the stress intensity factors. As a benchmark problem for this calculation, the asymptotic displacements are prescribed on the boundary of a circular domain. Excellent numerical results are obtained.

1.

Introduction

The mechanics of interface cracks between two joined materials is an important subject which has received much attention in the literature. The papers by J.R. Rice (1988) and J.W. Hutchinson (1990) presented a new view on the subject which delineated a philosophy for tackling the problem of a crack between two isotropic materials. One of the controversial issues which hindered progress was treatment of crack face contact which naturally spreads from the crack tip along the crack faces. The new approach was to neglect this region when it is sufficiently small and within the small scale yielding zone. In this situation, approximate bounds for the phase angle ψ (mode mixity) were determined by Rice (1988). It was found that there was a large range of ψ for which the contact region is sufficiently small. With this approach, interest in this subject was rekindled and a wealth of investigations followed. 183 T.-J. Chuang and J. W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 183–204. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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Following in this direction, Suo (1990) considered an interface crack between two dissimilar anisotropic materials for both non-oscillatory and oscillatory singularities. Investigations were carried out earlier for the case of a non-oscillatory singularity. In particular, Bassani and Qu (1989) and Qu and Bassani (1989) derived a condition for non-oscillatory behavior. In Suo (1990), the structure of the near tip field was determined for the oscillatory singularity; it is similar to that of a crack between two isotropic materials. Working independently, Ting (1986, 1990) determined the singularity for both the oscillatory and non-oscillatory cases. In addition, he developed the asymptotic stress and displacement fields (Ting, 1992). In order to employ a fracture mechanics approach for predicting the propagation of cracks between two anisotropic materials, stress intensity factors and a fracture criterion are required. In this investigation, the Stroh formulation is employed to derive the stress and displacement fields in the neighborhood of the tip of an interface crack between two transversely isotropic materials. An excellent treatise on these issues is provided by Ting (1996). The upper material is taken so that its symmetric plane is perpendicular to the x 1 –direction, whereas the symmetric plane of the lower material is perpendicular to the x 3 –direction. Of course, within each of these planes, there is no preferential direction. The symmetric plane of the lower material causes a mathematical degeneracy requiring an analysis which is more complicated than usual. In addition, a method for calculating stress intensity factors from finite element results is employed. This method has already been described and employed for isotropic bimaterials (see Wang and Yau, 1981, Shih and Asaro, 1988, Matos, et al., 1989, Nakamura, 1991, Nahta and Moran, 1993, Gosz, et al., 1998, and Banks-Sills, et al., 1999) and certain orthotropic bimaterials (Charalambides and Zhang, 1996). The M-integral, developed by Wang and Yau (1981), allows for the determination and separation of the stress intensity factors K 1 and K 2 for interface cracks between two isotropic materials. Because of the similar structure of the near tip fields for cracks between two anisotropic materials, the same integral formulation may be employed. This was done by Charalambides and Zhang (1996) for two general orthotropic materials. There are cases however, for which the near tip stress and displacement fields in that study may not be employed. For certain material symmetries, there is a mathematical degeneracy which invalidates these fields and requires a more complicated analysis. In this study, a particular material direction is taken which leads to this degenerate situation. In Section 2, the stress and displacement fields will be described for the particular materials chosen for study. The M-integral is presented in Section 3. A benchmark problem is carried out in Section 4 to demonstrate

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accuracy of the method. 2. Stress and Displacement Fields Relevant concepts related to interface fracture for both isotropic and anisotropic materials are presented here. In two dimensions and referring to Fig. 1, the in-plane stresses in the neighborhood of a crack tip at an interface are given by (1) where α, γ = 1, 2, i =

the complex stress intensity factor K = K 1 + i K2 ,

(2)

and the superscripts (1) and (2) are related to the real and imaginary parts

Figure 1.

Crack tip coordinates.

of Kr i∈ , respectively. In (1), following Ting (1996) (3) where (4) The 3 × 3 matrix

is given by (5) (6)

and (7)

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The subscripts 1 and 2 in (6) and (7) represent, respectively, the upper and lower material. Since Sj and L j are real and (8) knowledge of the left hand side of (8) is sufficient to determine (6) and (7). Instead of presenting more general expressions for the matrices Aj and B j which may be found in Ting (1996, pp. 170–172), the specific matrices for the upper and lower materials are presented in Appendix 1. Note that transversely isotropic materials are being studied here. It is observed that the notation employed here for ∈ in (3) differs from that of Rice (1988) and Hutchinson (1990) in which the fraction is inverted. The notation here is in keeping with Dundurs (1969). However, it is immaterial if one is consistent. Referring again to the expression for the stresses in (1), the functions and are presented in Appendix 2 for the two types of materials considered in this study. For two isotropic materials they are given in polar coordinates by Rice, et al. (1990) and in Cartesian coordinates by Deng (1993). The complex stress intensity factor in (2) may be written in non-dimensional form as (9) where L is an arbitrary length parameter and σ is the applied stress. The non-dimensional complex stress intensity factor may be written as (10) so that the phase angle (11) In (11)

(12) (13) The material parameters D 11 and D 22 always have the same sign; they are components of the matrix D in (6). The mechanical properties E A , E T , G A , G T , v A and v T are the usual material properties in the axial and

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transverse directions (namely, Young’s moduli, shear moduli and Poisson’s ratios); since the material is transversely isotropic, G T = E T /2(1+v T ). The parameter κ is given in Appendix 1 in eq. (44). The constants β j , j = 1, 2, 3 are related to the three complex eigenvalues of the elastic constants p j for the upper material (see Ting, 1996, pp 121–128), where p j = i β j for a transversely isotropic material with this material symmetry. For plane strain conditions, the stress components on the interface ahead of the crack tip are

(14) The crack face displacements in the vicinity of the crack tip are found to be

(15) where The interface energy release rate G i is related to the stress intensity factors by (16) where (17) Note that the subscript i in (16) represents interface and G i has units of force per length. It should be noted that inherently for any interface both K 1 and K 2 must be prescribed or equivalently G i and ψ. In describing an interface crack propagation criterion, one may prescribe a relation between K 1 and K 2 or what is commonly done, the critical energy release rate G i c is given as a function of the phase angle ψ . A possible generalization of the present problem is one in which the crack in Fig. 1 is rotated about the x 3 -axis while ensuring that the symmetry plane of the upper material remains perpendicular to the crack line. For this case, it may be observed from (4) that β remains invariant as was pointed out earlier by Ting (1986). Moreover, the complex stress intensity factor K can be defined as in (14) based upon a new crack tip coordinate system aligned along and normal to the crack line. The resulting stress and

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displacement fields can be transformed to the original x 1 -x 2 coordinate system using standard transformation matrices. 3. Area M-Integral In this section, the path independent M-integral is described. The Mintegral was introduced by Yau, et al. (1980) for separation of mixed modes in homogeneous bodies. It was converted to an area integral for isotropic bimaterials by Shih and Asaro (1988). It may be written as

(18) In (18), indicial notation is employed, the superscripts (1) and (2) represent two solutions and δ is the Kronecker delta. The mutual strain energy density W (1,2) of the two solutions is given by (19) The function q 1 is defined for finite element analysis as (20) where N m are the finite element shape functions of an eight noded isoparametric element and ξ and η are the coordinates in the parent element (for further details, see Banks-Sills and Sherman, 1992). The calculation of the M-integral is carried out in a ring of elements surrounding the crack tip (the area A in (18)). The elements within the ring move as a rigid body. For each of these elements q 1 is unity; so that, the derivative of q 1 with respect to x j is zero. For all elements outside the ring, q 1 is zero; so that, again the derivative of q 1 is zero. For elements belonging to the ring, the vector q 1m in (20) is chosen so that the virtual crack extension does not disturb the relative nodal point positions in their new locations; for example, a regular element with nodes at the mid-sides contains only mid-side nodes after distortion. The relationship (21) may also be obtained. In (18) and (21), problem (1) is that for which a solution is sought. Two auxiliary solutions are required in order to determine both and for this problem. These are denoted as (2a) and (2b).

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For solution (2a), choose = 1 and = 0. Such a solution does exist for some special loading. Equation (21) becomes (22) and from (18)

(23) The displacements required for solution (1) are taken from a finite element analysis of the problem to be solved; the stresses and strains are calculated from these. Asymptotic expressions for the stresses, strains and displacements for solution (2a) are employed. The stresses and displacements are presented in Appendix 2. For solution (2b ) , = 0 and = 1. Equation (21) becomes (24) and from (18)

(25) After calculating these integrals, (22) and (23), (24) and (25).

and

are found by equating

4. Benchmark Problem A benchmark problem is presented in this section. Here, a circular domain containing a crack along the interface between two transversely isotropic materials is analyzed under plane strain conditions (see Fig. 2). The material chosen here is a graphite/epoxy (AS4/3501-6) fiber reinforced material. The volume fraction of the fibers is about 65%. Although the graphite fiber is anisotropic, homogenization allows determination of the effective mechanical properties, some of which are presented in Table 1 (Pagano, 1999). In the upper material, the graphite fibers are aligned with the x1 –direction, while in the lower material they are along the x3–direction resulting in a mathematical degeneracy. Parameters necessary for calculation of the stress intensity factors and energy release rate are given in Table 2.

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L. BANKS-SILLS AND V. BONIFACE TABLE I. Effective mechanical properties of a graphite/epoxy fiber reinforced composite. property EA ET GA vA vT

value 138.2 GPa 10.4 GPa 5.5 GPa 0.3 0.55

TABLE II. Material parameters required for stress intensity factor and energy release rate calculation.

parameter

value

β1 β2 β3 ∈ D11 (GPa) – 1 D 22 (GPa) – 1

0.624 4.896 1.280 -0.028 0.231 0.312

For the geometry in Fig. 2, two analyses are performed. In the first case, K 1 is taken to be unity and K 2 is taken as zero. The asymptotic displacement field corresponding to these values (calculated from the expressions presented in Appendix 2) is applied to the outer boundary of the circular domain. Finite element analyses are carried out with the program ADINA (1999). The finite element mesh is exhibited in Fig. 3. There are 4,800 eight noded isoparametric elements and 14,641 nodal points. Quarter-point elements are employed at the crack tip. Although the singularity at the crack tip is a combination of square root and oscillatory, it was found by Banks-Sills, et al. (1999) that better results are obtained for interface cracks in bimaterial isotropic bodies when quarter-point elements are employed instead of regular eight noded isotropic elements. The result for K1 is found to be 1.0 + ∆ K 1 where 2 × 10 – 5 ≤ ∆ K 1 ≤ 3 × 10 – 4 for eight rings not

FRACTURE MECHANICS OF AN INTERFACE CRACK

Figure 2.

191

Crack in a transversely isotropic, bimaterial circular domain.

including the innermost two. For K 2 , the value was found to be between –2 × 1 0 – 5 and 6 × 1 0 –5 . As the second case, the displacement field corresponding to K 1 = 0 and K 2 = 1 is applied at the boundary. The results obtained by means of the M-integral are –8 × 10 – 5 ≤ K 1 ≤ 5 × 10 –5 and K 2 = 1.0 + ∆ K 2 w h e r e 3 × 10 – 4 ≤ ∆ K 2 ≤ 5 × 10 – 4. In this case, there is crack face overlap. The results of these two load cases demonstrate the accuracy of the stress and displacement fields presented in Appendix 2, as well as the Mintegral implementation. 5 . Summary

and

Discussion

The first term of an asymptotic expansion for both the stress and displacement fields has been developed for a crack along the interface of two transversely isotropic materials. The plane of symmetry for the upper material is perpendicular to the x 1 –direction, whereas that for the lower material is perpendicular to the x 3 –direction. Thus, the lower material involves a mathematical degeneracy. These fields have been employed as auxiliary functions in a conservative area M-integral. A benchmark problem was solved and shown to produce excellent results. Work is continuing in this direction for other cases with transverse isotropy. Some analytic solutions will be reported elsewhere. Acknowledgment The first author would like to give special thanks to Professor Anthony R. Ingraffea who provided a wonderful environment at Cornell University to carry out a large portion of this investigation while she was on sabbatical

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Figure 3. Finite element mesh of circular domain containing 14,641 nodal points and 4,800 eight noded isoparametric elements.

there. We would also like to thank Mr. Rami Eliasi for all his help with the finite element calculations. References Banks-Sills, L. and Sherman, D.(1992) On the computation of stress intensity factors for three-dimensional geometries by means of the stiffness derivative and J-integral methods, International Journal of Fracture 53, 1–20. Banks-Sills, L., Travitzky, N., Ashkenazi, D. and Eliasi, R. (1999) A methodology for measuring interface fracture properties of composite materials, International Journal of Fracture 99, 143–161. Bassani, J. and Qu, J. (1989) Finite crack on bimaterial and bicrystal interfaces. Journal of the Mechanics and Physics of Solids 37, 434–453. Bathe, K.J., (1999) ADINA – Automatic Dynamic Incremental Nonlinear Analysis System, Version 7.3, Adina Engineering, Inc. USA.

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Charalambides P.G. and Zhang, W. (1996) An energy method for calculating the stress intensities in orthotropic bimaterial fracture, International Journal of Fracture 76, 97–120. Deng, X. (1993) General crack-tip fields for stationary and steadily growing interface cracks in anisotropic bimaterials. Journal of Applied Mechanics 60, 183–189. Dundurs, J. (1969) Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading, Journal of Applied Mechanics 36, 650–652. Gosz, M., Dolbow, J. and Moran, B. (1998) Domain integral formulation for stress intensity factor computation along curved three-dimensional interface cracks, International Journal of Solids and Structures 35, 1763–1783. Hutchinson, J.W. (1990) Mixed-mode fracture mechanics of interfaces, in M. Rühle, A.G. Evans, M.F. Ashby, J.P. Hirth (eds.), Metal-Ceramic Interfaces, Pergamon Press, Oxford, 295–301. Matos, P.P.L., McMeeking, R.M., Charalambides, P.G. and Drory, M.D. (1989) A method for calculating stress intensities in bimaterial fracture, International Journal of Fracture 40, 235–254. Nahta, R. and Moran, B. (1993) Domain integrals for axisymmetric interface crack problems, International Journal of Solids and Structures 30, 2027–2040. Nakamura, T. (1991) Three-dimensional stress fields of elastic interface cracks, Journal of Applied Mechanics 58, 939–946. Pagano, N. (1999) Personal communication. Qu, J and Bassani J.L. (1989) Cracks on bimaterial and bicrystal interfaces. Journal of the Mechanics and Physics of Solids 37, 417–434. Rice, J.R. (1988) Elastic fracture mechanics concepts for interfacial cracks, Journal of Applied Mechanics 55, 98–103. Rice, J.R., Suo, Z., and Wang, J.-S. (1990) Mechanics and thermodynamics of brittle interface failure in bimaterial systems. in M. Rühle, A.G. Evans, M.F. Ashby, J.P. Hirth (eds), Metal-Ceramic Interfaces, Pergamon Press, Oxford, 269–294. Shih, C.F. and Asaro, R.J. (1988) Elastic-plastic analysis of cracks on bimaterial interfaces: part I–small scale yielding, Journal of Applied Mechanics 55, 299–316. Stroh, A.N. (1958) Dislocations and cracks in anisotropic elasticity, Philosophical Magazine 7, 625–646. Suo, Z. Singularities, interfaces and cracks in dissimilar anisotropic media, Proceedings of the Royal Society, London 427, 331–358. Ting, T.C.T. (1986) Explicit solution and invariance of the singularities at an interface crack in anisotropic composites, International Journal of Solids and Structures 22, 965–983. Ting, T.C.T. and Hwu, C. (1988) Sextic formalism in anisotropic elasticity for almost non-semisimple matrix N, International Journal of Solids and Structures 24, 65–76. Ting, T.C.T. (1990) Interface cracks in anisotropic bimaterials, Journal of the Mechanics and Physics of Solids 38, 505–513. Ting, T.C.T. (1992) Interface cracks on anisotropic elastic bimaterials–a decomposition principle, International Journal of Solids and Structures 29, 1989–2003. Ting, T.C.T. (1996) Anisotropic Elasticity–Theory and Applications, Oxford University Press, Oxford. Wang, S.S. and Yau, J.F. (1981) Interfacial cracks in adhesively bonded scarf joints, American Institute of Aeronautics and Astronautics Journal 19, 1350–1356. Yau, J.F., Wang, S.S. and Corten, H.T. (1980) A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity. Journal of Applied Mechanics 47, 335–341.

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Appendix 1 In this Appendix, the matrices A j , B j and B –j 1 which appear in (8) and are employed to calculate the oscillating part of the singularity ∈ in (3) are presented for the specific transversely isotropic materials studied here. They are related to the material properties. The subscript j represents the upper and lower materials, 1 and 2, respectively. The x 1 –direction is the axial direction of the upper material. The matrix A 1 is given by

(26)

where , j = 1,2,3, are normalization factors for the upper material which are not necessary in the calculation of (8). The constants β j , j = 1,2,3 are related to the three complex eigenvalues of the elastic constants p j (see Ting, 1996, pp 121–128), where p j = i β j for a transversely isotropic material with this material symmetry. They are given by

(27) (28) where are elements of the reduced compliance matrix, which for the present material are found to be

(29) (30) (31) (32) (33)

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195

The constants Q j are related to the material properties as

(34) (35) (36) (37) The material parameters E A , E T , G A , GT , vA and v T are the usual material properties in the axial and transverse directions (namely, Young’s moduli, shear moduli and Poisson’s ratios); since the material is transversely isotropic, G T = E T / 2 ( 1 + v T ). The matrix B 1 is given by

(38) Its inverse is given by

(39) In the lower material, the axial direction coincides with the x 3 –direction. The mechanical properties E A , E T , G A , G T , v A and v T are taken to be the same as for the upper material; but they are in different coordinate directions. It turns out that this material is mathematically degenerate. It has three identical complex eigenvalues p j = i where the subscript j = 1,2,3. To determine the stress and displacement fields, matrices alternative to A 2 and B 2 are required; these are A '2 and B ' 2 . Since (40) it is possible to calculate ∈ with the aid of (8). On the other hand, one may determine A 2 B 2– 1 without calculating the individual matrices (see Ting, 1996, p. 173). For brevity, only the primed matrices are presented. To obtain them, an orthogonalization procedure is employed; for details see Ting (1996, pp. 489–492) and Ting and Hwu (1988). They are found to be

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(41)

(42) and

(43) where (44) The normalization factors and are again unnecessary for determining both β and the stress and displacement fields. Nonetheless, for this special case of a mathematically degenerate material, they are seen to be

(45) (46)

Appendix

2

In this Appendix, the stress and displacement fields in the neighborhood of the crack tip are derived for the particular material combination considered in this investigation. Recall that both materials are transversely isotropic. In the upper material, the axial direction coincides with the x 1 –direction; whereas for the lower material, this direction coincides with the x 3 –direction. However, for any anisotropic material in the upper halfplane and mathematically degenerate material in the lower half-plane, the displacement field u ( j ) and the stress function Φ ( j ) are presented below ( j = 1,2 represents upper and lower material, respectively). Both the

FRACTURE MECHANICS OF AN INTERFACE CRACK

197

displacement and stress function are two-dimensional vectors in the x1 a n d x 2 –directions. For the upper half-plane,

(47) (48) where the diagonal matrix is (49) and h is a complex 3 × 1 vector to be determined. Equations (47) and (48) are a special case of equations found in Ting (1992). For the specific case at hand, transversely isotropic material with the axial direction coinciding with the x1 –axis, p j = i β j , j = 1, 2, 3. The bar over a quantity represents its complex conjugate. The matrices A 1 and B 1–1 a r e given in Appendix 1. For a mathematically degenerate anisotropic material in the lower halfplane, (50) (51) where the matrix F ( z ) is given by (52) and ()' represents differentiation with respect to z. For a transversely isotropic material with the axial direction coinciding with the x 3 –axis, p j = i ; hence, z ( 2 ) = z = x 1 + ix 2 . The vector h is determined by satisfying displacement continuity across the interface and is found to be (53) where (54)

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L. BANKS-SILLS AND V. BONIFACE

¿ From the above relations, explicit expressions for the stress and displacement fields are obtained. First of all, the stress components are related to the stress function vector Φ by

(55) (56) In the upper half-plane, the displacements are found to be

(57) and

FRACTURE MECHANICS OF AN INTERFACE CRACK

199

(58) where (59) (60) (61) (62)

(63) (64) (65) (66) (67) (68) (69) (70) (71) (72)

(73) (74)

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L. BANKS-SILLS AND V. BONIFACE

B 1 = cos 2 θ + β12 sin 2 θ B 2 = cos2 θ + β 22 sin2 θ .

(75) (76)

The in-plane stress components in the upper half-plane, denoted by the superscript (1), are given by

(77)

FRACTURE MECHANICS OF AN INTERFACE CRACK

201

(78)

(79) where (80) (81) (82) (83) For the lower half-plane, the displacements are found to be

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L. BANKS-SILLS AND V. BONIFACE

(84) and

(85) where

FRACTURE MECHANICS OF AN INTERFACE CRACK

203

N5 = cosh ∈ (π + θ ) N6 = sinh ∈ ( π + θ )

(86) (87)

M 9 = c o s (θ/2)

(88) (89)

M 10 = s i n (θ/2).

The in-plane stress components in the lower half-plane are given by

(90)

(91)

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L. BANKS-SILLS AND V. BONIFACE

(92) where M 11 = cos(3θ /2) M 12 = sin(3 θ /2).

(93) (94)

PATH-INDEPENDENT INTEGRALS RELATED TO THE J-INTEGRAL AND THEIR EVALUATIONS

SHIRO KUBO Department of Mechanical Engineering and Systems Graduate School of Engineering Osaka University 2-1, Yamadaoka, Suita, Osaka, 565-0871 JAPAN Abstract. In relation with Rice’s J-integral, local J vector, local J scalar, global J vector and global J scalar proposed by the present author are introduced, and their interrelations are discussed. It is shown that the local J vector is perpendicular to the crack front. The global J scalar is interpreted in terms of the energy release rate with crack extension. The relation between the global J scalar and the J-, M-, L-integrals are discussed, and extended expressions of the M- and L-integral are presented. The local and global J-integrals are applied to deduce J expressions for an axi-symmetric crack. A method for estimating the distribution of J-integral along crack front of a three-dimensional crack is described. Formulae for experimental evaluation of the J-integral and the modified J-integral for creep crack problems using displacement measurement are introduced. Analytical expressions for several specific cracks are described.

1 . Introduction The J-integral proposed by Rice (1968) has been playing a major role for evaluating the behavior of cracks under elastic-plastic conditions. The line integral expression of the J-integral was advantageously used in its numerical evaluation. For experimental evaluation of the J-integral from loaddisplacement curve, simple formulae proposed by Rice et al. (1973) promoted extensively the application of the J-integral. In this paper some integrals related to Rice’s J-integral and their interrelations are discussed. Formulae for experimental evaluation of the J-integral and the modified J-integral (C * -integral) for creep crack problems (Ohji et 205 T.-J. Chuang and J. W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 205–221. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

206

S. KUBO

al., 1974, 1976; Landes and Begley, 1976) using displacement measurement are introduced. Analytical expressions deduced by the present author for several specific cracks are described. 2. Definition of Local and Global J-Integrals for a Three-Dimensional Crack Consider a cracked three-dimensional body in the Cartesian coordinates x1 x2 x3 . The surface and the front of the crack are supposed to be smooth. Assume the existence of strain energy density W, which relates strain ∈ i j and stress σi j as, (1) The definition of infinitesimal strain is used: (2) where u i is displacement and j following comma denotes the partial differentiation with respect to x j . The traction Ti is defined in terms of σ j i and the outward unit normal vector v j a s , (3) The summation convention is used over the repeated index j. The crack surface is supposed to be traction free. No body force is applied in the body, and therefore the equilibrium equation is written as, (4) Local and global J vectors and scalars were proposed by Kubo et al. (1981, 1982). 2.1. LOCAL J VECTOR After Blackburn(1972) and Miyamoto et al. (1979) the local J-integral can be defined at point O on crack front as follows, by taking the energy balance of a plate of thickness B , which contains point O and is surrounded by surfaces S 1 to S 5 shown in Fig. 1: (5) Here F is defined by

INTEGRALS RELATED TO THE J-INTEGRAL

Figure 1.

207

Integration surface and path for local J-integral

(6) where (7) The integral is evaluated for the surface of the plate, and i i denotes the unit normal vector in the xi direction. The local J-integral J l o c al has three components and can be defined as a vector. It is easily shown that the local J-integral Jlocal is independent of the selection of surfaces used for evaluation, using the definition of strain energy density (Eq. (1)) and strain-displacement relationship (Eq. (2)) as is done for proving the path-independence of the J-integral (Rice, 1968). When the x 3 direction is taken to be in the thickness direction of the plate used for the energy balance, the local J-integral Jlocal defined by Eq. (5) can be rewritten as,

(8) where Γ is the contour of S 2 and is a path starting from the lower surface of the crack and arriving at the upper surface in the x1 x 2 plane, and A i s the surface surrounded by Γ and the crack in the plane. When curvilinear coordinates θ 1θ 2 θ 3 are used, F is given by (9)

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S. KUBO

Figure 2.

Integration surface for global J-integral

where g i is the contravariant base vector, and ”| i ” denotes the covariant derivative. 2.2. LOCAL J SCALAR The local J scalar can be defined as the absolute value of the local J vector: J local = | J l o c a l |.

(10)

2.3. GLOBAL J VECTOR The global J vector for the Cartesian coordinates x1 x2 x 3 can be defined by taking the integral of H over the surface S = S 1 + S 2 + S 3 + S 4 + S 5 surrounding the entire crack front as shown in Fig. 2. (11) When curvilinear coordinates θ1 θ 2 θ 3 are used, J g l o b a l is given by (12)

2.4. GLOBAL J SCALAR Introduce a vector field m = mi i i , which is differentiable and is not singular in the body. The global J scalar is defined by the following integral.

(13)

INTEGRALS RELATED TO THE J-INTEGRAL

209

Here V denotes the region surrounded by S and crack surfaces. It is easily shown that J global (m) is independent of the selection of surface S. When curvilinear coordinates θ ¹ θ ² θ ³ are used, J global (m) is given by,

(14) 3. Direction of Local J Vector When the x 3 direction is taken to be parallel to the tangential direction at point O on the crack front, the x 3 component of the local J vector J l o c a l is given as follows using the surfaces S 4 and S 5 defined in Fig. 1.

(15) Denoting the distance from crack tip in the plate by r, σi j ∈ ij has singularity of the order of r – 1. Stress σij has singularity of the order of r – 1 / 2 or weaker one and displacement u j has no singularity. The differentiation of these values with respect to x 3 does not change the order of singularity. These mean that the integrand of the first term on the right hand side of Eq. (15) has the singularity of the order of r – 1 . Then this integral approaches 0 when integral domain A shrinks to O. The second term also approaches 0 when A shrinks. Since J l o c a l is independent of the selection of integral domain A, (J l o c a l)3 is always equal to zero. J l o c a l is thus perpendicular to the crack front at each point of the crack front. 4. Interrelations Between J Scalars and Vectors 4.1. INTERRELATION BETWEEN LOCAL J VECTOR AND GLOBAL J VECTOR Consider layers of integral surfaces to cover a part of crack front ƒ shown in Fig. 3. On surfaces facing with each other, outward unit normal vector v i is opposite in direction, and therefore the contributions for these surfaces to F are cancelled with each other. The sum of integration of H over the surface of each layer is then equal to the integration of H over the cylindrical surface covering the crack front ƒ. From Eqs. (5) and (11) the following equation is thus obtained in the limit that the thickness B of the layer approaches 0. (16)

S. KUBO

210

Figure 3.

Figure 4.

Layers of integration surfaces

Integration surface very close to crack front

In this equation the integration is taken along the crack front ƒ. 4.2. INTERRELATION BETWEEN LOCAL J VECTOR AND GLOBAL J SCALAR Take the integration surface of J global (m) very close to the crack front as shown in Fig. 4. Since m has no singularity it can be taken as to be constant for an infinitesimally small amount of crack front df. Then the first term of the right hand side of Eq. (13) can be rewritten as (17) where ΓS is a path taken near the crack tip. The integrand of second term of the right hand side of Eq. (13) has the singularity of the order of r – 1 or weaker one, and therefore the term approaches to 0 as the integration surface S shrinks to the crack front. Therefore, the following equation relating the global J scalar with the local J vector is obtained. (18)

INTEGRALS RELATED TO THE J-INTEGRAL

211

4.3. INTERRELATION BETWEEN LOCAL J SCALAR AND GLOBAL J SCALAR When the angle between J l o c a l and m at each point of crack front is denoted by α, Eq. (18) is reduced to (19) where m denotes the absolute value of m. 4.4. INTERRELATION BETWEEN GLOBAL J VECTOR AND GLOBAL J SCALAR When vector m is constant along the crack front, Eq. (18) is reduced to, (20) Then using Eq. (16) the following relation is obtained. Jg l o b a l (m) = m · Jg l o b a l .

5.

(21)

Relation between Energy Release Rate and Global J Scalar

The potential energy II of the cracked body is given as, (22) where V T denotes the volume of the body, and U is the potential of external forces. Consider a sharp notch shown in Fig. 5. Suppose that the notch extended by m ∆ q and the volume of the body is reduced by ∆ V, where ∆ q is a scalar independent of location. Then the change in the potential energy is given by, (23) As was discussed by Budianski and Rice (1973) the first and the second terms cancel with each other due to the principle of virtual work. The change in the volume is given as, d V = –dS vi m i ∆ q,

(24)

212

S.KUBO

Figure 5.

Three-dimensional notch and its extension

where dS is a small element of notch surface SB . Then Eq. (23) is reduced to, (25) When S B is used for the integral surface for Jglobal (m), Eq. (13) gives, (26) Since the notch is stress free T j = 0, (27) Comparing Eq. (25) with Eq. (27) and noting that the directions of v i in these equations are opposite with each other, the following equation is obtained. (28) J global ( m) = – dII/ dq This equation states that J global ( m ) gives the release rate of the potential energy when notch extends by m at each point of the notch tip. 6. Relation Between Global J Scalar and J-, M- and L-Integrals 6.1. RELATION BETWEEN GLOBAL J SCALAR AND RICE’S J-INTEGRAL

Suppose that the vector m is equal to a constant c: (29) Then the global J scalar Jglobal ( m) = J global ( c ) is given as, (30)

213

INTEGRALS RELATED TO THE J-INTEGRAL This equation is rewritten as, Jglobal (c) = c ⋅ J global ⋅

(31)

For a two-dimensional crack with c1 = 1, c

2

= 0 , c 3 = 0,

(32)

J global (c) is reduced to Rice’s J-integral. 6.2. RELATION BETWEEN GLOBAL J SCALAR AND M-INTEGRAL

When the vector m is given as, (33) the global J scalar Jglobal (m) = J global (x) is expressed as,

(34) where δ i j denotes the Kronecker delta. For two-dimensional problems δii = 2 and for three-dimensional problems δ i i = 3. For nonlinear materials whose strain is proportional to the n-th power of stress, the strain energy density W is given by, (35) Then the integral over V in Eq. (34) is rewritten in terms of an integral over S using the divergence theorem, and J global (x) is reduced to, (36) Consider a linear elastic material for which n = 1. The integration in Eq. (36) is reduced to a line integral as follows for a two-dimensional crack using δ ii = 2. (37) For a three-dimensional crack, δ i i = 3 and then Eq. (36) is reduced to, (38)

214

S. KUBO

Equations (37) and (38) coincide with the expressions of the M-integral proposed by Knowles and Sternberg (1972). Equation (36) gives an expression of the M-integral extended for nonlinear materials. When cylindrical coordinates r θz are introduced, the components of m are written as, m1 = r, m 2 = 0, m3 = z , (39) the global J scalar J global (m) = J global (r + z) is written as,

(40) When spherical coordinates r φθ are introduced, the components of m are written as, m 1 = r, m 2 = 0, m 3 = 0, (41) and an expression of the global J scalar Jglobal ( m) = J global (r) in the spherical coordinates is written as,

(42)

6.3. RELATION BETWEEN GLOBAL J SCALAR AND L-INTEGRAL

When the vector m is given as, (43)

the global J scalar J global (e3 ik x k i i ) is expressed as, (44) using the divergence theorem, and the equilibrium equation (Eq. (4)) with the permutation symbol e ijk . When the cylindrical coordinates r θz are used, and the components of m are written as, m 1 = 0 , m 2 = –1, m 3 = 0,

(45)

INTEGRALS RELATED TO THE J-INTEGRAL

215

the global J scalar J global is written as, (46) When the spherical coordinates r φθ are introduced, the components of m are written as, m1 = 0 , m

2

=0, m

3

= –1,

(47)

an expression of the global J scalar J global in the spherical coordinates is written as, (48)

7. J-Integral for an Axi-Symmetric Crack As a simplest example of three-dimensional cracks, an axi-symmetric crack with radius a is discussed in this section. 7.1. J-INTEGRAL EXPRESSION BASED ON LOCAL J VECTOR

When the Cartesian coordinates are used, the local J vector at point O shown in Fig. 6 is in i 1 direction and is given by, (49) since the local J vector is perpendicular to the crack front as is discussed in section 3, and due to symmetry with respect to the crack plane. Rewriting the components of stress and displacement in terms of the cylindrical coordinates local J scalar Jlocal , which is equal to the magnitude of J local , is given by,

(50)

7.2. J-INTEGRAL EXPRESSION BASED ON GLOBAL J VECTOR

When the global J vector is evaluated for surface S 1 + S 2 in Fig. 7, (51)

216

S. KUBO

Figure 6.

Figure 7.

An axi-symmetric crack

An axi-symmetric crack cut into two halves along a symmetry plane

Since local J vector is constant in magnitude and is perpendicular to the crack front, the global J vector, which is equal to the integration of the local J vector along the crack front, is given as, J global = 2i2 a Jlocal .

(52)

From Eq. (51) the global J vector is given by the following equation for the cylindrical coordinated r θz.

(53)

INTEGRALS RELATED TO THE J-INTEGRAL

217

From Eqs. (52) and (53) the local J scalar J l o c a l is given by,

(54) This expression multiplied by 2 πa coincides with that proposed by Astiz et al. (1975). 7.3. J-INTEGRAL EXPRESSION BASED ON GLOBAL J SCALAR Suppose that the components of vector m is given as, m2 = m3 = 0

m 1 = ƒ(r),

(55)

in the cylindrical coordinates r θ z.

(56) From Eq. (19) relating the global J scalar and local J scalar, Jg l o b a l

=

2πa ƒ( a ) J l o c a l .

(57)

Then the local J scalar is expressed as,

(58) Equation (58) is reduced to Eq. (54) and Eq. (50) for ƒ(r) = 1 and ƒ(r) = 1/r, respectively. Equation (58) is a general expression of J-integral for an axi-symmetric crack including Eqs. (54) and (50) as special cases. Other expression can be obtained by employing other distributions of m. The equivalence of the J value for various expressions was confirmed by using the finite element analyses (Ohji et al., 1982). Since the value of local

218

S. KUBO

Figure 8.

A semi-circular surface crack

J scalar is independent of the selection of the expression used, expressions convenient for evaluation can be advantageously used. When Eq. (39) is used in the cylindrical coordinates r θz the local J scalar for the n -th power nonlinear material can be reduced to,

(59)

8 . Estimation of Distribution of J-Integral Using Global J Scalar The value of global J scalar J g l o b a l (m) can be obtained for various vector m. By combining these values the distribution of local J can be estimated. As an example, consider a semi-circular surface crack shown in Fig. 8. Cylindrical coordinates r θz are used. Since the local J vector is perpendicular to the crack front and the problem is symmetrical with respect to the crack plane, local J vector J l o c a l is in the r d i r e c t i o n . Jl o c a l = J l o c a l g1 ,

(60)

g1

where denotes the contravariant base vector in the r direction. The distribution of J l o c a l can be expressed using the Fourier series as, (61) Take the form of the vector m expressed as, m k = c o s k θg1,

(62)

where g 1 is the covariant base vector in the r direction. Then from Eqs. (18), (60), (61) and (62) Jg l o b a l ( m k ) = π ab k /2.

(63)

INTEGRALS RELATED TO THE J-INTEGRAL

219

J global ( m k ) is given by introducing m k in Eq. (14). Then from Eq. (63), the value of b k determining the distribution of J, is given by,

(64) The global J scalar method was applied to calculate the distribution of J-integral along crack front of a three-dimensional crack using the finite element stress strain analysis (Ohji et al., 1986a). 9. Simple Methods for Experimental Determination of J-Integral and Modified J-Integral Rice et al. (1973) proposed simple formulae for determining J values experimentally using load-displacement curve. Some extensions of the method were made (Kubo, 1975, Ohji et al., 1978). The method was successfully extended (Kubo, 1975, Ohji et al., 1978) for determining the modified J-integral J * (Ohji et al., 1974, 1976), which is equivalent to the C * integral (Landes and Begley, 1976), for creep crack problems. For example J* value for a plate with deep edge cracks can be given as, (65) where n is the creep exponent, σ net is the net section stress, and is the load-point displacement rate. When opening rate δ n of crack edge is used in place of , the J* value for shallow cracks as well as for deep cracks can be well approximated: (66)

10. Simple Formulae for Analytical Estimation of J-Integral Ohji et al. (1981) proposed simple formulae for estimating J values for a small crack in large plate. It was assumed that strain is proportional to the n -th power of stress. For mode III crack of size 2a, J value can be

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approximated by, (67) where net is the net section shear stress, and γnet is the corresponding shear strain. For mode III deep crack of ligament size b, J value is approximated by, (68) For a mode III center crack in a plate with crack length of 2a and ligament length of b, approximate J value can be obtained by combining Js and J d values given by Eqs. (67) and (68). (69) This equation is in analogy with Neuber’s rule for notch stress concentration. Similarly, for mode I shallow crack, J value is approximated by, (70) where σ net is the net section tensile stress, and ∈net is the corresponding strain. For mode I deep crack of ligament size b, J value is approximated by, (71) For a mode I center crack in a plate with crack length of 2a and ligament length of b, approximate J value can be obtained by applying Eq. (69) using J s and J d values given in Eqs. (70) and (71). For a mode III crack of length a emanating from a notch of radius ρ, J value is well approximated by the following equation (Ohji et al., 1986b). (72) Here max and γ max are the maximum shear stress and shear strain at notch root in the absence of the crack, and k being a constant dependent on the strain hardening exponent n. For n = 1, k = 1.5 and for n = 9, k ≈ 3 Acknowledgements The work is partly supported by the Ministry of Education, Science, Sports and Culture under Grant-in-Aid for Scientific Research.

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References Astiz, M.A., Elices, M., and Galvez, V.S. (1975) Proc. ICF4, Waterloo, Canada 3, 395. Blackburn, W.S. (1972) Int. J. Fracture 8, 343. Budianski, B., and Rice, J.R. (1973) J. Appl. Mech. 40, 201. Knowles, J.K., and Sternberg, E. (1972) Archive for Rat. Mech. Anal. 44, 187. Kubo, S. (1975) Study on Mechanics of Crack Initiation and Growth, Osaka University Doctoral Thesis (in Japanese) Kubo, S., and Ohji, K. (1981) J. Soc. Mater. Sci., Japan 30, 796 (in Japanese). Kubo, S., and Ohji, K. (1982) J. Soc. Mater. Sci., Japan 31, 32 (in Japanese). Landes, J.D., and Begley, J.A. (1976) Mechanics of Crack Growth, ASTM STP 590, 128. Miyamoto, H., and Kikuchi, M. (1979) U.S.- Japan Seminar on Fracture Mechanics of Ductile and Tough Materials and Its Application to Energy Related Structures, Hayama, Japan, 33. Ohji, K., Ogura, K., and Kubo, S., (1974) Preprint Japan Soc. Mech. Engrs. No. 740-11, 207 (in Japanese). Ohji, K., Ogura, K., and Kubo, S. (1976) Trans. Japan Soc. Mech. Engrs. 42, 350 (in Japanese). Ohji, K., Ogura, K., and Kubo, S. (1978) Trans. Japan Soc. Mech. Engrs. 44, 1831 (in Japanese). Ohji, K., Ogura, K., and Kubo, S. (1981) Trans. Japan Soc. Mech. Engrs., Ser. A 47, 400 (in Japanese). Ohji, K., Kubo, S., and Suehiro, S. (1982) Preprint Japan Soc. Mech. Engrs. No. 820-2, 57 (in Japanese). Ohji, K., Kubo, S., Suehiro, S., and Nishimura, K. (1986a) Trans. Japan Soc. Mech. Engrs., Ser. A 52, 1034 (in Japanese). Ohji, K., Kubo, S., Maeda, T. (1986b) Trans. Japan Soc. Mech. Engrs., Ser. A 52, 1300 (in Japanese). Rice, J.R. (1968) J. Appl. Mech. 35, 379. Rice, J.R., Paris, P.C., and Merkle, J.G. (1973) Progress in Flaw Growth and Fracture Toughness Testing, ASTM STP 536, 231.

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ON THE EXTENSION OF THE JR CONCEPT TO SIGNIFICANT CRACK GROWTH

B. COTTERELL AND Z. CHEN Institute of Materials Research and Engineering 3 Research Link, Singapore 117602 AND A. G. ATKINS Department of Engineering University of Reading Reading, BG6 6AY, UK

Abstract: A possible method of estimating the extra work required in real elasto-plastic fracture as compared with a hypothetical non-linear elastic specimen is discussed. It is shown how the extra work term can be used to calculate the specific essential work, R., from J R obtained by the standard ASTM method. Although the method presented apparently underestimates the extra work, it points a way forward to obtain the true crack growth resistance.

1.

