The Practical Use of Fracture Mechanics

The Practical Use of Fracture Mechanics

by

DAVID BROEK FractuREsearch Inc., Galena, OR, USA

Kluwer Academic Publishers Dordrecht / Boston / London

Library of Congress Cataloging in Publication Data

Brock, David. The pract,ical use of fr'actUJ'e mechanics.

Bibliography: p. 1. Fract,ure mechanics. 1. TiUe.

TA409.B773 1988 620.1'126 88-9336

ISBN-13: 978-0-7923-0223-0 e-ISBN-13: 978-94-009-2558-8 DOI: 10,1007/978-94-009-2558-8

Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands

Kluwer Academic Publishers incorporates the publishing programmes of

D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press

Sold and distributed in the U.S.A and Canada by Kluwer Academic Publishers,

101 Philip Drive, Norwell, MA 02061, U.S.A.

In all other countries, sold and distributed by Kluwer Academic Publishers Group,

P.O. Box 322, 3300 AH Dordrecht, The Netherlands.

first published in hardbound edition only in 1988 second printing with minor corrections in 1989

third printing 1991 reprinted 1994 reprinted 1996 reprinted 1997

hardback edition ISBN 90-247-3707-9 paperback edition ISBN 90-247-0223-0

printed Oil acid.ti·ee paper

All Rights Reserved

© 1989 by Kluwer Academic Publishers Softcover reprint of the hardcover 1st edition 1989

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical,

including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

Preface

This book is about the use of fracture mechanics for the solution of practical problems; academic rigor is not at issue and dealt with only in as far as it improves insight and understanding; it often concerns secondary errors in engineering. Knowledge of (ignorance of) such basic input as loads and stresses in practical cases may cause errors far overshadowing those introduced by shortcomings of fracture mechanics and necessary approximations; this is amply demonstrated in the text.

I have presented more than three dozen 40-hour courses on fracture mechanics and damage tolerance analysis, so that I have probably more experience in teaching the subject than anyone else. I learned more than the students, and became cognizant of difficulties and of the real concerns in applications. In particular I found, how a subject should be explained to appeal to the practicing engineer to demonstrate that his practical problem can indeed be solved with engineering methods. This experience is reflected in the presenta­tions in this book. Sufficient background is provided for an understanding of the issues, but pragamatism prevails. Mathematics cannot be avoided, but they are presented in a way that appeals to insight and intuition, in lieu of formal derivations which would show but the mathematical skill of the writer. A practicing engineer does care little about how a crack tip stress field is derived; he accepts that it can be done, as long as he can understand that the result must be of the form it is. His real concern is what it means for the solutions to practical problems.

Mathematical background is of use to future scientists, but few engineering students taking fracture mechanics courses will become researchers in fracture mechanics. My advice is that indeed very few should. Fracture mechanics has matured to a useful engineering tool as has e.g. buckling analysis. Certainly, it is not perfect, but no engineering analysis is. Not much buckling research is practiced today; the present number of researchers in fracture mechanics is far out of proportion to the remaining engineering problems.

Despite the acclaimed solid education of engineers, it is my experience in teaching fracture mechanics to literally hundreds of practicing engineers, that most have only a vague idea of such subjects as plastic deformation and design; to many Mohr's circle is an enigma; at most one in a class knows the stress concentration factor of a circular hole; fewer even remember yield criteria and their significance. For this reason Chapter 2 of the text discusses the effect of notches and local yielding and provides a simplified look at yield

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criteria. The treatment is necessarily compromising rigidity, but it serves the purpose of providing the insight without which fracture mechanics cannot be understood.

My research work covered fundamental fracture and fatigue mechanisms, experimental evaluation of criteria for fatigue, fracture, and combined mode loading, the development of engineering procedures for arrest analysis in stiffened panels, collapse conditions, and damage tolerance analysis in general. My engineering background however, has always prevailed and forced me to consider the practicality of procedures. This book reflects a lifetime of experience in research and practical applications. No subject is discussed on the basis of hearsay. Instead the basis is "hands-on" experience with virtually every issue from the fundamental to the practical.

I am aware of my shortcomings, prejudices and opinionations, but believe to be entitled to these on the basis of my engineering experience. This text reflects them, and I do not apologize. Too many "refinements" in engineering solutions pertain to secondary errors; they increase the complexity, but do not improve the solution. One does not improve the strength of a chain by improving the strong links. The weak links in the fracture mechanics analysis are the unknowns, not the procedures. This book is for engineering students and for engineers, who must solve urgent problems yesterday. Engineering solutions are always approximative, no matter what the subject is. Such is the nature of engineering. Necessary assumptions are far more influential than those due to limitations of fracture mechanics.

