Statistical Physics of Fracture, Breakdown, and Earthquake

Soumyajyoti Biswas got his PhD in 2015 for works carried out at Saha Institute of Nuclear Physics. He has been a postdoctoral fellow at the Institute of Mathematical Sciences, Chennai until March 2015. Presently he is a postdoctoral fellow at the Earthquake Research Institute, University of Tokyo.Purusattam Ray is professor in Physics at the Institute of Mathematical Sciences (IMSC), Chennai and an adjunct professor of Homi Bhabha National Institute (HBNI), Mumbai. He received his Ph.D. from Calcutta University in 1989. He was then SERC research fellow at the University of Manchester, England and subsequently Max Planck fellow at the MPI for Polymer Studies in Mainz and at the University of Mainz, Germany. He has made major contributions in the study of statistical physics of fracture. He annually organizes the International workshop on fracture and breakdown processes.Bikas K. Chakrabarti is a senior professor of theoretical condensed matter physics at the Saha Institute of Nuclear Physics (SINP), Kolkata, and a visiting professor of economics at the Indian Statistical Institute, Kolkata, India. He received his doctorate in physics from Calcutta University in 1979 (for research at SINP). Following postdoctoral positions at Oxford University and Cologne University, he joined SINP in 1983. His main research interests include physics of fracture, quantum glasses, etc., and the interdisciplinary sciences of optimization, brain modeling, and econophysics. He has written several books and reviews on these topics.

• Series Editors’ Preface XIIIPreface XVNotations XVII1 Introduction 12 Mechanical and Fracture Properties of Solids 72.1 Mechanical Response in Materials 82.1.1 Elastic and Plastic Regions 82.1.2 Linear Elastic Region 92.1.3 Nonlinear Plastic Region 102.2 Ductile, Quasi-brittle, and Brittle Materials 112.3 Ductile and Brittle Fracture 112.3.1 Macroscopic Features of Ductile and Brittle Fractures 122.3.2 Microscopic Features of Ductile and Brittle Fractures 143 Crystal Defects and Disorder in LatticeModels 173.1 Point Defects 173.2 Line Defects 183.3 Planar Defects 203.4 Lattice Defects: PercolationTheory 223.5 Summary 254 Nucleation and Extreme Statistics in Brittle Fracture 274.1 Stress Concentration Around Defect 274.1.1 Griffith’sTheory of Crack Nucleation in Brittle Fracture 304.2 Strength of Brittle Solids: Extreme Statistics 324.2.1 Weibull and Gumbel Statistics 324.3 Extreme Statistics in Fiber Bundle Models of Brittle Fracture 344.3.1 Fiber Bundle Model 344.3.1.1 Strength of the Local Load Sharing Fiber Bundles 354.3.1.2 Crossover from Extreme to Self-averaging Statistics in the Model 354.4 Extreme Statistics in Percolating Lattice Model of Brittle Fracture 374.5 Molecular Dynamics Simulation of Brittle Fracture 394.5.1 Comparisons with Griffith’s Theory 394.5.2 Simulation of Highly Disordered Solids 414.6 Summary 425 Roughness of Fracture Surfaces 455.1 Roughness Properties in Fracture 455.1.1 Self-affine Scaling of Fractured Surfaces 465.1.2 Out-of-plane Fracture Roughness 475.1.3 Distribution of Roughness: Mono- and Multi-affinity 495.1.3.1 Nonuniversal Cases 505.1.3.2 Anisotropic Scaling 545.1.4 In-plane Roughness of Fracture Surfaces 565.1.4.1 Waiting Time Distributions in Crack Propagation 605.1.5 Effect of Probe Size 625.1.6 Effect of Spatial Correlation and Anisotropy 655.2 Molecular Dynamics Simulation of Fractured Surface 665.3 Summary 686 Avalanche Dynamics in Fracture 696.1 Probing Failure with Acoustic Emissions 706.2 Dynamics of Fiber Bundle Model 746.2.1 Dynamics Around Critical Load 776.2.2 Dynamics at Critical Load 816.2.3 Avalanche Statistics of Energy Emission 816.2.4 Precursors of Global Failure in the Model 826.2.5 Burst Distribution: Crossover Behavior 846.2.6 Abrupt Rupture and Tricritical Point 856.2.7 Disorder in Elastic Modulus 876.3 Interpolations of Global and Local Load Sharing Fiber Bundle Models 886.3.1 Power-law Load Sharing 896.3.2 Mixed-mode Load Sharing 906.3.3 Heterogeneous Load Sharing 926.3.3.1 Dependence on Loading Process 936.3.3.2 Results in One Dimension 946.3.3.3 Results in Two Dimensions 966.3.3.4 Comparison with