Introduction

The two most accepted fracture mechanics concepts are the stress intensity factor for linear elastic materials established by Irwin (1957) and the J-integral established by Rice (1968) for elasto-plastic materials. As with any concept, both have been at times pushed beyond their regimes of validity. The J-integral is exact for a non-linear elastic (nle) material, but can be applied in many cases to an elasto-plastic (elp) material. In an elp material, the value of J-integral at initiation is equal to the specific essential work, R, performed within the fracture process zone (FPZ) at the tip of a crack. Provided the FPZ is small, J can be obtained from the potential energy or the non-linear strain energy release rate and is given by (1) where II is the potential energy of the system, Λ nle , is the non-linear strain energy, a is the crack length, B is the thickness of the specimen, and u is the displacement. In practice J is usually obtained from this energetic interpretation rather than from the original contour integral. 223

T.-J. Chuang and J. W. Rudnicki (eds.),

Multiscale Deformation and Fracture in Materials and Structures, 223–235. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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Figure 1. Schematic load-displacement curves for a real elp and hypothetical nle material

Schematic load-displacement curves for a real elp and a hypothetical nle specimen are shown in Figure 1. The nle specimen is assumed to be made from a hypothetical nle material that has the same stress-strain curve as the elp material during loading but which is completely elastic, and has the same specific work of fracture, R, which, if not a constant, has the same dependence on the crack extension, ∆a. Up to the initiation of a crack from a notch of length ai , the elasto-plastic deformation closely follows Hencky’s equations and is practically identical to the hypothetical nle curve. At I a ductile crack is initiated in both the elp and nle specimens. After initiation the load paths of the two specimens diverge. Depending upon the material and the specimen geometry, the load can either increase or decrease during propagation. Some elastic energy remains locked into the specimen on unloading because of the non-uniform plastic deformation and it is simplest to divide the work done up to the initiation of a crack (area OID) into a non-recoverable work, Γi (area OIE), and a recoverable elastic strain energy, Λ (area IDE). The non-recoverable work, Γi , is the plastic work combined with the non-recoverable elastic strain energy. The initiation value, Ji , of the J-integral is equal to the initiation value of the specific essential work of fracture, Ri . Often J is divided into a plastic part, Jp , and an elastic part, At crack initiation the plastic component, Jpi , can be obtained from Γ i and is given by (Sumpter & Turner 1976)

(2) where η is a factor that depends on the geometry, and b i is the remaining ligament at initiation. For the deep notch bend specimen η =2; for other geometries η, though not a constant, only weakly depends on the crack length. During crack propagation non-proportional unloading occurs behind the crack tip in a elp specimen, which cannot be modelled by a nle theory except under certain restrictive conditions. Hutchinson and Paris (1979) have shown that outside of the region of non-proportional loading, there is a region where the deformation is nearly proportional. If this region is well contained within the region dominated by the J-

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singularity, there exists an annular region where the Hutchinson-Rice-Rosengren equations hold (Hutchinson 1968; Rice and Rosengren 1968). Limitations on the slope of the crack growth resistance, as assessed by J, have been developed to try to ensure that crack growth is controlled by the J-integral (McMeeking & Parks 1979; Shih & German 1981; Hutchinson 1983). A further complicating factor is the dependence of even Ji on the degree of constraint which can only be accommodated by introducing a second parameter as for example the Q-stress suggested by O’Dowd and Shih (1991,1992). This paper will concentrate on the inconsistencies introduced by using a nle theory for propagation. Unless the plastic zone is well contained within an elastic stress field, the value of the propagation J-integral, JR , as determined by the standard method (ASTM 1987), is only independent of size and geometry for small crack extensions (Hancock et al. 1993; Joyce & Link 1995; Xia et al. 1995; Xia & Shih 1995). The philosophy for calculation of the plastic component, Jp , during crack propagation is to assume that the deformation along the path IA in Figure 1 is nle (Hutchinson and Paris 1979; Ernst et al. 1981; ASTM 1987). This feature, analyzing crack propagation in a real elp specimen as if it was nle, is one of the keys allows us to unlock the specific work of fracture, R, from JR calculated by the standard ASTM method. A simplified version of the integral equation formulated by Ersnt et al. (1981) can be obtained, if it is assumed that η is a constant, by the differentiation of Eq. (2) and is (3) where X is the load and u p is the plastic part of the displacement. In the Ernst et al. (1981) and the ASTM (1987) Standard methods, Eq. (3) is integrated numerically along the propagation path IA in Figure 1. However, the real elp displacements are larger than the equivalent nle displacements by an amount u extra and the load, X nle , on the equivalent nle load-displacement curve is not identical to the real elp load, X e l p , except for the special case of the double cantilever beam (DCB) geometry. The Ernst et al. (1981) method overestimates Jp by including some of the plastic work outside of the FPZ making many JR curves spurious (Cotterell & Atkins 1996). Thus the so-called crack resistance is often an artifact of the method of calculating JR . True crack growth resistance as assessed by the specific essential work probably can arise only from only two causes: shear lip formation, the original cause proposed for crack growth resistance (Kraft et al. 1961), and changes in the constraint which can be analysed by such methods as the strip FPZ model based upon a Gurson material (Xia & Shih 1995; Xia et al. 1995). Experimental evidence for the above statements has been given by Atkins et al. (1998) and is briefly summarised here. The crack propagation in one very special test geometry, the deep side-grooved double cantilever beam (DCB) specimen loaded by an end moment, is steady state from initiation and thus there should be no crack growth resistance because shear lip formation is suppressed by the side grooves and there is no change in constraint at the crack tip. Hence the specific essential work of fracture, R, must be a constant. Provided the beam crack length to height ratio is reasonably large, the crack propagation in an end-force-loaded DCB specimen is close to steady state. The load-displacement curves for two aluminum alloy DCB specimens, one with a notch length of 70mm with a crack propagated to a total length of 140mm and one whose notch length is 140mm, are shown

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in Figure 2. The initiation load for the specimen with the long notch and the load after propagation to the same total crack length for the short notch are identical, but the beam displacement for the second case is much larger. If the aluminum alloy were nle the loaddisplacement point for both initiation from the long notch and propagation to the same total crack length from the short notch would be identical. In this case J calculated for both specimens would be the same and equal to the specific essential work, R. However, when the specimen with the short notch is analyzed using the Ernst et al. (1981) technique, the resulting JR -curve indicates a spurious crack growth resistance (Atkins et al. 1998).

Figure 2.

Load-displacement curves for two elp aluminium specimens (After Atkins et al. 1998)

The DCB specimen is peculiar because the crack propagates into virgin material. The moment at the crack tip remains constant during crack propagation. Thus in an endforce-loaded DCB specimen, the force for crack initiation is identical to the force necessary to propagate a crack to the same length but, as seen in Figure 2, the displacement in the propagated crack is larger by an amount which we call uextra . The extra work performed by a epl specimen propagated from a notch length a i to a crack length a over that performed on a hypothetical nle specimen is given by the areas IA´ABB´I in Figures 1 and 2. The extra work is the nle work that is not recovered in the case of elp. In this paper we discuss a possible way of back calculating from a real elp load-displacement curve, the corresponding curve for a hypothetical nle specimen that does not include the extra work term. This virtual nle curve can then be analysed by the Ersnt et al. (1981) method to obtain the true value of JR which is the specific essential work of fracture, R. Chen (1997) has shown that, in the special case of the DCB specimen, it is possible to calculate the elp curve directly from the nle curve. Because of the peculiarity of the DCB specimen, the recoverable elastic energy in an elp specimen is identical to the recoverable linear elastic energy in the hypothetical nle specimen of the same crack length. The rate of external work in a elp specimen and in the hypothetical nle one are given by

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(4)

where w elp is the plastic work performed to create a unit crack extension, and Γnle is the non-linear part of the elastic strain energy. During propagation the bending moment is crack initiation. Thus during propagation

(5) In the hypothetical nle specimen non-linear elastic energy stored behind the crack tip is recovered as well as the linear elastic energy and hence

(6)

The displacement u elp can therefore be found exactly from u n le by integration of Eq (6) for the DCB specimen (Chen 1997). The question is can a similar technique be used on standard test geometries?

2.

A Posssible Link between ELP and NLE

We have shown how the scheme proposed by Ernst et al. (1981) includes in JR some of the plastic work performed outside of the FPZ. Here we discuss an approximate method for extracting R from conventional JR -curves obtained from specimens that are fully yielded at initiation. Consider first a nle specimen made from a hypothetical material defined in Section 1. Ahead of the crack tip there is a region of area S where the stresses are “plastic” (see Figure 3).¹ Assume for the moment that the “plastic energy” density within this zone is uniform and equal to γp so that Γn le = S γp . As the crack propagates both the size of the “plastic” region and the “plastic energy” density change. The rate of increase in “plastic energy” is given by (7)

the first term in Eq. (7) comes from the change in “plastic energy” density and the

¹ For a nle material the “plastic” region is defined as the region where the stress-strain relationship is non-linear.

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second term is numerically equal to the “plastic” energy recovered as the plastic zone shrinks since the material is nle. In an elp material the plastic work cannot be recovered and it is argued that this second term is numerically equal to the extra work that must be performed in a real elp specimen as compared with the hypothetical nle specimen. Thus for an elp specimen (8) where Uextra is the extra work performed in the real elp specimen. The extra work, assuming the area of the active plastic region is proportional to b ², is given by

(9) The plastic energy (or work) density for the real elp specimen cannot be exactly the same as that for the hypothetical nle specimen, but we argue that, since the active plastic region is one where the material is being loaded, the plastic energy will be approximately the same. Also critical is the assumption that the plastic energy density is uniform. Clearly the plastic energy density cannot be uniform at initiation because by definition at the boundary of the plastic region the stresses must be elastic and the plastic energy density zero. However, it is argued that if the plastic zone at the tip of the propagating crack is contained within the plastic zone at initiation then the plastic energy density becomes more uniform with crack growth and the accumulated error in the method is likely to be small. Figure 3 shows schematically how the plastic energy density develops with crack growth from an initial notch at position 1 through to a crack at position 3. The development of the plastic energy density along a line X-X is shown at the bottom of the figure. At initiation the plastic energy density is zero at the edge of the plastic zone. However, if the boundary of the active plastic zone is nested within the previous plastic zone, then the plastic energy density along the boundary will accumulate with crack growth and the plastic energy density along the active portion of the line X-X will become more uniform. It is also assumed that the shape of the plastic zone remains similar during crack propagation so that its area is proportional to b ². It is our intention to show how the specific essential work of fracture, R, may be obtained approximately from J R using the concept of extra work. For the moment consider a hypothetical nle specimen and assume that, up to initiation its loaddisplacement curve is identical to the real elp specimen. After the crack grown to a crack length a (point A´ in Figure 4), any unloading will be along A´O. The “plastic component” of the specific essential work Rp is given by (10)

JR CONCEPT TO CRACK GROWTH

Figure 3. Schematic development of plastic region and energy density

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Figure 4. Schematic load-displacement curve

where Γnle =area OA'C'O. Since the fracture is nle, no energy is dissipated except at the crack tip and hence the area OIA'O is given by (11) The accumulated “plastic” work of the hypothetical nle specimen (area OIA'C'O) can be obtained from Eqs. (10) and (11) and is

(12) From Eq. (9) the extra work , U extra , performed by the real elp specimen is given by

(13)

Hence the accumulated plastic work of the real elp specimen , defined by the area OIACO, is the sum of Eqs. (12) and (13) (14) Now our argument is that since J p is obtained in the standard ASTM method by pretending that the real elp specimen behaves as if it were nle, the accumulated work is also given by

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(15)

Hence the differential equation connecting Jp and R p i s (16) If η is a constant Eq. (16) can be rewritten as (17) Note that Eqs. (16) and (17) do not depend on any modelling of the load-displacement curves for either the true elp specimen or its equivalent nle specimen. We now explore some of the implications of the extra work concept. First let us assume that the elastic components of R and J are very small and look at the deep notch bend specimen where η=2. If the specific essential work of fracture, R, is constant, Eq. (17) implies that (18) where ∆ a is the crack extension. A size effect is apparent. However, most JR -curves are not linear. In the Section 3 we examine the effect of elastic deformation.

3. Elasto-plastic Crack Growth As the size of a specimen increases so does the proportion of elastic deformation. Hence specimen size has a large effect on the JR -curve. For the purposes of illustration we examine the behavior of an elasto-plastic deep-notch three point bend specimen with a notch ai /W=0.5 and assume that the load, X, in both the nle and elp specimens is given by

(19) The EPRI analyses use this separable variables power law (EPRI 1984). The exponent, n, of the plastic displacement, u p , can be identified loosely with the strain hardening exponent, and the squared on ligament, b, is consistent with the η-factor of 2 for deep notch bend specimens. The elastic displacement is obtained from the compliance, C(a/w), given in the ASTM Standard Method for Determining J-R Curves (ASTM E1 152-1987) so that

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(20) where S is the span and E the elastic modulus. The ratio S/W has been taken as 4. The inverse of the real problem has been examined, that is a specific essential crack growth resistance, R(∆ a), is assumed and the expected JR -curve calculated. However, in principle the real problem of calculating R from JR is no different. At initiation Ji=R i and JR are normalized by Ri . Dimensional analysis shows that the normalized JR d e p e n d s upon σ 0W/R i ,σ0 /E, R/Ri , and n . In this simple illustrative exploration a typical value of 0.1 has been chosen for n. 3 . 1 J R -CURVES FOR A CONSTANT ESSENTIAL FRACTURE WORK The specific essential work of fracture has an elastic, Re , as well as a plastic component, Rp . Using Eq.(9) and adding an elastic term, R is given by (21) where G e (a/W) has been obtained from the expression for the stress intensity factor given in ASTM E1 152-1987. The difference between the plane stress and plane strain elastic modulus has been ignored. For any crack length this equation has been solved to give u nle /W which then has been used to calculate the components R p and R e . Eq. (17) has then been integrated to give Jp . Making use of the fact that the ASTM method tacitly assumes nle behavior, u elp is given by

(22)

The elastic component J e and the elastic displacement have then been calculated from the load. JR and load-displacement curves are shown in Figures 5 & 6 for a range of specimen sizes based on a unit specimen where σ 0 W0 /Ri =10, σ0 /E=0.0005. For small specimens the elastic strains are very small and JR is a function only of a/W and is given

Figure 5. JR versus a/W curves

Figure 6. Load-displacement curves

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by Eq. (18). For larger specimens the crack growth resistance is less. In Figure 5, specimens larger than 10 times the unit size do not yield completely before initiation and are outside of the validity of the theory but the J R -curves have been included as they do show qualitatively how J R is size dependent. The load-displacement curves show a discontinuity in slope at initiation which is not observed in ductile specimens. However, if the R-curve is given some crack growth resistance, load-displacement curves without slope discontinuities can be obtained as is shown in the Section 3.2. 3.2

J R -CURVES FOR INHERENT CRACK GROWTH RESISTANCE

As discussed in the introduction there can be a real R crack growth resistance. In this exploratory paper, this crack growth resistance is modeled by the exponential expression

(23)

The analysis follows that of Section 3.1, except that now instead of a constant on the left hand side of Eq. (21) there is the expression contained in Eq. (23). In the JR and loaddisplacement curves shown in Figures 7 and 8, the two parameters chosen to define R in Eq. (23) are: ∆ R=4 and λ =5 the other parameters: σ 0 W0 /Ri =10, σ 0 /E=0.0005, and n=0.1, are the same as in Figures 5&6.

Figure 7. J R and R-curves

Figure 8. Load-displacement curves

Figure 7 shows both the crack growth resistance of the specific essential work of fracture, R, and J R. For very small crack growths JR is almost identical to the specific essential work of fracture, but the differences become marked for large crack growths. The difference between J R and R reflects the extra plastic work performed in real elp specimens over that in hypothetical nle specimens and does not reflect a real increase in fracture resistance. When some real crack growth resistance in the specific essential work of fracture is introduced, fracture initiation can occur prior to the attainment of the maximum load without a discontinuity in the load-displacement slope (see Figure 8).

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4.

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Comparison of Extra Work with Other Predictions

Figure 9.

JR -curves for 3-point bend A533B specimens, a/W=0.6.

Xia et al. (1995) have shown that accurate predictions of load-displacement and JR curves can be obtained by embedding a FPZ, in the form of a row of void-containing cell elements ahead of the crack, within a conventional elasto-plastic continuum. In particular Xia et al. (1995) were successful in modeling the experimental JR -curves obtained by Joyce & Link (1994) for A533B steel specimens of differing geometries. Xia et al. (1995) also have predicted the JR -curves for deep notch bend specimens whose depth ranges from 50 to 300 mm. There is complete yielding in specimens whose depth, W, is up to 200 mm. However the largest specimen, W=300 mm, is globally elastic. What we have done is to assume that the JR -curve for this largest specimen corresponds closely to the specific essential work of fracture, R. Using this assumed R-curve, we have predicted the JR -curves from Eq. (17), neglecting any elastic effects. These predicted JR curves are compared with those calculated by Xia et al. (1995) in Figure 9. Although the general trend of increased J R is shown, the predicted values are very significantly smaller.

5.

Conclusions

We have shown that the difference between a real elp specimen and its equivalent hypothetical nle counterpart, lies in an extra work term which is the energy that is not recovered in a real specimen as the plastic zone contracts. Hence because in the standard ASTM method J R is obtained by an analysis that assumes a nle deformation, it is possible to extract the specific work of fracture, R, from JR . The difficulty lies in expressing what is the plastic work accumulated in the active plastic zone. In this paper the inverse problem has been has been analyzed. A specific essential work

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of fracture, R, has been assumed and JR that would have been obtained from the ASTM standard method calculated. The general size effect on JR has been modeled, but the extra work, and hence JR, has been underestimated. Further work to determine a simple procedure to enable the specific essential work and the size dependence of JR to be accurately estimated is worthwhile since much effort has been expended on obtaining J R -curves that are size and geometry dependent.

References ASTM E 1152-87 (1987) Standard Test Method for Determining JR curves Atkins, A.G., Chen, Z. and Cotterell, B. (1998) The essential work of fracture and JR curves for the double cantilever beam specimen: an examination of elasto-plastic crack propagation, Proceedings of the Royal Society of London, A 454,815-833. Chen, Z. (1997) Elasto-Plastic Fracture Propagation in Cantilever Beams, Doctoral Thesis, University of Reading, Reading, UK. Cotterell, B. and Atkins, A.G. (1996) A review of the J and I integrals, International Journal of Fracture, 81, 357-372. EPRI (1984) Advances in Elastic-Plastic Fracture Analysis, Electric Power Research Institute. Ernst, H.A., Paris, P.C. and Landes, J.D. (1981) Estimations on the J-integral and tearing modulus from a single specimen test record, in Fracture Mechanics: 13th Conference, ASTM STP 743, R. Roberts (ed.), American Society for Testing and Materials, Philadelphia, 476502. Hancock, J.W., Reuter, W.G. and Park, D.M. (1993) Constraint and toughness parameterized, in Constraint Effects in Fracture, ASTM STP 1171, W.M. Hackett, K.H. Schwabe and R.H. Dobbs (eds.), American Society for Testing and Materials, Philadelphia, 21-40. Hutchinson, J.W. (1968) Singular behaviour at the end of a tensile crack in a hardening material, Journal of the Mechanics and Physics of Solids, 16, 13-31. Hutchinson, J.W. (1983) Fundamentals of the phenomenological theory of non-linear fracture mechanics, Journal of Applied Mechanics, 35, 1042-1051. Hutchinson, J.W. and Paris, P.C. (1979) in Elastic-Plastic Fracture, ASTM STP 668, J.D. Landes, J.A. Begley, and G.A. Clarke (eds.), American Society for Testing and Materials, Philadelphia, 37-51. Irwin, G.R. (1957) Analysis of stresses and strains near the end of a crack traversing a plate, Journal of Applied Mechanics, 24,361-364. Joyce, J.A. and Link, R.E. (1995) Effects of constraint on upper shelf fracture toughness, in Fracture Mechanics: 26th Volume, ASTM STP 1256, W.G. Reuter, J.H. Underwood, and J.C. Newman (eds.), American Society for Testing and Materials, Philadelphia, 142-163. Krafft, J.M., Sullivan, A.M. and Boyle, R.W. (1961) Effect of dimensions on fast fracture instability of notched sheets, In Proceedings Symposium on Crack Propagation, College of Aeronautics, Cranfield 8- 15. McMeeking, R.M. and Parks, D.M. (1979) On criteria for J-dominance of crack fields in large scale yielding, in Elastic-Plastic Fracture, ASTM STP 668, J.D. Landes, J.A. Begley, and G.A. Clarke (eds.), American Society for Testing and Materials, Philadelphia, 175-194. O’Dowd, N.P. and Shih, C.F. (1991) Family of crack-tip fields characterized by a triaxiality parameter - I Structure of fields, Journal of the Mechanics and Physics of Solids, 39,9891015.

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O’Dowd, N.P. and Shih, C.F. (1992) Family of crack-tip fields characterized by a triaxiality parameter - II Fracture applications, Journal of the Mechanics and Physics of Solids, 40,939963. Rice, J.R. (1968) Mathematical Analysis in the Mechanics of Fracture, in Fracture, H.Liebowitz (ed.), Academic Press, New York Vol II, 192-314. Rice, J.R. and Rosengren (1968) Plane-strain deformation near a crack tip in a power law hardening materials, Journal of the Mechanics and Physics of Solids, 16, 1-12. Shih, F.C. and German, M.D. (198 1) Requirements for a one parameter characterization of crack tip fields by the HHR singularity, International Journal of Fracture, 17,27-43. Sumpter, J. G. D. and Turner, C. E. (1976) Method for laboratory determination of JIC, in Cracks and Fracture, ASTM STP 601, ASTM, Philadelphia, 3-18 Xia, L. and Shih, C.F. (1995) I. A numerical study using computational cells with microstructurally based length scales, Journal of the Mechanics and Physics of Solids, 43, 233-259. Xia, L., Shih, C.F. and Hutchinson, J.W. (1995) A computational approach to ductile crack growth under large scale yielding conditions, Journal of the Mechanics and Physics of Solids, 43, 389-413.

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EFFECT OF T-STRESS ON EDGE DISLOCATION FORMATION AT A CRACK TIP UNDER MODE I LOADING G. E. BELTZ AND L. L. FISCHER Department of Mechanical and Environmental Engineering University of California Santa Barbara, CA 93106-5070, USA Abstract: We calculate the effect of the nonsingular stress acting parallel to a crack (the “T-stress”) on edge dislocation nucleation at a crack loaded in Mode I. We find that this leads to crack size effect – that is, for small cracks (of order 100 atomic spacings or less), the T stress causes the critical load for dislocation nucleation (expressed in terms of the applied stress intensity factor) to deviate from the classical T = 0 result. Specific results are discussed for the case of a finite crack subject to remote tension, where it is shown that the threshold for dislocation nucleation is reduced. 1. Introduction In the continuum modeling of atomic-scale phenomena at crack tips, an important feature of the asymptotic representation of the stress field is often overlooked. Following the work of Williams [1], the expansion of the stress field in cylindrical coordinates about the tip (see Figure 1) may be generally written as (1) where K is the well-known “applied” stress intensity factor, and Sij ( θ), Tij ( θ), etc. represent the angular variation of the field. The first term is singular in r, the second term remains finite in the vicinity of the tip, and the remaining terms vanish as r → 0 . Linear elastic fracture mechanics is based on the reasonable notion that fracture processes that occur close to the tip are only affected by the singular contribution; hence, only the first term of Equation (1) is acknowledged as a valid descriptor of the stress field in a vast majority of the fracture literature. The second term has a particularly simple form – that is, it can be shown [1] that T11 , T 33, and T13 (= T 31 ) must be constant in θ in order to satisfy the field equations of elasticity, and the remaining components must vanish in order to preserve the traction-free boundary condition on the crack faces. We will exclusively deal with plane strain for the remainder of this discussion, thereby rendering T11 the only component of interest. Then, the asymptotic representation of the stress field around a crack, ignoring terms of r1/2 and higher, takes the form 237 T.-J. Chuang and J. W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 237–242. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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The constant term T11 = T is commonly known as the “T-stress.” Several researchers have demonstrated that the T-stress has a significant influence on the shape and size of the plastic zone that develops at a crack in a ductile material [2-5] as well as the directional stability of an advancing crack in a brittle solid [5-8]. The latter effect can be qualitatively understood by imagining a slight upward perturbation of the crack path (see Figure 1): if T > 0, the crack tip will continue to undergo opening forces, and will continue to veer away from the x1-axis. If T < 0, the crack opening forces tend to decrease, such that the only way for the crack to continue to propagate is for it to remain on the x1 -axis. Here, we will focus on the effect of T-stress on the threshold for dislocation nucleation. The basic framework for analysis will be that due to Peierls [9] and Nabarro [10], as introduced by Rice [11] for dislocation formation at a crack. We directly calculate the critical stress intensity factor K crit for dislocation nucleation as a function of T, and comment on the situations for which this effect could be most significant. 2.

Peierls Framework for Dislocation Nucleation at a Crack

The Peierls framework for dislocation formation at a crack assumes that the dislocation/crack system can be thought of as two elastic semispaces separated by a common plane (the crack plane and slip plane) on which there is a discontinuous jump in the displacement fields [11]. There exists a periodic relationship between shear stress and slip displacement along the slip plane, with traction free surfaces along the crack plane, as shown schematically in Figure 1. Prior to dislocation nucleation, there

Figure 1. Schematic of crack and slip plane (dashed) inclined at angle θ. The T-stress gives a contribution to σ 11 in addition to the classical K field result. A positive T-stress decreases the resolved shear stress on the slip plane, while a negative T-stress increases the resolved shear stress.

is a distribution of slip discontinuity along the slip plane that ultimately reaches a point of instability with increased applied load and results in the nucleation of a dislocation.

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Using this theory, the critical stress intensity factor for various fixed values of the Tstress may be determined, thus establishing a locus of K and T values at which nucleation occurs. For all of the calculations presented here, a simple relationship due to Frenkel [12] between shear stress and slip displacement on the slip plane is assumed, (3) where τ is the shear stress, ∆ is the relative atomic displacement between two atomic planes, h is the interplanar spacing of those two planes, µ is the shear modulus, b is the Burgers vector, and γus is the unstable stacking energy (equal to µ b² /2π² h in the Frenkel model). As introduced by Rice [11], the continuum analog to ∆ (referred to as δ ) is thought of as ∆ extrapolated to a cut halfway between the slipping planes and is given by (4) From elastic considerations, the stress along the slip plane can be written as:

(5) where the first two terms on the right hand side give the pre-existing shear stress along the slip plane due to the applied load on the crack geometry (comprising the most singular term, scaled by K, as well as the constant term, proportional to T ), and the third term reflects the stress relaxation that occurs due to sliding along the cut. The kernel in the integral term represents the stress due to a dislocation positioned at s, integrated over the entire slip distribution consisting of a continuous array of infinitesimal Burgers vectors We seek a slip distribution δ (r) such that, for all r > 0, τ[ δ(r)] predicted by the linear elastic formulation, Equation (5), must equal τ [δ] provided by the atomic-based shear relation in Equation (3). Using a numerical procedure outlined by Beltz [13] and Beltz and Rice [14], we carry this out for incremental increases in K, at a fixed value of T, until an instability (i.e., dislocation nucleation) is attained. 3. Results and Discussion The principal results are shown in Figure 2, where the critical stress intensity factor for dislocation nucleation is plotted as a function of T-stress for several slip plane angles between 0° and 90°. It has been shown in earlier work [11,15] that the parameter √ 2 µ γu s /(l – ν ) is the natural normalization factor for Kcrit , as it represents the threshold for dislocation nucleation under the simplest geometry of a mode II shear

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Figure 2. Positive (or tensile) T-stress increases the critical stress intensity for nucleation, while a negative T-stress decreases the critical stress intensity for nucleation.

crack with a non-inclined slip plane. The vertical axis is compressed logarithmically in order to capture results for a wide range of slip plane inclination angles (ranging from 15° to 85°). The general trend is that positive values of T increase the threshold for dislocation nucleation, while negative values of T lower the threshold. The results for T = 0 are consistent with earlier work by Rice et al. [15]. For θ → 0, the critical load would of course become unbounded since the resolved shear stress driving dislocation formation would vanish. The T-stress effect is greatest for intermediate angles, and can cause variations in the critical load by up to approximately 50% for the T values considered here, depending on the sign of the T-stress. We note that the T-stress effect is weakest at angles near 0° and 90°, since the resolved shear stress due to the T-stress vanishes. As a practical matter, how large is the T-stress for a given geometry? To give some insight, we use a finite crack of length 2a centered in an infinite solid with remotely applied stress σ (see Figure 3). In this example, the applied stress intensity factor K is σ√π a , and T is – σ [3]. Combining and solving for T gives –K / √ πa . W e m u s t c h o o s e s o m e K characteristic of dislocation nucleation, say, α √2µ γ u s/(1 – ν ) , where α is some dimensionless parameter of order unity or moderately larger. Solving for the normalized T gives (6) Hence, we see that the T-stress effect may be most severe when the crack size is tens of atomic spacings or less. More specifically, we have plotted the reduction in the threshold for dislocation nucleation (as a percentage, from the uncorrected, or T=0, result) versus crack size for several values of θ in Figure 4. We take b = h and ν = 0.3.

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Figure 3. Finite crack of length 2a in an infinite solid, with remote tensile stress σ. The T-stress for this particular geometry is –σ ; thus, the threshold for dislocation nucleation is less than what would be calculated using the classical K-field result.

For small values of θ as well as for small values of the crack size, the T-stress effect is most significant. Clearly, one should proceed cautiously when applying continuum concepts to cracks as small as, say, 10 atomic spacings; however, we note that the Tstress effect can lead to appreciable reductions in Kc r i t for moderately longer cracks when the slip plane inclination angle is less than about 45°. This could lead to a substantial source of error if trying to reconcile atomistic results for short cracks with continuum-based predictions. If the crack is of some macroscopic dimension, then T ≈ 0 for this geometry, and the effect can be safely neglected. The results presented here are consistent with the crack size effect on dislocation nucleation as noted by Beltz and Fischer for mode III cracks [16] and Zhang and Li for mode I cracks [17]. We would like to dedicate this paper to Professor James R. Rice, with whom the first author (GEB) had the great fortune to have as a mentor beginning ten years ago. Jim’s high standards of scholarship, superb teaching, and supportive and modest personality benefitted all of us who’ve had the chance to work with him. 4.

References

1.

Williams, M.L.: On the stress distribution at the base of a stationary crack, J. Applied Mechanics 24 (1957), 109-114. Larsson, S.G., and Carlsson, A.J.: Influence of non-singular stress terms and specimen geometry on small-scale yielding at crack tips in elastic-plastic materials, J. Mech. Phys. Solids 21 (1973), 263-277. Rice, J.R.: Limitations to the small scale yielding approximation for crack tip plasticity, J. Mech. Phys. Solids 22 (1974), 17-26. Betegón, C., and Hancock, J.W.: Two-parameter characterization of elastic-plastic crack-tip fields, J. Applied Mechanics 58 (1991), 104-110. Fleck, N.: Finite element analysis of plasticity-induced crack closure under plane strain conditions, Engineering Fracture Mech. 25 (1986), 441-449. Cotterell, B., and Rice, J.R.: Slightly curved or kinked cracks, Int. J. Fracture 16 (1980), 155-169. Fleck, N.A., Hutchinson, J.W., and Suo, Z.: Crack path selection in a brittle adhesive layer, Int. J. Solids Structures 27 (1991), 1683- 1703.

2. 3. 4. 5. 6. 7.

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Figure 4. Reduction (from that of the T=0 result) of threshold for dislocation nucleation for a finite crack of length 2a subject to remote tension (see Figure 3).

8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Langer, J.S., and Lobkovsky, A.E.: Critical examination of cohesive-zone models in the theory of dynamic fracture, J. Mech. Phys. Solids 46 (1998), 1521-1556. Peierls, R.E.: The size of a dislocation, Proc. Phys.Soc. 52 (1940), 34-37. Nabarro, F.R.N.: Dislocations in a simple cubic lattice, Proc. Phys. Soc. 59 (1947), 256-272. Rice, J.R.: Dislocation nucleation from a crack tip: an analysis based on the Peierls concept, J. Mech. Phys. Solids 40 (1992), 239-271. Frenkel, J.: Zur theorie der elastizitätsgrenze und der Festigkeit Kristallinischer Körper, Zeitschrift Phyzik 37 (1926), 572-609. Beltz, G.E.: The Mechanics of Dislocation Nucleation at a Crack Tip, Ph.D. Thesis, Harvard University, Cambridge, Massachusetts, 1992. Beltz, G.E., and Rice, J.R.: Dislocation nucleation at metal-ceramic interfaces, Acta metall. mater. 40 (1992), S321-S331. Rice, J.R., Beltz, G.E., and Sun, Y.: Peierls framework for dislocation nucleation at a crack tip, in A.S. Argon (ed.), Topics in Fracture and Fatigue, Springer-Verlag, New York, 1992, pp. l-58. Beltz, G.E., and Fischer, L.L.: Effect of finite crack length and blunting on dislocation nucleation in mode III, Phil. Mag. A 79 (1999), 1367-1378. Zhang, T.-Y., and Li, J.C.M.: Image forces and shielding effects of an edge dislocation near a finite length crack, Acta metall. mater. 39 (1991), 2739-2744.

ELASTIC-PLASTIC CRACK GROWTH ALONG DUCTILE/DUCTILE INTERFACES

W. J. DRUGAN Department of Engineering Physics University of Wisconsin–Madison 1500 Engineering Drive Madison, WI 53706

Abstract: An analytical study is performed of the stress and deformation fields near the tip of a crack that grows quasi-statically along an interface between two generally dissimilar ductile materials. The materials are modeled as homogeneous, isotropic, incompressible, elastic-ideally plastic Prandtl-Reuss-Mises, and the analysis is carried out within a small-displacement-gradient formulation. The case of anti-plane shear deformations is considered first. We derive near-tip solutions for the full range of the ratio of the two materials’ yield stresses, and show that a near-tip family of solutions exists for each set of material properties; the implication is that far-field loading and geometrical conditions determine which specific near-tip solution governs in a particular problem. As a by-product of this analysis, we derive a new solution family for anti-plane shear crack growth in homogeneous material, one limiting member of which is the familiar Chitaley and McClintock (1971) solution. We also analyze the case of plane strain crack growth under applied tensile loading. Here, we account for curvature of inter-sector boundaries, in an attempt to obtain a complete set of solutions. When the material properties are identical, the solution family of Drugan and Chen (1989) for homogeneous material crack growth, which has an undetermined parameter in the near-tip field, is recovered. As the ratio of the two materials’ yield strengths, deviates from unity, the near-tip solution structure is found to change, but the near-tip fields are shown to continue to possess a free parameter for a substantial range of Below this range, a second solution structure develops for which the near-tip free parameter has a restricted range of freedom. Finally, a third near-tip solution structure develops for sufficiently low for which there are no free parameters. The implications of these results appear to be that as the plastic yield strength mismatch of the two materials becomes larger, far-field loading and geometry have increasingly weaker effects on the leading-order near-tip fields, until finally a mismatch level is reached beyond which far-field conditions no longer affect the leading-order fields. However, conclusions are complicated by the fact that the analysis also implies the radius of validity of the leading-order fields to decrease continuously with increasing yield strength mismatch (beyond a certain level), so that below some value, it will 243 T.-J. Chuang and J. W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 243-274. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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become necessary to retain more than one term to describe the physical near-tip fields. Although not specifically explored here, our analysis also allows comparison of the effects of changing elastic and plastic properties of the two materials on crack growth propensity, so that perhaps this analysis could assist in the optimization of interfacial fracture properties.

1.

Introduction

Interfaces between two ductile materials arise in numerous situations of practical importance. Composite materials is an obvious example; others include the interface between base and weld metals, or base metal and the heat-affected zone, in a weld; clad and coated materials; solder joints; and boundaries between different phases of the same material (e.g., an austenite/martensite boundary). When the materials involved possess at least some ductility, fracture of the interface – if it is sufficiently strong – would be anticipated to occur not by an immediate catastrophic propagation of a crack, but rather such would be preceded by a regime of slow, stable crack growth. A clear understanding of the mechanics of this part of the failure process seems essential to the overall characterization of ductile/ductile interface fracture. We shall employ asymptotic (near-tip) analysis of the infinitesimal-displacementgradient governing equations to obtain analytical solutions for the stress and deformation fields near a crack that propagates along the interface joining two (generally different) ductile materials. Each material will be modeled as homogeneous, isotropic, incompressible, elastic-ideally plastic Prandtl-Reuss-Mises. Two cases will be analyzed: crack growth under anti-plane shear, and under plane strain with nominally tensile applied loading. An interesting feature of the results is that, in both cases, a family of solutions is shown to exist at each ratio of the materials’ yield strength. For the anti-plane shear crack growth case, this is true for all values of the shear strength ratio, whereas for plane strain crack growth, this is true only for a range of yield strength ratios with one endpoint being the homogeneous material limit. Previous analytical studies have treated crack growth along the interface between a ductile material and a brittle one, the first being those of Drugan (1991) and Ponte Castañeda and Mataga (1991). Recent studies of the fields near a stationary crack on a ductile/ductile interface include those of Ganti and Parks (1997) and Sham, Li and Hancock (1999). Of course, pioneering work on the elastic-plastic analysis of stress and deformation fields near stationary and growing cracks in homogeneous materials is due to Rice (1967, 1968, 1974, 1982, 1987, etc.), Rice and Tracey (1973) and Rice et al. (1980) [with independent contributions from Slepyan (1974) and Gao (1980)], and the analysis presented here, as with many other studies in diverse and numerous subjects in solid mechanics, employs the elegant framework Jim Rice developed and builds on his remarkable physical and mathematical insights.

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Formulation

We will employ a small-displacement-gradient formulation to analyze quasi-static crack growth along an interface joining incompressible, homogeneous, isotropic elasticideally plastic materials modeled via an incremental (flow) theory. The crack geometry to be analyzed in all cases is illustrated in Figure 1. A Cartesian coordinate system is fixed in the body such that x1 points in the crack growth direction and x 3 is parallel to the (locally straight) crack front. A polar (r, θ ) coordinate system is centered at the crack tip and moves with it during crack growth. Angle θ i s m e a s u r e d counterclockwise from the x1 –axis, and the crack growth speed is The crack is shown growing along the interface between two generally different elastic-ideally plastic materials, having yield stresses in pure shear ki and elastic shear moduli Gi .

Figure 1. Cartesian x 1 , x 2 coordinate system is fixed in the body; polar r, θ system is centered at the crack tip and moves with it through the material during crack growth.

The governing equations within each homogeneous material are as follows. Equilibrium requires the stress tensor σ to satisfy (1) where ∇ is the usual vectorial differentiation operator so that ∇• denotes divergence, b is the body force vector per unit volume and a superscript T denotes tensor transpose. The rate of deformation tensor D is related to the material velocity vector v as (2)

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Elastic-plastic response in each material is assumed to be described` by the PrandtlReuss flow rule (3) where e and p denote elastic and plastic parts, respectively, G is the elastic shear modulus, s = σ – I trace( σ )/3 is the deviatoric stress tensor with I denoting the second-rank identity tensor; ≥ 0 is an undetermined parameter and a superposed dot denotes time rate at a fixed material point. Material is at yield when the Huber-Mises condition is met: (4)

where k is the (constant) yield stress in shear. Thus, in regions of ongoing plastic deformation, (3) and (4) must be satisfied with > 0; in material experiencing instantaneously elastic response, whether or not it has previously deformed plastically, (3) governs with ≡ 0 and (4) applies with ≤ replacing = . Rice (1982) argued, and Drugan (1985) proved, that these governing equations become analytically tractable in the limit as r → 0, even for general unsteady crack growth. Specifically, if b is assumed bounded everywhere and mild, physically sensible assumptions are made concerning the existence of certain quantities involving stress derivatives as r → 0, Rice (1982) and Drugan (1985) proved that (1) and (3) simplify to the following asymptotic forms, specialized here to the present problems in which all field quantities are assumed independent of x3 :

(5a)

(5b)

(5c)

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(6)

respectively, where R is an undetermined parameter having length dimensions. As will be reviewed explicitly in subsequent sections, in the cases of antiplane strain and plane strain, the asymptotic (r → 0) form of this governing equation set was shown by Rice (1982) to admit three types of solution – i.e., three types of near tip angular sector, in a given material. The construction of complete leading-order solutions for growing crack stress and deformation fields thus involves assembling sectors of these permissible types in such a manner that plastic deformation is everywhere nonnegative, that the yield condition is not violated in elastically deforming sectors, and that the appropriate inter-sector and boundary conditions are satisfied. Since we shall consider growing cracks, intersector boundaries (other than one lying on θ = 0) propagate quasi-statically through the material. Drugan and Rice (1984) analyzed restrictions on such quasi-statically propagating surfaces for a general class of elastic-plastic materials, inclusive of the model employed here. Recently, Drugan (1998a) provided rigorous confirmation of certain ad hoc assumptions they employed, and generalized their results to an even broader class of materials. Specialized to the present incompressible material model and the deformation modes considered here, these analyses show that the following important conditions must be satisfied across propagating intersector boundaries: (i) all components of the stress tensor must be continuous; (ii) only the material velocity components parallel to the moving surface, and their associated plastic shear strain components, may experience jumps, but only if all of the following conditions are met: (7a) (7b) (7c) Here, a superscript ∑ indicates that the (continuous) stress component is to be evaluated on the potential discontinuity surface; subscripts n and t refer to normal (in the propagation direction) and tangential (in the x1 , x 2 plane) directions to the moving surface whose normal velocity is c, and subscript δ stands for t and 3; [ ] denotes the jump in a quantity across the surface (value just ahead minus value just behind); and [ Λ] ≤ 0.

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In addition to the above inter-sector continuity conditions, we shall require the minimum necessary conditions across the line ahead of the crack ( θ = 0), so as to explore all possible near-tip solutions: namely, continuity of the traction and displacement vectors will be enforced. (Note that continuity of displacement across θ = 0 requires continuity of velocity across that line.) Finally, we shall impose the condition of traction-free crack faces.

3.

Anti-Plane Strain

3.1 GENERAL SOLUTION SECTOR TYPES AND CONTINUITY CONDITIONS We shall first provide a detailed – perhaps exhaustive – study of the anti-plane strain case, both because of its relative mathematical simplicity but also because observations and trends evident here will lend great insight into the more complicated plane strain case. In anti-plane strain, the only nonvanishing velocity component is v3 = v 3 (r, θ ) . From this, (1) – (3) show that the only nonvanishing components of σ and D are the 13 (=31) and 23 (=32) ones, which must also be independent of x3 . Thus, the yield condition (4) reduces to (8) As reviewed lucidly by Rice (1982), the leading-order in r as r → 0 forms of the governing equations, namely (2), (5c), (6) and (8), admit three types of solution – i.e., three types of near tip angular sector. These are summarized below, with the sign of σ θ 3 in the “centered fan” plastic sector chosen positive, as it will be in all our solutions. (i) “Centered Fan” Plastically Deforming Sector (9) (10) (11)

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(ii) “Constant Stress” Plastically Deforming Sector (12) (13)

(14) (iii) Elastically Deforming Sector (15)

(16) In the preceding equations, A, B, C, Q α 3 and V are undetermined dimensionless constants except that Q α 3 must be chosen such that (12) satisfies (8); subscript α has range 1, 2; G is the elastic shear modulus; ƒ( θ , t) is an undetermined function of integration; and t is time or some monotonically increasing parameter. Equations (13), (14) were derived by employing (6) and assuming that the velocity field in a “constant stress” plastically deforming sector is logarithmically singular in r, as it must be in the other sector types. This derivation is similar to its plane strain counterpart, which is given in the Appendix. Acceptable solutions to the near-tip growing crack fields must be assembled from the allowable solution sector types given above within each material, with the appropriate continuity/jump conditions enforced across inter-sector boundaries and the appropriate boundary conditions applied. For all intersector boundaries excluding θ = 0, the general conditions described in Section 2 specialize in the anti-plane strain case to the following: [σ α3 ] = 0

(17) (18)

Across θ = 0, we require continuity of σ23 (= σ θ 3 ) and of v 3 , and on θ = ± π we require σ 23 = 0.