The text is intended for the education of 'engineers'. At the same time it serves as a reference. For this reason there is some duplication and extensive cross­references are provided. This may be objectionable to the reader going through the text from A to Z, but it will be of help to those who read sections here and there. It is not perfect as no human effort ever is, and I shall welcome construc­tive criticism with regard to the engineering applications. My haste in accom­plishing things (enforced by the unfortunate situation that I have to make a living, while writing a book is an extraneous effort which is not very profitable) may be reflected in the text. Again, I am not apologizing, just explaining.

I am grateful to my wife, Betty, for putting up with my preoccupations and moods while writing this text, and for submitting all writing to a word-process­or. I am also thankful to my son Titus, who spent numerous hours in producing solutions to exercises and in drawing figures.

I dedicate this book to the memory of my father, Harm Broek. Many sons see their father as the ultimate example. So do I. His unfailing support has always been a driver of my ambitions.

Galena, Ohio, February 1988

Notice

Extensive computer software for fracture mechanics analysis was developed by the author of this book. This software is capable of - performing residual strength analysis in accordance with Chapters 3, 4 and lO

both for LEFM and EPFM. - performing fatigue crack growth analysis for constant amplitude, random

loading and semi random loading in accordance with Chapters 5 and lO, with or without retardation. There are options for various retardation models, rate equations and tabular rate data (Chapter 7).

- automatically generating semi-random stress histories on the basis of exceedance diagrams (Chapter 6) and performing clipping and truncation upon command.

- determining inspection intervals and cumulative probability of detection in accordance with the procedure discussed in Chapter 11, using the calculated crack growth curves, and accounting for specificity and accessibility.

- providing professional plots. An extensive library of materials data is included, as well as an extensive

library of geometry factors. Besides a pre-processor can generate geometry factors, using most of the procedures discussed in Chapter 8.

The above software is available for personal computers. Because of the large size of the software, it is split up in seven modules, each of which fIts in a personal computer. The modules communicate through disket files that are generated automatically. The software can be obtained from FractuREsearch Inc, 9049 Cupstone Drive, Galena, OH 43021, USA.

A much simplified version of the same software (also by the author) is available from the American Society of Metals (ASM), Metals Park, OH 44073, USA. This simplified version has no data library, no preprocessor for geometry factor, cannot do retardation, and does not generate semi-random stress histories.

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Contents

Preface v

Notice VB

Chapter 1. INTRODUCTION 1.1. Fracture control 1 1.2. The two objectives of damage tolerance analysis 3 1.3. Crack growth and fracture 8 1.4. Damage tolerance and fracture mechanics 15 1.5. The need for analysis: purpose of this book 17 1.6. Exercises 20

Chapter 2. EFFECTS OF CRACKS AND NOTCHES: COLLAPSE 22 2.1. Scope 22 2.2. An interrupted load path 22 2.3. Stress concentration factor 25 2.4. State of stress at a stress concentration 28 2.5. Yielding at a notch 31 2.6. Plastic collapse at a notch 35 2.7. Fracture at notches: brittle behavior 41 2.8. Measurement of collapse strength 44 2.9. Exercises 46

Chapter 3. LINEAR ELASTIC FRACTURE MECHANICS 48 3.1. Scope 48 3.2. Stress at a crack tip 48 3.3. General form of the stress intensity factor 52 3.4. Toughness 55 3.5. Plastic zone and stresses in plane stress and plane strain 57 3.6. Thickness dependence of toughness 61 3.7. Measurement of toughness 67 3.8. Competition with plastic collapse 70 3.9. The energy criterion 73 3.10. The energy release rate 75 3.11. The meaning of the energy criterion 79 3.12. The rise in fracture resistance: redefinition of toughness 79 3.13. Exercises 86

ix

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Chapter 4. ELASTIC-PLASTIC FRACTURE MECHANICS 4.1. Scope 4.2. The energy criterion for plastic fracture 4.3. The fracture criterion 4.4. The rising fracture energy 4.5. The residual strength diagram in EPFM: collapse 4.6. The measurement of the toughness in EPFM 4.7. The parameters of the stress-strain curve 4.8. The h-functions 4.9. Accuracy 4.10. Historical development of J 4.11. Limitations of EPFM 4.12. CTOD measurements 4.13. Exercises

