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 mono- graphs defining the current state of the field; others are accessible to final year under- graduates; 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, B OSTON, DORDRECHT,L ONDON, MOSCOW eBook ISBN 0-306-46952-9 Print ISBN 0-792-36718-9 ©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: http://www.kluweronline.com and Kluwer's eBookstore at: http://www.ebooks.kluweronline.com 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. vi EDITORS’ PREFACE FRACTURE: Arguably, Rice’s most well-known contribution is the 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 vii 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. DEFORMATION: Another major contribution of Rice has been the 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. viii 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 ix 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 J. W. RUDNICKI, Evanston, IL 25 August 2000