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3.2 SOLUTIONS FOR DUCTILE/DUCTILE INTERFACE CRACK GROWTH There appear to be two general near-tip solution configurations for anti-plane shear crack growth along a ductile/ductile interface between two elastic-ideally plastic materials that in general have different elastic and plastic properties. These are illustrated in Figure 2.

Figure 2. Near-tip solution configurations for anti-plane shear ductile/ductile interface crack growth. Material 1 lies above the crack line and has shear strength k 1, while Material 2 lies below and has shear strength k 2 . Sectors A, B, G, H are constant stress plastic; C, D are centered fan plastic; and E, F are instantaneously elastic.

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In both configurations of Figure 2, the boundaries θ1 and θ 2 are stress characteristics, in Figure 2(a) because of full stress continuity with centered fans, and in Figure 2(b) because these boundaries are sites of velocity jumps. This means that the stress states in the constant stress plastic sectors A and B have the following representations (the superscript denotes the appropriate sector): (19) (20) Thus, in both configurations, traction continuity across θ = 0 requires (21) This shows that θ1 and – θ 2 can only be equal for crack growth in homogeneous material, so that at least one of the constant stress sectors A or B is necessary in general for interfacial crack growth along yield-strength-mismatched interfaces. Incidentally, it is easy to see that a sub-yield elastic sector cannot border θ = 0 in any solution: nonviolation of the yield condition would require B = 0 in (15), rendering the stresses constant; then, full stress continuity with a necessarily at-yield trailing sector would result in the stress field in the hypothesized elastic sector being at yield. Proceeding with the analysis of the Figure 2(a) configuration: full stress continuity is required across θ1 and θ2, which is already satisfied by (19) and (20). From (18), because these boundaries are characteristics, i.e. have σr3 = 0, they are permitted to have jumps in v 3 , which indeed are necessary to avoid infinite velocities in Sectors A and B as these boundaries are approached. That is, (13) shows that as θ approaches θ 1 in Sector A or θ 2 in Sector B, v 3 becomes infinite unless V A = V B = 0. This choice being made, it is clear that leading-order velocity continuity is satisfied across θ = 0 (since velocities are zero at leading order in Sectors A, B). Observe that the v 3 jumps across θ1 and θ2 produce positive plastic work. Next, we enforce full stress and velocity continuity across the remaining intersector boundaries. As noted earlier, stress continuity is required; velocity continuity is required across θ 5 and θ6 because the condition σ r3 = 0 will not be met on these boundaries. Although this condition is met on θ3 and θ 4 , one can prove that the requirements of positive plastic work produced by a hypothesized velocity jump, together with the stress state in the elastic sector not violating yield, rule out a velocity jump across a propagating centered fan/elastic sector boundary [the plane strain version of this proof, due to Rice, is given in Appendix A of Drugan and Chen (1989)]. These continuity conditions lead to equations for the boundary angles and the constants appearing in (12) – (16) for Sectors E – H that do not involve the elastic shear moduli nor the yield strengths. The equations governing the boundary angles are:

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(22a, b)

with identical equations for angles θ4 a n d θ 6 , except that the last +1 in (22a) is replaced by –1. The unique physically meaningful solutions to these equations are: θ3 = – θ4 = 19.711224 °,

θ 5 = – θ 6 = 179.63341°;

(23)

Figure 2(a) has been labeled with these unique angles (although it is not drawn with these angles for purposes of illustration). The other constants appearing in (12) – (16) for Sectors E – H are, where a superscript again denotes the sector of applicability of the constant:

(24)

Thus, for the range of applicability of this solution family (about which, more shortly), all boundary angles except θ 1 a n d θ 2 are fixed and independent of the far-field loading and mismatch in material properties, as are the angular variations of the stress and deformation fields in all sectors except Sectors A and B. Different material properties change the amplitudes of the stress and deformation fields, by changing the amplitudes of k and G that appear in (9) – (16). To determine the range of material property mismatches for which this solution family can apply, we note that (21) and (23) must both be satisfied. The first interesting conclusion, then, is that, since the materials’ elastic shear moduli do not enter (21), this solution family appears capable of providing solutions for any elastic mismatch (at least the asymptotics suggests this). However, simultaneous satisfaction of (21) and (23) does limit the ratio of yield strengths for which this solution type applies, since (23) shows that –19.711224° ≤ θ 2 ≤ 0 ≤ θ 1 ≤ 19.711224 °. This with (21) shows that the solution configuration of Figure 2(a) can provide interface crack growth solutions for the yield strength ratio range: (25)

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One observes that for specific values of k2 /k1 inside this range, θ 2 is strictly specified in terms of θ 1 , but there is some freedom of choice of one of these. The specific solution must be determined from far-field conditions of the particular problem under consideration. When k 2 /k1 = 1.0622426 (0.94140449), this corresponds to θ 1 = 0, θ 2 = –19.711224 ° ( θ1 = 19.711224 °, θ 2 = 0): in these cases, say the first one for definiteness, Sector A has disappeared, as has Sector D. This limit case is illustrated in Figure 3.

Figure 3. Near-tip solution configuration for k2 /k1 = 1.0622426.

A s k 2 /k 1 increases above 1.0622426, the configuration shown in Figure 3 will still provide near-tip solutions; interestingly, the fields throughout Material 1 remain unchanged in their angular distribution, but in Material 2, angles θ4 and θ 6 , and hence the fields in Sectors B, F and H, change as k 2 /k1 increases. The equations governing the values of these angles and the parameters in the stress and velocity fields in these sectors are obtained by writing (21) with θ 1 = 0 and θ 2 = θ4 , and enforcing full stress continuity across θ4 and θ 6 , and velocity continuity across θ 6 . Note that θ 4 is now the site of a velocity discontinuity, which produces positive plastic work and corresponds to an elastic sector stress field that does not violate yield. The results are: (26)

(27)

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254

(28)

Solutions to (26) – (28) are listed in Table 1 for the full allowable range of 1.0622426 ≤ k 2 /k1 ≤ ∞ (reported as 0 ≤ k1/ k2 ≤ 0.94140449). θ4

θ6

AF

BF

CF

0.9414

–19.71

–179.633

1.057

0.3373

0.7038

0.9

–25.84

–179.655

1.051

0.3353

0.7143

0.8

–36.87

–179.766

1.006

0.3208

0.7639

0.7

–45.57

–179.868

0.9375

0.2986

0.8147

0.6

–53.13

–179.940

0.8514

0.2711

0.8613

0.5

–60

–179.980

0.7500

0.2388

0.9007

0.4

–66.42

–179.996

0.6339

0.2018

0.9342

0.3

–72.54

–179.9997

0.5025

0.1600

0.9615

0.2

–78.46

– 1 8 0–

0.3546

0.1129

0.9821

0.1

–84.26

– 1 8–0

0.1880

0.0598

0.9953

0.0

–90

–180

0

0

1

k 1/k2

Table 1. Values of the parameters for the solution configuration of Figure 3, for 0 ≤ k1/k 2 ≤ 0.94140449.

Interestingly, solutions for this range of k l /k2 can also be found with the near-tip configuration of Figure 2(b). The analysis of this configuration proceeds as follows: As explained earlier, the boundaries θ 1 and θ 2 are stress characteristics because they are the sites of velocity jumps; the stress states in Sectors A and B are therefore given by (19) and (20), and for the reasons explained earlier, V A = V B = 0. Thus, the velocities are zero in these sectors, and hence velocity continuity is satisfied across θ = 0. Full stress continuity is enforced across all intersector boundaries (except θ = 0,

ELASTIC-PLASTIC CRACK GROWTH ALONG INTERFACES

255

which already has traction continuity satisfied), and velocity continuity is enforced across θ3 and θ4 . The resulting equations are: (29)

(30)

(31)

(32)

(33)

Thus, for each value of the ratio k 1/k 2 ≤ 0.94140449, a range of solutions having this configuration can be found. The limitations are that θ1 ≥ 19.71 °

and

θ 2 ≤ −19.71°;

(34)

this is because otherwise the yield condition would be violated in the elastic sectors. For example, for the case k1 /k 2 = 0.8, (29) and (34) show that there will be a range of solutions with 19.71° ≤ θ1 ≤ 9 0 ° and –41.14 ° ≥ θ 2 ≥ –90°, with the other angles and constants being given by (30) – (33). A specific set of angles θ1, θ2, satisfying (29) and (34), apparently is determined by the far-field boundary and loading conditions of the specific problem under consideration. 3.3 NEW SOLUTION FAMILY FOR CRACK GROWTH IN HOMOGENEOUS MATERIAL We now specialize the preceding analysis to exhibit a new solution family for anti-plane strain crack growth in homogeneous materials. To parameterize the solutions within this family, it will be convenient to introduce a near-tip “mixity parameter” M as:

256

W. J. DRUGAN

(35)

The new solution family has solutions for all mixities in the range 0 < M ≤ 1. The solution family has the two configurations illustrated in Figure 2, except that now, of course, the yield strengths are the same in the upper and lower materials. This fact reduces (29) to the requirement that θ1 = –θ2 .

(36)

This requirement applies to both configurations of Figure 2. The remaining features of the solutions are very similar to those just analyzed for crack growth along an interface involving two different materials. Specifically: from (19), (20), (35), (36) and the fact that θ 3 = – θ 4 = 19.71°, we deduce that the solution configuration illustrated in Figure 2(a) applies for mixities in the range 1 ≥ M ≥ 0.7810. Notice that M = 1 is the Chitaley-McClintock (1971) solution, which has fully continuous stresses and velocities. As M decreases from this value to 0.7810, θ 1 (and – θ 2) increase from zero to 19.71 °. As discussed earlier, this only changes the solution fields in the range θ 2 ≤ θ ≤ θ1 , but the solution now has (admissible) velocity jumps across θ1 and θ 2 (and zero leading-order velocities within Sectors A and B), and a radial shear stress jump across θ = 0. The stress and velocity fields for solutions of this type are given by (9) – (16) and (19), (20) together with the angles and parameter values given by (23), (24) and the combination of (35) and (19). For mixities in the range 0.7810 ≥ M > 0, the solution adopts the configuration of Figure 2(b), but satisfying (36). As above, the angle θ1 (= θ 2 ) is determined from (19) and (35) for a given mixity, and the remaining angles and parameter values are given by (30) – (33), which are simpler now since the fields are antisymmetric about θ = 0. For the range of mixities just given, θ 1 sweeps through the range 19.71 ° ≤ θ 1 < 90 °. Which member of this solution family applies (i.e., which mixity) is determined by far-field loading and geometry. The small-scale yielding numerical finite element solutions of Dean and Hutchinson (1980) and Freund and Douglas (1982), together with the additional analysis of Drugan (1998b), seem to imply the near-tip presence of the M = 1 solution in the small-scale yielding case, but it seems that e.g. crack growth under general yielding conditions, or even under contained yielding conditions but with the second term in the Williams expansion for the surrounding elastic fields nonzero, may well produce others of the solution members derived here. Certainly, all results presented here are completely admissible solutions to the asymptotic (near-tip) governing equations.

ELASTIC-PLASTIC CRACK GROWTH ALONG INTERFACES

257

4. Plane Strain 4.1 GENERAL SOLUTION SECTOR TYPES AND CONTINUITY CONDITIONS In plane strain, the nonzero components of material velocity are v 1 = v l ( r, θ), v2 = v2 (r, θ ). From these, (1) – (3) show that the only nonvanishing components of s and D are the 11, 12 (=21) and 22 ones [and σ33 = ( σ 11 + σ 22 )/2], which must also be independent of x 3. Thus, the yield condition (4) reduces to

(37) As derived elegantly by Rice (1982), the leading-order in r as r → 0 forms of the governing equations (2), (5a), (5b), (6) and (37), admit three types of solution – i.e., three types of near tip angular sector. These are summarized below. (i) “Centered Fan” Plastically Deforming Sector (38)

(39) (40)

(41) (ii) “Constant Stress” Plastically Deforming Sector (42)

(43)

(44)

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W. J. DRUGAN

(45)

(46)

(iii) Elastically Deforming Sector (47) (48) (49) (50)

(51)

(52)

are undetermined In the preceding equations, A α , Bα , C, Cαβ , Pαβ a n d V dimensionless constants except that Pαβ must be chosen such that (42) satisfies (37); subscripts α, β, γ have range 1,2 and obey the summation convention; G is again the elastic shear modulus; f (θ, t ) is an undetermined function of integration; e is the natural logarithm base and δ αβ is the Kronecker delta. Equations (43)–(46) were derived by assuming that the velocity field in a “constant stress” plastically deforming sector is logarithmically singular in r, as it must be in the other sector types. A new, direct derivation of these is provided in the Appendix. Acceptable solutions to the near-tip growing crack fields must be assembled from the allowable solution sector types given above in each material, with the appropriate continuity/jump conditions enforced across inter-sector boundaries (including across the material interface) and the boundary conditions of zero tractions on the crack faces. For all intersector boundaries excluding θ = 0, the general conditions described in Section 2 specialize in the plane strain case to the following:

ELASTIC-PLASTIC CRACK GROWTH ALONG INTERFACES

259

[ σ αβ ] = [ σ 33 ] = 0

(53)

[ vθ ] = 0

(54) (55)

4.2 SOLUTIONS FOR DUCTILE/DUCTILE INTERFACE CRACK GROWTH Knowing that for plane strain crack growth in homogeneous materials there are two solution types – Mode I or tensile fields, and Mode II or shear fields – we anticipate two solution types for the interface crack growth problem also. We here focus on tensiletype solutions, anticipating that they will be the more important physically. By tensiletype solutions, we refer to solutions that reduce to the Mode I solutions when the properties of the two materials become identical; when these materials are different, these tensile-type solutions will of course not be symmetric or pure Mode I, as the ensuing analysis shows.

Figure 4. Solution configuration for plane strain ductile/ductile interface crack growth. Material 1 lies above the crack line, Material 2 below.

We begin with the analysis of the configuration shown in Figure 4 for the tensiontype case. This means that we choose σr θ positive in Sector C and negative in Sector D. Then, enforcing full stress continuity across θ 1 and θ 2 , as required by (53), the stress fields in Sectors A and B are, employing (38) and (42) in each material (again, superscripts denote the applicable sector):

W. J. DRUGAN

260

(56)

(57) Enforcement of traction continuity across θ = 0 thus requires (58)

(59)

The forms (56), (57) also permit specification of the velocity fields in Sectors A and B. First, for Sector A, (46) become S

= – cot 2θ 1 ,

b l =cot θ l ,

b 2 = – tan θ l ;

(60)

using these, (43) – (45) are, for Sector A:

(61)

(62)

(63)

These have exactly the same forms in Sector B, except that the subscripts of k, G, θ are changed from 1 to 2, and superscripts A are changed to B. Leading-order velocity continuity (i.e., analyzing only the logarithmically singular terms) across θ = 0 thus requires (when θ 1 ≠ 0 and θ 2 ≠ 0):

ELASTIC-PLASTIC CRACK GROWTH ALONG INTERFACES

261 (64)

(65) Now, in their analysis of tensile crack growth in homogeneous materials, Drugan and Chen (1989) derived a family of leading-order solutions for the growing crack stress and deformation fields by accounting for curvature of the boundaries that in Figure 4 have the angular locations θ 1 and θ 2 as r → 0. We shall perform an equivalently general analysis here. This means that the boundaries just mentioned will be taken to have the asymptotic forms, respectively, where m is an initially unspecified parameter:

(66a, b)

When the boundaries have these forms, there is an additional In(R/r) contribution to the Sector A velocity fields as r → 0 along the Sector A/C boundary from the terms in (61), (62), and similarly for Sector B; this introduces an additional (free) parameter m into the leading-order solution. We have chosen the same power m for the boundaries in the upper and lower materials because, as Drugan and Chen (1989) showed, boundary forms of type (66) lead to second-order stress field corrections proportional to (r/R) m , and we require traction continuity across θ = 0. We shall seek growing crack solutions with full stress and velocity continuity across all intersector boundaries (except for possible σrr jumps across θ = 0); the homogeneous material solutions found by Drugan and Chen (1989) have this property (except for the m → ∞ limiting case, as will also be true here). Thus, employing (66) in (61), (62) and (39), (40), full leading-order velocity continuity across the Sector A/C and B/D boundaries requires: (67) (68) (69) (70)

262

W. J. DRUGAN

We also enforce full stress and velocity continuity across all other remaining intersector boundaries; jumps in v r across θ 3 and/or θ 4 seem possible initially but are ruled out by the proof in Appendix A of Drugan and Chen (1989). The equation system resulting from all these additional continuity conditions can be summarized as follows. In Material 1: (71)

(72)

(73)

(74)

(75) (76) (77) (78) The equations and parameters in Material 2 ( θ < 0) are given by equations identical in form to (71)–(78), with the following changes: superscripts (D, F, H) replace (C, E, G); angles (θ2 , θ4 , θ 6 ) replace (θ 1, θ 3 , θ 5 ) (and of course negative solution values are sought); and the signs are changed of: the right sides of (71), the last sin term in (72), the last cos term in (73), and 2 θ 3 becomes –2 θ 4 in (74). Thus the total equation system to be solved is (58), (59), (64), (65), (67)–(70), (71)–(78), and the Material 2 versions of (71)–(78) just explained. There is one more unknown than the total number of equations, so that at least throughout the range for which the configuration of Figure 4 prevails, a one-parameter family of solutions is expected for each set of material properties. This is indeed found to be the case. Following Drugan and Chen (1989), we regard m as the undetermined parameter. This system is solved as follows: First, one solves (67)–(70) for in terms of V C, V D; substitution of these results into (64), (65) then permits solution of V C, V D in terms of angles and m only. The results are:

ELASTIC-PLASTIC CRACK GROWTH ALONG INTERFACES

263

(79)

(80)

(81)

(82) = G 1 / G 2 . Using (79)–(82) in (71), (74) and their Material 2 where = k l / k2 and counterparts, the six equations governing the six angles in Figure 4 are (58), (59), (72), (73) and the Material 2 counterparts of the latter two. Our approach was to begin with the two ductile materials being identical – the case for which Drugan and Chen’s (1989) homogeneous material solutions apply. Solutions for different ductile materials can then be obtained by slowly changing the values of yield stress and/or elastic moduli of the two materials, while tracking the solutions whose structure initially resembles that of Drugan and Chen (1989). We will summarize a few key results here to give a flavor of what the solutions show, focusing on the effect of changing the yield stress ratio between the two materials (i.e., we have chosen G 1 = G 2 in the solutions). For the purpose of obtaining specific results, we begin by choosing the value of Drugan and Chen’s (1989) near-tip characterizing parameter m = 1.24, the value Liu and Drugan (1993) showed it takes for homogeneous crack growth when small-scale yielding conditions prevail (i.e., the plastic zone size is small compared to the crack length and the distance to specimen boundaries). Figure 5 shows the leading-order (in distance, r, from the crack tip) near-tip solution structure for this m-value when the two materials are identical, as determined by Drugan and Chen (1989). The two families of orthogonal lines in plastic regions are stress characteristics; a material element with sides parallel and perpendicular to such a line has a stress state consisting of shear stress equal to yield stress in pure shear, plus a hydrostatic stress. Sectors A, B, G and H are “constant stress” plastic sectors, in which the Cartesian components of stress do not vary with position; Sectors C and D are “centered fan” plastic sectors, in which the stress state consists of the polar shear component of stress equaling the material’s yield stress in shear, plus a hydrostatic stress that varies linearly with angle; and Sectors E and F are instantaneously elastic, although containing material which has previously deformed plastically. The values of the inter-sector boundary angles, θ1 – θ 6, are given by the first row in Table 2, and Figure 5 is drawn with these angles.

264

W. J. DRUGAN

Figure 5. The near-tip solution configuration for homogeneous material, i.e., =1 with m = 1.24 (boundary angles drawn to scale).

Now recalling that = k l / k 2, i.e., the ratio of the upper material (x2 > 0) to lower decreases from 1 to about 0.657, material (x 2 < 0) yield stress, we found first that as the near-tip solution structure resembles that of Figure 5 except that the inter-sector boundary angles change smoothly with decreasing . Table 2 shows how these angles , and Figure 6 illustrates the near-tip configuration for = 0.7. change with Observe that the region of strongest plastic strain ing in each material, the centered fan sector, has drastically enlarged in the upper (lower yield stress) material, and dramatically condensed in the lower (greater yield stress) material. Also, the sector of purely elastic response has greatly enlarged in this lower material. For any of the values of given in Table 2, the solutions for the angles listed there can be employed to easily determine the values of all the other parameters appearing in the near-tip stress and deformation fields, via use of (79)–(82), (71), (74)–(78) and the Material 2 cou nterparts of these last six equations. Once these parameters have been calculated, the actual stress and velocity fields are given by (38)–(52), choosing the appropriate ones of these for each particular sector, and ensuring that the material properties correspond to whether the fields are in Material 1 or 2.

ELASTIC-PLASTIC CRACK GROWTH ALONG INTERFACES

265

θ1

θ2

θ3

θ4

θ5

θ6

1

45

–45

118.927

–118.927

156.860

–156.860

0.9

36.9723

–52.2066

121.053

–115.802

154.869

–159.451

0.8

26.3465

–59.5019

122.778

–109.666

153.098

–163.603

0.7

9.71432

–65.6553

124.288

–87.9485

151.422

–172.256

0.657166

4.31851

–65.2600

124.764

–65.2600

150.867

–176.434

Table 2. The near-tip inter-sector boundary angles (measured anticlockwise from x1, in degrees) as functions of

, for the near-tip configuration of Figures 5 and 6 (m = 1.24).

= 0.7 with m = 1.24 Figure 6. The near-tip solution configuration for (boundary angles drawn to scale).

W. J. DRUGAN

266

In addition to analyzing how the near-tip solution configuration alters as the yield stress ratio changes for the small-scale yielding value of m = 1.24, we examined whether the range of m for which solutions could be found at each value was altered by changing The importance of this is as follows: m is the only parameter undetermined by the asymptotic analysis in the leading-order (in r) asymptotic solution, and thus the near-tip fields experience differences in far-field loading, geometry and yielding extent only through changes in m. Furthermore, the next term in r in the asymptotic stress field derived by Drugan and Chen (1989) is proportional to (r / R) m , where R is a parameter with length dimensions that also depends on far-field conditions [look ahead to (92)]. Thus m also regulates how rapidly the leading-order stress fields alter with distance from the crack tip. For Mode I crack growth in homogeneous material, Drugan and Chen (1989) showed that near-tip solutions can be found for all m -values in the range 0 < m ≤ ∞. Our analysis of the present problem led to a fascinating result: For 1 ≥ ≥ 0.7, we find that near-tip solutions are possible for all m-values in the range of acceptable becomes slightly solutions in homogeneous materials, i.e., 0 < m ≤ ∞. However, as less than about 0.7, the range of m-values for which near-tip solutions can be found is dramatically narrowed, since the maximum admissible m-value is drastically decreased! This is illustrated in Table 3, which displays the maximum m for which an acceptable near-tip solution can be found, as a function of m max 0.7 to 1

∞

0.69354

27,5676.0

0.69

14.2562

0.68

4.39663

0.67

2.99652

0.65

2.18306

0.60

1.32282

0.591622

1.24

Table 3. Maximum m-value for which near-tip solutions can be found (continues in Table 5).

Returning now to our solutions for m = 1.24, observe from Table 2 that when reaches the value 0.657166, θ2 a n d θ 4 have coincided, meaning that Sector D, the centered fan sector in the lower material, has disappeared. The near-tip solution configuration has thus become that illustrated in Figure 7. There is still one more

ELASTIC-PLASTIC CRACK GROWTH ALONG INTERFACES

267

parameter than the number of equations constraining them for this configuration, since now the restrictive form of the velocity field in Sector D is no longer imposed.

Figure 7. Near-tip configuration for

= 0.657166, drawn to scale (m = 1.24).

Specifically, the Material 2 versions of (71) no longer apply, nor do (69) – (70), which were the requirements of velocity continuity across θ 2 . The new equations of velocity continuity across θ 2 are, employing (51), (52), the Sector B versions of (61), (62), and (66b): (83) Equations (81), which resulted from (67)–(68), still apply; using these in (64)–(65) and gives: solving the resulting equations for (84a)

(84b)

W. J. DRUGAN

268

Since θ 2 is a characteristic, its stress state can still be expressed as if it were a centered fan of zero angular extent, so that full stress continuity across θ2 still results in the stress field in Sector B being expressible as (57). Therefore, (58)–(59) still apply. The full equation set is thus (58), (59), (71)–(78), the Material 2 versions of (72)–(78) except that now θ 2 replaces θ4 , (81), (83) and (84). This is reduced to five equations for the five undetermined angles as follows: use (84) in (83) to express the in terms of V C . Use these and (71) in (74) and its Material 2 counterpart to express CC and C D in terms of V C. Substitution of these into (59) gives an equation that can be solved for VC involving only boundary angles; the result is:

(85) Then, employing (85) with (71), (83) and (84), allow (72), (73) and their Material 2 versions to be expressed purely in terms of the boundary angles, as is (58); these are the five equations to be solved. The solution of these equations shows that the configuration of Figure 7 persists until attains the value 0.591622, at which value θ1 has reached the value 0. Table 4 gives the near-tip sector boundary angles for corresponding to solutions with the configuration of Figure 7. θ1

θ2

θ3

θ5

θ6

0.657166

4.31851

–65.2600

124.764

150.867

–176.434

0.6

0.483300

–63.4319

124.749

150.885

–178.754

0.591622

0

–63.1361

124.749

150.885

–178.954

Table 4. Near-tip inter-sector boundary angles (measured anticlockwise from x1 , in degrees) as functions of for the near-tip configuration of Figure 7 (m = 1.24).

ELASTIC-PLASTIC CRACK GROWTH ALONG INTERFACES

269

≤ 0.591622, Sector A, the leading constant stress sector in the upper For material, has disappeared, so that the near-tip configuration is that shown in Figure 8.

Figure 8. Solution configuration for

≤ 0.591622, drawn to scale for the case of equality.

The analysis of this configuration proceeds as follows. Equations (57) are still valid, and (58), (59) apply with θ1 = 0; these become: (86) (87) Equations (64), (65) no longer apply; the new conditions of velocity continuity across θ = 0 are, using (39), (40) and the Sector B version of (61), (62): (88)

W. J. DRUGAN

270

(89)

Solving (88), (89) for

gives (90a, b)

The full equation set is thus (86), (87), (90), (71)–(78), the Material 2 versions of (72)–(78) except that now θ2 replaces θ 4, and (83). This is reduced to equations for the undetermined angles as follows: use (90) in (83) to express the in terms of V C . Use these and (71), in (74) and its Material 2 counterpart, to express C C and C D in terms of V C . Substitution of these into (87) gives an equation that can be solved for VC involving only boundary angles; the result is:

(91)

Then, employing (91) with (71), (83) and (90), allow (72), (73) and their Material 2 versions to be expressed purely in terms of the boundary angles, as is (86); these are the five equations to be solved. Observe, however, that there are only four remaining angles to be determined in the present solution configuration (Figure 8)! This means that the near-tip parameter m is no longer a free parameter for < 0.591622: only one near-tip solution can be found for each in this range. The solutions of the justdescribed five-equation set are reported in Table 5.

ELASTIC-PLASTIC CRACK GROWTH ALONG INTERFACES

271

m

θ2

θ3

θ5

θ6

0.591622

1.24

–63.1361

124.749

150.885

–178.954

0.55

0.939418

–61.6835

125.179

150.372

–179.603

0.5

0.713333

–60

125.529

149.948

–179.910

0.45

0.558467

–58.3718

125.776

149.643

–179.988

0.4

0.442601

–56.7891

125.960

149.413

–179.999

0.35

0.351256

–55.2437

126.101

149.236

–179.999993

0.3

0.276698

–53.7288

126.211

149.097

–179.999999995

0.25

0.214174

–52.2388

126.297

148.987

–180 + 2.1× 10–14

0.2

0.160548

–50.7685

126.365

148.901

–180 + 9.7× 10–25

0.15

0.113652

–49.3135

126.418

148.834

–180 + 9.3× 10–48

0.1

0.071932

–47.8696

126.459

148.781

–180 + 8.1× 10–116

0.05

0.034242

–46.433

126.490

148.741

–180 + 3.1× 10–497

Table 5. Near-tip inter-sector boundary angles and the unique m-value for which a solution exists, as functions of

≤ 0.591622, for the near-tip configuration of Figure 8.

One extremely important implication of these results is as follows: Drugan and Chen (1989) showed that in terms of a polar coordinate system centered at the moving crack tip, the stress field in a “centered fan” plastic sector has the structure (and this is anticipated to be the case in the other near-tip sectors):

(92) where s (0)( θ ; m), s (1)( θ; m ), …, are specified by the asymptotic analysis, except for the value of the characterizing parameter m (and additional constants in the higher-order

272

W. J. DRUGAN

terms). The results summarized in Table 5 thus show that for sufficiently large , the near-tip fields may be accurately described simply by the leading-order stress field [i.e., by just one term in (92)]. However, for values below a certain level, m has become sufficiently small that a leading-order analysis will certainly not suffice, so that accurate representation of physical near-tip stress fields will require retention of more than one term in (92). The value below which more than one term must be retained depends on the size scale of the microstructure of the material being modeled (since the radius of validity of the near-tip fields must be larger than this microstructural size, and the radius of validity of the leading-order term depends on m, and hence on ). The same conclusion applies for the near-tip deformation fields.

Acknowledgements Support of the National Science Foundation, Mechanics and Materials Program, under Grant CMS-9800157 is gratefully acknowledged.

References Chitaley, A. D. and McClintock, F. A. (1971), “Local Criteria for Ductile Fracture,” Journal of the Mechanics and Physics of Solids, Vol. 19, pp. 147- 163. Dean, R. H. and Hutchinson, J. W. (1980), “Quasi-Static Steady Crack Growth in Small Scale Yielding,” Fracture Mechanics, ASTM-STP 700, pp. 383-405. Drugan, W. J. (1985), “On the Asymptotic Continuum Analysis of Quasi-Static Elastic-Plastic Crack Growth and Related Problems,” Journal of Applied Mechanics, Vol. 52, pp. 60l-605. Drugan, W. J. (1991), “Near-Tip Fields for Quasi-Static Crack Growth Along a Ductile-Brittle Interface,” Journal of Applied Mechanics, Vol. 58, pp. 111-119. Drugan, W. J. (1998a), “Thermodynamic Equivalence of Steady-State Shocks and Smooth Waves in General Media; Applications to Elastic-Plastic Shocks and Dynamic Fracture,” Journal of the Mechanics and Physics of Solids, Vol. 46, pp. 313-336. Drugan, W. J. (1998b), “Limitations to Leading-Order Asymptotic Solutions for Elastic-Plastic Crack Growth,” Journal of the Mechanics and Physics of Solids, Vol. 46, pp. 2361-2386. Drugan, W. J. and Chen, Xing-Yu (1989), “Plane Strain Elastic-Ideally Plastic Crack Fields for Mode I Quasistatic Growth at Large-Scale Yielding – I. A New Family of Analytical Solutions,” Journal of the Mechanics and Physics of Solids, Vol. 37, pp. l-26. Drugan, W. J. and Rice, J. R. (1984), “Restrictions on Quasi-Statically Moving Surfaces of Strong Discontinuity in Elastic-Plastic Solids,” Drucker Anniversary Volume: Mechanics of Material Behavior, edited by G. J. Dvorak and R. T. Shield, Elsevier Scientific Publishing Company, Amsterdam, pp. 5973. Freund, L. B. and Douglas, A. S. (1982), “The Influence of Inertia on Elastic-Plastic Antiplane-Shear Crack Growth,” Journal of the Mechanics and Physics of Solids, Vol. 30, pp. 59-74. Ganti, S. and Parks, D. M. (1997), “Elastic-Plastic Fracture Mechanics of Strength-Mismatched Interface Cracks,” in Mahidhara, R. K. et al. (Eds.), Recent Advances in Fracture, The Minerals, Metals and Materials Society, pp. 13-25. Gao, Y. C. (1980), “Elastic-Plastic Field at the Tip of a Crack Growing Steadily in Perfectly Plastic Medium (in Chinese),” Acta Mechanica Sinica, Vo1. 1, pp. 48-56.

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Liu, N. and Drugan, W. J. (1993), “Finite Element Solutions of Crack Growth in Incompressible ElasticPlastic Solids with Various Yielding Extents and Loadings: Detailed Comparisons with Analytical Solutions,” International Journal of Fracture, Vol. 59, pp. 265-289. Ponte Castañeda, P. and Mataga, P. A. (1991), “Stable Crack Growth Along a Brittle/Ductile Interface. Part 1 – Near-Tip Fields,” International Journal of Solids and Structures, Vol. 27, 105. Rice, J. R. (1967), “Mechanics of Crack Tip Deformation and Extension by Fatigue,” Fatigue Crack Propagation, ASTM-STP 415, pp. 247-311. Rice, J. R. (1968), “Mathematical Analysis in the Mechanics of Fracture,” in Fracture: An Advanced Treatise, edited by H. Leibowitz, Academic Press, New York, Vol. 2, pp. 191-311. Rice, J. R. (1974), “Elastic-Plastic Models for Stable Crack Growth,” in Mechanics and Mechanisms of Crack Growth, edited by May, M. J., British Steel Corp. Physical Metallurgy Centre Publication, Sheffield, pp. 14-39. Rice, J. R. (1982), “Elastic-Plastic Crack Growth,” in Mechanics of Solids: The R. Hill 60th Anniversary Volume, edited by Hopkins, H. G. and Sewell, M. J., Pergamon Press, Oxford, pp. 539-562. Rice, J. R. (1987), “Tensile Crack Tip Fields in Elastic-Ideally Plastic Crystals,” Mechanics of Materials, Vol. 6, pp. 317-335. Rice, J. R., Drugan, W. J. and Sham, T-L. (1980), “Elastic-Plastic Analysis of Growing Cracks,” Fracture Mechanics, ASTM-STP 700, pp. 189-219. Rice, J. R. and Tracey, D. M. (1973), “Computational Fracture Mechanics,” in Numerical and Computer Methods in Structural Mechanics, edited by Fenves, S. J. et al., Academic Press, New York, pp. 585– 623. Sham, T.-L., Li, J. and Hancock, J. W. (1999), “A Family of Plane Strain Crack Tip Stress Fields for Interface Cracks in Strength-Mismatched Elastic-Perfectly Plastic Solids,” Journal of the Mechanics and Physics of Solids, Vol. 47, pp. 1963-2010. Slepyan, L. I. (1974), “Growing Crack During Plane Deformation of an Elastic-Plastic Body,” Mekhanika Tverdogo TeIa, Vol. 9, pp. 57-67.

Appendix:

Derivation of Logarithmically Singular Velocity Fields in a General Propagating Plane Strain Near-Tip Constant Stress Plastic Sector

Here we provide an asymptotic leading-order analysis of the velocity fields in a propagating constant stress plastic sector with an arbitrary (constant) stress state, i.e. one of the form (42) satisfying (37). First, employing (42), (6) simplifies to:

(Al)

To leading order as r → 0, this reduces to: (A2) which for the sake of the upcoming analysis we rewrite as: (A3a, b, c)

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Employing (2), (A3a, b) are: (A4a, b)

These may be combined to give a single partial differential equation for v1 :

(A5a, b)

The general solution to (A5a) is: (A6) and form this and (A4a) we find the general solution for v2 : (A7) In these solutions, ƒ and g are arbitrary functions of the indicated variables, and we have defined (A8) Finally, since we seek solutions with velocity fields that are logarithmically singular in r as r → 0, (A6) and (A7) become:

(A9a)

(A9b)

where B 1 and B2 are undetermined constants. From (A3c) and (A9a),

(A10)

STUDY OF CRACK DYNAMICS USING THE VIRTUAL INTERNAL BOND METHOD

PATRICK A. KLEIN Sandia National Laboratory HUAJIAN GAO Stanford University It is a great honor for us to contribute to this 60th Anniversary Volume in honor of Professor James R. Rice and to commemorate his profound contributions to the science of fracture. All of us are indebted to him for the leadership he has provided through the excellence of his scholarship as well as the warm and gentle way he has influenced all his students. H.G., 1999 Abstract. Most existing theories of fracture are based on small deformation constitutive models. These approaches are in contrast to the fact that extraordinarily large, nonlinear elastic deformations inevitably occur during brittle fracture. Though the classical approaches have proved successful in a wide range of applications, they may be inapplicable for or have proved incapable of explaining experimental observations in which nonlinear, hyperelastic material response is an essential feature of the phenomenon. The fracture path instabilities observed during dynamic propagation of cracks are among the phenomena that have not yet been thoroughly explained. Simulation approaches that incorporate a cohesive view of material are able to demonstrate the appearance of crack tip instabilities. Molecular dynamics and cohesive surface methods are among the methods that exhibit this behavior. A virtual internal bond (VIB) model with randomized cohesive interactions between material particles has been proposed as an integration of continuum models with cohesive surfaces and atomistic models with interatomic bonding. This approach differs from an atomistic model in that a phenomenological “cohesive force law” is assumed to act between “material particles” which are not necessarily atoms. It also differs from a cohesive surface model in that, rather than imposing a cohesive law along a prescribed set of discrete surfaces, a randomized network of cohesive bonds is statistically incorporated into the constitutive response of the material via the CauchyBorn rule, by equating the strain energy function on the continuum level to the potential energy stored in the cohesive bonds due to an imposed deformation. The approach could be viewed as an attempt to provide a more physical basis for the hyperelastic constitutive laws used in finite strain continuum mechanics. This approach allows the phenomenon of dynamic crack tip instabilities to be analyzed within a continuum framework. Direct simulation of crack growth without a presumed nucleation, growth, or branching criterion lends support to the theory that instabilities occur as a result of a local limiting speed, governed by the rate at which bond-breaking information can be transmitted to the material ahead of a propagating crack. 275 T.-J. Chuang and J. W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 275–309. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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1. Introduction Cracks propagating at high speeds in brittle materials display behavior that the fracture community has yet to find consensus in explaining. Much of the current debate involves the factors that determine the terminal velocity of propagating cracks and the roughened appearance of fracture surfaces produced under fast fracture. The experimental methods used to study dynamic fracture, some key experimental observations, as well as a review of a number of models that attempt to explain the observed phenomena are reviewed by Ravi-Chandar (Ravi-Chandar, 1998). The theoretical limiting speed for crack propagation is the Rayleigh surface wave speed c R . However, experimental studies of cracks propagating through brittle materials typically quote terminal crack speeds of approximately c R /2, while instabilities have been observed for crack speeds as low as cR /3. Detailed experiments of dynamic crack propagation through the brittle polymer Homalite100 appear in the studies of Ravi-Chandar and Knauss (Ravi-Chandar and Knauss, 1984a; Ravi-Chandar and Knauss, 1984b; Ravi-Chandar and Knauss, 1984c; Ravi-Chandar and Knauss, 1984d). They observe the wellknown “mirror-mist-hackle” appearance of the fracture surface. An image from their study (Ravi-Chandar and Knauss, 1984b) showing the fracture surface is reproduced in Figure 1.

Figure 1. “Mirror-mist-hackle” appearance of a fracture surface observed by Ravi-Chandar and Knauss (Ravi-Chandar and Knauss, 1984a; Ravi-Chandar and Knauss, 1984b; Ravi-Chandar and Knauss, 1984c; Ravi-Chandar and Knauss, 1984d).

