Probabilistic fracture mechanics and reliability ENGINEERING APPLICATION OF FRACTURE MECHANICS Editor-in-Chie/: George C. Sih G.C. Sih and L. Faria (eds.), Fracture mechanics methodology:-Evalua tion of structure components integrity. 1984. ISBN 90-247-2941-6. E.E. Gdoutos, Problems of mixed mode crack propagation. 1984. ISBN 90-247-3055-4. A. Carpinteri and A.R. Ingraffea (eds.), Fracture mechanics of concrete: Material characterization and testing. 1984. ISBN 90-247-2959-9. G.C. Sih and A. DiTommaso (eds.), Fracture mechanics of concrete: Structural application and numerical calculation. 1984. ISBN 90-247-2960-2. A. Carpinteri, Mechanical damage and crack growth in concrete: Plastic collapse to brittle fracture. 1986. ISBN 90-247-3233-6. J.W. Provan (ed.), Probabilistic fracture mechanics and reliability. 1987 ISBN 90-247-3334-0. Probabilistic fracture mechanics and reliability Edited by James W. Provan Associate Dean, Faculty of Engineering and Director, Fracture Control Laboratory, Department of Mechanical Engineering, McGill University, Montreal, Quebec H3A 2K6 Canada 1987 SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. _ Library of Congress Cataloging in Publication Data Probabilistic fracture mechanics and reliability. (Engineering application of fracture mechanics 6) Bibliography: p. Includes index. 1. Fracture mechanics. 2. Probabilities. 3. Reliability (Engineering) I. Provan, J. W. (James W.) II. Series. TA409.P747 1986 620.1'126 86-8445 ISBN 978-90-481-8297-8 ISBN 978-94-017-2764-8 (eBook) DOI 10.1007/978-94-017-2764-8 . Copyright © 1987 Springer Science+Business Media Dordrecht Originally published by Martinus Nijhoff Publishers, Dordrecht in 1987 Softcover reprint of the hardcover 1s t edition 1987 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photpcopying, recording, or otherwise, without the prior written permission of the pubiishers, Springer-Science+Business Media, B.V. Contents Preface XIII List of Contributors XV Chapter 1. Probabilistic .approaches to the matedal-related reliability of fracture-sensitive structures J. W. Provan 1.1 Introduction 1 1.1.1 Introductory remarks 1 1.1.2 Reliability: general considerations 3 1.1.3 Review of fatigue reliability models 4 1.1.3.1 The exponential distribution 4 1.1.3.2 The normal of Gaussian distribution 5 1.1.3.3 The log-normal distribution 6 1.1.3.4 The gamma distribution 7 1.1.3.5 The Weibull distribution 8 1.1.3.6 The Gumbel (extreme-value) distributions 10 1.1.3.7 The Birnbaum-Saunders distribution 12 1.1.3.8 Other reliability distributions 14 1.1.4 The hazard rate concept 14 1.2 P-S-N Analysis 15 1.1.2 Introductory remarks 15 1.2.2 P-S-N diagram 15 1.2.3 Reliability when the cycles-to-failure are dependent on the initial strength of the component 17 1.2.4 Time dependent stress-strength 19 1.2.5 Further considerations 19 1.3 Stochastic crack growth 22 1.3.1 Introduction 22 1.3.2 Stochastic crack propagation 22 1.3.3 Significantly weaker spot stochastic crack growth 26 v Contents 1.4 The micromechanics approach to fatigue failure 27 1.4.1 Introduction 27 1.4.2 The foundation of micromechanics 27 1.4.2.1 The micromechanic axioms pertaining to fatigue failure 27 1.4.2.2 Experimental and theoretical studies on the elastic response of metals 29 1.4.3 Fatigue crack initiation 31 1.4.4 Fatigue crack propagation 32 1.4.5 Fatigue crack experimental investigations 35 1.4.6 Comparison of theoretical and experimental results 36 1.5 A fatigue reliability law based on probabilistic micromechanics 37 1.5.1 Introductory remarks 37 1.5.2 The micromechanics fatigue reliability relation 38 1.5.3 An experimental investigation of fatigue reliability laws 39 1.6 Concluding remarks 44 Chapter 2. Probabilistic damage tolerance analysis of aircraft structures B. Palmberg, A. F. Blom and S. Eggwertz 2.1 Introduction 47 2.2 Basic assumptions 48 2.3 Load spectra 49 2.3.1 Specification of loads and environment 49 2.3.2 Acquisition of load spectra in service 50 2.3.3 Load sequences, truncations 52 2.3.4 Scatter considerations 53 2.4 Stress and stress intensity analysis 54 2.4.1 Structural