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,BOSTON, DORDRECHT,LONDON, 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