The direction of crack propagation is from left to right in the figure. The image shows that the crack surface becomes progressively rougher as the crack extends and the crack speed gets correspondingly higher. By the method of caustics, they are able to observe distinctly different behavior

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in the propagating crack front that helps explain the appearance of the fracture surface. In the initial stages of growth, the crack extends by the propagation of a single, curved front, similar to what is observed under quasi-static conditions. The fracture surface appears essentially smooth, leading to its description as the “mirror” zone. As growth progresses, the crack front appears to extend by the action of multiple, smaller crack fronts. The opaque appearance of the crack surface due to fine-scale roughness leads to its description as the “mist” zone. In the “hackle” zone, the scale of the multiple crack fronts increases, leading to a larger degree of fracture surface roughening. The emergence of crack tip instabilities under dynamic conditions was first predicted analytically in a study by Yoffe (Yoffe, 1951). Her analysis predicts branching at 60° when the propagation speed reaches approximately 0.6 c R . Though the result is significant in demonstrating the onset of fracture path instabilities, neither the crack speed nor the branching angle agrees well with experimental observations. Branching angles typically fall between 10° and 45°, while instabilities appear at speeds well below Yoffe’s predicted value. In particular, Fineberg and coworkers (Fineberg et al., 1991; Fineberg et al., 1992) make detailed measurements of crack propagation in PMMA, showing the onset of oscillatory behavior in the crack tip at speeds of roughly c R /3. Classical theories are unable to explain the development of crack tip instabilities at these speeds. It has been suggested by Gao (Gao, 1996; Gao, 1997) that the hyperelastic nature of crack tip deformation may be essential for understanding dynamic crack tip instabilities. Elastic softening due to hyperelastic stretching can drastically change the way acoustic waves propagate in a solid. The variation in the elastic properties with deformation referred to in this setting should not be associated with models of continuum damage. Damage models attempt to incorporate the effects of microstructural evolution in the behavior of a material, such as microcrack distributions (Kachanov, 1992) or the nucleation and coalescence of voids (Gurson, 1977). The elastic softening under consideration in the present context is an effect of the continuously varying stiffness of a nonlinear cohesive interaction. To illustrate this point, let us consider how a long wavelength signal propagates along an atomic chain stretched toward its cohesive limit. Before the chain is stretched, it behaves like an elastic bar along which a longitudinal wave can be transmitted where E denotes the initial stiffness (Young’s with speed equal to modulus) and ρ the density of the chain. As the chain is stretched near its cohesive (fracture) limit, the tangent stiffness vanishes while a finite tension builds up with magnitude equal to the cohesive stress Tmax . At this point, the chain behaves like an elastic string along which a transverse wave can be transmitted with speed equal to Note that the cohesive-state

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wave speed depends on the magnitude of the stress rather than the slope of the stress-strain curve. During this hyperelastic deformation, the slope of the force-displacement (or stress-strain) curve continuously decreases from its initial value corresponding to the elastic stiffness of the chain, and ultimately vanishes at the cohesive limit. Clearly, a linear theory assuming constant stiffness cannot be used to investigate the behavior of the chain near its failure point. Generalizing the above concept, Gao (Gao, 1996; Gao, 1997) has shown that a homogeneous, isotropic, hyperelastic solid exhibits this string-like behavior near its plane strain equibiaxial cohesive stress in that the cohesivestate wave speed is equal to where σ max is the cohesive stress and ρ is the density of the undeformed solid. The state of plane strain equibiaxial stress resembles the condition that a material particle experiences in front of a mode I crack tip. This observation comes directly from Irwin’s (Irwin, 1957) classic crack tip fields. This cohesive state is identified as the bottleneck state (Gao, 1996) for transmission of fract ure signals ahead of a mode I crack tip. The crack propagation velocity is limited by how fast elastic waves can transport strain energy ahead of the crack to sustain the bond breaking processes in the fracture process zone. From this point of view, the cohesive-state wave speed leads to the concept of local limiting fracture speed (Gao, 1996) which has provided an explanation for the “mirror-misthackle” instabilities. Assuming values of σmax= E /30 and a Poisson’s ratio of v = 1/4, the cohesive state wave speed is roughly 0.32 c R , which is in good agreement with Fineberg’s measurements. Gao’s (Gao, 1993) wavy crack model presents a multiscale view of a dynamic crack in which the global, or apparent, crack speed is driven near c R /2 to maximize the fracture energy absorbed by the advancing crack, while the microscopic tip speed may be significantly higher in response to local crack driving forces. A multiscale view may also be helpful in understanding the “mirror-mist-hackle” instabilities as the crack path begins to deviate in an attempt to circumvent the acoustic barrier presented by elastically softened material immediately ahead of the crack. Hyperelastic constitutive models that are capable of describing a material’s transition to fracture have not been available for studying these effects. As will be shown, the Virtual Internal Bond (VIB) model displays this cohesive state behavior and should therefore be capable of reproducing dynamic crack tip instabilities. The atomistic simulations of Abraham and coworkers (Abraham et al., 1994; Abraham, 1997) display dynamic crack instabilities reminiscent of the experimental results though at decidedly smaller length and time scales. The numerical studies of Xu and Needleman (Xu and Needleman, 1994), employing networks of cohesive surface elements, are also successful in producing crack tip instabilities that qualitatively match the experimentally

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observed behavior. In their study, the continuum elements are modeled using the Kirchhoff-St.Venant constitutive relations, that do not display any softening, while the cohesive surfaces are governed by a traction-separation potential with qualitative features similar to those discussed for the potential in Section 5. Their observation of instabilities may also be explained with the local limiting speed theory since the “composite” material composed of continuum elements with intervening cohesive surfaces has the effective properties of a cohesive continuum, though with mesh dependent anisotropic characteristics. 2. The Virtual Internal Bond model The general features of the VIB model (Gao and Klein, 1998), as well as more detailed study of its localization behavior (Klein and Gao, 1998; Klein, 1999), are described elsewhere. A brief review of the model is repeated here for completeness. The model attempts to incorporate the behavior of a spatial distribution of cohesive bonds that are presumed to act at the microstructural level. The link between the microstructure and continuumlevel measures of deformation is made using the Cauchy-Born rule. This approach is employed by Ortiz, Phillips, and Tadmor to develop constitutive models for single crystal materials (Tadmor et al., 1996). From a practical point of view, it is desirable to extend the procedure for embedding cohesive behavior beyond models for single crystal materials. We would like to develop a model to study fracture in noncrystalline materials, such as polymers or glasses, or in polycrystalline materials for which the elastic anisotropy is lost due to the homogenizing effect of many randomly oriented crystalline grains. The concept of homogenization is applied here to the interactions between particles. The resulting constitutive model embeds the cohesive nature of the interaction potentials, but is free of any orientational dependencies. The VIB model is developed within the framework of hyperelasticity. The current, or deformed, configuration of a body is described as x = ϕ (X), by a mapping ϕ of the undeformed configuration X. The arrangement of cohesive interactions among material particles is described by a spatial bond density function. The strain energy density is computed by integrating the bond density in space in a continuous analog to the sum over discrete lattice neighbors for the case of crystalline materials. The VIB form of the strain energy density function is (1)

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where is the undeformed representative volume, l is the deformed virtual is the volumetric bond bond length, U(l) is the bonding potential, density function, and is the integration volume defined by the range of influence of U. Depending on the range of influence of the bond potential function, the integration volume may not correspond with the . This difference may be illustrated for crystalline representative volume materials whenever the bond potentials extend beyond the lattice unit cell. This method was first alluded to by Gao (Gao, 1996) as a method for constructing an amorphous network of cohesive bonds by a spatial average. The deformed bond length l is computed from the Cauchy-Born rule, assuming the integration volume deforms homogeneously as described by a given In order to avoid questions as to whether the deformation gradient Cauchy-Born rule holds for the proposed microstructure, we consider only bond density functions that are centrosymmetric. Under this restriction, the deformation at the microstructural level must be homogeneous in order to maintain the symmetry present in the undeformed configuration. In extending this description from a lattice of discrete bonding interactions to a continuous bond distribution, the invariance in the deformation with respect to translation of our observation point is not limited to positioning the coordinate origin at crystal lattice sites. All points in the material act as centers of symmetry. From physical considerations, we assume the bonding function U will be relatively short range so that, although the bond density function describes the relative particle distribution of the entire body about the observation point, the integration volume will be on the order of the representative volume The undeformed virtual bond vector is represented as (2) where L is the reference bond length, and is a unit vector in the direction of the undeformed bond. Undeformed bonds are mapped to their deformed configuration l by (3) Making use of the right Cauchy-Green stretch tensor, the deformed bond length is (4) Expressed as a function of C, the deformed bond length transforms objectively by construction. The stress response and tangent moduli are computed from the strain energy density (1) using the relations from Green

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elastic theory (Marsden and Hughes, 1983). Employing (5) the (symmetric) 2nd Piola-Kirchhoff stress is (6)

and the material tangent modulus is (7)

The modulus (7) displays Cauchy symmetry as well as the usual major and minor symmetries of elasticity. This result that an amorphous solid with a random network of cohesive bonds satisfies the Cauchy relation seems to be a generalization of Stakgold’s theorem (Stakgold, 1950) to an amorphous solid. A material particle in an amorphous solid satisfies centrosymmetry in a statistical sense, and the cohesive force law corresponds to a two-body potential. Since we limit our study to centrosymmetric bond density functions, it is natural to express the strain energy density (1) in spherical coordinates as (8) where each bond is characterized by coordinates {L, θ , φ }. For conciseness, we have introduced the notation of a spherical average as (9) where L * represents the maximum distance over which particles interact. In spherical coordinates, the bond direction vector Ξ can be written as (10) The precise definition of D ( L, θ , φ) is that represents the number of bonds in the undeformed solid with length between L

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and L + dL and orientation between { θ, φ } and { θ + d θ, φ + d φ }. Using this notation, the components of the 2nd Piola-Kirchhoff stress and the material tangent modulus from (6) and (7) can be represented as (11) and (12) There are a few special cases with regard to the bond density function D (L, θ,φ): (1) The case (13) corresponds to a network of identical bonds of undeformed length L 0. The Dirac delta function is denoted here with δ D . A crystal lattice such as face-centered cubic with interactions limited to only first nearest neighbors can be represented as

(14) where D 0 is a scaling constant. (2) The case (15) represents a transversely isotropic solid. It will be convenient in the subsequent discussion to define a notation for the spatial average under the transversely isotropic case as (16)

(3) The case (17)

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yields an isotropic solid with fully randomized internal bonds. We will shortly see that the instantaneous response of this solid becomes transversely isotropic under equibiaxial stretching. The instantaneous response becomes generally anisotropic under finite deformations. The associated average for the isotropic case is (18) Selecting the radial bond distribution function as (19) where D 0 is a constant, implies that all of the bonds in the solid are of the same type, or initial length. The result is a model for amorphous material with nearest neighbor bonding only. With this bond distribution, the strain energy density from (8) and (9) becomes (20) (4) The case (21) yields a plane stress, isotropic solid. Unlike the standard plane stress approximation, this model is truly two-dimensional, displaying no outof-plane deformation or stress. The material can be thought of as being composed of a single sheet of cohesive bonds, or layers of noninteracting sheets, since all of the bonds lie in a single plane. Selecting the radial bond distribution function as (22) where D 0 is a constant, implies that all of the bonds in the solid are of the same type, or initial length. As was the case in three dimensions, the result is a model for amorphous material with nearest neighbor bonding only. For this case, the strain energy density has the especially simple form (23) Though this model may be physically unrealistic, its simplicity makes it useful for both theoretical and numerical investigations of the fracture properties resulting from the VIB approach.

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In the sections that follow, we will investigate the stress response and fracture properties exhibited by the VIB model. A majority of the discussion will center on the isotropic model defined above for three and two dimensions by cases (3) and (4), respectively. 3. Isotropically elastic properties at infinitesimal strain We are particularly interested in the properties of an amorphous network of cohesive bonds that is homogeneous and isotropic at small strain. The properties of the model at small strains can be used to fit the model parameters to particular materials. For this case, the bond density function is selected as D ( L,θ,φ, ) = D L (L ), and E, the Green Lagrangian strain, and S reduce to the strain and stress tensors of linear elasticity: E → ε a n d S → σ. Φ can be expanded up to the quadratic term as (24) where the first order term does not appear since the interaction potential is assumed to satisfy 〈 U '( L) 〉 = 0 to give a stress-free, undeformed state. This produces the generalized Hooke’s law (25) where the modulus under infinitesimal deformations is related to the cohesive bond distribution by

(26)

It can be directly verified that (27) is a fourth-rank isotropic tensor. The resulting elastic stiffness tensor is (28) with shear modulus equal to (29)

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The other elastic constants are (30) where λ and µ are the Lamé constants, v is Poisson’s ratio, and E is Young’s modulus. Since a variety of engineering materials display values of Poisson’s ratio within a range around l/4, the Cauchy symmetry exhibited by the VIB model does not represent a significant restriction of the formulation for application to practical analyses. The results for the plane stress, isotropic bond density function (21) are quite similar, but the relations for the elastic moduli (29,30) must be modified to account for the difference in spatial dimensions. The modulus in the generalized Hooke’s law (25) is given by

(31)

The elastic stiffness tensor has the same form given in (28). However, the shear modulus becomes (32) From Cauchy symmetry, the Lamé constants are equal (λ = µ), but the relations between the other elastic constants are different from the standard relations of three dimensional elasticity. From the Hooke’s law relations in strictly two dimensions, Poisson’s ratio, Young’s modulus, and the bulk modulus are related to the Lamé constants by and

(33)

Using (33), the elastic constants for the plane stress isotropic VIB model are and

(34)

4. Elastic properties under plane strain equibiaxial stretching The state of plane strain, equibiaxial stretching resembles the deformation a material particle experiences ahead of a mode I crack tip. The elastic

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properties under such a state of deformation at the cohesive limit of the material have been of key importance in the theory of local limiting fracture speeds (Gao, 1996; Gao, 1997). For plane strain equibiaxial stretching, the deformation gradient is (35)

Under this deformation, the bond length from (4), (36) becomes independent of the in-plane orientation angle ø. The components of the 2 nd Piola-Kirchhoff stress tensor can be calculated from (11) as

(37)

The components of the Cauchy stress are

(38)

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Note that the equibiaxial stretching results in equibiaxial stress. This is not necessarily true for a crystalline solid and indicates that our representation of an amorphous network of cohesive bonds does respond isotropically. The nonzero components of the material tangent modulus are obtained from (12) as

(39) and from the Cauchy relations

where (40) The components of the elastic modulus in (39) exhibit the symmetry of a transversely isotropic material. Therefore, under equibiaxial stretching the initially isotropic solid develops a transversely isotropic instantaneous response. When the equibiaxial stress reaches the cohesive limit σmax , the condition (41)

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or, equivalently (42) is satisfied. The following relations can be observed from (39): (43) which when combined with (42) indicate that all the in-plane components of C I J K L vanish, for

(44)

In order to understand the implications of these results, we introduce some additional concepts from the theory of wave propagation. In Lagrangian coordinates, the equation of elastodynamics at finite deformations is (45) where P = FS is the (nonsymmetric) 1st Piola-Kirchhoff stress tensor and ρ0 is the density of the undeformed material. A plane wave propagating through the continuum is described by (46) where u is the displacement of a material particle, k is the wave number, i is , c is the wave speed, and N is the propagation direction defined in the reference configuration. Inserting the plane wave description (46) into the equation of motion (45) yields (47) where (48) is the acoustical tensor. Nontrivial solutions of (47) require that the determinant of the coefficient matrix of a vanish, yielding the characteristic bulk wave speeds from the eigenvalues of q(N). The effective modulus B in (48) can be expressed in terms of the Lagrangian representation of stress and modulus as (49)

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Using the results for the stress and modulus at the cohesive limit (38,39,44), the speed of elastic waves propagating in the (X 1 , X 2 )-plane is (50) This result agrees with the previous analysis of Gao (Gao, 1996; Gao, 1997) but reaches the same conclusion from a different perspective. His analysis only assumes that the material possesses general cohesive properties and is not derived from the description of a particular constitutive model. Note that waves can travel with a different speed in an out-of-plane direction. This issue is not addressed here. The ratio (51) is the plane strain “Poisson’s ratio” at finite deformation. Gao (Gao, 1997) has shown that the cohesive state wave speed is given by (50) as long as Poisson’s ratio v e does not approach –1 as the cohesive stress is reached. In the present case, v e equals l/3 irrespective of the magnitude of the strain. The cohesive state wave speed is intimately connected to the concept of local limiting fracture speed which has been proposed as an explanation for the occurrence of the “mirror-mist-hackle” dynamic crack tip instabilities. The discrepancy between experimental observations of this speed and the established theory of dynamic fracture (Freund, 1990) has been discussed in previous work by Gao (Gao, 1993; Gao, 1996). A close examination of the mode I crack tip deformation suggests (Gao, 1996) that the state of plane strain, equibiaxial cohesive stress is the bottleneck state for transmission of bond breaking fracture signals ahead of a mode I crack. The crack propagation velocity is limited by how fast elastic waves can transport strain energy to material ahead of the crack tip to sustain the bond breaking processes in the fracture process zone. Prom this point of view, the cohesive state wave speed leads to the concept of local limiting fracture speed at which a straightforward propagation of the crack becomes unfavorable. The crack then attempts to choose an inclined growth direction through material with less elastic softening. Along an inclined direction, stress waves can propagate faster than in the straight-ahead direction where elastic softening has caused all of the in-plane moduli to vanish. As discussed by Gao (Gao, 1993; Gao, 1996), from a global point of view, the apparent crack motion is in a relatively “low inertia” state when the local crack branching occurs. As a result of the local-global inertia competition, the crack chooses to propagate along a wavy path, thus exhibiting the “mirror-mist-hackle” instability. Unlike other descriptions of fracture, this explanation incorporates not only the role of tractions generated as a results of crack opening displacements,

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but also the effect of deformation-induced variations in material response along the propagation direction. Taking some typical values for the cohesive strength relative to Young’s modulus and Poisson’s ratio as σ max = E /30 and v = l/4, we estimate that (52) where c R is the Rayleigh surface wave speed. This is precisely the speed at which crack tip instability is observed experimentally (Fineberg et al., 1991). The present calculation with amorphous cohesive bonds provides further support for the theory of local limiting fracture speed. It also shows that by introducing the essential features of a cohesive view of material, the present model exhibits the correct cohesive state wave speed and should be suitable for use in numerical simulations of dynamic fracture. 5. A model interaction potential The preceding discussions have not made reference to any particular cohesive potential. For the purpose of demonstration, we introduce the phenomenological cohesive force law (53) The defining characteristics of this potential are illustrated in Figure 2. The potential has a “well” of depth U 0 = U (L) = – A B 2 . This quantity can be related to the fracture energy through a J-integral analysis (Klein and Gao, 1998). The response of the potential for bond lengths around the location of this minimum is determined by the parameter A. The “stiffness” at the minimum is U "(L) = A . In the left portion of Figure 2, the potential

Figure 2.

Characteristics of the phenomenological cohesive force potential.

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U is shown with a quadratic potential exhibiting the same energy, force, and stiffness at l = L. The quadratic potential defines the response of a “linear spring”, for which the force always varies proportionally with the stretch in the bond. Compared with the quadratic potential, the cohesive potential displays greater stiffness for compressive deformations and the characteristic loss of convexity in tension. The second parameter B determines the cohesive properties of the potential. The maximum force is generated for a stretched length of In terms of the 2nd Piola-Kirchhoff stress (6), we see that the cohesive strength displayed by the VIB model is not determined by the force U ′ (l ), but rather by the ratio U ′(l ) / l . The maximum value for this ratio occurs for (54) from which we find (55)

6.

Failure

indicators

The promise of cohesive modeling is the ability to produce fracture in numerical simulations without requiring an imposed failure criterion. Failure occurs as a natural consequence of the cohesive formulation, which embeds a finite strength and fracture energy in the material model. Although we are not required to impose a failure criterion, we do need to identify a failure indicator. With classical fracture theory, conditions for crack propagation are posed as a threshold criterion that clearly defines the point at which failure occurs. With a cohesive view of material, failure proceeds in a much more gradual manner. The material does not simply fail, but exhibits a complete history of response from the undeformed state through the cohesive limit to complete failure. In selecting a failure indicator, we choose to associate it with a threshold condition that corresponds to reaching the cohesive limit rather than a cutoff condition when the stresses being sustained by the material are deemed insignificant. Hill (Hill, 1962) has described the loss of strong ellipticity of the strain energy density function as an indication of the loss of stability of a solid. The analysis is related to the eigenvalues of the acoustical tensor q(N) (48). The loss of strong ellipticity coincides with a loss of uniqueness in the solutions of the governing equation of elastodynamics (45). The discontinuous modes of deformation that become admissible with the loss of

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strong ellipticity appear across a characteristic surface, which we describe with the normal N * in the undeformed configuration. With the loss of strong ellipticity produced by models of plasticity exhibiting softening, these characteristic surfaces are associated with slip, or shear, bands. In the present context, we associate these characteristic surfaces with highly localized bands of deformation that become new crack faces as the material reaches complete failure. We also study the behavior of a second failure indicator that is related more directly with the cohesive strength than is the analysis of the acoustical tensor. Since the bond density functions of the VIB model are defined with respect to the undeformed configuration, the 1s t Piola-Kirchhoff stress P is related to the force generated by the network of virtual bonds. The maximum values of P corresponds to the largest forces that can be sustained by the bonds of the virtual microstructure. The onset of failure indicated by these two criteria as a function of the state of deformation is shown in Figure 3 for the plane stress, isotropic VIB model. The configuration of the imposed deformation is shown in

Figure 3. (a) Geometry of the loading configuration and (b) a comparison of failure indicators for the plane stress, isotropic VIB model as a function of the state of deformation.

Figure 3(a). The deformation is characterized by E 1 / E2 , the ratio of the principal values of the Green strain, where E 1 and E2 are the strains along and perpendicular to the primary loading direction, respectively. The normal to the characteristic surface associated with the loss of strong ellipticity is shown as N*. Figure 3(b) illustrates how the two criteria differ in indicating the onset of failure as a function of the state of deformation for B / L = 0.05 using the model interaction potential from Section 5. The

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stretch in the primary loading direction at the point of failure is designated as λ *. The deformation states for uniaxial stress (P2 = 0) and uniaxial strain (E2 = 0) are indicated in the figure. For E 2 / E 1 < 0.3, the loss of strong ellipticity clearly occurs at a smaller stretch than the stretch required to reach the maximum value of P 1 . In this regime, the maximum value of P 1 attained at λ * in the figure is unreachable since homogeneous states of deformation become unstable once the acoustical tensor is non-positive definite. As the imposed deformation becomes more nearly equibiaxial, the critical stretches for the two criteria converge. For E2 / E 1 > 0.3, the two curves in the figure are nearly indistinguishable. For the equibiaxial case, both criteria indicate the onset of failure at λ* = B /L, the stretch for which the bond force is maximal. Figure 4 shows the orientation of the characteristic surface with normal N* as a function of the state of deformation. The angle θ * is measured from the axis of primary loading, as indicated in Figure 3(a). States of

Figure 4.

Orientation of the localized band as a function of the deformation.

deformation corresponding to uniaxial stress and strain are indicated in the figure. The key result of these calculations is that the characteristic surface associated with the loss of strong ellipticity is oriented perpendicular to the primary loading direction for imposed deformations approaching the equibiaxial state, E 2 / E1 > 0.5 in the figure. From the asymptotic crack tip solutions of linear elasticity, we know that the state of equibiaxial stretching resembles the deformation a material particle experiences in front of a mode I crack tip. Therefore, it is encouraging that the direction of crack growth indicated by N* matches our expectations for the direction of propagation

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of a mode I crack. Under mixed-mode conditions, classical fracture theory requires that we postulate a crack propagation direction. Typically, one prescribes crack propagation in the locally mode I direction, the orientation for which K I I = 0. Alternatively, one could select the direction corresponding to the largest hoop stress σ θ θ surrounding the crack tip. Since the mechanisms of material failure are not well understood, the most appropriate choice for the direction of crack propagation is also unclear. With the VIB model, the fracture characteristics under mixed-mode conditions is embedded in the constitutive behavior. Once the parameters in model have been selected, no additional criteria need to be imposed in order to reproduce a wide variety of fracture phenomena. As shown in Figures 3 and 4, the loss of strong ellipticity appears to be an appropriate indicator of the onset of failure. For loading with mode I character, it agrees with our expectation for failure to initiate as the stress in the material approaches the cohesive limit and for crack propagation to occur in the direction perpendicular to the direction of the largest stress. 7. Numerical simulations In order to demonstrate the capability of the VIB model to reproduce dynamic crack tip instabilities, we present two-dimensional simulations using the plane stress, isotropic VIB model and three-dimensional simulations using the VIB model in principal stretches (Klein, 1999). Details of the VIB model in principal stretches will not be described here, but we note that expressing the model in principal stretches greatly improves computational efficiency for three-dimensional problems. For these calculations, the VIB model is used within an updated Lagrangian finite element formulation. The equations of motion are integrated in time using an explicit, central difference scheme from the classical Newmark family of methods. The geometry of the finite element models is shown in Figure 5. As noted in the figure, the three-dimensional model possesses considerably more degrees of freedom although its overall dimensions are smaller. The two-dimensional model contains 159,175 nodes in 158,705 elements for a total of 318,106 degrees of freedom, accounting for the nodes with prescribed kinematic boundary conditions. The three-dimensional model contains 1,305,702 nodes in 1,260,000 elements for a total of 3,865,800 degrees of freedom. The models are loaded by symmetrically prescribed velocity boundary conditions for the nodes along the upper and lower edges of the domains. The boundary nodes are accelerated to the prescribed velocity over a time of 0.2 µ s, essentially simulating an impact load. Additional boundary conditions are prescribed to prevent rigid body motion. For the three-dimensional simulations, the remaining surfaces are traction-free. The material parameters

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Figure 5. Geometry and dimensions of the two- and three-dimensional finite element models used to study dynamic crack tip instabilities. The size of the three-dimensional model is shown on the two-dimensional model with a dashed outline.

are selected to be representative of PMMA, the material used in the experiments of Fineberg (Fineberg et al., 1991; Fineberg et al., 1992), with E = 3.24GPa, v constrained by Cauchy symmetry as described in Section 3, and ρ = 1200 kg/m3 . These values produce dilatational and shear wave speeds of cd = 1740 m/s and c s = 1000 m/s, respectively, and a

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Rayleigh wave speed (Freund, 1990) given by (56) The fracture energy is selected as 350 J/m2 , and the cohesive strength is selected as E /30. A J-integral analysis can be used to show that the fracture energy depends on mesh size with the current form of the VIB model (Klein and Gao, 1998). The given cohesive parameters dictate that elements in the central region of height 1.0 mm have dimension of h = 9.2 µ m. A threeparameter potential could be created to allow the elastic properties, fracture energy, and cohesive strength to be selected independently. However, this approach represents only a partial solution to the well-known difficulties associated with simulating strain localization. For the two-dimensional simulations, four-noded quadrilateral elements are used throughout the model. The central region is meshed with a regular arrangement of square elements with dimension 9.2 µ m while the elements in the outer region increase in size with distance away from the central zone. For the three-dimensional simulations, the entire domain is discretized into a regular, structured grid of eight-noded hexahedral “bricks” with dimension 9.2 µ m. A pre-crack of length 0.5mm ensures that propagation is initially directed along the centerline of the model, though its path is not prescribed in any way. The simulation results show that crack growth, instabilities, and branching emerge naturally from the properties of the VIB constitutive model. For the two-dimensional simulations, the apparent length of the crack is monitored by tracking the elements in which the acoustical tensor (48) is no longer positive definite. Checking this condition requires searching all wave propagation directions to see if there are any for which the wave speeds vanish. Since the models are constrained to prevent rigid body translations, the crack length is taken as the greatest distance along X1 between the original crack tip position and the centroids of localized elements in the undeformed configuration. The time step for the explicit integration scheme is selected so that ∆t cd / h = 1/3, where h is the minimum element size. With the current state of our simulation procedures, the two-dimensional simulations are better-suited to quantitative study than are the threedimensional simulations. Working in two dimensions, we are able to make the computational domains larger, reducing the effect of boundaries on the crack behavior, and analysis of the acoustical tensor requires considerably less effort. For these reasons, quantitative analysis of the two-dimensional simulations is presented, while the results of the three-dimensional simulations are more qualitative. We present results of the two-dimensional simulations for three values of the prescribed boundary velocity, υ BC =

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{2.6,5.2,10.4} m/s. The crack length as a function of the simulation time for all three cases is shown in Figure 6. The time to the initiation of crack

Figure 6. Apparent crack length over time for three different values of the imposed boundary velocity.

growth decreases as the impact velocity increases, presumably in response to the time required to generate the critical driving force at the initial crack tip. For the impact velocities of 2.6m/s and 5.2m/s, the figure shows that the crack moves forward and then stops before accelerating to the terminal velocity. Based on the dimensions of the model and the dilatational wave speed in the material, the initial loading wave reaches the crack tip from the boundaries after roughly 1.35 µ s. An average terminal velocity is calculated for each case from the slope of the crack length curves δ a ( t ) beyond the initial transient behavior. The terminal velocity increases from 0.48cR to 0.53 cR as the impact velocity increases from 2.6 to 10.4m/s. Figure 7 shows the crack morphologies for the three cases. Each point along the fracture path marks an element for which the acoustical tensor condition for localization is satisfied at one or more of the element integration points. The images in the figure actually correspond to superpositions of the fracture path history over the entire simulation time. Since the VIB model is entirely elastic and does not incorporate any irreversibility in the fracture processes, secondary branches along the fracture path “heal” after the leading edge of the crack has advanced far enough to unload the material in its wake. The fracture paths indicate that the cracks initially propagate straight ahead along the symmetry line of the domain. In each case, the first deviation from straightforward propagation is marked by a symmetric

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Figure 7. Fracture patterns for impact velocities of (a) υ BC υ BC =5.2m/s, and (c) υBC = 10.4m/s.

= 2.6m/s, (b)

branch at roughly 10-15° to the initial crack plane. This first branch occurs sooner in time and at a shorter crack length as the impact velocity is increased. The subsequent fracture path becomes more irregular as one of the two branches arrests while the other continues to propagate. Once the symmetry of the original propagation has been disrupted, the branches become more irregular. In both Figures 7(b) and (c), the branching angles become noticeably larger as the crack advances, reaching a maximum angle of approximately 35° in (b), and reaching almost 55° in (c). The initial, straight ahead propagation of the crack is “mirror”-like, while the branching at later times forms a “hackle” zone. What is not evident from the fracture paths shown in Figure 7 is that some indication of instability appears ahead of the crack tip significantly earlier than the occurrence of the first branch, representing the “mist” mode of propagation. Figure 8 shows the arrangement of elements in which the acoustical tensor condition is met at four instants leading to the first branch in the fracture path for the case of υBC = 5.2 m/s. Initially (a), all elements displaying localization lie along a straight path extending from the pre-crack. After 5.1 µ s (b), the first evidence of localization in elements above and below the symmetry line appears. Based on the local limiting speed theory of dynamic crack tip instabilities, the crack has reached a speed at which the strain softened material immediately ahead of the crack is unable to maintain a sufficient rate of energy transfer, and the crack has begun to probe alternate propagation directions. Between 5.1 and 5.9 µs after impact (c), the crack continues to accelerate and the acoustical barrier ahead of the

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Figure 8.

299

The onset of branching for υ B C = 5.2m/s.

crack tip enlarges, evolving to an extended region of “damaged” material. Since deformations in the VIB model are strictly reversible, the material recovers as the tip moves away, leaving no indication of this extended region in the subsequent fracture path. At some time before 6.1 µ s (d), the crack tip reaches a critical state, and the first true branch appears in the crack path. The sequence of Figures 8(a)-(d) bears resemblance to the “mirror-misthackle” progression of crack face roughness observed in experiments. The small scale roughness in the numerical results is too fine to be resolved with the current mesh dimensions, but the transition to larger scale roughness is clearly displayed. The crack length data in Figure 6 can be numerically differentiated in order to calculate the apparent crack velocity as a function of time. The crack velocity, normalized by the Rayleigh wave speed cR , for υ BC = 5.2 m/s is shown in Figure 9. As is evident from the crack length data, the crack moves forward at approximately 2.0 µ s after the impact occurs, arrests, and then accelerates quickly to an average terminal velocity of approximately 0.5 c R . As observed by Fineberg (Fineberg et al., 1992), the crack accelerates quickly, but continuously, instead of initiating at the terminal velocity. As the crack accelerates and rapid branching begins, the crack velocity becomes more irregular. This result qualitatively matches the experimental measurements as well as the numerical results obtained by Xu and Needleman (Xu and Needleman, 1994). Markers A-D indicate the times at which the crack is pictured in Figures 10-13. The four points correspond to (A) initial stages of propagation in the straight-ahead direction; (B) the point at which the small scale instabilities shown in Figure 8(b) appear; (C) the point shown in Figure 8(d) at which the first clear branch appears; and (D) propagation at the terminal velocity. Figures 10-13 show two views of the propagating crack at the four points indicated in Figure 9. The upper plot (a) in each figure shows the

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Figure 9. Apparent crack velocity δ over time for an impact speed of υBC = 5.2 m/s with markers A-D indicating the times at which the crack is pictured in Figures 10-13.

distribution of the instantaneous shear wave speed cs for a wave traveling in the X 1 -direction. The contours are normalized by the shear wave speed in the undeformed material (c s ) 0 . These figures are intended to show how the highly deformed material undergoing fracture is affecting the transfer of crack driving energy to the region ahead of the tip. The shear wave speed also provides a clear indicator of the growing crack. The fracture path is more difficult to identify in the stress plots since elements are not removed from the simulation after the cohesive stress is reached. Stresses, though small, remain continuous across the crack faces, making the crack path difficult to locate. The lower plots (b) show the distribution of the crack opening stress σ22 , normalized by the cohesive stress σ c . These plots show how the structure of the crack tip stress fields change as the crack propagates through the specimen. Figure 10 shows the crack 4.6 µ s after impact as it accelerates from the arrested state to a speed of approximately 0.3 cR . In Fineberg’s (Fineberg et al., 1992) results, the crack begins to display oscillatory behavior at this speed. Neither the shear wave speed distribution (a) nor the distribution of σ 22 (b) shows any signs of crack tip instability. The stress field, with a strong stress concentration marking the current tip position, displays the expected shape similar to the classical tip fields of Irwin (Irwin, 1957). Figure 11 shows the crack at 5.1 µ s after impact, the time at which the acoustic barrier shown in Figure 8(b) has just started to form. The initial shape of a growing deformation-softened region is evident at the tip of the crack. The crack speed at this instant hovers around 0.4 c R . Both the shear

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Figure 10. 4.6µ s speed normalized by X 1 -direction and (b) stress σ c ,both shown

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after impact: (a) the distribution of the instantaneous shear wave the initial shear wave speed ( c s ) 0 of a wave propagating in the the distribution of the σ 22 component normalized by the cohesive over the deformed domain.

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Figure 11. Contours of (a) the acoustical shear wave speed and (b) the opening stress 5.1 µs after impact.

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Figure 12. Contours of (a) the acoustical shear wave speed and (b) the opening stress 6.2µ s after impact.

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Figure 13. Contours of (a) the acoustical shear wave speed and (b) the opening stress 8.3 µs after impact.

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wave speed and the stress distribution display waves being emitted from the tip region in a manner that is not present in Figure 10. Figure 12 shows the crack tip shortly after the initial branch in the fracture path. Both plots in the figure still display a high degree of symmetry. Elastic waves in the wake of the moving tip are even more evident. In the stress plot, the two tips are so close to each other that the combined stress field has a shape which resembles the field of a single tip, though the extent of the highly stressed material is much larger. Clearly, the crack tips are interacting so strongly that crack propagation criteria relying on classic K -field analyses are inapplicable. The final plots in Figure 13 show the crack in a late stage of the simulation. Several branches are clearly visible. Both plots display a degree of chaotic behavior as each tip individually seeks a fast fracture path. The highly stressed region ahead of the multiple crack tips extends over a tremendous area, and no resemblance to the classical crack tip fields remains. Figures 14 and 15 show the results of the three-dimensional simulations. In each figure, the upper plot (a) shows the distribution of the minimum instantaneous shear wave speed, while the lower plot (b) shows the surface within the domain at which the minimum shear wave speed has dropped to half of the corresponding value in the undeformed material. Figure 14 shows the results before the onset of the first branch. A pronounced barrier is visible ahead of the propagating crack. Although the geometry and boundary conditions possess a degree of planar symmetry, the crack is clearly threedimensional in character. The regions of the crack front approaching the free surfaces trail behind the region of the crack front nearer the center of the domain. Figure 14 shows results shortly after branching has initiated. The first true branches initiate on the free surfaces and propagate inward. The three-dimensional simulations terminate before extensive branching is observed due to difficulties with “mesh tangling”, elements with negative volume produced by extreme deformations. 8.

Conclusions

The simulations of dynamic crack propagation in this section show qualitative agreement with experimental observations of fast fracture. The timeaveraged, terminal velocities of crack propagation are clustered around c R / 2, though the instantaneous velocity fluctuates rapidly in the range 0.30.8 c R . Initial instabilities to straight-ahead propagation appear for crack speeds as low as 0.4 cR , speeds that cannot be predicted using classical fracture analysis. The results provide support for the local limiting speed explanation for the onset of crack tip instabilities. The VIB constitutive model displays a continuous reduction in stiffness as the material is stretched

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Figure 14. impact.

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The acoustical barrier in three dimensions before branching 2.7

µs after

toward the cohesive limit. This elastic softening creates an unstable condition at a rapidly propagating crack tip because the straight-ahead fracture path is blocked by an acoustic barrier to the energy transfer needed to sustain the fracture process. In order to circumvent this barrier, the crack seeks alternate fracture paths, leading to tip oscillations that eventually result in larger scale branching. Notably, the local limiting speed theory

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Figure 15.

307

The onset of branching in three dimensions 3.2 µs after impact.

does not require the action of micro-cracks, material inhomogeneities, or wave reflections from boundaries in order to explain the appearance of instabilities in the fracture path. Although these effects may contribute to fracture surface roughening in certain cases, they are not strictly required if one adopts a hyperelastic view of the near tip deformations.

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References Abraham, F.: 1997, ‘On the transition from brittle to plastic failure in breaking a nanocrystal under tension (NUT)’. Europhysics Letters 38, 103–106. Abraham, F., D. Brodbeck, R. Rafey, and W. Rudge: 1994, ‘Instability dynamics of fracture: a computer simulation investigation’. Physical Review Letters 73, 272–275. Fineberg, J., S. Gross, M. Marder, and H. Swinney: 1991, ‘Instability in dynamic fracture’. Physical Review Letters 67, 457–460. Fineberg, J., S. Gross, M. Marder, and H. Swinney: 1992, ‘Instability in the propagation of fast cracks’. Physical Review B 45, 5146–5154. Freund, B.: 1990, Dynamic Fracture Mechanics. New York: Cambridge University Press. Gao, H.: 1993, ‘Surface roughening and branching instabilities in dynamic fracture’. Journal oƒ the Mechanics and Physics oƒ Solids 41, 457–486. Gao, H.: 1996, ‘A theory of local limiting speed in dynamic fracture’. Journal oƒ the Mechanics and Physics of Solids 44, 1453–1474. Gao, H.: 1997, ‘Elastic waves in a hyperelastic solid near its plane strain equibiaxial cohesive limit’. Philosophical Magazine Letters 76, 307–314. Gao, H. and P. Klein: 1998, ‘Numerical simulation of crack growth in an isotropic solid with randomized internal cohesive bonds’. Journal of the Mechanics and Physics of Solids 46, 187–218. Gurson, A.: 1977, ‘Continuum theory of ductile rupture by void nucleation and growth: PART I’. Journal of Engineering Materials and Technology 99, 2–15. Hill, R.: 1962, ‘Acceleration waves in solids’. Journal of the Mechanics and Physics of Solids 10, 1–16. Irwin, G.: 1957, ‘Analysis of stresses and strains near the end of a crack traversing a plate’. Journal of Applied Mechanics 24, 361–364. Kachanov, M.: 1992, ‘Effective elastic properties of cracked solids: critical review of some basic concepts’. Applied Mechanics Review 45, 304. Klein, P.: 1999, ‘A Virtual Internal Bond Approach to Modeling Crack Nucleation and Growth’. Ph.D. thesis, Stanford University. Klein, P. and H. Gao: 1998, ‘Crack nucleation and growth as strain localization in a virtual-bond continuum’. Engineering Fracture Mechanics 61, 21–48. Marsden, J. E. and T. J. R. Hughes: 1983, Mathematical Foundations of Elasticity. New York: Dover Publications, Inc. Ravi-Chandar, K.: 1998, ‘Dynamic fracture of nominally brittle materials’. International Journal of Fracture 90, 83–102. Ravi-Chandar, K. and W. Knauss: 1984a, ‘An experimental investigation into dynamic fracture: I. crack initiation and arrest’. International Journal of Fracture 25, 247–262. Ravi-Chandar, K. and W. Knauss: 1984b, ‘An experimental investigation into dynamic fracture: II. microstructural aspects’. International Journal of Fracture 26, 65–80. Ravi-Chandar, K. and W. Knauss: 1984c, ‘An experimental investigation into dynamic fracture: III. on steady state propagation and branching’. International Journal of Fracture 26, 141–154. Ravi-Chandar, K. and W. Knauss: 1984d, ‘An experimental investigation into dynamic fracture: IV. on the interaction of stress waves with propagating cracks’. International Journal of Fracture 26, 189–200. Stakgold, I.: 1950, ‘The Cauchy relations in a molecular theory of elasticity’. Quarterly of Applied Mechanics 8, 169–186. Tadmor, E., M. Ortiz, and R. Phillips: 1996, ‘Quasicontinuum analysis of defects in solids’. Philosophical Magazine A 73, 1529–1563.

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Xu, X.-P. and A. Needleman: 1994, ‘Numerical simulations of fast crack growth in brittle solids’. Journal of the Mechanics and Physics of Solids 42, 1397–1434. Yoffe, E.: 1951, ‘The moving Griffith crack’. Philosophical Magazine 42, 739–750.

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CRACK TIP PLASTICITY IN COPPER SINGLE CRYSTALS

JIN YU Department of Materials Science and Engineering Korea Advanced Institute of Science and Technology P.O. Box 201, Chongryang. Seoul, Korea AND J.W. CHO Technical Center Deawoo Heavy Industries Co Inchun, Korea

Abstract: Crack tip fields in ductile crystals were studied using diffusion bonded copper single crystals for the two orientations studied by Rice. An optical microscope, stylus profilometer, and X-ray were used to study slip traces on specimen surfaces, surface profiles and lattice rotations, respectively. The plastic zone developed as an assemblage of fan-shaped sectors, the details of which depended on the crystal orientation and the latent hardening behaviors of the crystal. Resultantly, deformation fields of the two orientations were substantially different from each other and also from theoretical predictions as well. Etch pit observations of the specimen interior showed slips on the secondary (or tertiary) systems to meet the compatibity requirement, and that crack tip plastic sectors found on specimen surfaces are reasonably valid in the specimen interior as well, particularly for the B orientation. A simple plane strain model based on exclusive latent hardening could explain many features of experimental results observed on specimen surfaces and specimen interiors reasonably.

1. Introduction In our previous work [1], two high symmetry orientations of the f.c.c. crystals studied by Rice [2] were investigated using diffusion bonded Cu single crystals. The plastic zone developed as an assemblage of fan-shaped sectors for both specimens, but observed slip traces and sector positions on specimen surfaces differed markedly between the two orientations and also from the theoretical predictions as well [2-4]. Operations of slips on coplanar slip planes (CSP) were mutually exclusive, and caused necking in one orientation but protrusion in the other. The disparities were ascribed to the extensive work hardening common in f.c.c. crystals which introduced complex problems such as latent hardening and anisotropic expansion of the yield surface. Following experiment by Shield [5] based on the surface strain measurement by a microscopic Moire showed identical crack tip plastic sectors 311 T.-J. Chuang and J.W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 311-329. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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found by Cho and Yu [1], and finite calculation by Mohan et al. [6] suggested that both finite deformation and lattice rotation strongly influence the structure of the solution. Recently, Cuitiño and Ortiz [7] took into account the latent hardening behavior of copper using a forest hardening model [8], and showed that slip activities differed markedly between the specimen interior and the surface indicating that the surface observations provided only indirect measures of the interior. In the present analysis, slip trace observations on specimen surfaces were detailed at several loading stages, and elementary observations of the crack tip displacement fields and the lattice rotation on specimen surfaces were reported. Then, slip traces in the specimen interior were analyzed through etch pit observations on {111} planes using specimens with very low dislocation density. A simple model based on the Tresca yielding and the latent hardening is proposed to explain slip traces on the specimen surfaces and interior.

2. Experimental Procedure High-purity Cu single crystals (99.999%) with a dislocation density of 5 × 1010m –2 were grown by the Bridgman method and cut into the two crystal orientations shown in Fig. 1. In the A orientation, the crack plane is (010), the crack front lies along [101], and the crack propagation is along the [101] direction. The B orientation, obtained by rotating the A crystal 90° clockwise around the [101] axis, - has the crack plane on the (101) and crack propagation along the [010] direction. According to Rice [2], the yield locus is unaltered by this operation and therefore the two orientations have identical stress and deformation fields within the small displacement gradient formulation neglecting the lattice rotation effects. For both orientation, (111) and (111) are CSPs with a zone axis along [101] which can intersect the whole crack front line, while (1 11) and (111) are non-coplanar slip planes (NSP) which can cross the crack front at points. For the sake of convenience, the z=0 plane was set at the specimen middle plane as shown in Fig. 1. A cracked bend specimen was made by cutting a single crystal into two pieces, and joining them along the original cut planes after slight tapers were made. The crack tip radius was typically less than 5 x 10 –6 m , and tilt and rotation angle across the bonded plane were typically less than ± 1°. Specimens with low dislocation density (10 9 m –2) were prepared for the etch pit observation of the specimen interior by cyclic annealing [9], and cut along the {111} plane parallel to the crack front but not intersecting the crack tip. Hereafter, A1/B 1 and A 2/B2 denote specimens with high and low dislocation densities, respectively. The 3 point bend tests were conducted under a loading rate of 1.67 × 10–6 ms –1 , and slip traces on the specimen surface were observed with an optical microscope after unloading. Variations of the through thickness displacement (u z ) were measured using a stylus profilometer with a resolution of 0.1µm. Displacements on specimen surfaces were measured by recording the positions of inclusions and etch pits before and after the bend tests, while the lattice rotations in the surface layer were measured using the back reflection Laue method with a resolution of 0.5º.

