Fiber Bundle Model

Alex Hansen earned his Ph.D. from Cornell University in 1986. He was then Joliot-Curie fellow at the Ecole Normale Supérieure in Paris, followed by postdoctoral positions at the Universities of Cologne and Oslo. In 1992, Hansen became a CNRS scientist at the University of Rennes 1 in France. From 1994, he has been professor of physics at the Norwegian University of Science and Technology in Trondheim, Norway. Hansen is member of the Norwegian Academy of Science and Letters, the Royal Norwegian Society of Science and Letters and theNorwegian Academy of Technical Sciences. He is Dr. honoris causa at the University of Rennes 1. Hansen chairs at present the Commission on Computational Physics of the International Union of Pure and Applied Physics, where he is also a vice president.Per Christian Hemmer earned his doctorate from the Norwegian Institute of Technology in 1959. He was then NORDITA fellow in Copenhagen, followed by a postdoctoral position at The Rockefeller Institute, New York. In 1959 he was appointed professor of theoretical physics at the University of Trondheim, now the Norwegian University of Science and Technology. He has been dean of the faculty for general sciences at the university. Hemmer has been secretary of the Commission on Statistical Physics of the International Union of Pure and Applied Physics, as well as a vice president of the union. He is member of the Royal Norwegian Society of Science and Letters, the Norwegian Academy of Science and Letters, and the Norwegian Academy of Technical Sciences.Srutarshi Pradhan completed his Ph.D. work at Saha Institute of Nuclear Physics, Kolkata, India in 2004 and earned his doctorate from Jadavpur University, Kolkata, India . He was then a postdoctoral fellow at the Norwegian University of Science and Technology for 4 years (2004-2007) with a fellowship from the Norwegian Research Council. In 2008 he was appointed as a research scientist at SINTEF Petroleum Research, Trondheim, Norway. Pradhan has been promoted to senior scientist position at SINTEF Petroleum Research in 2012.

• Preface XIII1 The Fiber Bundle Model 11.1 Rivets VersusWelding 11.1.1 What Are Models Good For? 31.2 Fracture and Failure: A Short Summary 41.3 The Fiber Bundle Model in Statistics 51.4 The Fiber Bundle Model in Physics 61.5 The Fiber Bundle Model in Materials Science 81.6 Structure of the Book 82 Average Properties 112.1 Equal Load Sharing versus Local Load Sharing 112.2 Strain-Controlled versus Force-Controlled Experiments 122.3 The Critical Strength 162.4 Fiber Mixtures 222.5 Non-Hookean Forces 242.5.1 Fibers with Random Slacks 242.5.2 Elastic–Plastic Model 253 Fluctuation Effects 273.1 Range of Force Fluctuations 283.2 The Maximum Bundle Strength 303.3 Avalanches 323.3.1 The Burst Distribution 333.3.1.1 The Forward Condition 353.3.1.2 The Backward Condition 363.3.1.3 Total Number and Average Size of Bursts 373.3.2 Asymptotic Burst Distribution: 5/2 Law 403.3.2.1 Asymptotic Burst Distribution: Nongeneric Cases 423.3.3 Inclusive Bursts 453.3.4 Forward Bursts 473.3.5 Avalanches as RandomWalks 493.3.5.1 The Exact RandomWalk 493.3.5.2 Asymptotic Burst Distribution via RandomWalks 513.3.6 Energy Release 533.3.6.1 High-Energy Asymptotics 533.3.6.2 Low-Energy Behavior 553.3.7 Failure Avalanches for Stepwise Load Increase 573.3.7.1 UniformThreshold Distribution 583.3.7.2 General Threshold Distribution 594 Local and Intermediate Load Sharing 634.1 The Local-Load-Sharing Model 644.1.1 Redistribution of Forces 664.1.2 Determining the Failure Sequence 674.1.3 Bundle Strength 684.1.4 Failure of First and Second Fibers 694.1.4.1 Other Threshold Distributions 754.1.4.2 Localization 764.1.5 Hole Size Distribution 784.1.5.1 Defining the Hole Size Distribution 784.1.5.2 Hole Size Distribution for Equal Load Sharing 794.1.5.3 Hole Size Distribution