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ANALYTICAL FRACTURE MECHANICS
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ANALYTICAL FRACTURE MECHANICS David J. Unger Department of Mechanical Engineering and Engineering Mechanics Michigan Technological University Houghton, Michigan
ACADEMIC PRESS
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This book is printed on acid-free paper. (~ Copyright 9 1995 by ACADEMIC PRESS, INC. All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any inforrnation storage and retrieval system, without permission in writing from the publisher. A c a d e m i c Press, Inc. A Division of Harcourt Brace & Company 525 B Street, Suite 1900, San Diego, California 92101-4495
United Kingdom Edition published bv Academic Press Limited 24-28 Oval Road, London NWI 7DX Library of Congress Cataloging-in-Publication Data Unger, David J. Analytical fracture mechanics / by David J. Unger p. cm. Includes bibliographical references and index. ISBN 0-12-709120-3 (acid-free paper) 1. Fracture mechanics. 2. Mechanics, Analytic. TA409.U64 1995 620.1'126--dc20
I. Title. 95-2193 CIP
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To my mother and father, Stella and Anthony Unger
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Contents
Preface xi Acknowledgments
xiii
Introduction 1.1 Equations of Continuum Mechanics 1 Equilibrium 1 Strain-Displacement 4 Change of Volume 4 Compatibility of Strains 4 1.2 Equations of Elasticity 5 1.3 Equations of Plasticity 6 Strain Hardening 9 Material Stability 11 Incremental Strain-Stress Relationships 12 Flow Theory versus Deformation Theory 16 1.4 Plane Problems of Elasticity Theory 16 Cartesian Coordinates 17 Polar Coordinates 19 Kolosov Equations 21 Boundary Conditions 23 1.5 Linear Elastic Fracture Mechanics 23 Mode I 25 Mode II 28 Mode III 29 1.6 Strip Models of Crack Tip Plasticity 34 Mode I Small-Scale Yielding Strip Model 39 Mode III Small-Scale Yielding Strip Model 41 vii
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1.7 Exact Elastoplastic Solutions for Mode III 45 Isotropic Hardening 52 1.8 Plane Strain Problems Involving Plastic Theory 63 Plane Strain 63 Prandtl-Hill Solution 75 Failure Criteria 82 Power Law Hardening Materials under Plane Strain 90 1.9 Plane Stress Problems Involving Plastic Material 94 Tresca Yield Condition 95 Mises Yield Condition 99 Power Law Hardening Materials under Plane Stress 105 1.10 Numerical Solutions of the Mode I Elastoplastic Problem 105 Numerical Solutions 108 1.11 Miscellaneous Mathematical Topics 114 Complete Solutions 114 Monge-Ampere Family of Partial Differential Equations 118 Order Symbols 120
1 On the Continuance of an Analytical Solution across the E l a s t i c - P l a s t i c Boundary of a M o d e I Fracture Mechanics Problem 1.1 Elastoplastic Stress Analyses for Modes I and III 122 Mode III 122 Mode I 127 Parabolic-Hyperbolic Plastic Boundary 136 Uniqueness and Continuity of Stress 137 Stress Discontinuities 142 1.2 Developable Surfaces 143 1.3 Strain Rates for Plane Stress under the Tresca Yield Condition 147 1.4 Mode I Displacements 150 1.5 Speculations Concerning an Analytical Mode I Elastoplastic Solution 163 Color Plates 168
2 Plastic Zone Transitions 2.1 A Finite-Width Dugdale Zone Model for Mode III Elastic Solution 172 Plastic Solution 178
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Recovery of Previous Solutions 183 Crack Tip Opening Displacement 185 Comments 186 2.2 An Energy-Dissipation Analysis for the Transition Model 2.3 Effective Crack Length for the Transition Model 193 2.4 Fracture Assessment Diagrams 197
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3 Environmental Cracking 3.1 Hydrogen-Assi