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Figure 1. Two crystal orientations studied by Rice [2]; (a) A and (b) B. The { 111} planes noted by solid lines are CSP, and those marked by dotted lines are NSP.

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3. Experimental Results 3.1. Load-Displacement Curves Single crystal tension specimens were loaded along the [010] and the [101] axes, and resultant stress-strain curves are presented in Fig. 2(a). Both specimens showed almost no stage I hardening, and transitions to stage II hardening occured at a tensile stress of around ~10 MPa. If this value is taken as the yield stress( σy), the plane strain limit loads (Po) of the cracked three point bend specimens was found to be 1.12 × 10 4 and 1.43 × 104 N / m for the A and B orientations, respectively [10]. The load-displacement (P-∆) curves of the three point bend specimens A1 and B 1 are shown in Fig. 2(b). Numbers in the P- ∆ curves denote serial unloading and reloading stages, and Bauschinger effects were negligible. Note that the applied load far exceeds the limit load even at the loading stage 1. The P-∆ curves of the A 1 and B1 specimens showed only slight differences because the effect of crystal anisotropy became smaller under the triaxial stress state present near the crack tip. If the specimens were treated as isotropic, the work hardening coefficient n (in ε = Aσ n) was deduced to be 2 from Fig. 2(b) using I1’yushin’s theorem [10]. Here, σ and ε refer to the uniaxial stress and strain, respectively. 3.2. Slip Traces on Specimen Surfaces 3.2.1. A 1 specimen Evolution of the crack tip plastic zone at the four loading stages indicated in Fig. 2(b) are presented in Figs. 3(a) - (d), and surface slip traces after the fourth loading are schematically described in Fig. 3(e). The fan shaped plastic sectors with well defined sector boundaries developed from the load stage 1. With further load, more slip lines appeared on the specimen surface and eventually developed into slip bands. In general, plasticity in the sector VI was the least active except that in sector V, and the CSP traces in the sector VI which were observed only after the loading stage 3 were not counted here. Slip traces on the (111) and (111) CSPs were confined to the sectors II and III, respectively, while those on the (111) and (111) NSPs extended over several sectors. Operations of slips on CSPs were mutually exclusive in sectors II and III, and slips on the (111) plane of the sector II was not observed very near the crack tip –4 for r ≤ 8 × 10 m even after the fourth loading. A close examination of the sector boundary β showed that each slip trace on the (111) plane in sector III is blocked by another slip trace on the (111) plane in sector II, or the other way around. Spacings between parallel slip traces in sectors II and III increased with the distance from the crack tip, but decreased with further loading. Solid arrows in Fig. 3(e) show dominances of the (111)[011] over the (111)[110] in sector II, and the (111)[110] over the (111)[011] in sector III as necking occured, which implies that pairs of slip - systems giving effective in-plane shear along the [121] or [121] directions under plane strain deformation were not equally activated on specimen surfaces. Slip traces on NSP developed strongly in sectors I and II from the loading stage 1; however, those in the sector III became clear only after loading stage 3. In sectors I and IV, only NSP slips were operative, and it was not possible to distinguish the (111) and (111) NSP slip traces, which were later shown to be mutually exclusive ( cƒ. Fig. 8 ).

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Figure 2. (a)The stress strain curves from the uniaxial tension tests with load along [010] and [101], and (b)load-displacement curves of the 3 point bend specimens A1 and B 1 . Numbers in the curves denote sequential unloading and reloading stages.

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Figure 3. Developments of the crack tip plastic sectors of the A1 specimen after the (a) 1st, (b) 2nd, (c) 3rd, (d) 4th loading stages indicated in Figure 2. and (e) a schematic diagram of (d).

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Sector boundaries maintained constant angles at the crack tip during the –4 sequence of loading, but deflected backward for r > 6× 10 m ( β line) due to remote stresses. If near tip fields dominated the region with r ≤ 6×10 m, only CSP slip traces in sectors III and NSP slip traces in sector I~III appeared to form initially by the crack tip stress fields. Noticeable slip traces were not found in sector V, and the elastic sectors noted by Rice [2] are a possibility. –4

3.2.2. B1 specimen Here, only CSP slip traces, (111) traces in sector II and (1 1) traces in sector III developed from loading stage 1, and the main features remained the same with further loading. Figure 4 shows the crack tip plastic zone after the 4th stage loading and positions of sector boundaries differ slightly from those of Shield [5], presumably due to differences in notch radius, amount of bending, etc. The β boundary deflected backwards with the increase of load for r ≥ 6 × 10

–4

m, and

thus only CSP slips appeared to be activated by the near tip fields (r ≤ 6 × 10 m ) . However, subsequent etch pit analysis of the specimen interior (cf. Fig. 9.) showed activation of slips on the NSPs to meet the compatibility requirement at sector boundaries, and the results should be interpreted as more active NSP slips in the A specimen than in the B specimen. As in the A 1 specimen, slips on CSPs were mutually exclusive and expected to operate under the plane strain condition too. It was not clear from this observation alone whether sectors I and IV are elastic sectors noted by Rice [2], or sectors of low plastic strain. After the 3rd loading stage, slight NSP traces on the ( 11) and (11 ) planes parallel to the y axis appeared in the sector IV. Note that mechanically introduced scratch lines of Fig. 4(b) were deflected by 6~15° at sector boundaries indicating strain discontinuity there. Further analysis showed that the rotation around the z-axis based on the scratch line analysis was 3~4 times larger than the lattice rotation found by X-ray, which suggests that the rotation by slip accounted for most of the rotation observed here [11]. Here, based on the slip line observation on specimen surfaces which had the stress state near the plane stress, postulates were made of those in the specimen interior which was close to the plane strain state. Under plane strain deformation, the –4

effective slip vectors on the (111) and (1 1) planes can be taken as the

and

and the resolved shear stress (RSS) τ = b i σ ij n j

(1)

is not affected by the in-plane stress σzz because b z = n z = 0. Here, n and b are unit vectors normal to the slip plane and along the slip direction, respectively. Thus, variations of σzz along the specimen thickness direction do not influence RSS of CSP slips because nz equals zero, and CSP slips active under the plane stress condition are expected to be the same under the plane strain too. In contrast, NSP slips are expected to diminsh in the specimen interior because nz ≠ 0 and b z≠0 for slips on the ( 11) and (11 ) planes, and RSS decreases with σzz .

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Figure 4. Crack tip plastic sector of the B1 specimen (a) after the 4th loading stage; (b) an enlarged photograph of (a) showing scratch lines; and (c) a schematic diagram of (a).

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319

3.3. Near Tip Surface Profiles By running the stylus profilometer parallel to the crack line, out of plane displacements (uz) on specimen surfaces were measured and presented in Figs. 5(a) and (b). For the sake of convenience, the maximum necking point was set as uz = 0. 3.3.1. A1 specimen Necking dominated the surface profiles of the near-tip regions, and was much more severe here than in the B1 specimen. Since necking can be efficiently accommodated by operations of slips on the NSPs, the amount of shear and corresponding RSS on the NSPs are assumed to be greater in the A 1 specimen. The presence of the δ boundary was quite clear, but the α boundary was hardly observable. The maximum necking occurred at the β boundary where CSP slips intersect, and slip systems compatible with the necking profile were (1 1)[110] in sector III and (111)[01 ] in sector II. Therefore, in addition to the operation of NSP slips, necking is caused by the asymmetric operations of CSP slips near specimen surfaces. 3.3.2. B1 Specimen In contrast to the A1 specimen, slips on two adjacent CSPs caused local protrusion here; and directions of slip in sectors II and III of the specimens A1 and B1 , which are compatible with the surface profiles, are marked with arrows in Figs. 3 and 4. The necking profile observed here corresponded to more frequent slip along [011] than along [110] on the (1 1) plane {or more frequent slip along [0 1] than along [1 0] on the (111) plane}. If only CSPs are active slip planes and the two slips along face diagonal directions on each CSP are equally activated, the specimen undergoes plane strain deformation and u z = 0. Thus, necking is caused primarily by slips on NSPs and unequal activation of slips on CSPs made supplementary necking or protrusion. Overall, the B specimen was much closer to the plane strain deformation and thereby more suitable for the study of crack tip plasticity. 3.4. Displacement Fields Near The Crack Tip. Using the work hardening coefficient (n) of the two specimens deduced from Fig. 2 (b), the displacement fields near the crack tip were calculated using the HRR solution [17,18] and marked as vectors with flat ends in Fig. 6. These were compared with experimentally measured displacements of etch pits, which were denoted as vectors with dots and arrows at the end for A 1 and B1 specimens, respectively. Start and end points of vectors corresponded to positions before and after bending ( 4th stage), and both specimens were permanently bent about 3.7° after the test. Displacement vectors of the A1 specimen leaned backward (toward higher θ ) compared to those of the B 1 , while calculated vectors based on the HRR solution fell in-between. Note that vector magnitudes increased with θ, but that differences between specimens diminished with θ (cf. encircled areas 1, and 2 in the figure). Having different displacement fields, A1 and B 1 specimens were expected to have different stress and strain fields and show different slip traces which were not just the rotated product of each other.

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Figure 5. Surface profiles measured by running stylus profilometer at constant y for (a) A1 and (b) B1 specimens. u z was set zero at the maximum necking point of each scan

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Figure 6. Displacement vectors near the crack tip of A 1 and B 1 specimens after the 4th stage loading denoted with solid dots and arrow tips, respectively. Calculated HRR displacements vectors denoted with flat ends.

Figure 7. Locations on the specimen surfaces studied with X-ray for the lattice rotation measurements of (a) A 1 and (b) B 1 specimens (30×)

3.5. Lattice Rotations Near The Crack Tip Rotations of crystal lattices after the fourth loading at various spots on the specimen surfaces marked in Fig. 7 are summarized in Table 1 3.5.1. A 1 specimen All the sectors except for sector I (spot ) showed clockwise rotations around the z axis (ω z ) by 3~4°, which was close to the plastic bend angle during the 3 point bend test. Rotations around the y axis ( ω y ) were much smaller ; ω y = 0.3~0.5° for the spot and -0.5~-0.7° for the spot where the positive sign of the rotation corresponded to the clockwise rotation. The clockwise rotation in sector I and the anticlockwise

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rotation in the sector V were consistent with the observed necking behavior in Fig. 5(a). For other spots in sectors II, III, and IV, ω x and ω y were not discernable by this method. 3.5.2. B1 specimen As in the A1 specimen, ω z = 3~4° for all sectors except for the sector I (spots and ), and ω x and ω y were much smaller than ωz . The absence of rotation in sector I ahead of the crack tip is consistent with the calculation by Mohan et al. [6]. For the rotation around the x and y axes, ω y = 0.5~1° for the spot in the sector III, and ω x = 0.5~1° for the spot on the α boundary, which are consistent with he necking profile shown in Fig. 5(b). It is interesting to note that ω x and ωy decreased with distance from the crack tip ( r ) within a sector while ω z was almost independent of r. This appears partly inconsistent with the result by Mohan et al. [6] which showed decreases of the rotation angle with r except the sector I. Presumably, ω z was mainly caused by the plastic bending during the bend test, while ω x and ω y were related to the local necking or protrusion affected by crack tip fields. TABLE 1. Lattice rotation angle near the crack tip measured by X-ray A 1 specimen

ωx

ωy

ωz

I

0

0.3~0.5°

0

sector

spot

II

ND

ND

3~4°

III

ND

ND

3~4°

IV

DN

ND

3~4°

V

0

-0.5~0.7°

3~4°

ωx

ωy

ωz

II

0.5~1°

0

3~4°

III

0

0.5~1°

3~ 4°

B1 specimen sector

spot

0

I

IV

3~4° ND: not detectable

3.6. Specimen Interior Observation The A2 and B2 specimens were cut along the {111} plane not interesting the crack tip, and dislocation etch pits of the specimen interior were investigated. Figure 8(b) shows mosaics of dislocation etch pits of the A2 specimen on the (111) cut plane which is 0.43 mm apart from the crack tip. The region to the left of the bonded interface corresponded to the upper part of the crystal (i.e. y > 0, and z > 0), and vice versa. Since α and β boundaries were clearly discernable and almost parallel to

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Figure 8 (a) A 2 specimen cut along the (111) plane 0.43 mm apart from the crack tip; (b) dislocation etch pits on the (111) plane for z > 0.

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the bonded interface, the positions of sector boundaries α and β remained almost constant regardless of the plane stress or plane strain conditions. In Figure 3, slip traces were not found in sector II for r ≤ 0.8 mm, but a reasonably high density of dislocation etch pits were found in sector II indicating the occurrence of slips on the and/or NSP slip systems to meet the compatibility requirement. The β constant regardless of the plane stress or plane strain conditions. In Fig. 3, slip boundary appeared most distinct due to the high density of sessile dislocations produced from dislocation interactions on the two CSPs. The figure clearly shows inclined slip traces due to the operation of NSP slips, slips on the left ( y > 0 and z > 0) and right side ( y < 0 and and z > 0 ) of the bonded interface, respectively (see next). The NSP slips extended over sectors (I~IV) on the specimen surface, but were very much diminished in the specimen interior. Thus, the crack tip plastic zone was much larger on the specimen surface than in the specimen interior because RSS on the NSP was larger for the plane stress than the plane strain conditions. The reason why only slips on NSP depend sensitively on the stress state is related to the fact that RSS depends on σ z z for the NSP slips but not for CSP slips. In the fatigue crack growth experiment using a Cu single crystal with the A plane in sector III and orientation, Neumann [14,15] found slip traces on the on the and planes in section IV on the specimen surfaces, but only traces of sector III in the specimen interior, which is consistent with the observation of the present work. A significant difference between the two studies is that dislocations were generated mainly at the crack tip in Neumann’s case by the excessive work-hardening during the fatigue precracking, but at the near tip dislocation sources in the present case. Note that operations of NSP slips were also mutually exclusive. Among the four NSP slip systems which can cause nonzero u z , only slip operated in the region y > 0 and z > 0, and there was a mirror symmetry in the slip operation with respect to the y = 0 and z = 0 planes. The selection of a NSP slip system in a given region was dictated by the compatibility to the macroscopic deformation and the preferential slip initiation in the highly stressed regions. Accordingly, slip propagation from the specimen surface into the bulk and from the near tip region into the far field region were favored, which were both along the directions of decreasing RSS. plane, 0.15 mm In the case of the B 2 specimen, a cut was made along the away from the crack tip as shown in Fig. 9(a), and the resultant etch pits are shown in Fig. 9(b). Unlike the A2 specimen, which showed active NSP slip traces near the specimen surface, inclined slip traces coming from the slips on the NSPs were not found, and the etch pit densities were more or less constant through the thickness. Since sector boundaries were parallel to the bonded interface, the specimen underwent basically the plane strain deformation by CSP slips throughout the thickness, which was not affected by σzz . Etch pits in the sector II were much stronger than those in the sector III because of the intersection of the (111) slip with the observation plane{ plane}, and possible formation of Lomer -Cottrell locks. In addition, from numerous etch pits ascribable to dislocations on the secondary slip systems, we assumed that NSP slips also operated in sectors II and III, but to a vanishing degree. to the CSP slips.

CRACK TIP PLASTICITY IN Cu SINGLE CRYSTALS

Figure 9 (a) B 2 specimens cut along the plane 0.15 mm apart form the crack tip; (b) etch pits on the cut plane for z>0; and (c) a magnified version of (b).

325

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Figure 10 Crack tip plastic sectors constructed using exclusive latent hardening for the (a) A orientation under plane stress, (b) A under plane strain, (c) B under plane stress, and (d) B under plane strain condition.

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4. Discussion In the previous section, it was shown that the A and B specimens showed quite different crack tip plasticity primarily due to the anisotropic expansion of the yield locus with the latent hardening [16,17]. In order to understand the crack tip slip behaviors of copper single crystals, a simple model based on exclusive latent hardening is proposed here with the following assumptions: 1. Once plastic flow occurs in the primary slip system with the largest Schmid factor, exclusive latent hardening suppress slip activities in subsequent slip systems. 2. Slips are initiated in regions of higher RSS and propagate into regions with lower RSS, in conformity to the macroscopic plastic flow. Using the crack tip stress fields based on anisotropic elasticity [18], contours of the critical resolved shear stress (CRSS) were calculated for all the slip systems of the A and B specimens under the plane stress and plane strain conditions, and the primary slip traces were marked in Fig. 10. Note that the CRSS contours of the slip systems producing plane strain deformation, slip systems BII and BV, DI and DVI, AIII and CIII, are more or less the same in size under the plane stress or plane strain conditions, while those producing non-plane strain deformation, slip system AVI and AII, CI and CV, are much larger under the plane stress condition. Note also that the non-plane strain CRSS contours are much larger in the A specimen, which explains why NSP shear and the degree of necking are much larger in the A specimen. In the case of the A specimen, the plane stress prediction is quite different from Fig. 3 except extensive NSP slips traces, while the plane strain prediction can reasonably describe CSP slip traces observed in the sectors II and III, and NSP slip traces in sector I and IV. Needless to say, coexistence of NSP and CSP slips in sectors II and III could not be predicted due to the exclusive latent hardening assumption. In the case of the B specimen, plane stress and plane strain predictions are not much different, and agreements are generally much better. CSP slip traces in sectors II and III are common to the plane stress and plane strain predictions, which partly explain the parallel α, β and γ lines throughout the specimen thickness in Fig. 9. Overall, the plane strain prediction was closer to what was found in Fig. 4. The reason why the plane strain predictions make better estimates of surface slip traces observed for both specimens can be related to the necking which introduces nonzero σx z , σ y z , and σ z z, thereby deviating the stress state from the plane stress state substantially. According to Cuitiño and Ortiz [7], stress states on the specimen surface and interior middle plane differ markedly from the plane strain field. 5. Conclusions 1. The crack tip plasticity developed in the fan-shaped sectors with well defined sector boundaries, but the two orientations studied by Rice[2] showed quite different deformation fields; The B specimen showed only CSP sectors, while A specimen showed CSP and NSP sectors. Operations of slips on CSPs were mutually exclusive and the same was true of NSPs, even though CSP and NSP slips operated simultaneously in sectors II and III of the A specimen. NSP slips were quite active on the A specimen surface due to the large RSS and necking. Also, operations of slips on CSPs caused local necking in the A specimen but protrusion in B. Displacements were continuous at sector boundaries but not the displacement

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gradient, which suggests constant plastic strain within a sector but strain discontinuity at sector boundaries. 2. Etch pit observations of near tip displacement on specimen surfaces again confirmed that the two orientations have quite different crack tip fields. Generally, displacement vectors of the A specimen pointed toward higher angle and differences between the two specimen diminished with θ. Subsequent X-ray measurement showed that both specimens had ω z = 0 in sector I but ω z = 3~4° in all other sectors which was close to the permanent bend angle after the test, suggesting that most of the rotation near the crack tip was caused by slip. Rotation of the lattice due to necking was typically smaller than 1°. 3. Etch pit observations of the specimen interior showed that crack tip sectors found on specimen surfaces were reasonably valid in the specimen interior as well, particularly for the B specimen. In the case of A specimen, NSP slips developed near the surface but diminished in the specimen center with decreasing σ z z . Sectors showing only single slip traces on the specimen surface, for example sector II of the specimen B, revealed secondary slip traces attesting the limitations of the experimental method used here. 4. A plane strain model based on exclusive latent hardening could explain experimental observations of the primary slip traces and sector boundaries on specimen surfaces and interior reasonably well.

6. References 1. Cho, J.W. and Yu, J.: Near crack tip deformation in copper single crystals, Phil. Mag. Lett. 64 (1991), 175-182 2. Rice, J.R.: Tensile crack tip fields in elastic-ideally plastic crystals, Mechanics of Materials 6 (1987) 317-335 3. Saeedvafa, M. and Rice, J.R.: Crack tip singular fields in ductile crystals with taylor power-law hardening .2. –plane-strain, J. Mech. Phys. Solids 37 (1989), 673-691 4. Rice, J.R., Hawk, D.E. and Asaro, R.J.: Crack tip fields in ductile crystals, Int. J.Fracture 42 (1990), 301-321 5. Shield, T.W.: An Experimental study of the plastic strain fields near a notch tip in a copper single crystal during loading, Acta Mater. 44 (1996), 1547-1561 6. Mohan, R.,Ortiz, M. and Shih, C.F.: An analysis of cracks in ductile singlecrystals .2. –mode-I loading, J. Mech. Phys. Solids 40 (1992) 315-337 7. Cuiti ño, A.M. and Ortiz, M.: Three-dimensional crack-tip fields in four-pointbending copper single-crystal specimens, J. Mech. Phys. Solids 44 (1996) 863-904 8. Cuitiño, A.M. and Ortiz, M.: Computaional modeling of single-crystals, Modeling Simul. Mat. Sci. Eng. 1 (1993) 225-263 9. Kitajima, S., Ohta, M. and Tonda, M.: Production of highly perfect copper crystals with thermal cyclic annealing, J. Cryst. Growth 24/25 (1974) 521-526 10. Kanninen, M.F.: Advanced Fracture Mechanics, Oxford University Press., N.Y., 1985 11. Yu, Jin, unpublished work, 1991 12. Rice, J.R. and Rosengren, G.F.: Plane strain deformation near a crack tip in a power-law hardening material, J.Mech. Phys. Solids. 16 (1968) 1-12 13. Hutchinson, J.W.: Singular behaviour at the end of a tensile crack in a hardening material, J. Mech. Phys. Solid 16 (1968) 13-31

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14. Vehoff, H. and Neumann, P.: In situ sem experiments concerning the mechanism of ductile crack growth, Acta Metallurgica, 27, (1979) 915-920 15. Neumann, P., Fuhirott, H. and Vehoff, H.: Experiments concerning brittle, ductile, and environmentally controlled fatigue crack growth, in J.T. Fong (ed.), Fatigue Mechanisms, ASTM STP 675, (1979) 371-395 16. Jackson, P.J. and Basinski, Z.S. : Latent hardening and the flow stress in copper single crystals, Can. J. Phys. 45, (1967) 707 17. Basinski, S.J. and Basinski Z.S.: Chapter 16, Plastic deformation and work hardening, P. 261 in Dislocations in Solids, Vol. 4, ed. F.R.N. Nabarro, NorthHolland, 1983 18. Paris, P.C. and Sih, G.C.: Stress analysis of cracks, in Fracture Toughness Testing, ASTM STP 381, (1965) 30-83

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NUMERICAL SIMULATIONS OF SUBCRITICAL CRACK GROWTH BY STRESS CORROSION IN AN ELASTIC SOLID

Z. TANG AND A.F. BOWER Division of Engineering Brown University Providence, RI 02912

AND T.-J. CHUANG Ceramics Division

National Institute of Standards and Technology Gaithersburg, MD 20899-8521 Abstract: A front-tracking finite element method is used to compute the evolution of a crack-like defect that propagates by stress driven corrosion in an isotropic, linear elastic solid. Depending on material properties, loading, and temperature, we observe three possible behaviors for the flaw: (i) gross blunting at the crack tip; (ii) stable, quasi-steady state notch-like growth; and (iii) unstable sharpening of the crack tip. The range of material parameters and loadings that cause each type of behavior is computed. Our results also confirm the existence of a threshold stress level (known as the fatigue limit) that leads to crack sharpening and ultimately to catastrophic fracture. Contrary to earlier predictions, however, our simulations show that the fatigue threshold is determined not only by the driving force for crack extension but also by the kinetics associated with the chemical reaction at the crack tip. Our results suggest that the fatigue threshold is likely to decrease as temperature is reduced. Finally, we have computed the steady state crack tip velocity as a function of applied load in the regime of steady state crack growth. Our predicted crack growth law is in good qualitative agreement with experiment, but uncertainties in material data make quantitative comparison difficult. 1 . Introduction Advanced ceramics, fiber reinforced composites and optical glasses are exploited in the design of devices and components by various industries, ranging from aerospace applications to computer hardware. The durability of ceramics and glasses in service is therefore a major concern. Experiments suggest that the lifetimes of many components are limited by subcritical crack growth (Zhou and Curtin, 1995). Under sustained loading conditions, two classes of crack growth are observed, depending on the stress, temperature and material. At elevated temperatures, the most common form of failure is by crack growth along interfaces or grain boundaries. In contrast, at room temperature, or in a corrosive environment, transgranular fracture is the dominant mechanism of failure. In amorphous materials such as glass, 331 T.-J. Chuang and J. W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 331–348. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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subcritical crack growth is the main mode of failure at all temperatures. The focus of this paper is subcritical transgranular crack growth in brittle elastic solids. In general, ceramics and glasses are notable for their resistance to a hostile environment. Nevertheless, if a stressed ceramic component is exposed to chemical attack, it may suffer from a form of delayed fracture known as static fatigue. In some materials this behavior has been attributed to a process involving chemical dissolution of material from the region near the tips of small pre-existing cracks in the solid (Wiederhorn, 1975, White, et al., 1986, Simmons and Freiman, 1986, Gehrke, et al., 1990,). In this case, the loss of material causes cracks to progressively sharpen and increase in length until catastrophic fracture occurs. Experiments have revealed that, for these materials and ambient environment, the time to failure is a strong function of the applied stress. In particular, if the applied stress lies below a threshold value, known as the fatigue limit or stress corrosion limit, failure can be avoided. It is clearly desirable to determine this limit, and in situations where the stress must exceed the corrosion limit, to determine the rate of crack growth as a function of the applied stress. In the literature, the latter is often expressed empirically as v = AK n or v = B exp( CK) from the experimental data (see for example, Wiederhorn et al. 1974, Freiman et al. 1985). The main objective of the present work is to present a physics-based model to describe the subcritical crack growth behavior within a material subject to chemical dissolution. Charles and Hillig (1962) were the first to develop a micromechanical model of crack growth by corrosion. They considered the behavior of an elliptical cavity in an isotropic, linear elastic solid, using a model based on absolute reaction-rate theory to characterize the rate of dissolution of material from the crack flanks (Hillig and Charles 1965). The essential feature of this model is that the rate of material loss from a surface is influenced by both the stress acting tangent to the surface and also by surface curvature. Stress generally tends to increase the rate of material loss, while the curvature of a concave surface reduces it. The competition between these two effects may be characterized by a dimensionless parameter (1) where σ tip is some measure of the stress near the cavity tip, γ is the free energy per unit area of the unstressed surface, and κ tip is the surface curvature near the crack tip. For large Σ , the effects of stress dominate over curvature, so that the ellipse tip propagates more rapidly than the flanks. This causes the ellipse to sharpen, and eventually results in the formation of a crack which triggers brittle fracture. For small Σ , the effects of stress are negligible. Material at the tip of the ellipse then dissolves more slowly than material near the flanks, and the ellipse is blunted, eventually evolving to a rounded cavity. The critical value of Σ that discriminates between blunting and sharpening gives the static fatigue limit for the solid. Charles and Hillig estimated the critical Σ by assuming that the crack remains elliptical throughout its evolution. Chuang and Fuller (1992) extended the Charles-Hillig (1962) model to compute the initial rate of dissolution of material from the entire surface of the ellipse.

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They showed that an initially elliptical flaw is unlikely to remain elliptical throughout its growth, and instead predicted four possible regimes of behavior for the crack: (a) gross blunting, where the rate of dissolution of material from the crack flanks exceeds the rate of removal near the tip, so that the ellipse approaches a circular shape; (b) enhanced blunting, where the material just adjacent to the crack tip is removed faster than material at the crack tip itself, resulting in blunting near the apex; (c) necking, where the material removal rate is a minimum just adjacent to the crack tip, resulting in a neck-like crack forming near the apex and (d) gross sharpening, where material is removed most rapidly near the crack tip. In addition they showed that, for a typical reaction theory based consititutive law of corrosion, a second material parameter m plays an important role in governing the behavior of the crack. This parameter will be defined and discussed in more detail in Section 4: for now it is sufficient to note that m quantifies the nature of the corrosion law. In general, the expressions for both the driving force for material removal, and also the associated activation energy, contain linear and quadratic terms in stress. For large m, the linear term dominates, while for small m the quadratic term is dominant. Chuang and Fuller’s (1992) computations suggest that there exists a threshold value for m, which controls a transition from enhanced blunting behavior (regime b) to neck-like crack growth (regime c). Existing micromechanical models are thus based either on a simple geometrical description of the crack, or draw conclusions based on the initial rate of material loss from the crack surface. In this paper, we use a numerical technique to compute in detail the evolution of a crack propagating by stress driven corrosion. We consider a large, plane, linear elastic solid which contains a crack-like notch near its center. We adopt Hillig and Charles’ (1965) constitutive law to describe the rate of material loss from the crack surface as a function of stress and curvature. The finite element method is used to solve the coupled equations of linear elasticity and those governing material loss from the crack surface, while a front tracking method is devised to track the evolution of the crack’s geometry with time. Our results confirm many of the predictions of existing models: we observe a transition from crack blunting to sharpening at a critical value of Σ ; we find that m has a strong influence on the behavior of the crack, and observe most of the features of crack evolution predicted by Chuang and Fuller (1992). However, some surprising new insights emerge from our simulations. In particular, we find that the transition from blunting to sharpening is determined not only by the driving force for material removal, but also by the kinetics of this process, so that there is no single pair of values for Σ and m which lead to crack sharpening. Instead, the critical combination of Σ and m depends on a third dimensionless material parameter Φ , which is a function of temperature. The implication of this result is that the fatigue threshold for a given material is likely to vary with temperature: our results suggest that a decrease in temperature will decrease the fatigue threshold. Secondly, our results show that the initial rate of material loss from the crack surface is not a good predictor of its subsequent behavior. Consequently, the four regimes of behavior proposed by Chuang and Fuller (1992)

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Fig. 1. Idealized geometry used to study the growth of a stress corrosion crack in an elastic solid, showing a typical finite element mesh. The crack is elliptical, with ratio of semiaxes b / a = 0.01. Each element shown is a six noded triangle. are observed only for a vanishingly short time. Enhanced blunting (regime b) quickly evolves to gross blunting (regime a); and gross sharpening (regime d) is never observed - crack sharpening is always accompanied by the formation of a neck near the crack tip (regime c). Instead, we observe three types of crack growth: (i) Gross blunting; (ii) Stable, quasi-steady state notch like crack growth; and (iii) Sharpening, accompanied by the formation of a neck near the crack tip. We find that regime (i) will occur in all materials, provided that the applied load is sufficiently low. In contrast, regime (ii) exists only in materials in which m, exceeds a critical threshold. In such materials, the crack will blunt at low loads, or sharpen to propagate as a self-similar notch at higher loads. The crack would presumably continue to grow in this manner until the stress near the crack tip exceeds the ideal strength of the solid. In materials with m below the critical threshold, the ellipse appears to sharpen without limit, and rapidly forms an ideal crack. 2 . Model

Description

We idealize a typical ceramic component as a planar, isotropic, linear elastic solid with Young’s Modulus E and Poisson’s ratio v, Fig 1. The solid is assumed to contain a single crack like flaw, with characteristic length 2a, near its center. In this paper, we will report results only for an initially elliptical cavity, but we have obtained similar results for a notch-like flaw with constant tip curvature and flat sides. The solid is loaded by a uniform remote stress σ ∞ acting perpendicular to the major axis of the flaw, thereby inducing a displacement field ui (x j ) and stress distribution σ i j (x j ) within the solid. The displacement and stress fields are related

STRESS-ASSISTED CORROSIVE CRACK GROWTH.

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by the usual linear elastic constitutive law (2) and the stress must satisfy the equilibrium equations α ij , j = 0. In subsequent discussions, we will assume that the displacements u i are small, implying that the change in shape of the cavity surface due to elastic distorsion of the solid is negligible. We will, however, account rigorously for large changes in shape of the reference configuration as material is dissolved near the crack tip. The defect surface is assumed to be exposed to an unspecified, chemically reactive species, which progressively removes material from the solid. We use Hillig and Charles’ (1965) constitutive law to characterize the resulting rate of loss of material. In developing this model, it is assumed that chemical reactions at the solid/vapor interface limit the rate of material removal, so that it is not necessary to account explicitly for processes involving diffusion of material to or away from the reaction site, nor is it necessary to model adsorption or desorption of chemically reactive species at the surface. The reaction is driven by a difference in chemical potential between the material near the solid surface and the reaction product, and the rate of reaction is determined by a combination of the driving force and the activation energy associated with the chemical process. In the absence of stress, this causes a flat surface to recede at uniform rate v0 , which is generally a function of temperature. Surface curvature and stress modify the chemical potential of atoms near the void surface, and also influence the activation energy. The recession rate of a curved, stressed surface is therefore expressed as (3) Here, v n is the normal velocity of the surface in the unstressed reference configuration (v n is negative because the surface is receding), R is the gas constant and T is temperature. In addition, (4) where σ = σ i j ti t j is the tangential stress at the solid surface; V m is the molar volume of the solid; α is a dimensionless phenomenological constant such that αV m = ( ∂ φ/ ∂ σ )σ =0 is the activation volume for the reaction; β is a second dimensionless constant, which accounts for both a quadratic term in stress in the Taylor expansion of the activation energy about σ = 0 and also for the strain energy released as atoms are removed from the surface; E' = E /(1 – v 2 ) is the plane strain modulus; γ is the energy per unit area of a stress free surface and κ is the surface curvature, defined so that κ > 0 for a concave surface. Additional details concerning the derivation of this kinetic law may be found in Hillig and Charles (1965). 3 . Numerical

procedure

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We have used the finite element method to solve the equations outlined in the preceding section. It is convenient to divide the calculation into three steps. Assume that the shape of the crack or notch at time t = 0 is known. The first step is then to determine the distribution of stress in the solid at time t = 0. Next, we determine the material lost from the void surface during a subsequent interval of time ∆ t. Finally, the reference configuration is updated, and a new distribution of stress is computed for time t = ∆ t. The computation is repeated to determine the evolution of the crack as a function of time. The standard finite element method for linear elastic solids is used to compute the distribution of stress in the solid. We also use a finite element procedure to calculate the change in shape of the void surface as material is removed by corrosion. Let ∆ h(s) denote the depth of material lost during a time interval ∆ t at position s on the void surface. ¿From the preceding section, we have that (5) Due to the presence of surface curvature in the expression for φ , it is difficult to integrate this equation with respect to time using an explicit Euler scheme. We therefore use a semi-implicit method, noting that the change in curvature of the surface during a time interval ∆ t may be estimated as (6) We use this estimate in a general Euler time integration scheme (7) where 0 ≤ θ ≤ 1 is a parameter controlling the time integration. For θ = 0, (7) reduces to a standard explicit forward-Euler scheme, while choosing θ = 1 corresponds to a semi-implicit scheme with a first order predictor for curvature. Choosing θ > 0 has a marked impact on the stability of the algorithm, allowing time step sizes to be increased by several orders of magnitude without loss of accuracy. Our tests show that θ = 1 leads to the best numerical stability, while the best accuracy appears to occur around θ = 0.5. The accuracy is relatively insensitive to θ , however, since small time steps must be taken to ensure that the stress field is updated correctly. We have used θ = 1 in all the computations reported here. Combining (6) and (7) and writing the result in weak form then leads to

(8)

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Here, δ h(s) is a twice continuously differentiable test function of arc length around the void surface, and S denotes the void surface. Eq (8) must be satisfied for all admissible δh(s). The usual finite element procedure is used to obtain a discrete form of (8): the variations of δ h(s) and ∆ h( s ) are interpolated between discrete points on the void surface by means of piecewise cubic Hermitian interpolation functions, which allows the surface integrals to be expressed in terms of a finite set of values of ∆ h, δ h and ∂ ∆h/∂s, ∂ δh/δs. The condition that (8) must hold for all δh then leads to a sparse, unsymmetric, system of linear equations to be solved for the discrete values of ∆ h and ∂∆ h/∂s. These results then form the basis for a finite element solution for the shape of the corrosion crack as a function of time. At time t = 0, the initial shape of the crack is specified by a set of ‘control points’ on the void surface. The geometry of the solid is then interpolated between these points, using cubic parametric splines. The analysis begins by generating a mesh of six noded, triangular finite elements to fill the solid. We have found that the advancing front method of Peraire et al (1987) is particularly effective for this purpose. The algorithm allows one to generate meshes with an arbitrary variation of element size: in our computations we use the error estimate of Zhu and Zinkiewicz and Zhu (1987) to generate a nearly optimal mesh at each time step. A typical finite element mesh is illustrated in Fig.1: the mesh contains approximately 3500 elements and 15000 degrees of freedom. The smallest element near the crack tip has a height of approximately –6 10 a, where a is the crack length. For a crack length of 10µm, this corresponds to a spacing between nodes of only 0.01nm. We then proceed to compute a finite element estimate for the nodal values of displacement, and subsequently use a variational recovery scheme to project values of stress from the integration points within each element to the nodes. The nodes that lie on the void surface are then used to generate a one-dimesional finite element mesh to solve (8). Finally, nodal values of ∆ h and ∂ ∆ h/∂s on this mesh are used to compute a new spline representation for the void surface. The procedure is repeated to determine the history of crack propagation. 4. Results

and

Discussion

To discriminate between the various regimes of behavior of the crack, we adopt the following dimensionless measures of stress, material properties and crack geometry (9) Here, is the crack tip stress intensity factor, κ 0 is the initial crack tip curvature, γ is the surface energy, E' is the plane strain modulus, V m is the molar volume of the solid, R is the gas constant and T is temperature. Finally, α and β are the two dimensionless parameters appearing in the corrosion law (3–4). Note that one may also define an additional (but not independent) dimensionless parameter (10)

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which is a function only of the material properties and is independent of applied loading. In addition, we introduce the dimensionless time measure (11) Provided that the conditions necessary for the applicability of linear elastic fracture mechanics are met, the values of λ and Φ , together with any two of the parameters Σ , Γ , or m, completely characterize the behavior of the crack. It is straightforward to appreciate their physical significance: Σ quantifies the relative effects of crack tip curvature and crack tip stress on the rate of material removal by corrosion: for large Σ , stress dominates, tending to cause rapid crack growth, while for small Σ crack growth is retarded by the influence of curvature. Similarly, Γ can be loosely thought of as the ratio of crack tip energy release rate to the Griffiths toughness Large positive values of Γ imply a large driving force for crack growth. However, because the chemical reaction at the crack tip may either provide an additional thermodynamic driving force for crack growth, or may involve additional energy dissipation as heat, the condition need not be satisfied for the crack to advance, and crack growth is possible for all values of Γ . Indeed, since the kinetic parameter β may be negative, for some materials it is possible that Γ < 0. The dimensionless parameter Φ describes the kinetics of crack growth: a large value for Φ implies a large change in crack tip velocity with stress or curvature. Finally, the material parameter m quantifies the relative

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Fig. 2.

339

Sequences of crack surface profiles for Φ = 0.24, m = ∞ , λ = 10 4 for two load levels: (a) Blunting occurs for Σ = 0.8; the time interval ∆ t
= 2 × 10 –4 . (b) Sharpening, followed by stable notch like growth occurs for Σ = 4.0; ∆ t
= 4.8 × 10 –5 .

magnitudes of the linear and quadratic terms in the driving force for stress driven material removal. For m < 0, the linear term tends to increase the rate of crack growth with stress, while the quadratic term retards growth. For m = 0, the linear term has no contribution, reducing the corrosion law to the form used by Wilkins and Dutton (1976). For m > 0, both linear and quadratic terms in the expansion tend increase the rate of crack growth, and in the limit m → ∞ the linear term in the corrosion law dominates. Typical values for the material parameters in our model are listed in Table 1. We now proceed to investigate the behavior of the crack for the range of physically reasonable values of the dimensionless parameters. For simplicity, we will consider first the case Γ = 0 ( m → ∞ ), wherein the linear term in stress in the expression for the driving force for crack growth dominates over the quadratic term. Fig. 2 shows the behavior of the crack for two different levels of applied stress Σ , and for an intermediate value of the kinetic parameter Φ . The figures each show a sequence of profiles of the crack surface (in the undeformed configuration), at equally spaced intervals of dimensionless time t
. For low values of Σ , the crack blunts; while if Σ exceeds a critical threshold, the crack sharpens, forming a neck in the process. Further insight into the behavior of the crack can be gained by examining the distribution of surface curvature κ and the rate of loss of material from the region near the crack tip. Fig. 3 shows a sequence of graphs of normalized surface velocity as a function of arc length near the crack tip ( s = 0 corresponds to the crack tip; the arc length is normalized by initial crack length a ); Fig.4 shows a corresponding sequence of surface curvature. Observe that in Fig 3a, the initial velocity of the crack tip is less than the velocity of points just adjacent to the tip. This simulation therefore lies in the regime classified as ‘enhanced blunting’ by Chuang and Fuller: the tip initially propagates more slowly than the crack flanks. Our results show,

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Fig. 3. Sequences of normalized crack surface velocity for Φ = 0.24, m = ∞, λ = 10 4 at two load levels: (a) Blunting for Σ = 0.8; time interval ∆ t
= 2 × 10 –4 ; (b) Sharpening to steady notch like growth for Σ = 4.0; ∆ t
= 4.8 × 10 –5

Fig. 4. Sequences of crack surface curvature for Φ = 0.24, m = ∞ , λ = 10 4 at two load levels: (a) Blunting for Σ = 0.8; time interval ∆ t
= 2 × 10 –4 ; (b) Sharpening to steady notch like growth for Σ = 4.0; ∆ t
= 4.8 × 10 –5 . however, that this blunting behavior lasts for only a short time, and soon gives way to gross blunting, where the entire tip region propagates more slowly than the crack flanks. This is accompanied by a progressive decrease in crack tip curvature, Fig 4a. Figs 2b, 3b and 4b show the behavior of the crack for a high value of Σ. Observe that in these results, the initial velocity of the crack surface is a maximum at the crack tip - thus placing the simulation in Chuang and Fuller’s (1992) regime (d): gross sharpening. In fact, we have not observed gross sharpening in any of our

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Fig. 5 Variation of (a) crack tip velocity and (b) curvature, for various material parameters and load levels. simulations: instead crack sharpening is always accompanied by the formation of a neck near the crack tip, as shown in Fig. 2b. Figs 3b and 4b show the corresponding variation of crack surface velocity and curvature. The crack tip velocity, curvature and stress all increase, but the crack tip curvature increases more rapidly than the stress, so that after a transient period the curvature and velocity distributions approach a quasi-steady state. Further evidence for this behavior is presented in Fig. 5, which shows the crack tip velocity and curvature for various combinations of Σ, m and Φ. For Σ = 4, Φ = 0.24, and m = ∞, the crack tip curvature and velocity appear to approach steady values. We take this as an indication of stable, quasi-steady notch like crack growth, which will continue until stress levels near the crack tip approach the ideal strength of the solid and so trigger unstable fracture. This form of crack growth should be contrasted with unstable sharpening, wherein the crack tip curvature and velocity increase without limit, leading to rapid failure. Since we do not observe either enhanced blunting or gross sharpening in our simulations, we will not follow Chuang and Fuller’s (1992) characterization of the behavior of the crack. Instead, we will classify the behavior of the crack in one of three regimes: (i) Gross blunting; (ii) stable, quasi-steady notch like crack growth; (iii) unstable sharpening, wherein the crack tip curvarure and velocity both increase without limit. We turn next to examine the influence of kinetics on crack growth behavior. Fig 6 shows the behavior of the crack for identical values of remote stress Σ and material parameter m = 0, but two different values of Φ. Since Φ influences the relative magnitudes of the crack’s surface velocity at its tip and flanks, in this case increasing Φ has a qualitatively similar effect to increasing Σ. For low values of Φ, the crack always blunts; while for high values of Φ the crack tip sharpens. An important consequence of this observation is that the critical value of Σ required to trigger crack sharpening depends on Φ. Since Φ is temperature dependent, our simulations suggest that the fatigue threshold will vary with temperature. Lower

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Fig. 6. Sequences of crack surface profiles for Σ = 2.66, m = ∞, λ = 10 4 for increasing rate parameters (a) Blunting for Φ = 0.24; time interval ∆ t
= 2.2 × 10 –4 (b) Sharpening for Φ = 0.48; ∆ t
= 5.0 × 10–5 .