models 54 2.4.2 Available handbook and literature results 58 2.4.3 Disturbances such as residual stresses, contact stresses, friction 60 2.5 Imperfections 62 2.5.1 Classification of imperfections 62 2.5.2 Quality control 62 2.5.3 Equivalent initial flaw sizes 63 2.5.4 US military damage tolerance requirements 67 2.6 Crack growth 68 2.6.1 Constant amplitude crack growth rate 68 2.6.f Variable amplitude crack growth rate 70 2.6.3 Variability in crack growth rate 72 2.6.4 Predicting fatigue crack growth 73 2.6.5 Stochastic modelling of crack growth 77 2.7 Fracture mechanics and residual strength 86 2.7.1 Material behaviour 86 VI Contents 2.7.2 Linear elastic fracture mechanics 89 2.7.3 Nonlinear fracture mechanics 92 2.7.4 Scatter in fracture toughness 95 2.7.5 Probability of failure 96 2.8 Inspection during service life 102 2.8.1 Damage sources and inspection procedures 102 2.8.2 Probability of crack detection by NDI methods 102 2.8.3 Length of intervals 106 2.8.4 US military damage tolerance requirements 107 2.8.5 Crack size distribution after multiple inspections 108 2.9 Structural safety 113 2.9.1 Discussion of USAF damage tolerance requirements 113 2.9.2 Residual strength of structures 115 2.9.3 Safety analysis 116 2.9.4 Numerical example 118 2.10 Concluding remarks 128 Chapter 3. Aircraft structural reliability and risk analysis F. H. Hooke 3.1 Introduction 132 3.1.1 The reliability concept 133 3.2 Basic reliability and risk mathematics 135 3.2.1 Mixed population with different risks 137 3.3 Physical aspects of structural failure 139 3.3.1 Loading actions 139 3.3.2 Structural behaviour 140 3.4 Mathematical-statistical model representing a real structural situation 142 3.5 Reliability without inspections - the safe life situation 146 3.5.1 Risk of static ultimate load failure 147 3.5.2 Risk of failure with deteriorating strength 148 3.5.3 Partitioning the risk 149 3.5.4 Averaging the instantaneous risk 152 3.5.5 Truncation of the strength - time curve 152 3.5.6 Structures with initial cracks 154 3.5.7 Structures with multiple failure modes and locations 155 3.6 Risk of failure with inspectable structures 156 3.7 Illustrative examples 157 3.7.1 Safe life situation - ultra high strength material 158 3.7.2 Safety-by-inspection situation: typical aluminium alloy material 160 3.7.2.1 Distribution of strength with virgin strength preserved 160 3.7.2.2 Change of strength distribution with time- deteriorating strength 161 3.7.2.3 Safety by inspection 162 VII Contents 3.8 Acceptable risk 164 3.9 Reliability of reliability estimates 165 3.9.1 Confidence intervals related to the sampling of H 166 3.9.2 Confidence regions related to the sampling of (11og H 166 3.9.3 Confidence intervals related to the sampling of Uo 167 3.9.4 Confidence in relation to extrapolation of the spectrum 167 3.9.5 Confidence related to safe-by-inspection structures 168 3.10 General discussion 168 Chapter 4. Stochastic crack growth models for applications to aircraft structures J. N. Yang, W. H. Hsi, S. D. Manning, and J. L. Rudd 4.1 Introduction 171 4.2 Stochastic models for fatigue crack propagation 173 4.2.1 Stochastic crack-propagation model 173 4.2.2 Fatigue crack growth data in fastener holes 174 4.2.3 Lognormal crack growth rate model and analysis procedures 176 4.2.4 Lognormal random process model 181 4.2.5 Lognormal white noise model 182 4.2.6 Lognormal random variable model 183 4.2.7 Correlation with experimental test results 186 4.2.7.1 General lognormal random process model 186 4.7.2.2 Lognormal random variable model 189 4.3 Second moment approximation 191 4.3.1 Mean and standard deviation of W(t) 194 4.3.2 Weibull approximation 195 4.3.3 Gamma and other approximations 196 4.3.4 Correlation between second moment approximations and experimental test results 197 4.4 Fatigue crack propagation in center-cracked specimens 198 4.4.1 Synergistic sine hyperbolic crack growth rate function 201 4.4.2 Stochastic models and second moment approximations 202 4.4.3 Correlations with experimental test results 203 4.5 Factors affecting probabilistic prediction of fatigue crack