Chapter 5. CRACK GROWTH ANALYSIS CONCEPTS 5.1. Scope 5.2. The concept underlying fatigue crack growth 5.3. Measurement of the rate function 5.4. Rate equations 5.5. Constant amplitude crack growth in a structure 5.6. Load interaction: Retardation 5.7. Retardation models 5.8. Crack growth analysis for variable amplitude loading 5.9. Parameters affecting fatigue crack growth rates 5.10. Stress corrosion cracking 5.11. Exercises

Chapter 6. LOAD SPECTRA AND STRESS HISTORIES 6.1. Scope 6.2. Types of stress histories 6.3. Obtaining load spectra 6.4. Exceedance diagram 6.5. Stress history generation 6.6. Clipping 6.7. Truncation 6.8. Manipulation of stress history 6.9. Environmental effects 6.10. Standard spectra 6.11. Exercises

88 88 88 90 93 97 98

102 106 109 112 116 118 121

123 123 123 126 130 133 136 145 149 157 163 165

168 168 169 175 176 180 192 195 198 204 205 205

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Chapter 7. DATA INTERPRETATION AND USE 208 7.1. Scope 208 7.2. Plane strain fracture toughness 209 7.3. Plane stress and transitional toughness, R-curve 212 7.4. Toughness in terms of J and JR 214 7.5. Estimates of toughness 215 7.6. General remarks on fatigue rate data 218 7.7. Fitting the da/dN data 222 7.8. Dealing with scatter in rate data 232' 7.9. Accounting for the environmental effect 236 7.10. Obtaining retardation parameters 238 7:11. Exercises 241

Chapter 8. GEOMETRY FACTORS 243 8.1. Scope 243 8.2. The reference stress 244 8.3. Compounding 247 8.4. Superposition 249 8.5. A simple method for asymmetric loading cases 255 8.6. Some easy guesses 258 8.7. Simple solutions for holes and stress concentrations 260 8.8. Simple solutions for irregular stress distributions 267 8.9. Finite element analysis 271 8.10. Simple solutions for crack arresters and multiple elements 274 8.11. Geometry factors for elastic-plastic fracture mechanics 278 8.12. Exercises 279

Chapter 9. SPECIAL SUBJECTS 282 9.1. Scope 282 9.2. Behavior of surface flaws and corner cracks 282 9.3. Break through: leak-before-break 290 9.4. Fracture arrest 293 9.5. Multiple elements, multiple cracks, changing geometry 300 9.6. Stop holes, cold worked holes and interference fasteners 311 9.7. Residual stresses in general 316 9.8. Other loading modes: mixed mode loading 319 9.9. Composites 327 9.10. Exercises 329

Chapter 10. ANALYSIS PROCEDURES 332 10.1. Scope 332 10.2. Ingredients and critical locations 332

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10.3. Critical locations and flaw assumptions 10.4. LEFM versus EPFM 10.5. 10.6. 10.7. 10.8.

Residual strength analysis Use of R-curve and JR-curve Crack growth analysis Exercises

Chapter II. FRACTURE CONTROL ILl. Scope 11.2. Fracture control options 11.3. The probability of missing the crack 11.4. The physics and statistics of crack detection 11.5. Determining the inspection interval 11.6. Fracture control plans 11.7. Repairs 11.8. Statistical aspects 11.9. The cost of fracture and fracture control 11.1 O. Exercises

Chapter 12. DAMAGE TOLERANCE SUBSTANTIATION 12.1. Scope 12.2. Objectives 12.3. Analysis and damage tolerance substantiation 12.4. Options to improve damage tolerance 12.5. Aircraft damage tolerance requirements 12.6. Other requirements 12.7. Flaw assumptions 12.8. Sources of error and safety factors 12.9. Misconceptions 12.10. Outlook 12.11. Exercises

334 339 345 353 355 361

362 362 362 369 373 377 379 384 385 387 389

391 391 391 393 395 397 404 408 410 417 420 422

Chapter 13. AFTER THE FACT: FRACTURE MECHANICS AND FAILURE ANALYSIS 424

13.1. Scope 13.2. The cause of service fractures 13.3. Fractography 13.4. Features of use in fracture mechanics analysis 13.5. Use of fracture mechanics 13.6. Possible actions based on failure analysis 13.7. Exercises

424 425 428 430 436 440 440

Chapter 14. APPLICATIONS 14.1. Scope 14.2. Storage tank (fictitious example) 14.3. Fracture arrest in ships 14.4. Piping in chemical plant (fictitious example) 14.5. Fatigue cracks in railroad rails 14.6. Underwater pipeline 14.7. Closure

Chapter 15. SOLUTIONS TO EXERCISES

SUBJECT INDEX

xiii

443 443 443 447 462 465 476 483

485

515