Fig. 7. A fracture mechanism map showing the range of values of Σ and Φ required to cause crack blunting or sharpening, for m = ∞. In this case a sharpening crack stabilizes to steady notch like growth. temperatures increase Φ and therefore decrease the fatigue threshold. Of course, the rate of crack growth is also reduced if temperature is reduced, so that unstable fracture will be kinetically limited at very low temperatures. We have conducted several simulations to map the critical combinations of Σ and Φ that will cause the crack to sharpen. The result is shown in Fig. 7. Our computations suggest that there is a critical load level below which crack

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Fig. 8. Sequences of normalized crack surface velocity for m = 0, λ = 10 4 at two load levels: (a) Blunting for Γ = 4.55, Φ = 0.12; time interval ∆ t
= 10 –4 (b) Unstable sharpening for Γ = 4.55, Φ = 0.24; ∆ t
= 7 × 10 – 6 . sharpening will not occur for any Φ. This critical stress appears to coincide with Chuang and Fuller’s estimate for the fatigue threshold Σ c r , shown as a dashed line in Fig. 7. We have also investigated the role of the material parameter m in governing crack growth behavior. As an example, we next present results for m, = 0, wherein the quadratic term in stress in (4) dominates over the linear term. For this case the load level must be parameterized by the dimensionless group Γ, since Σ = 0 for all stress levels. Typical results for two values of Γ and an intermediate value of Φ are presented in Fig. 8. Qualitatively, the behavior of the crack is similar to the results presented for m = ∞. For low loads, the crack blunts, while for high loads, the crack sharpens. However, in this case a sharpening crack never stabilizes: instead, the crack tip curvature and velocity both increase to the limit of the resolution of our finite element method. Evidence for this assertion is presented in Fig. 5, which shows variations of crack tip curvature and velocity for various combinations of material parameters and applied load levels. For m = 0, both the crack tip curvature and velocity appear to increase without limit if the load Γ exceeds the fatigue threshold. Our estimate for the critical combinations of Φ and Γ which lead to unstable crack sharpening are shown in Fig. 9. As for m = ∞, we note that the fatigue threshold is generally a function of Φ, and consequently is a function of temperature. There appears to be a critical value for Γ below which blunting always occurs, irrespective of the value of Φ. However, in this case the threshold does not appear to coincide with Chuang and Fuller’s (1992) estimate of the fatigue threshold. We have conducted several further simulations to investigate crack growth behavior for arbitrary m values. Our results are summarized on Fig. 10, which shows fracture mechanism maps for several m values. As before, Σ parameterizes the magnitude of the applied load, while Φ is primarily dependent on material

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Fig. 9. A fracture mechanism map showing the range of values of Γ and Φ required to cause crack blunting or sharpening, for m = 0. In this case sharpening continues without limit to form an ideal crack.

Fig. 10. A fracture mechanism map showing the range of values of Γ and Φ required to cause crack blunting or sharpening, for various m. For m exceeding between 2.5 and 3.3, a sharpening crack stabilizes; for m below this range sharpening is unstable. properties associated with the rate of corrosion. Recall that setting m = 0 gives a corrosion law with only quadratic terms in stress; while for m → ∞, the linear

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Fig. 11. Transient variation of crack tip velocity during convergence to steady-state notch like growth, for two initial conditions. Both results have Λ = 37.8 and so eventually converge to the same velocity. term dominates. We find that reducing m tends to reduce the critical Σ and Φ that will ensure crack blunting. For high values of m, exceeding these values will cause the crack to sharpen, but the sharpening will stabilize to produce stable notch-like crack growth at constant velocity and crack tip curvature. If m falls below a value of between 2.5 and 3.3, then we see no tendency for the crack to stabilize. Instead, the crack continues to sharpen, with a corresponding increase in crack tip velocity, until our simulations can no longer reliably resolve the crack tip. In the regime of stable notch-like crack growth, the variation of crack tip velocity with applied load is of particular interest. Dimensional considerations indicate that the steady state crack tip velocity and curvature may be expressed as (12) where G and F are functions to be determined, κ s s denotes the steady-state crack tip curvature, and (13) Fig. 11 illustrates this trend. The figure shows the variation of crack tip velocity with time, for two cracks with identical values of crack driving force Λ = 37.8, but different initial conditions ( Σ = 2.48, Φ = 6.1) and ( Σ = 4.98, Φ = 1.5), respectively. In both cases m = M = ∞ and λ = 10 4 . After an short transient period, both cracks propagate with the same crack tip velocity, regardless of the initial conditions.

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Fig. 12.(a) Steady state crack tip velocity and (b) curvature as a function of applied load, for m = M = ∞. The scatter in the numerical data is caused by small fluctuations in the numerical solution due to variations in finite element mesh size. We have calculated the steady state crack tip velocity and curvature as functions of load parameter Λ, for the particular case m = M = ∞. Results are shown in Fig. 12, and suggest that our numerical data may be approximated by a crack tip velocity law of the form (14) This is in remarkable qualitative agreement with experimental observations, which are generally fit by v = B e x p (CK I ), where B and C are empirical constants that depend on temperature and nature of the corrosive environment (Wiederhorn et al 1974, Wiederhorn 1975). Quantitative agreement is less satisfactory, however. (Wiederhorn Experiments indicate that B ~ 10 –21 ms –1 and –1 et al, 1974), while data listed in Table 1 suggest that B ~ 10 –12 ms a n d C ~ This discrepancy may be partly due to errors in values for material properties listed in the table: our predictions are particularly sensitive to variations in γ and α. There are several other explanations, however. Using data in Table 1, our calculations predict steady state crack tip curvatures of the order 1011 m –1 , and the crack tip stress is correspondingly high. The validity of a continuum linear elastic solution in this regime is questionable. In addition, our stress-corrosion law is somewhat speculative, and requires experimental verification. These are promising areas for future study. 7. Conclusions We have used a numerical technique to predict the evolution in shape of a crack like defect propagating by stress driven corrosion. Our computations predict three possible types of behavior for the crack: (i) gross blunting, where the crack evolves

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towards a rounded profile, (e.g. Fig. 3a); (ii) stable, notch like crack growth, where the crack initially sharpens, but approaches a steady self-similar profile and propagates with constant tip curvature and tip velocity (e.g. Fig. 3b); and (iii) unstable sharpening, where the crack tip curvature, stress, and velocity appear to increase without limit (e.g. Fig. 8b). In general, the behavior of the crack is determined by a dimensionless load factor Σ, two material parameters m and Φ , and a shape factor λ, defined in (9). The various regimes of behavior are plotted as a function of these parameters in Figs 7, 9 and 10. For sufficiently low values of applied load, the crack always blunts, irrespective of the value of Φ or m. For larger applied loads, the flaw will either sharpen to form a stable notch that propagates with constant velocity and tip curvature, or else will sharpen without limit to form an ideal crack. The former behavior occurs in materials with m exceeding a threshold value between m crit = 2.5 and 3.3; in materials with m < m crit , the crack sharpens in an unstable manner. Our simulations confirm the existence of a critical level of applied stress which must be exceeded to cause crack growth. Contrary to earlier predictions, however, our results suggest that the fatigue threshold is determined not only by the driving force for crack extension but also by the kinetics associated with the chemical reaction at the crack tip. The critical values of Σ or Γ required to cause sharpening are thus a function of m and Φ, as illustrated in Figs 7–10. Since the critical stress depends on Φ, which is in turn temperature dependent, our results imply that the fatigue threshold decreases as temperature is reduced. Finally, we have used our computations to calculate the crack tip velocity as a function of applied load, in the regime of steady state notch like crack growth. Our results are illustrated in Fig. 12, and are in excellent qualitative agreement with the standard empirical crack growth law v = B e x p (CKI ). Our calculations appear to underestimate values of C and overestimate B, however. This may partly be due to inaccuracies in values used for material data, but may also be due to the limitations of a linear elastic continuum analysis. 6. Acknowledgements This work was supported by the NIST/ATP membrane program and the MRSEC program of the National Science Foundation under award DMR-9632524 with Brown University. 7. References Charles, R.J. and Hillig, W.B., (1962), The Kinetics of Glass Failure by Stress Corrosion, in Symposium on Mechanical Strength of Glass and Ways of Improving it, Union Scientifique Continentale du Verre, Charleroi, Belgium, pp.511-27. Chuang, T.-J. and Fuller, E.R., (1992), Extended Charles-Hillig Theory for Stress Corrosion Cracking of Glass, J. Am. Ceram. Soc., 75 (3) pp.540-45.

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Freiman, S.W., White, G.S. and Fuller, E.R., Jr.(1985), Environmentally Enhanced Crack Growth in Soda-Lime Glass, J. Am. Ceram. Soc., 68 (3) pp.108-112. Gehrke, E., Ullner, C. and Hahnert, M., (1990). Effect of Corrosive Media on Crack Growth of Model Glasses and Commercial Silicate Glasses Int. J. Glass Sci. Tech. 63 (9) pp.255-65. Hillig, W.B. and Charles, R.J., (1965), Surfaces, Stress-Dependent Surface Reactions and Strength, in High Strength Materials, V.F. Zackaray. Wiley & Sons, New York, pp.682-705. Michalske, T.A., (1983), The Stress Corrosion Limit: Its Measurement and Implications, in Fracture Mechanics of Ceramics, Vol. 5, ed. R.C. Bradt, A.G. Evans, D.P.H. Hasselman and F.F. Lange, Plenum Press, New York, pp.277-289. Peraire, J., Vahdati, M., Morgan, K., and Zienkiewicz, O.C., (1987), Adaptive Remeshing for Compressible Flow Computations, J. Comp. Phys. 7 2 , 449-466. Simmons, C. J. and Freiman, S.W., (1981), Effect of Corrosion Processes on Subcritical Crack Growth in Glass, J. Am. Ceram. Soc.,64 (11) pp.683-86. White, G.S., Greenspan, D. C. and Freiman, S.W.,(1986), Corrosion and Crack Growth in 33% Na2 O–67% SiO2 and 33% Li2 O–67% SiO2 Glasses, J. Am. Ceram. Soc., 69 (1) pp.38-44. Wiederhorn, S.M., (1975), Crack Growth as an Interpretation of Static Fatigue, J. Non-Cryst. Solids, 19(1), pp. 169-81. Wiederhorn, S.M., Evans, A.G., Fuller, E.R. and Johnson, H., (1974), Application of Fracture Mechanics to Space-Shuttle Windows, J. Am. Ceram. Soc., 57, pp. 319-323. Wilkins, B.J.S. and Dutton, R., (1976), Static Fatigue Limit with Particular Reference to Glass, J. Am. Ceram. Soc., 5 9(3-4), pp.108-12. Zienkiewicz, O.C. and Zhu, J.Z., (1987), A Simple Error Estimator and Adaptive Procedure for Practical Engineering Analysis, Int. J. Numer. Meth. Engng, 24, 337-357. Zhou, S.J. and Curtin, W.A., (1995), Failure of Fiber Composites: A lattice Green Function Model, Acta Metall. Mater. 43 (8), pp.3093-3104.

ENERGY RELEASE RATE FOR A CRACK WITH EXTRINSIC SURFACE CHARGE IN A PIEZOELECTRIC COMPACT TENSION SPECIMEN

Anja Haug Materials Department University of California Santa Barbara, California 93106 USA AND Robert M. McMeeking Department of Mechanical and Environmental Engineering University of California Santa Barbara, California 93106, USA

Dedicated to James R. Rice on the occasion of his 60th birthday. Abstract: The fracture behavior of the piezoelectric material PZT-4 in a compact tension specimen is modelled. The influence of the electrical field and mechanical load on the energy release rate and the mode mixity ratio is considered. Free charge accumulation on the crack surface is enforced in the boundary conditions and a finite element analysis is employed. The results are discussed in comparison with the results from McMeeking [1] of a crack free of extrinsic charges. It is found that the free charge on the crack surface diminishes the influence of the electric field on the energy release rate. Consequently, it may be difficult to deduce the true values of the crack tip stress intensity factor and the crack tip field intensity factor in an experiment without knowing the charge condition on the crack surface.

1. Introduction The influence of electric field and mechanical loading on a piezoelectric compact tension specimen is investigated. The methology makes use of the J-integral and so this paper is very appropriate for a collection of articles dedicated to Jim Rice. In addition, Jim’s influence on the second author of this paper (R.M.M.) has been extensive from his days as a graduate student to the present time. The education, mentorship and guidance that Jim provided has been of enormous benefit and R.M.M. will always be deeply grateful for this. McMeeking [1] has already addressed the subject of the paper in the situation where the surface of the crack is free of extrinsic charge and he has predicted the effect 349 T.-J. Chuang and J.W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 349–359. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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of the electric field on the J-integral [2, 3] in that case. However, it is common for free charge in the atmosphere to be attracted to the intrinsic charge layers on the surfaces of polarized bodies. It is of interest to investigate the effect on the J-integral of this free charge accumulated on the crack surface. The approach of McMeeking [1] is followed to provide a numerical analysis of a piezoelectric compact tension specimen of a specific configuration and material as used in some experiments by Park and Sun [4]. This involves finding a stationary value for the functional Ψ given by (1) where W, the electrical enthalpy, is (2) sij is the strain tensor, E i is the electric field, Di is the electric displacement, A is the planar area of the specimen and ∆ is the displacement relative to the crack plane of one of the loading points in the direction of its prescribed applied force F (Figure 1). The interior of the crack is considered to be a sub-region of the domain A, so that the energy stored in the crack by the electric field contributes to the total electric enthalpy of the specimen. In this way, the effect of the crack opening on the capacitance and piezoelectricity of the specimen is accounted for. It follows that the surfaces of the crack are not components of the perimeter of the specimen but instead the crack surfaces are treated as interfaces interior to the domain of the calculation. Appropriate continuity conditions are enforced across these interior interfaces and this will be described below. The perimeter S surrounding the problem domain is therefore exterior to the rectangle BCGH as shown in Figure 1 and does not include the crack surfaces. The mechanical boundary condition applied to this exterior perimeter is that the traction T i is zero. The electrical boundary conditions consist of zero free charge on the sides CG and HB of the specimen and a specified value of the potential at electrodes BC and GH glued to the top and bottom. The final condition is that F is specified.

Figure 1. A compact tension specimen

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The constitutive laws connecting the electric displacement, stress, electric field and strain are (3) (4) where C ijkl is the tensor of elastic stiffness at fixed electric field, e ijk is a tensor of piezoelectric coefficents and εij is the dielectric permittivity tensor at fixed strain. The stationary values of Ψ are obtained by simultaneous variation of the displacements u i and of the potential φ. Stationary values of Ψ generated by variations of u i and φ unconstrained other than by the boundary conditions are associated with exact solution of the governing equations of piezoelectricity for the problem including the boundary conditions. These equations have been summarized by McMeeking [1]. However in this work as in Ref. [1], the finite element method is used and so an approximate solution is achieved. The geometry of the problem analyzed is shown in Figure 1. The crack has length a=6.9 mm, the ligament from the load points to the back face is given by b=a+c=20.6 mm. The height of the specimen is 2h=19.1 mm and it has thickness t=5.1 mm. The load points are positioned at a distance d on either side of the crack where d is 4.5 mm. The loading device is idealized as a pin of zero diameter exactly fitting into a vanishingly small hole in the specimen and subject to a load F. This is in contrast to the real component which has a pin of finite diameter [4]. The material is piezoelectric, poled in the positive x3 -direction and transversely isotropic about the poling axis. The specimen is assumed to lie in the x1-x 3 plane as shown in Figure 1. The needed relationships for plane strain of the specimen are: (5) (6) (7) (8) (9) where C i j , ei j and εi j are coefficients from equations (3) and (4) stated consistent with Voigt notation. The potential on the top surface HG is -V and on the bottom surface BC is V. The electric displacement D1 is zero on the left and right sides BH and CG, a condition justified by the high dielectric permittivity of the piezoelectric material compared with that of air. Only the top half of the specimen is analyzed (Figure 1). On the symmetry line ahead of the crack, OD, the shear traction and the vertical displacement u3 are both zero. The potential is zero along the entire line AD. The crack surface is also free of traction and the electric field is continuous across the crack. The treatment of the opening of the crack, the electrostatic energy in the crack and the total charge layer on the crack surface are described below.

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2. Finite Element Analysis The finite element equations were solved as a nonlinear system in which the geometric effect of the crack opening on capacitance was taken into account. As in the treatment by McMeeking [1], other geometric and material nonlinearities are ignored. In the finite element code used for the calculations, 4-node plane isoparametric elements with a 2 by 2 rule for integrating the stiffness matrix are utilized. The mesh is shown in Figure 2.

Figure 2. Finite element mesh for the analysis

The condition imposed on the upper crack surface represents continuity of the electric field and is given by (10) where δ c(x 1) is the crack opening displacement at position x1 on the crack and is equal to 2u 3 and E 3 (x 1 ) is the electric field in the x3 direction in the material adjacent to the crack surface. This result comes about because it has been assumed that the total surface charge density on the crack is zero, which requires that in the x3 direction the electric field in the crack is equal to the electric field in the material, i.e. there is no jump in the electric field. The condition implies that enough free charge in the atmosphere is attracted to the surface of the crack to neutralize any intrinsic surface charge induced by material polarization. Therefore, it is assumed that the atmosphere carrying the free charges can penetrate the crack and that accumulation of free charge onto the surfaces takes place sufficiently fast to quickly neutralize the polarization charges. Since σ 33 i s zero on the crack surface, equations (6) and (9) can be combined to give at the crack surface (11)

Given unit thickness, a nodal charge equivalent to D3 at node i on the crack surface can be computed by multiplying D3 by L i , which is an effective length of crack surface associated with node i. The effective length Li is taken to be half the distance between nodes i-1 and i+1, defined to be the two neighbors of node i on the crack

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surface with node i+1 to the right of node i. Equation (10) and (11) are then combined to provide (12) where Q i is the nodal charge for node i, direction,

is the displacement of node i+1 in the x 1

is given a similar interpretation for node i-1, φi is the potential of node i

and u 3i is the displacement of node i in the x3 direction. It should be noted that to achieve the expression in equation (12), the approximation (13) has been used to estimate the strain s11 at node i. An iterative approach is used to solve the finite element equations, including the condition represented by equation (12). At each iteration, the finite element equations are solved and the residuals at each finite element node are computed. The residuals for most nodes are simply the un-neutralized nodal free charge and unbalanced nodal forces. However, for nodes on the crack surface, the electric residual for node i is given by (14)

where Q i is the current nodal charge for the node i at this iteration and is computed directly from the finite element equations given that the displacements and potential at each node has been calculated for this iteration. The iteration is carried out by a Newton method to drive the residuals for each -5 node towards zero. Convergence to solutions which change by less than 10 o f t h e existing nodal potentials and displacements occurs after only 2 iterations. To initiate iteration the potential on the crack surface is taken to be zero.

3. J-integral The crack tip energy release rate is equal to the J-integral where J gives the reduction of potential energy of the specimen per unit area of crack advance [2, 3]. The form suitable for piezoelectric materials is (15) where n i is the normal vector to the contour Γ which completely encloses the crack tip [1]. As a result, J is the sum of the contributions in the crack where the contour Γ passes through it and the contribution from the contour in the piezoelectric material. The contribution to J arising from the segment of the contour within the crack is - W δ c , since

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E 1 and σ i j are both zero in the crack. This contribution is therefore the negative of the product of the electrical enthalpy

and the crack opening displacement δc

where the contour passes through the crack. Let JM be the contribution to J from the contour through the piezoelectric material. It follows that (16) where φ c is the potential on the surface at the point where the contour integral enters the crack. JM is computed by the domain integral method as described in [1]. This numerical evaluation of J is done after the converged finite element solution is obtained. The following intensity factors are also calculated [1] in the numerical evaluation, defined in the following way. The Mode I stress intensity factor K I is such that the asymptotic behavior of the tensile stress ahead of the crack on the crack plane is (17) where x 1 is the distance from the crack tip. The electric displacement mode intensity factor K D is such that the asymptotic behavior of the D 3 -component of the electric displacement ahead of the crack on the crack plane is (18) and the electric field mode intensity factor KE is such that the asymptotic behavior of the E 3 -component of the electric field ahead of the crack on the crack plane is (19)

4. Results The material properties are chosen to be consistent with PZT-4 [1,4]: Elastic Constants 10 (Pa): C 11 = 13.9 x 10 10 , C 12 = 7.78 x 10 , C 13 = 7.43 x 10 10, C 33 = 11.3 x 10 10 , C 44 = 2.56 x 1010 , Piezoelectric Coefficients (C/m2 ): e31 = -6.98, e 33 = 13.84, e 15 = 13.44, -9 Dielectric Permittivities (F/m): ε11 = 6.00x 10 , ε 33 = 5.47 x 10 -9 . The following results are only valid for the specified geometry (Figure 1) and material data except that values can be generalized if ratios of parameters are held fixed [1]. In Figure 3 the energy release rate is plotted against the applied electric field Ea which is computed as V/h. The electric field is made dimensionless by multiplication by and the energy release rate is normalized by G(F,0), its value at zero applied electric field. Results are plotted in Figure 3 for 7 values of the applied load F as

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indicated in the figure. The results show that an applied electric field reduces the energy release rate slightly if modest mechanical loads are applied whether the field is positive or negative. The crack tip energy release rate is independent of the direction of the electric field. Applying a mechanical load of 5 kN and an electric field in the range ±2.3 MV/m in this particular compact tension specimen causes G to differ from G(F,0) by less than 1.6%.

Figure 3. Crack tip energy release rate for a piezoelectric compact tension specimen in the presence of an applied electric field and a mechanical load divided by the crack tip energy release rate at the same mechanical load but without the applied electric field. The results are shown as a function of the applied electric field and each curve represents a different level of mechanical loading.

Figure 4A and Figure 4B also give the energy release rate, but now plotted against the applied electrical field divided by the applied mechanical load. (This parameter is also made dimensionless by multiplication by suitable quantities). The effect of electric field is almost undetectable for higher forces on the scale used in Figure 4A. However on the range and scale used in Figure 4B, the results indicate that at a given ratio of electric field to mechanical load, the electric field reduces the energy release rate by a bigger fraction when the mechanical load is high. It should be noted that the results indicate that generally an electric field of a few MV/m is required to cause a change to the energy release rate comparable with 10%; a field of 2.5MV/m and a load 1kN cause a difference of 7.5%. The value of the energy release rate in the absence of applied electric field is found to be (18) where t is the thickness of the specimen. This is identical to the results found by McMeeking [1].

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Figure 4. Crack tip energy release rate for a piezoelectric compact tension specimen in the presence of an applied electric field and a mechanical load divided by the crack tip energy release rate at the same mechanical load but without the applied electric field. The results are shown as a function of the applied electric field divided by the mechanical load and each curve represents a different level of mechanical loading. A and B show the results in different ranges and to different scales.

In Figure 5 the ratio K E /K I is plotted versus the ratio of applied electric field to applied mechanical load. Appropriate normalization is used. It can be seen (if one looks closely at the figure) that there is a non-zero value of K E e q u a l t o a b o u t even when there is no electric field applied to the specimen. This is equivalent to a value of K D equal to about 1.1

Thus neither form of the

electrical intensity factors are zero when the applied electric field is absent. However, the electric displacement intensity factor is almost exactly what is expected from the nonzero electric displacement induced on the uncracked ligament by the mechanical load in the absence of applied electric field due to the piezoelectric effect. The nonzero electric field intensity factors at zero applied electric field therefore arises from this consideration.

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Figure 5. Crack tip mode mixity for a piezoelectric compact tension specimen in the presence of an applied electric field and a mechanical load. Mode mixity is defined as the crack tip electric field intensity factor divided by the Mode I crack tip stress intensity factor. Each curve represents a different level of mechanical loading.

In Figure 5 all curves are indistinguishable which means that a fixed ratio of applied electric field to applied load always results in the same K E /K1 ratio. The results for K E and K1 can be summarized by (19)

(20) where the slight nonlinearity in the results has been ignored. The result in equation (19) is identical to that found by McMeeking [1].

5. Discussion In McMeeking’s investigation of the energy release rate for a compact tension specimen [1], he found that when the capacitance of the space in the interior of the crack is accounted for, the applied electric field has much less influence on the energy release rate than when the space in the crack is considered to be impermeable to the electric field. In this paper, we have now found that when the capacitance of the crack interior is accounted for and it is assumed that free charge in the atmosphere accumulates quickly on the surface of the crack, attracted there by the intrinsic charge layer induced by polarization, the effect of the applied electric field on the energy release rate is reduced

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even more. This insight can be confirmed by comparing Figures 3 and 4 with equivalent diagrams in Ref. [1]. However, the effect of the applied electric field is not completely negligible in this regard and we find that the energy release rate is still influenced noticeably by the applied electric field. To a good approximation, the applied electric field has no influence on the Mode I stress intensity factor and this result is independent of the behavior of free charge on the crack surface; e.g. KI is the same whether free charge accumulates on the free surface or not. Moreover, we find that with the conditions imposed in our analysis of the compact tension specimen, the electric field mode intensity factor is (to a good approximation) linearly dependent on both the applied load and the applied electric field. This is in contrast to the results given in Ref. [1] (where free charge does not accumulate on the crack surface) in which the electric field mode intensity factor was found to be linearly dependent on the applied electric field but also has strong nonlinear dependence on the applied load. This behavior observed in the absence of free charge accumulation on the crack was brought about by the crack opening induced by the applied load which changes the crack’s capacitance. The results of our new calculations with free charge accumulating on the crack surface indicate that the free charge diminishes the effect of the changing capacitance as the crack opens due to increasing applied load. However, the free charge does not make the crack invisible to the electric field and intensification of the electric field around the crack tip occurs, with the intensity factor proportional to the applied electric field. It will have been observed that there is a non-zero electric field mode intensity factor when the applied electric field is zero and the applied force is non-zero. This behavior occurs whether free charge accumulates on the crack surface or not and was also observed in the results of McMeeking [1]. This feature can be attributed to a piezoelectric effect that is observed to require a non-zero singular electric field and electric displacement at the crack tip even when there is no applied electric field. In this sense, the linear dependence of the electric field mode intensity factor on K I observed in equation (20) is parasitic on the presence of a stress singularity at the crack tip. It is of interest that a conjugate effect (i.e. a non-zero Mode I stress intensity factor at zero applied force when the applied electric field is non-zero) is absent or at least negligible in the numerical results we have obtained. We concede that our results do not shed any light on why positive electric fields transverse to the crack in poled PZT4 encourage crack growth and negative ones discourage crack growth [4-12]. In the results in our paper, the energy release rate is quadratic in the applied electric field. This implies that if fracture toughness is independent of the ratio of K E / KI then both negative and positive electric fields should discourage crack growth. However, as argued in Ref. [1], it is much more likely that the effective fracture toughness is dependent on the ratio KE /K I providing a possible explanation of the influence of the sign of the electric field on crack growth. The effective fracture toughness is the sum of intrinsic and extrinsic contributions, where the extrinsic contributions include the effect of intergranular residual stresses caused among other sources by domain reorientation during poling [5, 6] and shielding effects due to the development of depolarized and repolarized regions of material near a growing crack tip [1, 9, 10, 12]. Due to the possibility of nonlinear depolarization and polarization

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rotation, the extrinsic contribution to the fracture toughness will certainly be influenced by the direction of the electric field relative to the initial polarization of the piezoelectric material. Therefore, the dependence of fracture on the sign of the electric field can be attributed to these phenomena. The results presented in this paper may have a role to play in the resolution of this issue since it is known that free charge does accumulate on the polarized surfaces of ferroelectrics [13]. However, the contribution of these results will be in quantifying properly the stress and field intensity factors for a cracked specimen in a given experiment. If free charge accumulation occurs quickly on the crack surface in such an experiment, the field and stress intensity factors must be calculated along the lines given here so that the toughness can then be identified properly. It should be noted that the time in which the polarization charge is neutralized is not addressed in this work. However, we assume that the charge neutralization process is sufficiently fast that in an experiment it would have occurred prior to the testing of the specimen.

Acknowledgement This research was supported by Grant 9813022 from the National Science Foundation. References [1] R.M. McMeeking, Crack tip energy release rate for a piezoelectric compact tension specimen, Engineering Fracture Mechanics, 64 (1999) 217-244. [2] J.R. Rice, A path independent integral and the approximate analysis of strain concentration by notches and cracks, Journal of Applied Mechanics, 35 (1968) 379-386. [3] G.P. Cherepanov, Mechanics of Brittle Fracture, McGraw-Hill, New York, 1979, p. 98. [4] S.B. Park and C.T. Sun, Fracture criteria for piezoelectric ceramics, Journal of the American Ceramic Society, 78 (1995) 1475-1480. [5] R.C. Pohanka, R.W. Rice and B.E. Walker, Jr., Effect of internal stress on the strength of BaTiO3 , Journal of the American Ceramic Society, 59 (1976) 71-74. [6] R.W. Rice and R.C. Pohanka, Grain-size dependence of spontaneous cracking in ceramics, Journal of the American Ceramic Society, 62 (1979) 559-563. [7] K.D. McHenry and B.G. Koepke, Electric field effects on subcritical crack growth in PZT, Fracture Mechanics of Ceramics, 5 (1983) 337-352. [8] A.G. Tobin and Y.E. Pak, Effect of electric fields on fracture behavior of PZT ceramics, in Smart Materials (V.K. Vardan, ed.) Proc. SPIE 1916 (1993) 76-86. [9] G.A. Schneider, A. Rostek, B. Zickgraf and F. Aldinger, Proceedings of the 4th International Conference on Electronic Ceramics and Applications, (1994) 1211-1216. [10] H.C. Cao and A.G. Evans, Electric-field-induced fatigue crack growth in piezoelectrics, Journal of the American Ceramic Society, 77 (1994) 1783-1786. [11] C.S. Lynch, W. Yang, L. Collier, Z. Suo and R.M. McMeeking, Electric field induced cracking in ferroelectric ceramics, Ferroelectrics, 166 (1995) 11-30. [12] C.S. Lynch, Fracture of ferroelectric and relaxor electro-ceramics: influence of electric field, Acta Materialia, 46 (1998) 599-608. [13] B. Jaffe, W.R. Cook and H. Jaffe, Piezoelectric Ceramics, Academic Press, London and New York (1971).

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MICROMECHANICS OF FAILURE IN COMPOSITES An Analytical Study

Asher A. Rubinstein Department of Mechanical Engineering Tulane University New Orleans, LA 70118

Abstract: An analysis of fracture resistance mechanisms in several composite systems is presented. A description of the basics of the analytical method developed specifically for the analysis of fracture development in composites is presented as a unified approach to different composite systems. The method was applied to several composite systems, including composites formed from a brittle matrix reinforced by unidirectional fibers, composites consisting of a brittle matrix reinforced by ductile particles, and a metal matrix reinforced by ceramic fibers. The reinforcement mechanisms in these composites are based on the formation of a system of restrictive forces imposed on the crack surfaces by reinforcing components. The region where these restrictive forces are activated is represented as a line process zone. A classical fracture mechanics modeling technique was employed, using the process zone concept and small or large-scale analysis. The distinctive characteristic of the described method is an explicit consideration in the analysis of the discrete distribution of the reinforcing components within the composite. The developed methodology allows one to obtain analytical solutions to the representative elasticity problems and to investigate detailed micromechanical aspects of the process.

1. Introduction Most of the development of analytical methods for the analysis of fracture development in composites was done in application to ceramic matrix composites (CMC). The development of these composites has become a topic of significant research effort in industrial and academic laboratories. The purpose is to take advantage of the thermomechanical properties of ceramics and to compensate for their low fracture toughness. The basic ideas explaining the fracture process and effectiveness of fiber reinforcement were developed using mechanical models of the process. The development of the methodology for analysis of these composites and relating the composite microstructure to the fracture resistance was done by Aveston, Cooper and Kelly (1971), Rose (1987), Budiansky et al.(1988), Budiansky and Amazigo (1989), 361 T.-J. Chuang and J. W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 361–384.  2000 Kluwer Academic Publishers. Printed in the Netherlands.

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Marshal and Cox (1987), Pagano and Dharani (1990), Rubinstein and Xu (1992), Rubinstein (1993,1994), Budiansky and Cui (1994), and Meda and Steif (1994a, 1994b) for fiber reinforced ceramics; Budiansky et al.(1988), Sigl et al.(1988), Erdogan and Joseph (1988), Bao and Hui (1990), and Rubinstein and Wang (1996, 1998a) for particulate-reinforced composites, and Rubinstein and Wang (1998b) and Wang (1998) for metal matrix composites 1 . Although the developed models sometimes reflect different aspects of the modeled material and employ different analytical and numerical techniques, they all have a common feature. The basic physics of the fracture resistance mechanism is the formation by the reinforcing components of a crack opening constraint in the form of a bridging zone. The common methodology of composite failure modeling is based on relating the macro loading parameters to the governing microscale factors, incorporating into the analysis the effective fiber constraint on crack surface separation. The analytical approach may differ in consideration of fiber action, using smeared fiber action formulation or considering a discrete fiber or particle distribution within the bridging zone. The crucial item for determining the strength and fracture resistance of the composite is the value of local stress intensity factors acting on internal and surface micro- and macrocracks. The models have to include the basic material information. The critical information for CMC fracture process modeling is the effective force imposed by the reinforcing components on the surfaces of the developing cracks. The fibers or reinforcing particles act as bridges connecting the crack surfaces and, thus, restricting the crack opening. The relationship between the force induced by the reinforcing components and the magnitude of corresponding crack opening displacement plays a key role in developing the fracture resistance mechanism in the composite. Naturally, this information should be obtained from experimental observations, and it should correspond to a specific composite system. A number of factors influence this relationship, and a number of procedures for obtaining the data could serve as the basis for determination of the fiber pullout - force relationship, F(u), for a specific composite system, or ductile particle deformation pattern within the bridging zone. Several of these relationships and their effect on fracture resistance development were investigated. The analytical approach described in this paper is based on a discrete distribution of the reinforcing components within the bridging zone; this is a distinctive quality of the method. In this approach, a model is developed based on an exact analytical solution of the corresponding problem that represents the processes taking place during the crack growth. Using this approach, all fracture mechanics parameters can be monitored at any intermediate step as the crack tip progresses between the reinforcing particles or fibers. The main advantage of this methodology, as compared to the methods based on smearing of the fiber or particle action within the bridging zone, is obtaining actual values of all fracture mechanics parameters rather than their average values. The numerical procedure required for parametric study of the model also appears to be simpler and more direct. In both cases of discrete fiber or particle distribution and smeared reinforcing action within the bridging zone, the process zone is treated as a line zone ahead of the crack, and the elastic properties of the composite outside of the 1

The reference list on the subject is not complete; only principal references are cited.

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process zone are assumed to be homogeneous, although not necessarily isotropic. In the following sections, the basics of the analytical formulation as applied to cases of reinforcement by unidirectional fibers, ductile particle reinforcement of ceramic matrix, and reinforcement in metal matrix composites will be outlined, and several examples of the results will be described. The emphasis here is on the similarities in mathematical formulation as applied to different physical systems.

2. Basics of the Modeling Scheme The process of fracture resistance development requires a three dimensional description. Considering the periodic distribution, the analysis is formulated for a two dimensional plane which is placed through the centers of the particles, or fibers, perpendicular to the crack plane and aligned with the loading direction. The developing crack front takes the wavy form, as observed by Botsis and Shafiq (1992). The maximal values of the stress intensity factors along the crack front will be on the trailing portions of the crack front. The analytical formulation described here corresponds to the family of planes intersecting locations of maximal stress intensity along the crack front. The stress intensity factors controlling the crack growth and the most significant effects associated with the reinforcement are taking place in this plane. Therefore, focusing attention on this plane is justifiable. The stress state, however, is not exactly in the category of plane stress or plane strain types. Because the problem is periodic, we assume the stress state to be close to the plane strain case, and therefore consider it as such. This assumption is commonly used in the modeling of the fracture process in composites. The analytical formulation of the model is based on the classical description of the stress state in an elastic body in terms of analytic potentials (Muskhelishvili (1963)). The conventional definition of the analytic stress potentials, φ(z) and ψ (z ), is given by relationships (1); using standard notations, µ is a shear modulus, κ = 3-4v for plane strain or κ = (3-4v)/(1+ v) for generalized plane stress, and v is Poisson’s ratio. The complex displacement is given in (1) in the form of a gain between two points A and B on the complex plane, and the complex force in (1) is given as a resultant of traction on an arc connecting these points.

(1)

Position the crack and the bridging zone along the line y=0 on the complex plane and consider Mode I type loading. Under these conditions and Mode I loading the symmetry of the problem allows one to relate the unknown stress potentials as

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for a small scale analysis. The small scale approach limits the size of the bridging zone as compared to the size of the crack and any other geometrical parameter of the problem, thus assuming the crack to be infinitely large. In case of finite crack size and a comparable bridging zone, relationship (2) changes as given in equation (3) (Rubinstein (1994)) with σ ∞ being the applied stress. (3) Thus, the problem is reduced to one unknown function. Consideration of Mode I - type loading does not limit the generality of the analysis. The corresponding results for Mode II and Mode III may be obtained from the results derived for Mode I. Both of these cases also are reduced to one unknown function with the same boundary conditions as in Mode I. For example, in case of Mode II, the relationship between the complex stress potentials becomes (Rubinstein (1985)), (4) Generally, the stress potential φ'(z) for Mode II type loading can be obtained using the solution for Mode I by substituting -iK II( ∞ ) for K I( ∞ ) in potential φ'(z), where i is the imaginary unit. Solutions for different fracture modes are similar up to the conditions on the fibers or other reinforcing components under consideration. These conditions usually change under different modes of fracture. For example, if we consider fiber pullout versus force on the fiber relationship, it is unreasonable to expect this relationship to remain unchanged under these two different conditions. The system of forces acting on the fiber and conditions on the fiber-matrix interface change. Therefore, in case of mixed mode loading conditions on the reinforcing component, the resulting effect depends on the specific relationship between the loading modes. Usually, Mode I is considered as a dominant fracture mode in reinforced composites. The components manufactured from these composites are usually designed to have the reinforcement aligned in the direction of the maximal load. The approach to the analysis of brittle matrix reinforcement presented here is based on a micromechanical consideration. The action of each reinforcing component is considered as part of the system of activated reinforcing elements within the bridging zone, and yet they are considered as discretely spaced elements. The distribution of these elements within the bridging zone may take an arbitrary form, and the analysis presented here may accommodate it. However, in most cases this distribution is considered to be periodical; these cases will be presented below. This is also typical for other models which are based on substitution of the actions of the discretely distributed reinforcing elements by the set of continuously distributed forces within the bridging zone. Consideration of the periodically distributed reinforcing elements, using the methodology presented here, is strictly a matter of convenience; this methodology can

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be applied to any systematic or random distribution. To describe the boundary conditions, consider here small scale models only. The large scale cases are developed in a similar manner by Rubinstein (1994). For Mode I small scale conditions, the complex potential behavior at infinity has to be specified as

(5) Additional conditions for the analytic function φ'(z) have to be stated along the x-axis. Position the crack and the bridging zone along the x axis, y=0, with the bridging zone from x=0 to x=c, and the crack on - ∞ < x < 0. The remote Mode I loading is assumed to be applied along the y axis. The reinforcing components with thickness a are distributed with period p, and the first component within the bridging zone is on the interval 0 < x < a. The thickness of the component could be the fiber diameter or the particle diameter. Formulation of specific boundary conditions taking place on these intervals for different types of composites are outlined in the following subsections. Using equations (1) and (2), the shear stress on y=0 is vanished and the normal stress and the displacement are (6)

2.1 UNIDIRECTIONAL FIBER REINFORCEMENT The boundary conditions along y=0 state the stress free crack surface, x < 0, and the crack ligaments between the fibers. On the ligaments representing the fibers we state the condition of a constant displacement. This condition was determined from consideration of the stress state around a cylindrical fiber under the force pulling it out from the matrix. Under this stress state, the matrix deformation around the fiber will assume a cylindrical symmetry, and, therefore, the displacement along the rim of the fiber-matrix interface on the surface will be constant. Importantly, the stress state in the matrix in the vicinity of the fiber in the plane of consideration corresponds to a nearly undeformed fiber-matrix interface. These facts led us to the statement of the constant displacement on the ligaments representing fibers. The constant displacement statement also enforces the fact that each fiber is pulled out from the matrix on its specific constant amount along the crack surface intersecting with the fiber-matrix interface. Any solution to a boundary value problem with more complicated boundary conditions still will have a solution given below as a homogeneous part of any more general solution. Thus, using equations (6), the boundary conditions for a bridging zone with N fibers are

f

(7)

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Figure 1. Bridging zone formed by unidirectional fibers.