for Local Load Sharing: Localization 804.1.6 Estimating the Strength of the Local-Load-Sharing Model 824.1.6.1 Lower Bound for the Largest Hole 834.1.6.2 Competing Failure Mechanisms 854.1.6.3 UniformThreshold Distribution 864.1.7 Force and Elongation Characteristics 884.1.8 Burst Distribution 904.2 Local Load Sharing in Two and More Dimensions 934.2.1 Localization 944.2.2 Similarity with the Equal-Load-Sharing Fiber Bundle Model 954.2.3 Burst Distribution 994.2.4 Upper Critical Dimension 994.3 The Soft Membrane Model 1014.4 Intermediate-Load-Sharing Models 1044.4.1 The γ-Model 1054.4.2 The Mixed-Mode Model 1064.5 Elastic Medium Anchoring 1074.5.1 Size Equals Stiffness 1094.5.2 Localization in the Soft Clamp Model 1094.5.3 Asymptotic Strength 1104.5.4 Fracture Front Propagation 1125 Recursive Breaking Dynamics 1155.1 Recursion and Fixed Points 1165.2 Recursive Dynamics Near the Critical Point 1195.2.1 Universality 1205.2.1.1 General Threshold Distribution 1245.2.2 Postcritical Relaxation 1255.2.2.1 UniformThreshold Distribution 1265.2.2.2 General Threshold Distribution 1295.2.3 Precritical Relaxation 1315.2.3.1 UniformThreshold Distribution 1315.2.3.2 General Threshold Distribution 1335.2.4 Critical Amplitudes 1346 Predicting Failure 1376.1 Crossover Phenomena 1386.1.1 The Avalanche Size Distribution 1386.1.1.1 Burst Avalanches at Criticality 1416.1.2 Energy Bursts 1426.1.3 The Crossover Phenomenon in Other Systems 1446.1.3.1 Earthquakes 1446.1.3.2 The Fuse Model 1486.2 Variation of Average Burst Size 1526.3 Failure Dynamics Under Force-Controlled Loading 1526.4 Over-Loaded Situations 1556.4.1 Breaking Rate of Loaded Fiber Bundles 1556.4.1.1 Uniform Distribution 1556.4.1.2 Weibull Distribution 1576.4.1.3 Large Overload Situation 1586.4.2 Energy Emission Bursts 1596.4.2.1 Uniform Distribution 1616.4.2.2 Weibull Distribution 1626.4.2.3 Large Overload Situation 1646.4.3 Energy Burst Pattern 1667 Fiber Bundle Model in Material Science 1697.1 Repeated Damage andWork Hardening 1707.1.1 The Load Curve 1717.1.2 Repeated Damages 1717.1.3 Damages Ending in Complete Bundle Failure 1747.2 Creep Failure 1757.2.1 A Model for Creep 1767.2.2 A Second Model for Creep 1807.2.2.1 Damage Accumulation 1807.2.2.2 Time Evolution 1817.2.2.3 Healing 1827.3 Viscoelastic Creep 1837.4 Fatigue Failure 1867.4.1 A Fatigue Experiment 1867.5 Thermally Induced Failure 1887.5.1 Failure Time for a Homogeneous Fiber Bundle 1887.5.2 Failure Time for Low Thermal Noise 1897.6 Noise-Induced Failure 1907.7 Crushing: The Pillar Model 1948 Snow Avalanches and Landslides 1978.1 Snow Avalanches 1978.2 Shallow Landslides 199Appendix A Mathematical Toolbox 203A.1 Lagrange’s InversionTheorem 203A.2 SomeTheorems in Combinatorics 204A.2.1 Basic SelectionTheorems 204A.2.2 A Distribution with Restrictions 205A.3 Biased RandomWalks 206A.3.1 Probability of No Return 206A.3.2 Gambler’s Ruin 207A.4 An Asymmetrical Unbiased RandomWalk 209A.5 Brownian Motion as a Scaled RandomWalk 211Appendix B Statistical Toolbox 213B.1 Stochastic Variables, Statistical Distributions 213B.1.1 Change of Variable 213B.1.1.1 A Useful Interpretation 213B.1.2 The Characteristic Function 214B.1.3 The Central LimitTheorem 215B.2 Order Statistics 216B.2.1 Ordering the Variables 216B.2.2 The Average of the mth Ordered Variable 217B.3 The Joint Probability Distribution 217B.3.1 Extreme Statistics 218B.3.2 The Largest Element: The Three Asymptotic Distributions 218B.3.2.1 The Gumbel Distribution 219B.3.2.2 The Second and Third Asymptotes 220B.3.3 The Smallest Element: TheWeibull Distribution 221Appendix C Computational Toolbox 223C.1 Generating Random Numbers Following a Specified Probability Distribution 223C.1.1 When the Cumulative Probability May Be Inverted 223C.1.2 Gaussian Numbers 224C.1.3 None of the Above 225C.2 Fourier Acceleration 225References 229Index 233