propagation 206 4.5.1 Fatigue crack growth analysis procedures 206 4.5.2 Equal number of data points for each specimen 207 4.5.3 Data processing procedures 208 4.6 Conclusions and discussions 209 Chapter 5. Durability of aircraft structures S. D. Manning, J. N. Yang, and J. L. Rudd 5.1 Introduction 213 5.2 Durability design requirements 214 5.2.1 Analytical requirements 215 5.2.2 Experimental requirements 216 VIII Contents 5.3 Durability analysis criteria 216 5.3.1 Durability critical parts criteria 216 5.3.2 Economic life criteria/guidelines 217 5.4 Durability analysis methodology 219 5.4.1 General description 220 5.4.2 Assumptions and limitations 220 5.4.3 Initial fatigue quality (IFQ) model 221 5.4.3.1 IFQ model equations for Case I (b > I) 224 5.4.3.2 IFQ model equations for Case II (b = 1) 225 5.4.4 Durability analysis procedures 226 5.5 Durability analysis details 227 5.5.1 EIFS distribution 228 5.5.2 Test/fractographic guidelines 229 5.5.2.1 Test guidelines 230 5.5.2.2 Guidelines for fractographic data 232 5.5.3 Fractographic data pooling concepts 232 5.5.4 Determination of EIFSD parameters 234 5.5.4.1 General concepts and guidelines 235 Qr 5.5.4.2 Estimation of 236 5.5.4.3 Estimation of a.;, P; and 8; 237 5.5.4.4 Determination of a. and QP 238 5.5.4.5 Estimation of EIFSD parameters and evaluation of goodness-of-fit 239 5.5.5 Statistical scaling of P parameter 245 5.5.6 Probability of crack exceedance 247 5.5.6.1 Service crack growth master curve 247 5.5.6.2 Crac~ exceedance predictions 249 5.5.7 Formats for presenting durability analysis results 251 5.6 Durability analysis demonstration 251 5.6.1 Fighter lower wing skins 251 5.6.2 Complex-splice specimens subjected to bomber load spectrum 259 5.7 Comparison of deterministic and probabilistic approaches for durability analysis 263 5.7.1 Deterministic crack growth approach 263 5.7.2 Probabilistic approach 264 5.7.3 Conceptual comparisons 264 5.7.3.1 Durability analysis based on DCGA 265 5.7.3.2 Durability analysis based on probabilistic approach 265 5.7.4 Conclusions 266 5.8 Summary and concluding remarks 267 Chapter 6. The reliability of pressurized water reactor vessels R. F. Cameron, G. o. Johnston, and A. B. Lidiard 6.1 Introduction 269 IX Contents 6.2 Statistics of pressure vessel failure 272 6.2.1 Nuclear primary circuit 274 6.2.2 Non-nuclear vessels 275 6.3 General physical aspects 277 6.3.1 Cracks - their causes, detection and repair 281 6.3.1.1 Causes and incidence of cracks 282 6.3.1.2 Detection of cracks by non-destructive examination 283 6.3.1.3 Repair of cracks 283 6.3.2 Material toughness and failure mechanisms 283 6.3.3 Crack growth by fatigue 287 6.3.4 Transient loadings 290 6.4 Mathematical formulation 291 6.4.1 General expressions for the failure integral 293 6.4.2 Deterministic crack growth 296 6.4.3 Effect of previous loading: the 'cold hydrotest' 298 6.5 The distribution functions of physical quantities 299 6.5.1 The initial crack-size distribution function 299 6.5.1.1 Cracks arising in manufacture 300 6.5.1.2 Efficiency of the detection of cracks 302 6.5.2 Fracture toughness and flow stress 304 6.5.3 Fatigue crack growth and the transfer function 306 6.6 Applications 307 6.6.1 U.S. and European L.W.R.pressure vessel calculations 308 6.6.2 Survey of results and conclusions 309 6.6.2.1 Absolute failure rates 309 6.6.2.2 Dependence of failure rate upon time in servite 311 6.6.2.3 Sensitivity to the crack incidence function 312 6.6.2.4 Sensitivity to fracture toughness 314 6.6.2.5 Sensitivity to stress-intensity function and failure condition 315 6.6.2.6 Sensitivity to crack growth rates 316 6.6.2.7 Relative effects of different transients 317 6.6.2.8 Relative contributions of different regions of the vessel 317 6.6.2.9 In-service inspection 318 6.7 Conclusion 318 Appendix 6.1 Failure criterjon in elastic-plastic fracture mechanics (the R6 method) 320 Appendix 6.2 320 Appendix 6.3 Example calculation of ~ 321 Chapter 7. Applications of PFM in the nuclear industry to reactor pressure vessel, main coolant piping and steel containment R. Wellein 7.1 Introduction 325 x