The corresponding geometry is illustrated in Figure 1. In addition to these conditions a physical condition relating the fiber pullout displacement, u, and the force on the fiber, F, has to be introduced into consideration. The relationship F(u) has to be obtained from an experiment. Marshal and Price (1991), and Carter, et al (1991), reported a practically linear relationship between the fiber pullout displacement and the force on the fiber, at least during the initial stage. On the other hand Mumm and Faber (1995), reported a parabolic relationship. Analytical models used both relationships. Of course, these data correspond to different composite systems. However, comparing results of the analysis for equivalent situations, the difference in the pattern of F( u) does not seem to be as important as the corresponding numerical values of this function. Results shown here correspond to a linear function, F( u), with a parameter λ representing interface properties. (8) Subscript k (k = 1, 2,…, N ) indicates the particular fiber to which relationship (8) is applied. This equation completes the set of conditions for the problem. The stress potential was found in the form

(9)

The N constants d k appearing in equation (9) were found numerically by enforcing condition (8) and solving a system of linear equations. Function (9) was formed on the basis of the result of the Keldysh - Sedove problem (Muskhelishvili (1963)). Computation of the force and displacement components for equation (8) was conducted

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using the Gauss - Chebyshev quadrature. Solution (9) is a solution to a homogeneous problem; it is a dominating part of any other solution with different boundary conditions on the ligaments representing the fibers. 2.2 PARTICULATE REINFORCEMENT OF A BRITTLE MATRIX. The toughening mechanism in these composites is based on an attraction of the crack tip to particles of a lesser stiffness. Some aspects of the crack attraction to defects were investigated by Rubinstein (1986). The ductile particles are forming plastic bridges as the crack front passes through. The extensibility of these bridges depends on the particle ductility and the strength of the particle-matrix interface. The interface strength controls the development of the particles’ shape during the plastic deformation and thus influences the tri-axial stress state within the particles. A detailed numerical study of developing particle shapes was reported by Tvergard (1992, 1995), the experimental observations were reported by Venkateswara Rao et al. (1992), and plastic deformation development in constrained long wires was reported by Ashby et al. (1989). The theoretical analysis presented here departs from the traditional methodology developed for these materials by Budiansky et al. (1988), Erdogan and Joseph (1989), and Bao and Hui (1990). The presented analysis, Rubinstein and Wang (1996,1998), departs from the traditional approach of smearing the action of the particles over the bridging zone using continually distributed forces. The particles here are considered to be distributed periodically, with period p, the active cross section of the particle k within the line bridging zone is between points ak and b k , when k is counted from the beginning of the bridging zone at x=0; this geometry is illustrated in Figure 2. The normal stress along the crack line in the plastically deforming particle k is σk , when the deformation within the particle is fully plastic, and FY when the plastic zone initiates. The particles are assumed to be ideally plastic with yield stress σY . The particles from a strain hardening material may be included in this analysis without additional effort by adjusting values σ k according to the extent of deformation of each particle. In the presented work, the intent of considering the different values of the plastic stress on each particle was to include the tri-axiality effect. Thus, the boundary conditions for the stress potential, φ'( z ), for the corresponding plane problem are

(10)

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Figure 2. Bridging zone formed by ductile particles.

The shape evolution of initially spherical particles during the intense plastic deformation is approximated as a neck of parabolic profile, (11),

(11)

which connects the intact portions of the particle-matrix interfaces, yk here is the vertical coordinate of the intersection of the neck with the intact portion of the spherical particlematrix interface, and x k is the corresponding horizontal coordinate. Parameter A is introduced as a characteristic of the interface strength. A weak interface corresponds to A=0, and a strong interface corresponds to high values of A. Several examples of the particle profiles with different interface strength are given in Figure 3. The radius of the narrow cross section, in this case is given by equation (12).

(12)

So a k = pk-r k , and b k =pk+r k . Additionally the condition of a constant particle volume during the plastic deformation is enforced. Using average crack opening displacement, u k , on k-th particle, this condition on particle k is given by equation (13)

(13)

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Figure 3. Particle deformation shapes for different interface strength parameter A. A=0 corresponds to weak interface, A=1, .., 10 - intermediate, and A=50 - strong interface.

The solution of the boundary value problem (10) is given by equation (14).

(14)

Integrating function (14), one finds the corresponding potential (15) and displacement using equation (6).

(15)

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The values of rk , a k , b k and u k were found numerically by solving the nonlinear system of equations (11), (12) and (13) simultaneously, thus basically solving a system of nonlinear algebraic equations, rather than dealing with nonlinear integral equations as traditional methods would require. A special case, when A =0, corresponds to a cylindrical shape of deforming particle; the computations are significantly simplified in this case because equations (12,13) can be solved analytically. Generally, the numerical procedure involves evaluation of the current crack tip position and adapts according to its position. Thus, depending on whether the crack tip is located between the particles, or within the particle, or depending on the extent of plastic zone development, a different algorithm is applied, Rubinstein and Wang (1998). Those algorithms are based on specific conditions for the crack growth appropriate for the considered interval. If the crack tip is located between the particles, the constant stress intensity factor equal to the critical value for the matrix, K mIC has to be maintained; if the crack tip is located within the particle, different conditions, depending on the particle size, must be applied. To relate the particle size and other mechanical parameters of the composite system a parameter k was introduced (16).

(16)

In fact this parameter relates the particle size to the Dugdale plastic zone size in the material with the same yield stress under an acting stress intensity factor equal to the matrix toughness, K mIC . When k=1, the diameter of the undeformed particle is equal to this Dugdale zone size, and in this case, as well as in case k<1 when the crack tip reaches the particle, the plastic yield within the particle will develop throughout the particle without necessarily increasing the remote loading. In other cases the plastic zone will initially develop, but to continue throughout the particle, the remote load has to be increased. In all cases it was found that the point of highest resistance to future crack advancement is the instant when the crack tip advances from the particle into the matrix, Rubinstein and Wang (1998). 2.3 METAL MATRIX REINFORCED BY CERAMIC FIBERS Reinforcement of a metal matrix by means of aligned fibers or ceramic layers can be very effective in improving the fracture resistance of base metals, Bloer, et al. (1996). The model of the fracture resistance mechanism development in these composites is developed here. The aim is to determine the relationships between the composite parameters and fracture resistance effectiveness. This method outlined above was advanced to allow nonlinear behavior of the matrix within the bridging zone, making it applicable to a broad class of composite materials. As in previous cases, a discrete distribution of the reinforcing components is considered, and the model is based on an exact analytical solution of the corresponding boundary value problem. Within the line process zone, the following effects are taking place. The intact fibers

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(or elastic layers) bridge the failed, partially failed, or plastically deformed matrix. We consider the area between the fibers where the matrix may experience plastic deformation. The intensity of plastic deformation may be significant to initiate void formation on that interval. The interval is classified as partially failed if the void size is less than the space between the fibers. If the void reaches the size of the space between the fibers, we consider this region as failed. The plastic flow of the matrix is constrained by the elastic fibers. Therefore, the spread of the plastic deformation from the crack plane cannot be anticipated to be significant. This justifies our assumption of a line-like plastic zone, in a way similar to the Dugdale model for a single crack (Dugdale (1960)). As a criterion of the matrix failure, or as a void initiation criterion, we use a critical value for the crack opening displacement, δ c. 2.3.1 Physical Aspects and Assumptions of the Model. The physics of the process zone formation and development requires a special consideration. A few aspects of the reinforcement mechanism are not typical for other fiber-reinforced composites. One of these aspects deals with special properties of the relationship between fiber pullout displacement and the force on the fiber, and another aspect deals with the process of void formations between the fibers. In both cases the simplest physically admissible conditions are applied. The fiber pullout relationship is assumed to be linear, in the following form: λ

(Fi

– F 0i

)= B i

i = 0,1,…, N

(17)

Here λ is again the pullout coefficient, Fi and B i are force and displacement respectively on the (i +1)th fiber, and F0i is the threshold force on this fiber. One must introduce the threshold force, F0i , on the fiber, to assure consistency of the physical processes in this composite system. The threshold force will simply prevent matrix separation before plastic deformation in the surrounding fiber area takes place. We introduce F0i as a force on the (i+1)th fiber produced by the surrounding stress field, which initiates the plastic flow around that fiber. In practical terms, we use the value of the force corresponding to the stress state when the plastic zone tip just passes the fiber. This value of the threshold force is physically suitable, although it is not the only possible definition. There is no specific experimental information on what value should be used for the threshold force. Another aspect of the model requiring special treatment is the void formation process. The initiation of the void formation is assumed to be the state when the crack opening displacement within the plastically deformed area reaches the critical value. The expected location is on the interval between the fibers, somewhere near the center of the interval. The yield stress acting within this region acts as surface traction on the crack surfaces. As soon as the void is initiated, the traction is removed from the void’s newly formed free surface. The removal of the surface traction from the crack surfaces increases the crack opening displacement and, in turn, stimulates void growth. The void size is determined by the critical value of the crack opening displacement, which is maintained at the ends of the void. This is a typical instability of the physical process,

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which must be reflected in the model and in the numerical implementation of the process. Thus, there can be no continuous quasi-static void growth from infinitesimal size to any intermediate size. The only possible static states are no void or a void of a finite equilibrium size. The model reflects this aspect of the physical process, and the numerical scheme is specifically designed to accommodate it. The numerical scheme is described in detail by Wang (1998). The same criterion using the critical crack opening displacement at the tip is applied for determination of the main crack position. The resulting nonlinear model represents all basic physical aspects of nonlinear behavior of the material within the process zone, including the instabilities associated with void formation and the development of traction-free crack surfaces. The model demonstrates the effectiveness of the plastic yield on the fracture properties of composites.

Figure 4. The process zone geometry and corresponding boundary conditions.

2.3.2 Analytical Development The analytical formulation of the model is conducted in terms of a two-dimensional problem as in previous sections. The region surrounding the process zone is treated as elastic material, and the analysis is conducted using the method outlined above. The plane of consideration is the plane perpendicular to the crack plane and crack front, and passing through the centers of periodically spaced fibers. In the case of a layered composite, it is a plane perpendicular to the layers. The symmetry conditions are used again, reducing the problem to one unknown analytic potential φ'(z). The boundary conditions for this potential are set along the crack line, z = x + i0, as follows: (a) on the crack and void surfaces Re φ'(x) = 0, traction free surface condition; (b) on the segments where the plastic yielding is active Re φ'(x) = σ 0 /2, where σ 0 i s the yield stress; (c) on the fibers and in front of the process zone Imφ'(x) = 0, stating constant or zero displacement;

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(d) as z → ∞ : φ'(z) → K ∞ /2(2 π z)1/2 , the remote loading condition. The above boundary conditions and geometry of the process zone are illustrated in Figure 4. For convenience of development of the numerical procedure, the case of periodically spaced fibers, or layers, is considered with period p, and uniform fiber thickness a. This is not an essential assumption; the solution can be easily extended to a more general case. The function φ'(z) represents a combination of both cases considered in previous sections. The function was constructed using the Keldysh-Sedov theorem, Muskhelishvili (1972). According to this theorem, the boundary conditions outlined above determine the function uniquely up to the type of singular behavior at the ends of the intervals determining the real and imaginary parts of the function. Setting this behavior in a form suitable for the physical conditions of the problem, we construct the complex potential as

(18)

The function R(z) in (2) is defined as

(19) The process zone starts at the crack tip position c t ; it includes N fibers and the plastic zone in front of the N th fiber ending at point d t . The integrals in (18) are taken over all intervals representing the plastically deformed area and the segments representing voids. It does not include segments representing fibers. The switching function Tj (t) in (18) is introduced for computational convenience as Tj (t) = 1 on segments with active plastic yield, T j (t) = 0 on segments representing voids formed within the plastic region. The limits of the integral are specified in general terms here to include all the appropriate intervals. The process zone with the boundary conditions is illustrated in Figure 4. The function φ'(z), as given by equation (18), contains N+2 constants Cm , and unknown limits of the void boundaries on each interval. The first part of (18) represents the homogeneous solution of the corresponding boundary value problem. To find the constants and the voids’ boundaries, one uses the set of equations (17) which apply to each segment representing the fibers. The net force on the fiber is computed as

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(20)

The crack opening displacement is calculated using the following integral:

(21)

B i in set of equations (17) is the total crack opening displacement calculated for a fiber i which corresponds to x = a + pi in equation (21). An additional set of conditions is set to determine the position of the crack tip, c t , and the sizes and positions of the voids on each interval between the fibers, g i and h i , where i corresponds to a specific interval within the process zone. These conditions are for all i: (22) The displacements in (22) have to be computed using equation (21). If the crack opening displacement within the space between the fibers on a particular interval does not reach the critical value, no void will form. The final condition to be satisfied is the condition of bounded stress at the initiation of the plastic zone in front of the process zone. This condition is: (23)

Equations (17), (22) and (23) with formulae (18-21) form a system of nonlinear equations which were solved simultaneously to obtain constants Cm, positions ct , d t , and g i with h i. Details of the development of the numerical scheme are described by Wang (1998).

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3. Fracture Resistance of Composites. The numerical procedure developed on the basis of the described methodology can provide a broad range of information about the processes taking place within the bridging zone of the composites and a complete stress field at any point outside the bridging zone. Development of fracture resistance in composites and its dependence on internal microstructure were prime targets of the analysis. A few numerical examples of the developed models are given below. The emphasis of the selected examples is on the effects associated with the discrete distribution of the reinforcing components.

Figure 5. Fracture resistance curves for fiber reinforced ceramics. The length of the bridging zone is given in terms of the number of participating periodic cells.

3.1 CERAMIC MATRIX COMPOSITES. In Figure 5, the development of fracture resistance curves in ceramics reinforced by unidirectional fibers is presented. The data are given as a function of a number of periods with active fibers participating in the bridging zone and as a variation of the dimensionless fiber pullout parameter λ . Additionally, the cases of different fiber spacing are shown in Figure 5. This dependence on the fiber spacing aspect ratio is an important composite fabrication parameter which controls a variety of properties. The fracture resistance is measured as an ability of the composite to prevent matrix cracking. In these terms, a composite with the fibers positioned extremely close to each other will have no or very little fracture resistance. This result is demonstrated in Figure 6, where

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fracture resistance is given as a function of the fiber spacing aspect ratio for a given number of fibers participating in the bridging zone. A similar pattern was observed in the analysis of the composite strength, Rubinstein ((1994). In the cases discussed here of fiber reinforced brittle matrix, a linear relationship (8) was used describing the relationship of fiber pullout displacement to the force on the fiber. However, this is not a limitation of the described method. Several nonlinear cases were investigated for the simplified case of one bridging fiber, Rubinstein (1993). The advantage of considering that case was availability of a complete closed form analytical solution. The results demonstrated that this nonlinear pattern of this relationship plays an important role only when the crack tip is located very close to the fiber. Thus, it would apply to cases of very high volume fraction of fibers. Otherwise the linear pattern gives very similar results.

Figure 6. Resistance curves vs. fiber spacing aspect ratio a/p; the length of the bridging zone is constant.

3.2 PARTICULATE-REINFORCED CERAMICS The results illustrating the behavior of the particulate-reinforced ceramic matrix are based on the analytical scheme which considers discrete particle distribution. However, to relate the obtained results with other methods based on smeared continuous force distribution within the bridging zone, the average stress versus average displacement are given in Figure 7. The data in Figure 7 were obtained using a discrete particle distribution. The fracture resistance of particulate-reinforced composites depends on a number of parameters: particle spacing, particle size and ductility in relationship to matrix toughness, and the strength of the particle-matrix interface, to name a few. The developed analytical and computational schemes allow one to investigate every aspect

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of the processes taking place during the crack growth and formation of the bridging zone. The particle size and yield stress form a parameter (16), k, which if related to the matrix toughness determine whether the plastic zone will develop all the way through the particle immediately after the crack tip has approached it, or the particle may partially yield and require a significant additional load in order for the crack front to pass through. The particle spacing is important for fracture resistance development. The closer the particles are positioned, the more energy is absorbed due to plastic deformation.

Figure 7. Average bridging stress as a function of crack opening displacement and the interface strength parameter

Figure 8. Fracture resistance curves for particulate reinforced ceramics.

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The particle-matrix interface plays a special role in this process. The quality of the interface determines the pattern of the plastically deforming particles. The maximal length of the deformed particles controls the potential length of the bridging zone, and the changes in diameter of the particles determine the load carrying capacity of the particles. Thus, it is not surprising that the composites with a weak interface exhibit higher fracture resistance, Rubinstein and Wang (1998). These results are consistent with experimental observations reported by Venteswara Rao et al (1992). However, the very weak interface will allow the crack to pass through by debonding the particlematrix interface before the particles exhibit substantial plastic deformation. These cases were not included in the presented analysis. The limit of the effectiveness of the interface has to be evaluated. The fracture resistance data corresponding to a weak interface are presented in Figure 8. The data in Figure 8 include dependence of the fracture resistance on particles’ spacing aspect ratio d/p; these data are converted to particles’ volume fraction as well.

Figure 9. Fracture resistance curves.

3.3 METAL MATRIX COMPOSITES. To demonstrate the effectiveness of metal matrix reinforcement, the fracture resistance curves were computed. The fracture resistance curves shown in Figure 9 are normalized by the stress intensity factor K0 . This is a stress intensity factor which would generate a Dugdale zone of a maximal size for a specified critical value of the crack tip opening displacement, δc , in the matrix without reinforcement. The critical value of the crack opening displacement, δc, and dimensionless fiber pullout parameter, Λ, are important

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parameters controlling the degree of fracture resistance development. Λ is defined in equation (24). (24) Because the numerical values of these parameters are not available for the material systems described in the literature by Bloyer, et al. (1996), we may compare our results with the experimental data qualitatively only. The data depicted in Figure 9 represent three cases, corresponding to two sets of value of Λ, and two sets of critical values of crack opening displacements. The R-curves in Figure 9 are given as functions of the position of the leading end of the process zone normalized by the period of the composite system structure. This form of presentation is important because it provides information on the extent of process zone development or damage surrounding the crack tip. However, the experimental measurements would most likely be conducted as a function of the crack tip position. Although qualitatively the results will appear to be similar, quantitatively they may differ. The developed model can supply information for both cases. In Figure 9, the line breaks correspond to the instances when the front end of the line plastic zone passes reinforcing fibers. Only relatively short bridging zones were investigated. Reinforcement of the metal matrix appears to be very effective in improving fracture resistance and in constraining the plastic flow typically surrounding the crack tip in ductile material. The values of parameter Λ were chosen without a relationship to a particular composite system. As discussed in previous sections, the fiber pullout displacement has to be compatible with the displacement due to the plastic flow and constraint by the surrounding elastic material. Contrary to fiber reinforced ceramics, not all values of Λ are admissible to a particular composite system. The limitations of these values and the relationship of these limitations to other composite parameters are not completely understood at this time. The modeling technique employed in this investigation is new and makes available for evaluation certain features of the physical process that are not available if other methods of analysis are employed. For example, if the method of smearing of the active forces within the bridging zone is applied, and on that basis the method of integral equations is employed, the basic relationship of the average bridging stress within the process zone versus crack opening displacement must be specified. These relationships may be derived from a simplified cell model. The method developed for this study allows us to obtain the actual relationship within the investigated process zone, simply by following one period cell after it has entered the process zone. The resulting relationship is depicted in Figure 10. This relationship, as was expected, is highly nonlinear, and contains softening regions due to void formation within the process zone. The softening regions are shown as straight dashed lines; however, this is not the actual pattern of the process. As discussed earlier, at the moment a void is initiated, the system loses stability. The next stable point is the instant when the void reaches equilibrium size. The space between

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these two states was connected on the graphs with straight lines to maintain continuity of each considered case. Accommodation of these unstable intervals during the process zone development represented a challenge for development of the numerical procedure for this analysis.

Figure 10. The average stress versus average COD relationship within the process zone.

In Figure 11, a typical crack opening displacement pattern within the bridging zone and its vicinity is shown. The fibers are indicated on Figure 11; the black regions represent the plastically deforming regions. The lighter areas on the fibers indicate the portion of the fiber pulled out from the matrix according to equation (17). The voids depicted in Figure 11 were developed according to the outlined process. The computational procedure required evaluation of the crack opening displacement at every intermediate step. Observation of the development of the process zone was very helpful in understanding the physical process, and as a controlling factor for the numerical procedure. As was observed, the voids due to the described unstable growth process have a tendency to spread over the space between the fibers rather quickly, leaving only narrow regions adjacent to the fibers where plastic yield takes place. The example presented in Figure 11 corresponds to the case of a/p = 0.1, Λ = 0.2, and δ c /p = 0.2.

4. Concluding Remarks A unified approach to the failure analysis of different composite systems was presented. The methodology outlined here offers a powerful tool for micromechanical analysis of composite materials. Using this approach, one may obtain very detailed information

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about the processes taking place within the process zone of the composite. Variation of any micromechanical parameter may be accurately evaluated at any intermediate step of the process zone development. One of the specific advantages of this methodology is explicit participation in the analysis of the spacing of the reinforcing components. The interaction effects of these components participated in the stress analysis in their vicinity. These details may be used for determination of potential debonding, in the case of fiber composites, or crack path deflection along the fiber-matrix interface. In the case of particulate reinforcement, the quality of the particle-matrix interface plays a principal role in the developed analysis, which allows one directly to examine the associated effects. All details of the ongoing processes were accounted for within the process zone of the metal matrix composites. The considered processes included elastic deformation of the fibers, the relationship between fiber pullout and force on the fiber, plastic deformation of the matrix, formation of voids and associated instabilities. All these processes are taking place on each period of the fiber spacing within the process zone. The method does not have a limit for the number of periodically spaced fibers considered.

Figure 11. Crack opening displacement within the process zone and its vicinity resulting from numerical simulation. Black regions indicate plastically deformed regions.

In all considered cases, exact solutions to the corresponding boundary value problems were formed, and the constant parameters involved in the solutions were obtained using computational schemes. The computations were organized in an errorcontrolled environment wherein the maximal acceptable cumulative error was specified in advance. The numerical procedures formulated for the described approach are simple and

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based on the solution of a system of algebraic equations with the number of unknowns equal to the number of reinforcing components involved in the developed process zone. This is contrary to traditional schemes based on solution of integral equations where the numerical discretization of the interval usually requires a substantial number of nodes in order to obtain reliable accuracy of the results. The presented analysis was developed using methods of two dimensional elasticity. The results obtained here are projected to a three dimensional problem characterizing composite behavior. This is a legitimate projection for the following reasons. The analysis of the bridging mechanism consists of two parts. These are the stress analysis of the region surrounding the bridging zone, and the analysis of the constraint imposed by the composite reinforcing components. The two parts must be matching, consistent, and compatible. The stress analysis developed here represents an exact solution of the corresponding problem applicable to a specific plane passing through the centers of the fibers or inclusions, and which is perpendicular to the crack front. This plane is a plane of symmetry for the considered periodic problem. The average strain over the period in the direction perpendicular to the considered plane is zero. Therefore, the plane strain equations can be applied to obtain an average result. The crack front, on the other hand, is not straight. It develops a wavy pattern as the crack front progresses from one reinforcing array to the next. The stress intensity factor acting within the considered plane plays a decisive role in determining the future crack progress. The plane problem implies a straight crack front and, thus, the results of the analysis may be used as a conservative prediction of fracture resistance development. The stress functions (9), (14) and (18) represent the stress state on the described plane, and they, of course, could be directly applied to the fracture analysis of layered composites. On the other hand, the constraint on the crack opening displacement imposed by the reinforcing components was constructed specifically for the considered composite materials. The displacement restriction on the fibers for fiber reinforced composites, second equation in system (7), the particle shape development for particulate reinforced composites, equations (12) and (13), and equation (17) for metal matrix composites, are based on local cylindrical symmetry. In the case of particulate composites, maintaining constant volume of the plastically deformed material in each spherical particle restricts the crack opening displacement. Thus, the crack opening displacement restrictions brought in through the conditions on the fibers or the plastic particles are the key components relating the stress functions to the specific cases considered here. To apply the developed analysis to layered composites, one would have to restructure the displacement conditions imposed by the reinforcing components. The presented analytical method of composite analysis relates essential composite design parameters with composite performance in terms of fracture resistance, thus relating essential micromechanical processes taking place on microscale with composite performance. This approach can be very useful in optimal composite design. Acknowledgment This work was supported by National Aeronautic and Space Administration, Air Force Office of Scientific Research, and in part by Kyoto University Foundation through a visiting scholar fellowship.

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References Ashby, M. E., Blunt, R. J. and Banister, M. (1989) FIow characteristics of highly constrained metal wires. Acta. Metal. 37 (7), 1847-1857. Aveston, J. Cooper, G. A. and Kelly, (1971) A. Single and multiple fracture, Conference on The Properties of Fiber Composites, National Physical Laboratory, Guildford, Suerey, ICP Science and Technology Press, pp. 15-26. Bao, G., and Hui, C.-Y. (1990) Effects of interface debonding on the toughness of ductile-particle reinforced ceramics. Int. J. Solids Structures. 26 (5/6), 631-642. Bloyer, D. R., Venkateswara Rao, K. T. and Ritchie, R. O., (1996) Resistance-Curve Toughening in Ductile/Brittle Layered Structures: Behavior in Nb/Nb3 Al Laminates. Materials Science and Engineering A, 216, 80-90. Botsis, J. and Shafiq, A. B., (1992) Crack growth characteristics of an epoxy reinforced with long aligned glass fibers, Int. J. Fracture, 58, R3-R11. Budiansky, B. Amazigo, J. C., and Evans, A. G. (1988) Small scale crack bridging and the fracture toughness of particulate-reinforced ceramics. J. Mech. Phys. Solids. 36 (2), 167-167. Budiansky, B. and Amazigo, J. C. (1989) Toughening by aligned, frictionally constrained fibers. J. mech. Phys. Solids 37, 93-109. Budiansky, B, and Cui, Y. L. (1994) On the tensile strength of a fiber-reinforced ceramic composite containing a crack-like flaw, J. mech. Phys. Solids, 42 (1), 1-20. Budiansky, B. Hutchinson, J. W. and Ewans, A. C. (1986) Matrix fracture in fiber reinforced ceramics. J. mech. Phys. Solids, 34, 167-189. Carter, W. C., Butler, E. P. and Fuller, E. R. Jr., (1991) Micro-mechanical aspects of asperity-controlled friction in fiber-toughened ceramic composites. Scripta Met. 25, 579-584. Dugdale, D. S., (1960) Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids, 8, 100-104. Erdogan, F. and Joseph, P. F. (1989) Toughening of ceramics through crack bridging by ductile particles. J. Am. Ceram. Soc. 72 (2), 262-270. Marshall, D. B. and Cox, B. N. (1987) Tensile fracture of brittle matrix composites: influence of fiber strength. Acta Met., 35, 2607-2619. Marshal, P., and Price, J., (1991) Fibre/matrix interface property determination. Composites, 22 (1), 53-57. Meda, G. and Steif, P. S. (1994a) A detailed analysis of cracks bridged by fibers - I. Limiting cases of short and long cracks. J. Mech. Phys. Solids. 42 (8), 1293-1321. Meda, G. and Steif, P. S. (1994b) A detailed analysis of cracks bridged by fibers - II. Cracks of intermediate size. J. Mech. Phys. Solids. 42 (8), 1323-1341. Muskhelishvili, N. I.., (1963) Some basic problems of the theory of elasticity. Noordhoff, Groningen, Holland. Muskhelishvili, N. I., (1972) Singular Integral Equations. Noordhoff, Groningen, Holland. Mumm, D. R. and Faber, K. T., (1995) Interfacial debonding and sliding in brittle matrix composites measured using an improved fiber pullout technique. Acta Metall. Mat. 43 (3), 1259-1270. Pagano, N. J. and Dharani, L. R. (1990) Micromechanical models for brittle matrix composites. Fiber reinforced ceramic composites. Materials, processing and technology. (K. S. Mazdiyasni, ed.) pp. 4062. Rose, L. R. F. (1987) Crack reinforcement by distributed springs. J. mech. Phys. Solids 35, 383-405. Rubinstein, A. A., (1985) Macrocrack interaction with semi-infinite microcrack array, International Journal of Fracture, 28, 113-119. Rubinstein, A. A., (1986) Macrocrack - microdefect interaction. J. Applied Mechanics, 53, 505-510. Rubinstein, A. A., (1987) Semi-infinite array of cracks in a uniform stress field. Engineering Fracture Mechanics, 26 (1), 15-21. Rubinstein, A. A., (1993) Micromechanical analysis of the failure process in ceramic matrix composites. Journal of Engineering for Gas Turbines and Power, Transactions of the ASME, 115, 122-126. Rubinstein, A. A., (1994) Strength of fiber reinforced ceramics on the basis of a micromechanical analysis. J. Mech. Phys. Solids. 42 (3), 401-422.

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Rubinstein, A. A., (1997) Micromechanical Approach to Failure Process in Composites, Advances in Fracture Research. (B. L. Karihaloo, Y-W. May, M. I. Ripley and R. O. Ritchie, Editors) ICF9 Sydney, Australia,. 2, 631- 642. Pergamon. Rubinstein, A. A., (1998) Fracture Analysis of Composites by a Micromechanical Approach. Composites Science and Technology, 58, 1785 - 1792. Rubinstein, A. A. and Xu, K. (1992) Micromechanical model of crack growth in fiber reinforced ceramics. J. Mech. Phys. Solids. 40 (1), 105-125. Rubinstein, A. A., and Wang, P., (1996) Failure development in particulate composite, Durability and Damage Tolerance of Composite Materials. AD-Vol. 51/MD-Vol. 73, Proceeding of the ASME Aerospace and Materials Divisions, (Editors: W. S. Cham, M. L. Dunn, W. F. Jones, G. M. Newas, P. V. D. McLaughlin and R. C. Wethehold) Book No. G01026-1996, ASME, 415-425. Rubinstein, A. A., and Wang, P. (1998a) The fracture toughness of particulate-reinforced brittle matrix, J. Mech. Phys. Solids. 46 (7), 1139-1159. Rubinstein, A. A., and Wang, P. (1998b) Micromechanics of failure in metal matrix composites. Transactions of the CSME, .22 (4B). 457-465. Sigl, L. S., Mataga, P. A., Dalgleish, B. J., McMeeking, R. M., and Evans A. G. (1988) On the toughness of brittle materials reinforced with a ductile phase. Avta Metall. 36 (4), 945-953. Tvergard, V., (1992) Effect of ductile particle debonding during crack bridging in ceramics. In. J. Mech. Sci. 34 (8), 635-649. Tvergard, V., (1995) On the micromechanics and fracture of ceramics. Fracture of Brittle Disordered Materials: Concrete, Rock and ceramics. Edited by G. Baker and B. L. Karhaloo, F & FN Spon, London, UK, 361-375. Venkateswara Rao, K. T., Soboyejo, W. O., and Ritchie, R. O., (1992) Ductile-phase toughening and fatiguecrack growth in Nb-reinforced molybdenum disilicide intermetallic composites. Metallurgical Transactions A, 23A, 2249-2257. Wang, P., (1998) Micromechanical Analysis of Failure Development in a Class of Composite Materials, Ph. D. Dissertation, Tulane University.

J-INTEGRAL APPLICATIONS TO CHARACTERIZATION AND TAILORING OF CEMENTITIOUS MATERIALS VICTOR C. LI Department of Civil and Environmental Engineering, University of Michigan Ann Arbor, MI, 48109-2125

1. Introduction The J-integral introduced by Professor James R. Rice (Rice, 1968) has found extensive applications in a broad variety of engineering materials. In the last decade, fracture characterization of concrete and deliberate tailoring of fiber reinforced cementitious composites have made great strides. The J-integral has played an important role in these advances. This article reviews the contributions of the J-integral in three distinct but related areas: (a) Characterization of the fracture process of cementitious materials, (b) Testing methodology for the tension-softening constitutive relation in cementitious materials, and (c) Design of cementitious composites with ultra high ductility. Cementitious materials include a broad variety of engineering materials mostly used in civil engineering applications. These include the ubiquitous concrete made of a composition of aggregates with cement as binder. When the stone aggregates are replaced by sand particles, the composite is referred to as a mortar. Fiber reinforced concrete (FRC) is concrete containing a small amount of fiber, typically less than a few percent by volume, and usually in discontinuous form. In recent years, the trend in high performance cementitious composites has been in the use of mortar as the matrix reinforced with an increasingly broad choice of fiber types. In the early days, high performance cementitious composites are synonymous with high fiber content composites. With improved understanding of the micromechanisms responsible for multiple cracking and pseudo strain-hardening, some of these high performance cementitious composites can now be engineered with only two percent or less of fibers, making them viable economically and processing-wise for use in large scale structural applications. In all of these cementitious materials, a common theme is that the aggregates, sand particles or fibers serve as bridging elements when cracks traverse the cement matrix. The fracture process and mode of failure are strongly influenced by the properties of the bridging tractions working against crack opening and extension. The non-linear fracture process in concrete is widely accepted in the engineering community, and more accurate prediction of fracture load in concrete elements is now possible. There is a gradually expanding, although still somewhat limited, adoption of fracture based safe design of concrete structural elements. The application of high performance fiber reinforced cementitious composites in load carrying structures is emerging. A rapid growth in this area is expected in the next few years, especially in Japan. 385 T.-J. Chuang and J. W. Rudnicki (eds.), Multiscale Deformation and Fracture in Materials and Structures, 385–406. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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A detailed account of the fracture processes in cementitious materials can be found in Li and Maalej (1996a,b). The early use of the J-integral for toughness characterization of concrete was proposed by Halvorsen (1980) and Mindess et al (1977). This article is written in honor of Professor James R. Rice, on the occasion of celebrating his 60 t hbirthday.

2. Fracture Models for Concrete and Fiber Reinforced Cementitious Composites Hillerborg (1976, 1983) was one of the first to recognize the importance of aggregate bridging in concrete and fiber bridging in FRC in the fracture processes of these material. By including the often large scale process zone as an extension of the traction free crack, the word “fictitious crack” was coined. In order to predict or simulate crack propagation in these materials, fracture models are needed. Whatever the source of crack face traction is, it is convenient to consider the process zone as the near tip crack segment containing a line of “springs” tying the crack faces. While any spring will invariably resist crack opening, the amount of energy absorption and many macroscopic fracture behavior will depend on the detail behavior of these springs. In general, the springs can be linear or non-linear, hardening or softening. In the case of softening, it can be a result of the same physical process leading to crack tip extension and therefore the presence of the process zone implies cancellation of the crack tip singularity. We refer to such process zone as having ‘cohesive’ behavior. It is also possible to have the springs and crack tip extension as a result of distinctly different physical processes, as in the case of a fiber reinforced cement. In this case, the spring actions are associated with fiber bridging, whereas the crack tip extension is a result of breaking down of the cement material. We refer to such process zone as having ‘bridging’ behavior. For a bridged crack, the presence of the bridging zone can co-exist with a crack tip singularity. Such distinction between a cohesive crack and a bridged crack was first recognized by Cox and Marshall (1994). Cohesive crack models have been considered in a variety of contexts. Barenblatt (1962) assumed the cohesion on the crack faces to be provided by the forces resisting the separation of the layers of atoms in metals. Rice (1980) studied rock break down at ends of earthquake shear faults. Hillerborg (1983) considered aggregate and ligament bridging in cracked concrete producing a tension-softening behavior. The bridged crack model appears most appropriate for fiber reinforced cementitious materials. Consider a crack with a process zone of arbitrary size (Fig. 1). Traction in the process zone takes a general relationship between crack face traction versus crack opening σ ( δ). The spring law can exhibit hardening and softening, with spring force falling to zero at a critical crack opening δt*. A relationship between such a general spring law and the crack driving force J can be derived by adopting the contours shown in Figure 2, and invoking the path independent property of the J-integral.

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Figure 1. Line-Spring Concept of Process Zone Governing Crack Growth

Figure 2. J-integral Contours Chosen for Process Zone Analyses

The result for Mode I is

(1) where δ is the relative crack opening displacement. We consider three crack models corresponding to different crack tip and spring behavior. 2.1 COHESIVE CRACK MODEL When the presence of the cohesive zone is a direct result of the crack tip break down process, as is often assumed to be the case in concrete, the crack tip singularity must vanish, i.e. (2) Equation (1) then implies

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(3) Figure 3 illustrates the tension-softening stress profile σ (x) in the process zone. Note that the stress profile is continuous as it makes the transition from intact material ahead of the fictitious crack tip (δ =0) to the tension-softening material behind the crack tip.

Figure 3. (a) Stress Profile σ (x) in Cohesive Crack Model and (b) Corresponding σ ( δ ) Relationship. Note Crack Opening δ1 at Physical Crack Tip

The integral above can be re-written as

(4)

if Q is taken outside the process zone, since for δ > δ t, σ = 0 so that the integration for δ > δ t can be truncated. Because the traction free crack must propagate when the crack mouth opening exceeds δt *, Eqn. (4) affords a definition of the critical J value, or

(5)

Equation (5) denotes an upper limit of J with no restriction on the size of the cohesive zone. It corresponds to the critical value of non-linear fracture parameter J when traction free crack extension initiates. Hence

J = Jc

(6)

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with Jc defined in (5) can be considered a crack initiation condition in large scale cohesive zone crack model. For the special case of small scale ‘yielding’, i.e. if the process zone is small compared to all other characteristic dimensions in the problem, then JΓ Q G, the standard energy release rate crack driving force parameter.

Further, at imminent

δt*), G propagation (δ t G c. This affords a physical interpretation of Gc in terms of the inelastic behavior of the process zone material, i.e.,

(7)

or graphically, Gc represents the area under the σ – δ spring law (Fig. 3b). 2.2 BRIDGED CRACK MODEL For the bridged crack model, the presence of the process zone does not cancel the crack tip singularity. This is the case of fiber reinforced cementitious composites, in which the crack tip singularity can be associated with the fracture toughness of the cement, while fiber bridging provides the spring tractions on the crack wake. Figure 4 illustrates the bridging stress profile σ (x) in the process zone. Note that the stress profile is discontinuous as it makes the transition from intact material ahead of the fictitious crack tip to the bridging material behind the crack tip.

Figure 4. (a) Stress Profile σ (x) in Bridged Crack Model and (b) Corresponding σ (δ) Relationship.

Assuming small scale yielding of the material ahead of the bridging zone, the contour Γp can be shrunk onto the crack tip (but remains in the K-dominant zone, assuming that it tip exists), and we have J ΓP G c , where G ctip denotes the toughness of the crack tip material independent of the bridging zone processes. Equation (1) then becomes

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(8) Again, since σ becomes zero when δ t exceeds δ t *, a critical value of J can be defined for extension of the traction free crack: (9) For large scale bridging then, crack extension initiates when

J = Jc

(10)

with Jc defined as in (9). In the case of small bridging length compared to all other length scales in the problem (small scale ‘bridging’), Jc and Gc coincides and

(11)

2.3 EMBEDDED PROCESS ZONE MODEL The Bridged Crack Model discussed above assumes small scale ‘yielding’ for the crack tip material. However, this does not have to be the case. The recently developed highly ductile Engineered Cementitious Composites (Li and Hashida, 1993) is a good example. In such materials, the fiber bridging zone is embedded inside a volume of material undergoing inelastic deformation (See also Figure 11 in Section 4). This suggests an Embedded Process Zone Model shown schematically in Figure 5 together with a nonlinear stress-strain curve depicting the behavior of the inelastic zone (shaded area) embedding the process zone.

Figure 5. (a) Embedded Process Zone Model, with Inelastic Behavior in Shaded Volume of Material Represented by Non-Linear σ – ε Curve in (b)

Equation (1) then gives

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(12) where Jm represents the inelastic energy absorption inside the volumetrically distributed inelastic zone. When both the off-crack-plane inelastic zone and the on-crack-plane (cohesive) process zones are fully developed, the maximum value of J is reached. Thus (13) For most materials best described by the Bridged Crack Model, the crack tip toughness is usually much smaller than the energy consumed in the bridging zone. For materials that can be described by the Embedded Process Zone Model, experimental measurements have indicated a Jm comparable in magnitude to the energy absorbed in the process zone (second term on the right hand side of (13)) (Maalej et al, 1995a). Again, for large scale process zone embedded inside an inelastic zone, the fracture criterion in terms of J will be J = Jc (14) with Jc defined as in (13). In the case of small process zone length and small inelastic off-crack-plane zone compared to all other length scales in the problem, Jc and Gc coincides and (15) identical to the energy release rate of (11). In this limit, J m = G ctip. 3. J-Based Fracture Testing in Tension-Softening Material It can be seen from the above discussion on fracture process characterization that the σ (δ) curve plays an important role as constitutive relation of the line-springs in the fracture process zone. It is therefore important to have experimental methodology or micromechanics based modeling to determine σ( δ). In the following, an experimental technique (Li et al, 1987; Leung and Li, 1989; Hashida et al, 1993; Li et al, 1994) for the determination of σ ( δ ) taking advantage of the J-integral is briefly reviewed. Micromechanics based modeling of σ(δ) for various fiber reinforced cementitious materials can be found in Li and co-workers (1992, 1995, 1996, 1997). The experimental technique to be discussed is based on the Compliance Test, first used by Landes and Begley (1972) for elastic-plastic metallic materials, and utilizes the interpretation of J as the difference in potential energy for a differential change in crack length: (16)

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where B is the specimen thickness. Because crack length change is often difficult to measure, we use a pair of specimens identical in every respect except for a small difference in crack length. The test can be carried out with any specimen geometry, and the resulting value of J c should in principle be the same. The compact tension specimen can be a convenient choice. Suppose the two specimens have initial crack lengths of a1 and a2 , where a2 is slightly larger than a1. For a valid test, a 2-a 1 should be smaller than all other dimensions in the specimen, including the thickness B. The load P and load point displacement ∆ are measured for each specimen (Fig. 6a). Due to process zone growth, the P – ∆ curve can be nonlinear. Obviously the specimen with longer crack will have larger displacement value for a given load level (more compliant). For any fixed ∆, the area between these two curves may be interpreted as (17a) so that (17b) Since a1, a2 and B are known a priori , J can be calculated for each value of ∆. This is shown in Fig. 6b as a J – ∆ curve. The plateau value of J is interpreted as the Jc value associated with the full development of the process zone. During specimen loading, the crack tip opening displacement is also monitored. Thus a triplet of Load P, Load-point displacement ∆, and crack tip displacement δt is recorded during the test, and the specimens are loaded to beyond the peak load into the softening regime. Figure 6c shows a ∆ versus δt correlation curve.

Figure 6a. P – ∆ Record for an FRC Specimen (after Li and Ward, 1989)

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Figure 6b: J – ∆ Record

Figure 6c. ∆ –

δt Record

The correlation between ∆ and δ t were used by Li and co-workers to deduce the σ−δ curve. By using the relationship between J and σ(δ) (Equation (4)) for tension-softening materials, the σ−δ curve (Fig. 7) can be obtained by differentiation. That is, (18)

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Figure 7. Deduced Tension-Softening Curve Figure 7 also shows the tension-softening curve of the same FRC material obtained from a uniaxial direct tension test. The good comparison of the two σ– δ curves suggests the accuracy of the J-based testing technique. The J-based method is particularly suitable for brittle materials with sharp dropping σ– δ curves for which the direct tension test may be difficult to carry out due to load instability beyond peak. The J-based testing technique was originally developed for mortar and concrete (Li et al, 1987; Teramura et al, 1990), but has since been applied to fiber reinforced composites (Leung and Li, 1989; Rokugo et al, 1989; Li et al, 1994; Hashida et al, 1994), and rocks (Chong et al, 1989; Hashida, 1990). The J-based technique has been extended to σ–δ relationship determination for materials in which the crack tip singularity is not canceled (Li et al, 1994).

