Multiscale Deformation and Fracture in Materials and Structures

MULTISCALE DEFORMATION AND FRACTURE IN MATERIALS AND STRUCTURES SOLID MECHANICS AND ITS APPLICATIONS Volume 84 Series...

Author: T-J. Chuang | J.W. Rudnicki

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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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©2002 Kluwer Academic Publishers New York, Boston, 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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EDITORS’ PREFACE

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

EDITORS’ PREFACE

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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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EDITORS’ PREFACE

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

EDITORS’ PREFACE

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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

49

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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TABLE OF CONTENTS

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

311

TABLE OF CONTENTS

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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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BIOGRAPHY OF J. R. RICE

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

1. 2.

3.

4. 5.

6. 7.

8.

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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.

xxviii 13.

14. 15.

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21. 22.

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24. 25.

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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.

29.

30. 31.

32. 33.

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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.

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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.

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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.

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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.

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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.


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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.

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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.


164. 165.

166.

167. 168.

169.

170.

171.

172.

173.

174.

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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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


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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

YIELD AND FAILURE OF SHEET METALS

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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


5

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.


11

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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W.Y.CHIEN, ET AL.

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.


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.


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.

PLASTICITY THEORY FOR ALUMINUM SHEETS

19

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


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)


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.


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


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.

30

B. CHEN ET AL.

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.

J.-S. WANG

32

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

HYDROGEN EMBRITTLEMENT IN Fe SINGLE CRYSTALS

33

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.

34

J.-S. WANG

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.


35

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)

36

J.-S. WANG

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.


37

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.

J.-S. WANG

38

(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 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


43

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].

Vehoff H. and Rothe W., Gaseous hydrogen embrittlement in FeSi- and Nisingle crystals, Acta metall., 1983, 31, 1781-1793. Vehoff H. and Klameth H.-K., Hydrogen embrittlement and trapping at crack tips in Ni-single crystals, Acta metall., 1985, 33, 955-962. Miyata K; Igarashi M., Effect of ordering on hydrogen embrittlement of Ni-Cr alloy, Mater Trans JIM 1996, 37, 703-710 Walston W.S; Bernstein I.M; Thompson A.W., The effect of internal hydrogen on a single-crystal nickel-base superalloy, Met Trans A, 1992, 23A, 13131322. Vehoff H., and Neumann P., Crack propagation and cleavage initiation in Fe2.6%Si single crystals under controlled plastic crack tip opening rate in various gaseous environments, Acta metall., 1980, 28, 265-272. Vehoff H., Rothe W., and Neumann P., The influence of gaseous hydrogen on the fracture process in Fe-2.6%Si single crystals, in Proceedings of ICF5, Vol. 1, pp. 265-271, Cannes, 1981. Katz Y; Chen X.; Lii M.J.; Lanxner M.; Gerberich W.W., The anisotropic nature of local crack stability in bcc crystals, Eng Frac Mech, 1992, 41, 541567. Terasaki F; Kawakami T; Yoshikawa A; Takano N., Mechanism of crack propagation due to hydrogen embrittlement in iron single crystals stressed along [001] axis, Rev de Metall-Cahiers D, 1998, 95, 1519-1529.

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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.

46 [29]. [30] [31].

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[34]. [35]. [36]. [37]. [38].

[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.


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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H. RIEDEL AND B. BLUG

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.

A MODEL FOR SOLID STATE SINTERING

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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


54

(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)


55

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


57

(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 ( 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.

P. M. ANDERSON AND X.J. XIN

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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.

STRESS TO TRANSMIT A SCREW DISLOCATION

101

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).


103

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.


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


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,


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

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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

ELASTIC SPACE WITH AN INCLUSION

125

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


127

(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”:


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)


(2.32)


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:


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)


(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)


135

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

(3.10)


(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.

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(3.13)

where


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)


137

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)


(4.6)

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

138

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)


139


(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)


141

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”:


143 (5.16)


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.


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-

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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/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

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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.


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,


Figure 1:

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

by (2) Thus, H ≈ H tan, for 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 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 =


297

{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

298


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


Figure 8.

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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


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

301

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.


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


Figure 15.

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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.

308


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.


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).

317


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).


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


321

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.


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).

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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.


327

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


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.

STRESS-ASSISTED CORROSIVE CRACK GROWTH

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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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Z. TANG, A. BOWER AND T.-J. CHUANG TABLE 1: Representative values for material properties

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


Fig. 2.

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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).


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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A. A. RUBINSTEIN (2)

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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A. A. RUBINSTEIN

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

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