4. Steady State Cracking and Strain-Hardening Design Although most cementitious materials are considered brittle (e.g. cement), or quasibrittle (e.g. concrete and FRC), it is possible to design cementitious composites with extremely ductile behavior. One such material, known as Engineered Cementitious Composite (ECC for short), exhibits tensile strain capacity up to 7.5% (Li et al, 1996). The design of such materials is based on the J-integral analyses of steady state cracking and the micromechanics of fiber bridging. In order to achieve the desirable pseudo-strain hardening behavior, two criteria must be satisfied (Li and Leung, 1992; Li et al, 1996): (i) steady state cracking criterion, and (ii) first crack criterion, which requires the first cracking stress to be lower than the maximum fiber bridging stress. Additional cracks (multiple cracking) can then form on further loading. The steady state criterion has been studied by a number of researchers, (see, e.g. Marshall and Cox (1988); Li and Wu, (1992); and Li and Leung, (1992)). In fiber composites, the extension of a matrix crack is accompanied by fiber bridging across the

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crack flanks. As the matrix crack extends, the bridging zone increases in length. During crack opening, the bridging stress increases as fiber/matrix interfaces debond and the debonded segments of fibers stretch (hardening spring behavior). When the bridging stress increases to the magnitude of the applied load, the crack flanks flatten to maintain the constant applied stress level (Li and Wu, 1992). This load level is termed the steady state cracking stress σss . Based on a J-integral analysis of a steady state crack, Marshall and Cox (1998) showed that (19) where J tip refers to the crack tip toughness. In most fiber reinforced cementitious composites with less than 5% fiber volume fraction, J tip can be approximated as the cementitious matrix toughness. The steady state stress σss and the flattened crack opening δss are related via the bridging law σ( δ). The right hand side of (19) is known as the complementary energy of fiber bridging, and corresponds to the shaded area above the σ( δ ) curve in Fig. 8. For steady state cracking to occur at all, the steady state cracking stress must be less than the maximum bridging stress σ0 in the bridging law. That is, (20) Equation (19) and (20) together imply

(21)

Figure 8. Complementary Energy Concept of Fiber Bridging

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Equation (21) provides a general condition for transition from quasi-brittle to strainhardening failure mode, and highlights the importance of the total complementary energy (right hand side of (21)) in composite design. For Eqn. (21) to be useful in fiber, matrix and interface tailoring, it will be necessary to determine the bridging law σ(δ) specific for a given composite system. In fiber reinforced cementitious composites in which the fibers are randomly oriented and in which pull-out (rather than fiber rupture) are expected, the bridging laws developed by Li and Leung (1992) can be summarized in the following form:

(22)

where bridging stress

is the crack opening corresponding to the maximum

(23)

Corresponding equations for cases where fibers can rupture and for cases where fibers are of variable diameters can be found in Maalej et al (1995b), and Li and Obla (1996). In Eqs. (22) and (23), Vƒ, Lƒ, dƒ, and E f are the fiber volume fraction, length, diameter and Young’s Modulus, respectively. τ is the fiber/matrix interface friction, and the snubbing factor (24)

where f is a snubbing coefficient which must be determined experimentally for a given fiber/matrix system (Li et al, 1990). The snubbing coefficient raises the bridging stress of fibers bridging at an angle inclined to the matrix crack plane, appropriate for flexible fibers exiting the matrix analogous to a rope passing over a friction pulley. Finally, η = (VƒEƒ )/(VmEm), where V m and Em are the matrix volume fraction and Young’s Modulus, respectively. The condition for steady state cracking expressed in Eqn. (21) can now be interpreted as a critical fiber volume fraction above which the composite will show pseudo strain-hardening. Using (22) in (21), this critical fiber volume fraction can be defined in terms of the fiber, matrix and interface parameters (Li and Wu, 1992):

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Equation (25) is important for composite design. It provides guidelines for tailoring the crit microparameters such that V ƒ is minimized (Li, 1998). Strain-hardening composites can then be designed with the minimum fiber content.

Figure 9. Uniaxial Tensile Stress-Strain Curve of an ECC with CO2 gas plasma treated PE fibers

In what follows, we describe the mechanical properties for an ECC. Unless otherwise stated, the ECC referred to contains two volume percent of polyethylene fibers. Using Eq. (25) and appropriate parametric values (see Li, 1998), the critical fiber volume fraction is estimated to range between 0.5% and 1%. Hence a composite with 2% fiber should satisfy the condition of pseudo strain-hardening, and exhibit high strain capacity after first cracking. Figure 9 shows the stress-strain curves from uniaxial tension tests. The ECC strain hardens to an average strain at peak stress εcu approximately equal to 5.6 % (about 560 times the strain capacity of the unreinforced matrix). For this composite, real-time observation showed that multiple cracking occurred with many sub-parallel cracks across the specimen during strain-hardening. Beyond peak stress, localized crack extension occurred accompanied by fiber bridging. The multiple cracking pattern of a specimen is shown in Figure 10.

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Figure 10. Multiple Crack Pattern of an ECC at About 4.2% Tensile Strain

The total fracture energy of ECC was determined (Li and Hashida, 1993; Maalej et al, 1995) by means of the J-based technique (Eqn. 17b) and using a set of DCB specimens with different notch lengths. Concurrently with the tests, damage evolution on the specimen surface was recorded using a camera. Figure 11 presents the damage evolution recorded at various loading stages. It is particularly noted that an extensive microcrack damage zone spreads around the notch tip before the localized crack starts to grow. Significant energy absorption is therefore expected from the off-crack-plane volumetric inelastic deformation process. The total fracture energy measured for this ECC was 27 kJ/m2 ,with approximately over half of this energy consumed in the inelastic damage process occupying an area of 1150 cm 2 around the crack tip, and the rest coming from the pull-out of fibers on the crack wake. For the ECC, the Embedded Process Zone Model discussed in Section 2.3 is most appropriate in describing its fracture process.

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Figure 11. DCB Specimen Showing Evolution of Notch Tip Inelastic Zone at Four Loading Stages.

The notch-sensitivity of ECC has been examined with double edge notched specimens. Test results are shown in Figure 12, which plots the peak load as a function of the reduced section of the notched specimens. The data of the notched specimen lying near (and actually slightly above) the linear line suggests that these composites are notchinsensitive. The surface of the notched specimen (Fig. 13) shows multiple cracks typical of strain hardening fiber reinforced composites. Although the ultimate localized fracture is in the reduced section, multiple cracking spreads along the full length of the specimens prior to final failure. These results suggest that highly damage tolerant structural behavior can be achieved.

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Figure 12. Nominal Failure Load of Double Edged Notch Specimen

Figure 13. Damage Pattern of DEN Specimen

The ability of ECC to deform non-linearly with strain-hardening in tension combined with high damage tolerance suggests its use in concrete elements which require bolt jointing. An experimental study (Li and Kanda, 1998) was carried out to determine the

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response of an ECC slab under circular indentor load (Fig. 14). For this ECC material, 1.25% of PVA fiber was employed. In uniaxial tension the strain capacity was 5%. As control, a similar slab with plain mortar was tested under the same load configuration. Three different size indentors were used. Figure 15 shows the load-deformation (indentor deflection) curves for the ECC and the mortar specimens (used as control). Each specimen type was loaded with three bearing sizes expressed as a percentage of slab surface area. While the load capacity in each case is comparable, it is clear that the deformation capacity of the ECC slab is about one order of magnitude higher than that of the mortar slab at failure. Figure 16 shows the failure pattern of a mortar specimen, which fractures brittlely into several pieces as expected. The corresponding failure of the ECC specimen is much more ductile. Even as the indentor penetrates the surface of the slab, the surrounding material undergoes inelastic damage with no fractures (Figure 17a). Figure 17b gives an enlarged view of the indentor punch. The results of this test confirms the notion that the strain-hardening and damage tolerance of ECC can be very effective in alleviating the high stress concentrations experienced by structural elements whenever steel and concrete materials come into contact with each other. Such elements may include concrete embedded steel anchors, and connections in hybrid concrete/steel structural members.

Figure 14. Geometry of indentor and ECC/Mortar Slab

Figure 15. Load -Deformation Curves for (a) Mortar, and (b) ECC Slab

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Figure 16. Fracture Failure

of A Mortar Specimen

Figure 17. (a) Ductile Indented Pattern of A ECC Specimen, and (b) Enlarged View

ECC is now being investigated for structural applications (Li and Kanda, 1998; Fukuyama et al, 1999; Parra-Montesinos and Wight, 2000). A recent study (Fischer and Li, 2000) on exploiting the strain-hardening and multiple cracking behavior of ECC in highly earthquake resistant building systems involves fully reversed cyclically loaded flexural members. These flexural members are made of ECC reinforced with longitudinal steel or FRP rods. Figure 18 shows the deformed shape of an ECC/aramidFRP element at 10% interstory drift. Microcracks less than 200µm are formed along the full length of the element. The corresponding hysteretic loops indicating large deformation capacity but with low residual deformation (especially for drift less than 5%) are also shown. In contrast to typical concrete/steel element behavior, no spalling of the ECC matrix or buckling of the axial reinforcements are observed. Corresponding tests with ECC/steel elements show extremely high energy absorption behavior (Fischer and Li, 2000). These characteristics can be used to design building systems with high safety as well as minimal post-earthquake repair needs.

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Figure 18. (a) FRP Reinforced ECC Flexural Element at 10% drift and (b) load deformation behavior

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5 . Conclusions and Further Discussions The J-integral has played an important role in the understanding of the mechanics and the engineering of the composites of cementitious materials. This article reviews the use of the J-integral in fracture process characterization, tension-softening curve determination, and microstructural tailoring of cementitious materials. Advancements in these areas provide important tools for failure analyses of structures of cementitious materials, as well as tools for cementitious composite design for safe structures. Before closing, we note an often misdirected criticism of the application of the Jintegral to tension-softening materials. The criticism is directed at the fact that since the J-integral rests on the assumption of non-linear elasticity, and since the process zone material softens inelastically, the application of the J-integral to tension-softening materials cannot be valid. The apparent paradox is resolved if it is understood that the Jintegral contour (see, e.g. the contours ΓQP+ and Γ QP- in Fig. 2) is placed in the elastic material adjacent to a line of springs (representing the softening material) which can unload. Then the unloading of the springs (softening branch of the σ– δ curve) during crack opening causes a corresponding elastic unloading of the material in which the contour is placed. This can be better envisioned with a tensile specimen of a quasibrittle material. Once a localized fracture zone is formed, the material in the fracture zone unloads (inelastically), while to maintain equilibrium, the material outside the fracture zone, which remains elastic also unloads, but unloads elastically. This phenomenon was nicely illustrated with concrete specimens in uniaxial tensile experiments carried out by Petersson (1981). The strain/displacement gage across the fracture zone shows inelastic unloading (stress drop with increase in crack opening), but the gages outside the fracture zone unloads elastically (decreasing stress with decreasing strain deformation retracing the elastic loading). Hence for the application of the Jintegral to tension-softening materials, there is no violation of the requirement of the Jintegral as long as the unloading of the material outside the line-spring is elastic.

References Barenblatt, G. I. (1962) The mathematical theory of equilibrium cracks in brittle fracture, Advanced Applied Mechanics 7, 55-125. Chong, K.P., Li, V.C., and Einstein, H.H. (1989) Size effects, process zone, and tension softening behavior in fracture of geomaterials, International J. of Engineering Fracture Mechanics 34(3), 669-678. Cox, B. and Marshall, D. (1994) Concepts in the fracture and fatigue of bridged cracks, Overview No 111. Acta Meta. Mate. 42, 341-363. Fischer, G. and Li, V.C. (2000) Structural composites with ECC, to appear in the Proceedings of the ASCCS2000. Fukuyama, H., Y. Matsuzaki, K. Nakano, and Y. Sato (1999) Structural Performance of Beam Elements with PVA-ECC, in H. Reinhardt and A. Naaman (eds.), Proc. of High Performance Fiber Reinforced Cement Composites 3 (HPFRCC 3), Chapman & Hall, pp. 53 l-542. Halvorsen, G.T. (1980) J-integral study of steel fiber reinforced concrete, International J. Cement Composites, 2(1) 13-22. Hashida, T. (1990) Evaluation of fracture processes in granite based on the tension-softening model, In S.P. Shah, S.E. Swartz & M.L. Wang (eds), Micromechanics of Failure of Quasi-Brittle Materials, Elsevier Applied Science, London, 233-243. Hashida, T., Li, V.C., and Takahashi, H. (1994) New development of the J-based fracture testing technique for ceramic matrix composites, J. American Ceramic Society 77(6), 1553-1561.

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Hillerborg, A. (1983) Analysis of One Single Crack, in F.H. Wittmann (ed.) Fracture Mechanics of Concrete, Elsevier Science Publisher, B.V., Amsterdam, pp. 223-250. Hillerborg, A., Modeer, M., and Petersson, P. E. (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research 6, 773782. Landes, J.D. and Begley, J.A. (1972) The Effect of Specimen Geometry on J I C, in Stress Analysis and Growth of Cracks, ASTM STP 514, ASTM, Philadelphia Leung, C.K.Y., and Li, V.C. (1989) Determination of fracture toughness parameter of quasi-brittle materials with laboratory-size specimens, J. Materials Science 24, 854-862. Li, V.C. (1992) Post-crack scaling relations for fiber reinforced cementitious composites”, ASCE J. of Materials in Civil Engineering, 4(1), 41-57. Li, V.C. (1998) Engineered Cementitious Composites – Tailored Composites Through Micromechanical Modeling, in N. Banthia, A. Bentur, A. and A. Mufti (eds.) Fiber Reinforced Concrete: Present and the Future, Canadian Society for Civil Engineering, Montreal, pp. 64-97. Li, V.C., Chan, C.M., and Leung, C.K.Y. (1987) Experimental determination of the tension-softening curve in cementitious composites, J. Cement and Concrete Research 17-3, 441-452. Li, V.C. and Leung, C.K.Y. (1992) Steady state and multiple cracking of short random fiber composites, ASCE J. of Engineering Mechanics 118(11), 2246-2264. Li, V.C. and Hashida, T. (1993) Engineering ductile fracture in brittle matrix composites, J. of Materials Science Letters 12, 898-901. Li, V.C. and Kanda, T. (1998) Engineered cementitious composites for structural applications, ASCE J. Materials in Civil Engineering 10(2), 66-69. Li, V.C. and Maalej, M. (1996a) Toughening in cement based composites, Part I: Cement, mortar and concrete, J. of Cement and Concrete Composites 18(4), 223-237. Li, V.C. and Maalej, M. (1996b) Toughening in cement based composites, Part II: Fiber reinforced cementitious composites, J. of Cement and Concrete Composites 18(4), 239-249. Li, V.C., Maalej, M., and Hashida, T. (1994) Experimental determination of stress-crack opening relation in fiber cementitious composites with crack tip singularity, J. Materials Science 29, 2719 - 2724. Li, V.C., Mihashi, H., Wu, H.C., Alwan, J., Brincker, R., Horii, H., Leung, C., Maalej, M., and Stang, H. (1996) Micromechanical models of mechanical response of HPFRCC, in A.E. Naaman and H.W. Reinhardt (Eds.) High Performance Fiber Reinforced Cementitious Composites, RILEM Proceedings 31, pp. 43-100. Li, V.C. and Obla, K. (1996) Effect of fiber diameter variation on properties of cement based matrix fiber reinforced composites, Composites Engineering International Journal Part B 27B, 275-284. Li, V.C. and Ward, R. (1989) A novel testing technique for post-peak tensile behavior of cementitious materials, in H. Mihashi, H. Takahashi, and F.H. Wittmann (eds.) Fracture Toughness and Fracture Energy – Test Methods for Concrete and Rock, Balkema, Rotterdam, pp. 183 – 195. Li, V.C., Wang, Y., and Backer, S. (1990) Effect of inclining angle, bundling, and surface treatment on synthetic fiber pull-out from a cement matrix, J. Composites 21(2), 132-140. Li, V.C. and Wu, H.C. (1992) Conditions for Pseudo strain-hardening in fiber reinforced brittle matrix composites, J. Applied Mechanics Review 45(8), 390-398. Li, V.C., Wu, H.C., and Chan, Y.W. (1996) Effect of plasma treatment of polyethylene fibers on interface and cementitious composite properties, J. of American Ceramics Society 79(3), 700-704. Lin, Z. and Li, V.C. (1997) Crack bridging in fiber reinforced cementitious composites with slip-hardening interfaces, J. Mechanics and Physics of Solids 45(5), 763-787. Maalej, M., Hashida, T., and Li, V.C. (1995a) Effect of fiber volume fraction on the off-crack-plane fracture energy in strain-hardening engineered cementitious composites, J. Amer. Ceramics Soc. 78(12), 33693375. Maalej, M., Li, V.C., and Hashida, T. (1995b) Effect of fiber rupture on tensile properties of short fiber composites, ASCE J. Engineering Mechanics 121(8), 903-913. Marshall, D.B. and Cox, B.N. (1988) A J-integral method for calculating steady-state Matrix Cracking Stresses in Composites, Mechanics of Materials 7, 127-133. Mindess, S., Lawrence, Jr., F.V. and Kesler, C.E. (1977) The J-integral as a fracture criterion for fiber reinforced concrete, Cement and Concrete Research, 7, 731-742. Parra-Montesinos, G.J., and J.K. Wight (2000) Behavior and Strength of RC Column-to-Steel Beam Connections Subjected to Seismic Loading, to appear in the Proceedings of the 12th WCEE. Petersson, P-E, (1981) Crack Growth and Development of Fracture Zones in Plain Concrete and Similar Materials, Report TVBM-1006, Lund, Sweden, 174pp.

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Rice, J. R. (1968) A path-independent integral and the approximate analysis of strain concentration by notches and cracks. J. Applied Mechanics 35, 379. Rice, J.R. (1980) The mechanics of earthquake rupture, in A.M. Dziewonski and E. Boschi (eds.) Physics of the Earth’s Interior, Italian Physical Society/North Holland, Amsterdam. Rokugo, K., Iwasa, M., Seko, S., and Koyanagi, W. (1989) Tension-softening diagams of steel fiher reinforced concrete. In S.P. Shah, S.E. Swartz & B. Barr (eds.), Fracture of Concrete and Rock, Recent Developments, pp. 513-522. Teramura, S., Normura, N., Hashida, T., and Mihashi, H. (1990) Development of a core-based testing method for determiningfracture energy and tension-softening relation of concrete, in S.P. Shah, S.E. Swartz and M. L. Wang (eds.), Micromechanics of Failure of Quasi-Brittle Materials, Elsevier Applied Science, London, pp. 463-473.

Author Index

Abouaf, M. 50 278 Abraham, F. 82 Agrait, N. 358 Aldinger, F. 391 Alwan, J. 361,367 Amazigo, J.C. Anderson P. M. 31,41,87,89 Antonellini, M.A.161 Argon, A. S. 10,90 50 Arzt, E. Asaro, R.J. 184 Ashby, M.F. 50,53,61,65,367 Askenazi, D. 184,190 Astiz, M.A. 217 Atkins, A.G. 223,225 Aveston, J. 361 Bader, S.D. Baker, S. Ball, P. Banister, M. Banks-Sills, L. Bao, G. Barenblatt, G.I. Barlat, F. Barnett, S.A. Bartelt, M.C. Barthel, E. Basinski, Z.S. Basinski, S.J. Bassani, J. Baud, P. Baudin, P. Bayle, B. Beck, W. Begley, J .A. Beltz, G. E.

108 396 109 367 183-184, 188, 190 362,367 386 17-21,26 88 108 72 319,327 319,327 184 170 50 31 43 206,219,391 41,87,90,96,103,237, 239 Bernstein, I. M. 31,44

Betegon, C. Biot, M.A. Birnbaum, H. K. Biwa, S. Blackburn, W.S. Bloyer, D.R. Blug, B. Blunt, R. J. Bockris, J. O’M. Boniface, V. Bordia, R.K. Botsis, J. Bouvard, D. Bowden, F. P. Bower, A.F. Boyce, M. C. Boyle, R.W. Brinker, R. Brodbeck, D. Budiansky, B. Bullough, R. Butler, E.P. Byerlee, J.D.

238 108 31 71,74 206 370,379 49 367 43 183 50 363 50 82 108,331 19 225 391 278 211,361,367 112 366 164,177

Cahn, J.W. 108,112 Cammarata, R.C. 112 Cao, H.C. 358 Carlsson, A.G. 238 Carter, W.C. 366,108 Chambreuil, A. 31 Chan, C.M. 394 Charalambides,P.G. 184 Charles, R. J. 332-333,335 Chen, B. 17 Chen, X.F. 43 Chen, C.H. 168 Chen, X. 31,44 Chen, Z. 223,225-227 Chen, L.Q. 109,114,117

408

AUTHOR INDEX

Chen, X.-Y. 243,261-263, 266,271 Cheng, Y. 89 Chenot, J.L. 50 Cherepanov, G.P. 350,353 1,3,8,9 Chien, W. Y. Chitaley, A.D. 243,256 Cho, J.W. 311-312 Chong, K.P. 394 Chu, C. -C. 10 Chuang, T.-J. 51,107,112,331-333, 339-343, Cleary, M.P. 164,177-178 Clememens, B.M.88 Cocks, A.C.F 50,81,108 Collier, L. 358 Comsa, G. 108 Cook, W.R. 359 Cooper, G.A. 361 Corten, H.T. 188 Cotterell, B. 223,225,238 Coussy, O. 171 Cox, B. 386,394 Cox, B. N. 362 Craft, D. 108 Cui, Y.L. 361 Cuitino, A.M. 312,327 Curtin, W.A. 331 Dalgleish, B.J. David, C. Dawson, P.R de la Figuera, J. Deng, X. Desrues, J. Dharani, L.R. Diercks, V. Dimaggio, F.L. Dimiduk,D.M. Dolbow, J. Doremus, P. Drory, M.D. Drugan, W.J.

362 161 50 108 186 161,168 362 108 161,169 88 184 50 184 243-244,246-247, 256,261-263,266,271

Du, Z.Z. Duesbery, M.S. Dugdale, D.S. Dundurs, J. Dutton, R.

50 90,93 371 89,186 338

Einstein, H.H. Eisele, U. Eliasi, R. Elices, M. Erdogan, F. Ernst, H.A. Eshelby, E.J. Evans, A.G.

394 50 184,190 217 362,367 226-227 123 346,358,361-362,367

Faber, K.T. Ferreira P. J. Fineberg, J. Finno,R.J. Fischer, L.L. Fischer, G. Fleck, N. A. Foecke, T. Freiman, S.W. Frenkel, J. Freund, L. B. Fuhirott, H. Fukai, Y. Fukuyama, H. Fukuyama, S. Fuller, Jr. E.R

366 31 277,290,295,299-300 169-170 237,241 402 72,81,238 43 332 90,239 89,108-109,112,289 324 43 402 31 332-333,339-343

50 217 244 108,275,277-280,289290,296 Garagash, D. 168 Gehrke, E. 332 George, J. 108 Gerberich W.W. 31,43-44 German, M.D. 225 Gaebel, R. Galvez, V.S. Ganti, S. Gao, H.

AUTHOR INDEX Gill, S.P.A. Gillia, O. Glas, F. Gong, X. Gosz, M. Graf, A. Greenspan, D.C. Greenwood, J. A Greskovich, C. Gross, S. Gurland, J. Gurson, A. L. Gurtin, M.E. Guyer, J.E.

108 50 109 109 184 2 332 72,74 65 277,290,295,299-300 9 l-10, 17-29,277 112 109

Hack, J. E Hahnert, M. Hall, E.O. Halvorsen, G.T. Han, C. Hanada, S. Hancock, J.W. Harris, W.W. Hashida, T. Haug, A. Hazzledine, P.M. Head, A.K. Hector,Jr.L.G. Herbek, J. Herring, C. Hibbitt, D. H. Hill, R. Hillerborg, A. Hillert, M. Hilliard, J.E. Hillig, W .B. Hirth, J. P.

43 332 88 386 169 31 225,238,244 169 390-391,394,396,398 349 88-89 89,98-99 152 108 107 7 1,2,11,18,71-72,75,291 386 59 108,112 332-333,335 32-35,42, 89-90, 101, 105 41 9 391 2,18 31

Hoagland R. G Hom, C. L. Horii, H. Hosford, W. F. Hu, Z.

409

Huang, Y. 17 Huang, H. -M. 1,10-11,13-14 Hui, C.-Y. 362,367 Hull, D. 107 Hurtado, J.A. 89 Hutchinson, J.W. 10,72,81,183,186, 224-225, 238 Hwang, R.Q. 108 Igarashi, M. Im, J. Imbault, D. Irwin, G.R. Issen, K.A. Iwasa, M.

31,44 10 50 223,300 161 394

Jackson, P. J. Jaffe, H. Jaffe, B. Jagota, A. Jiang, J.S. Johnson, H. Johnson, K. L Joseph, P.F. Joyce, J.A.

327 359 359 50 108 346 7 l-72.74,80-82,113 362,367 225,233

Kachanov, M. 123,277 Kagawa, K.I 51,107 Kanda, T. 400,402 Kanters, J. 50 Kanwal, R.P. 123 Karafillis, A. P. 18-19 Karapetian, E. 123 Katz, Y. 31,43-44 Kawakami, T. 31 90,93 Kaxiras, E. Keer, L.M. 123 Kelly, A. 361 Kern, K. 108 Kesler, C.E. 386 Khachaturyan, A.G. 108-109 Kikuchi, M. 206 Kimura, H. 41

410

AUTHOR INDEX

Kitajima, S. 312 Kiuchi, K 43 Klameth H.-K 31,44. Klein, P.A. 275,279,,290,294,296 Knauss, W. 276 Knowles, J.K. 214 Koehler, J. 152-153 Koepke, B.G. 358 Koike, S 43 Koyanagi, W. 394 Kozak, V 50,55,56 Krafft, J.M. 225 Kraft, T 50 Krznowski, J.E. 89 205-206,217,219-220 Kubo, S. Kuhlmann-Wilsdorf, D. 152 Kuhn, L.T 50,55,65,108 Kung, H. 88 Landes, J.D. 206,219,226-227,391 Lanxner, M 31.44 Larsson, S.G. 238 Larsson, R. 161 Lawerence, Jr. F.V. 386 Leung, C.K.Y. 391,394,396 Levine, L.E. 145,147-149 Li, J. 244 Li, J.C.M. 241 Li, D. 108 Li, V.C. 385-386,390-391 Liao, K. -C. 2,3,13,19,25,27,29 Lii, M.J 31,43-44 Link, R.E. 225,233 Lothe, J. 89-90,101,105 Lu, W. 107,109,115 Lufrano, J 43 Lur’e, A.I. 123 Lynch, C.S. 358 Maalej, M. 386,391,394,396 MacEwen, S. R. 17 Mader, S. 153 Maeda, T. 220

Magnin, T 31 Majorana, C.E. 168,170 Marder, M. 277,290,295,299-300 Marshal, P. 366 Marshall, D.B. 362,386,394 Martin, R.J.III 168 Mataga, P.A. 244,362 Matos, P.P.L. 184 Matsui, H 41 Matsuzaki, Y. 402 Maugis, D 72 McBreen, J 43 McClintock, F.A. 243,256 McHenry, K.D. 358 McHugh, P.E 50 37 McLean, D McLellan, R. B 43 McMahon, C. J 31 McMeeking, R.M 9,43,50,55, 65,108,184, 225,349-51 Mear, M. E. 10 Meda, G. 362 Merkle, J.G. 219 Mesarovic, S. Dj. 71-72 Mihashi, H. 391 Mikeska, K.R 50 Mindess, S. 386 Miyamoto, H. 206 Miyata, K 31,44 Modeer, M. 386 Mohan, R. 312 Mokni, M 161,168 Mollema, P.N. 161 Mooney, M.A. 169 Moran, B. 184 Morgan, K. 337 Mori, K 50 Mullins, W.W. 107 Mumm, D.R. 366 Mura, T. 87,89-90,96,98-99,103 Murdoch, A.I. 112 Muskhelishvilli, N.I. 363,366,373

AUTHOR INDEX Nabarro, F.R.N. Nahta, R. Nakamura, T. Nakano, K. Namazue, H. Needle, A. Neumann, P. Ng, K.-O Niebus, H. Nishimura, K. Noggle, T. Nur, A

238 184 184 402 43 2, 9-10, 18, 107, 278, 299 31,44,324 109 108 220 152-153 164, 177

Obla, K Ogbonna, N. Ogura, K. Ohji, K. Ohta, M. Olevsky, E.A Olsson, W.A Ortiz, M. O’Dowd, N.P.

394,396 81 205 205206,217,219-220 312 50 161 279,312,327 225

Pacheco, E.S. 87,89-90,96,98-99,103 Pagano, N.J. 362 Pak, Y.E. 358 Pan, J. 1,2,3,108 Paris, P. 2 19,224,226-227 Paris, P.C. 319,327 Park, S.B. 351,354 Parks, D.M. 225,244 Parra-Montesinos, G.J. 402 Parteder, E 50,58 Paterson, M.S. 163 Pearson, J. 108 Pearson, J. M 81 Peierls, R.E. 238 Peraire, J. 337 Peric, D. 160 Petch, N. J. 88. Petersson, P.E. 386,404 Phillips, R. 279

411

Pierce, D. 10 Plankensteiner, A.F Pohanka, R.C. 358 Pohl, K. 108 Pompe, W. 109 Pond, Sr.R.B. 153 Ponte, C. P. 244 Price, J. 366 Prigogine, I. 108 Qu, J. Rafey, R. Rahman, M. Raisson, G Rao, S.I. Ravi-Chanda,K. Rayleigh, J.W.S. Reuter, W.G. Rice, J.R.

50,58

184

278 123 50 88-89 276 108 225 11,18,31-35,37,39, 41, 51,72,87,90, 96, 103, 107, 112, 159, 182-183, 186, 205, 207, 211, 219, 223, 225, 238-240, 244, 246-248, 311, 314, 317, 327, 346, 350, 353-355, 357-358, 362, 366-367, 370, 376, 378-379, ~386, 394-398,400,402 Rice, R.W. 358 Riedel, H 49-50,54-58 Rimmer, D.E. 107 Ritchie, R.O. 367,370,378-379 Robertson I. M 31 Rodel, J 50 Rokugo, K. 394 Rose, L.R.F. 361 Rosengren, G.F. 225 Rosolowski, J.H 65 Rostek, A. 358 Rothe, W 31,44

AUTHOR INDEX

412 Royburd, A.L. Rubinstein, A.A. Rubio, G Rudge, W. Rudnicki, J.W. Runesson, K.

109 361-362,364-365 82 278 159,162,167-176 160-162,171,178-179

311 Saeedvafa, M. 2,10 Saje, M. 168 Sanavia, L. 161,169 Sandler, I.S. 402 Sato, Y. 108 Schatz, A. 50 Scherer, G.W Schneider, G.A. 358 37 Schoeck, G. 168 Schrefler, B.A. Schubnel, A. 170 Seah M. P 37 Seko, S. 394 123 Selvadurai, A.P.S. Sevostianov, I. 123 Shafig, A.B. 363 244 Sham, T.-L. 123 Sharma, D.L. Shen, H. 43 117 Shen, J. Sherman, D. 188 311 Shield, T.W. 184,225,312 Shih, C. F. 89 Shilkrot, L.E. 145,147 Shim, Y. 50 Shinagawa, K 88 Shinn, M. 112 Sieradzki, K. 362 Sigl, L.S. 319,327 Sih, G.C. 51,107 Sills, L.B Simmons, D.C. 332 Soboyejo, W.O. 367,378 Sofronis P 31,43 109 Speck, J.S. Srolovitz, D.J. 89,108

Stakgold, 1. Stang, H. Stauffer, D. Steif, P.S. Sternberg, E. Stingl, P Storåkers, B Sture, S. Suehiro, S. Sugimoto, H. Sullivan, A.M. Sumpter, A. Sun, D.-Z. Sun, C.T. Sun, Y. Suo, Z. Suzuki T Svobada, J. Swinney, H.

281 391 147-148 362 214 50 71,73-75 160 217 43 225 224 50 35 1,354 41, 90,93 107-109, 115, 184, 238,358 43 50,52-57 277,290,295,299-300

Tabor, D. 82 Tadmor, E. 279 Takano, N. 31 31 Takasugi, T. Tang, Z. 331 Tang, S. C. 1,3, 17 108 Taylor, J.E. Terasaki, F. 31 Thompson A.W. 31,44 Thomson, R. M. 37,145,147-149 Timothy, S. P. 81 Ting, T.C.T. 184,187 Tobin, A.G. 358 Tonda, M. 312 Tong, W. 152 43 Toribio, J. 18,244 Tracey, D. M. 184,190 Travitzky, N. Triantafyllidis, N.2 Tromans, D. 31 Turner, T. 224 2,3,9, 18, 367 Tvergaard, V.

AUTHOR INDEX Ullner, C.

332

Vahdati, M. 337 Vanderbilt, D. 109 Vardoulakis, I. 171 Vardoulakis, I.G. 169 31,44,324 Vehoff, H. Venkateswara Rao, K.T. 367,370 82 Vieira, S Viggiani, G. 169 Voorhees, P.W. 109 Walker, B.E. 358 Walston W.S. 31,44 Wang, P. 362,367,370,372,378 Wang J.-S 3l-32,35,39,41-42 Wang, S.S. 184,188 Wang, Y. 114,396 Weiland, H. 152-153 White, G.S. 332 Wiederhorn, S.M. 332,346 Wieserman, L.F. 152 Wight, J.K. 402 Wilkins, B.J.S. 338 Williams, M.L. 237 Willis, J.R. 112 Wilsdorf, H. 152 Wittig, F 50 Wong, T.F. 161,170 Wu, H.C. 391,394-395-396

413

Wu, P. D. Wu, C.H.

17 112

Xia, L. Xia, Z. C. Xin, X.J. Xu, G. Xu, K. Xu, X.-P.

225,233 17 87 90 362 278,299

Yamamoto, H. Yang, W. Yau, J.F. Yoffe, E. Yokogawa, K. Yoshikawa, A. Yu, J.

2 358 184,188 277 31 31 41,311-312

71 Zdunek, A. B. Zeppenfield, P. 108 Zhang T.-Y. 43,241 184 Zhang, W. Zhou, S.J. 331 Zhu, W. 161 Zhu, J.Z. 337 Zickgraf, B. 358 Zienkiewicz, O.C. 337 Zipse, H. 50,52-53,56,58 Zureick, A.H. 123

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Subject Index

ABAQUS 7,50,58,65 Acoustic waves 277 barrier 306 Adhesion 71,72,74-76, 8l-83 map 81-83 Alloy FeSi 31 Amplification factor 147 Analysis upper bond 2,24 failure localization 10 thermodynamic 32 finite element, 3D 1,3,6,13-14, 43,50, 190, 295,351-352 Hall-Petch 88 J-integral 290 asymptotic 252,271,273 Anisotropic parameter R 3 Atomic diffusion 110 force microscopy 152,156 mobility 37 Avalanche point 150 Bands 152,154 Bridged crack model 389 Burgers vector 40,90,239 Cauchy principal value 97 symmetry 285,295 Cauchy-Born rule 279 Cell walls 147 physics 151 Cellular structure 147 Ceramic fiber 361 particulate-reinforced 376

Chemical potential 35 Cleavage Griffith 42-43 Cohesive force law 275 crack model 387 models, 279-284,287 Compact tension specimen 349 Compaction, 160-161, 168-170 Composite ceramic matrix 361,375 metal matrix 378 Constitutive laws 145,151 Contact 71-85 sticking 80-81 frictionless 77 scaling 72-76 maps 71,78-83 elastic-plastic 71-85 Continuum damage model 277 Coplanar slip 312,317,319,324,327 Crack branching 277,296,298 propagation 276,278 Crack growth 225-27,232-33,235 plane strain 259 tensile 261 resistance 234-35 anti-plane shear 244,250 Crack tip 31,349 plasticity 311 Crack propagation 225-27,230,243-44 Crack dynamics 275 Crack opening 352 Crack bifurcation, 298 Crack growth rate 39 Crack opening displacement 381 Crack initiation 33,35

416

SUBJECT INDEX

Crack opening displacement 387 386 Crack fictitious Crack cohesive 386 386 Crack bridged 151 Critical system self organizing Crystal orientation 313 399,401 Damage tolerance Damage evolution 14,398 Deformation anti-plane shear 243 25,43,381 plastic Dielectric permittivity 354 Diffusion, 168, 167, 174. bulk 53,61 grain boundary 53,61 surface 53,61 hydrogen 44 Diffusivity 114 Dilatant, 161, 166-168, 171 Dilatational wave 295,297 Dislocation nucleation 37,237 emision 41,96, 237 Volterra 89,99 screw 87,89 Displacement field crack tip 319,321 Dissipation plastic work 23 Domain reorietation 358 Double cantilever beam 226-28 Ductile particles 368 Effect shielding 42 Elastic energy 110 123, 143 space field 123-143 constant 354 Elastic-plastic, 171, 175 Elasto-plastic 223-33,235,237 Electrical enthalpy 350 Ellipsoid 124 Ellipticity 292,294

Embedded Atom method 40-41,88,93 process zone model 390,398 Embrittlement, interfacial 32 Hydrogen 31 Energetics 110 Energy stacking fault 87 surface 112 elastic 110-111 density 229-230 release rate 37,349,354 stacking 40 stacking fault 96,100-102,239 unstable stacking 239 Epi1ayer108,110,112,116 Etch pit 311,325,328 factor character 150 Failure prediction 2,13-14 criterion 291 indicator 291 Fiber pull-out 381,396 reinforcement 365 bridging 386 Fiber-reinforced MMC 370 Field intensity factor 358 Finite element formulation Lagrangian 294 Finite strain 275,286 Forming limit 2 Fracture 349 process zone 31,223,226,228, 236-37 interface 244 specific work of 224,226,237 mechanics 43,385

SUBJECT INDEX Fracture extra work 227,229-32,236-37 398 energy resistance 375 essential work of 223,22527,230-33,235-37 Frankel model 239 Function bond density 285 relaxation 155 yield 20,27 367 Gauss-Chebyshev quadrature 36 Gibbs-Duhen relation Global J scalar 208,217-218 Global J vector 208,215 Grain coarsening 57 Grain growth 54 Graphie/epoxy 189 Green body 49 Hilert distribution 60 57,59 Hillert Law 284-85 Hooke’s law 402 Hysteretic loop Inclusion rigid ellipsoidal 123 Indentation 71-85 71,78-83 maps 80-81 pre-stressed 77-78 master curves 123 Inhomogeneity Instability 165, 167, 174, 275,277-78, 300, 381 290 Interaction potential Interface non-slipping 87 bimaterial 87 90 Cu/Ni epitaxial 88 crack 183 183 Isotropic transversely J-integral 205,350,353,385 388,393-395 205,219 J-integral modified JR-curve 226,229,232-37 Keldysh-Sedove problem 366,373

417

Kinetics segregation 38,44 L-integral 205,212,214 Lame’s coefficients 126 Langmuir-McLean model 37 Latent hardening 311,327-28 Lattice rotation 154, 321-22 387 Line-spring concept Load-displacement 230,232,234-36 Load-displacement curve 314-15, 392 Loading mode I 363-64 Loading, axisymmetric 3 Local J scalar 208-210 Local J vector 205-206,209-210,215 Localization, 160-162,171, 173, 175-177 Localization flow 3 Localized ordering 152 Lomer-Cottrell locks 148,324 M-integral 183,188,205,212-213 Mass transport 110 Material piezoelectric 349,351,354 composite 385 design 385,402 cementitious 385 layered 244 homogeneous 243-244,255,259,261 elastic-plastic 243 tension softening 392 characterization 385 testing 398-403 McLean isotherm 41 Mechanism lock breaking 148 source 148 fracture resistance361 Mobility grain boundary 54 Model virtual internal bond 275, 279, 284, 291- 292, 294,296 constitutive 5l-58 371 Dugdale 311 Moire Molecular dynamics 275

418 Nanophases Newton method Non-coplanar slip

SUBJECT INDEX

108 353 312, 317, 319, 324, 327 Non-recoverable work 225 Nonlinear elastic 223-33, 235,237 Notch sensitivity 399 Numerical algorithms 294 Partculate reinforcement 367 Pattern formation 146 Peierls-Nabarro Model 238 Peierls-Nabarro approach 40 Percolation 145 Periodic pattern 108 Phase refining 109-110 Phase coarsening 109-l10 Plastic anisotropy 2 work accumulated 231 flow 2 zone 238 Plasticity rigid-perfect 7 PMMA 277,295 Polarization 358 Pore pressure 160-163, 165, 167, 172-174, 177-180 Pore fluid 160, 162, 166, 168, 172 Potential function 124 Powder technology 49 Powder silicon carbide 58 Process simulation 50 Process zone 387,389 Rayleigh wave 299 Recoverable work 225,228,237 Rice-Thomson model 40 Rock, 160-161, 163, 167-168, 169 Rotation rigid particle 123- 142 Scaling law 149 Self-organization 109 Shear in-plane 7 Shear wave 295,300,302-304 Sheet aluminum 17-18,21,29 Sheet metal failure 3

Sheet forming 1 Sheet metal rate-sensitive 3 Sheet cell model 4,26 Similarity 71-76 Single crystal 31,37 copper 311 iron 42 Sintering aid 65 Slip 152-153 active system 40 bands 153-154 elementary unit 152 line 152 plane 40,101 trace 312 Small scale bridging 390 yielding 256,266 Soil, 160, 161, 163, 169, 177 Solid porous plastc 2 Solid Surface 108 Solute segregation 34 Solution upper bond 4 Spanning cluster 149 Sphere rigid 140 Spheroid oblate 132 Spheroid prolate 136 Stiffness, relation 143 tangent 277 Strain anti-plane 248 cluster 147 energy density 283,291 Green 292 Green Lagrangian 284 hardening 3 Strain rate macrscopic 21,25 plastic 20-21 shear 21 Strength yield 19, 243

SUBJECT INDEX Stress Cauchy 286 critical shear 87 27-28 effective eqibiaxial 287,289 intensity factor 354 mean 27 Peierls 40 Piola-Kirchhoff 28l-82,288,291-92 potentials 363 plane 727 327 resolved shear sintering 51,56 113 surface yield 20 yield uniaxial 27 92 Stress-displacement relation 145 Stress/strain relations Stress-separation curve 32 Stress-strain law, universal 153 Stress-strain law 152 Stress-strain curve 278 Stretching, eqibiaxial 285 Stroh formulation 183, 194 Stylus profilometer 311-12 113-l14 Surface stress Surface profile 320 Surface energy 111-112 T-stress 237 Temperature, ductile vs.brittle transition 39,41 Tension hydrostatc 7 5,7 biaxial uniaxial 7,25-26

419

Tensor elastic stiffness 284 Theory dilation plasticity 17,19 Gurson type 18,21 density functional 93 continuum plasticity 18 44 Thermodynamics segregation Threshold percolation 148,150-151 Time localization 153 Transition ductile/brittle 37 Translation rigid particle 123- 142 Transport 146 Universality class 148 Unzipping event 148 Viscosity shear 54 Viscosity bulk 55 Void coalescence 9,14,18 Void growth 9,14,18 Void nucleation 2,18 237 Williams expansion X-ray 311,321 Yield criterion anisotropic 1,3,27,29 Gurson’s 2-3,7-8,13-14 Hill’s 1-3,11,13, 2 Yield surface curvature effect 10 Zone mist 277-78,289,298 deformation-softened 300 Dugdale cohesive 378 bridge 362,364,366 plastic 326 hackle 277-78,289,298 plastic 367 361 process