Mixed Load Sharing Model 1016.4 RandomThreshold Spring Model 1016.5 Summary 1077 Subcritical Failure of Heterogeneous Materials 1117.1 Time of Failure Due to Creep 1117.1.1 Fluctuating Load 1127.1.2 Failure Due to Fatigue in Fiber Bundles 1197.1.3 Creep Rupture Propagation in Rheological Fiber Bundles 1227.1.3.1 Modification for Local Load Sharing Scheme 1267.2 Dynamics of Strain Rate 1297.3 Summary 1348 Dynamics of Fracture Front 1358.1 Driven Fluctuating Line 1358.1.1 Variation of Universality Class 1408.1.2 Depinning with Constant Volume 1428.1.3 Infinite-range Elastic Force with Local Fluctuations 1448.2 Fracture Front Propagation in Fiber Bundle Models 1468.2.1 Interfacial Crack Growth in Fiber Bundle Model 1468.2.2 Crack Front Propagation in Fiber Bundle Models 1498.2.3 Self-organized Dynamics in Fiber Bundle Model 1518.3 Hydraulic Fracture 1618.4 Summary 1639 Dislocation Dynamics and Ductile Fracture 1659.1 Nonlinearity in Materials 1659.2 Deformation by Slip 1659.2.1 Critical Stress to Create Slip in Perfect Lattice 1669.3 Slip by Dislocation Motion 1679.4 Plastic Strain due to Dislocation Motion 1699.5 When Does a Dislocation Move? 1709.5.1 DislocationWidth 1709.5.2 Dependence on Grain Boundaries in Crystals 1719.5.3 Role of Temperature 1719.5.4 Effect of Applied Stress 1729.6 Ductile–Brittle Transition 1729.6.1 Role of Confining Pressure 1729.6.2 Role of Temperature 1739.7 TheoreticalWork on Ductile–Brittle Transition 17410 Electrical Breakdown Analogy of Fracture 17710.1 Disordered Fuse Network 17810.1.1 Dilute Limit (p → 1) 17910.1.2 Critical Behavior (p → pc) 18010.1.3 Influence of the Sample Size 18110.1.4 Distribution of the Failure Current 18210.1.4.1 Dilute Limit (p → 1) 18210.1.4.2 At Critical Region (p → pc) 18310.1.5 Continuum Model 18310.1.6 Electromigration 18410.2 Numerical Simulations of Random Fuse Network 18510.2.1 Disorders in FailureThresholds 18710.2.2 Avalanche Size Distribution 18810.2.3 Roughness of Fracture Surfaces in RFM 19110.2.4 Effect of High Disorder 19310.2.5 Size Effect 19610.3 Dielectric Breakdown Problem 19710.3.1 Dilute Limit (p → 1) 19810.3.2 Close to Critical Point (p → pc) 19910.3.3 Influence of Sample Size 19910.3.4 Distribution of Breakdown Field 20010.3.5 Continuum Model 20010.3.6 Shortest Path 20110.3.7 Numerical Simulations in Dielectric Breakdown 20110.3.7.1 Stochastic Models 20110.3.7.2 Deterministic Models 20210.4 Summary 20511 Earthquake as Failure Dynamics 20711.1 Earthquake Statistics: Empirical Laws 20711.1.1 Universal Scaling Laws 20911.2 Spring-block Models of Earthquakes 21411.2.1 Computer Simulation of the Burridge–Knopoff Model 21511.2.2 Train Model of Earthquake 21911.2.3 Mapping of Train Model to Sandpile 22111.2.3.1 Mapping to Sandpile Model 22211.2.4 Two-fractal Overlap Models 22311.2.4.1 Model Description 22411.2.4.2 GR and Omori Laws 22511.3 Cellular Automata Models of Earthquakes 22711.3.1 Bak TangWiesenfeld (BTW) Model 22811.3.2 Zhang Model 23211.3.3 Manna Model 23411.3.4 Common Failure Precursor for BTWand Manna Models and FBM 23711.3.4.1 Precursor in BTWModel 23811.3.4.2 Precursor in Manna Model 24011.3.4.3 Precursor in Fiber Bundle Model 24011.3.5 Olami–Feder–Christensen (OFC) Model 24011.3.5.1 Moving Boundary 24211.4 Equivalence of Interface and Train Models 24611.4.1 Model 24811.4.2 Avalanche Statistics in Modified Train Model 25011.4.3 Equivalence with Interface Depinning 25311.4.4 Interface Propagation and Fluctuation in Bulk 25511.5 Summary 26112 Overview and Outlook 265A Percolation 269A.1 Critical Exponent: General Examples 269A.1.1 Scaling Behavior 270A.2 Percolation Transition 270A.2.1 Critical Exponents of Percolation Transition 272A.2.2 Scaling Theory of Percolation Transition 273A.3 Renormalization Group (RG) Scheme 274A.3.1 RG for Site Percolation in One Dimension 276A.3.2 RG for Site Percolation in Two-dimensional Triangular Lattice 278A.3.3 RG for Bond Percolation in Two-dimensional Square Lattice 279B Real-space RG for Rigidity Percolation 281C Fiber Bundle Model 285C.1 Universality Class of the Model 285C.1.1 Linearly Increasing Density of Fiber Strength 285C.1.2 Linearly Decreasing Density of Fiber Strength 286C.1.3 Nonlinear Stress–Strain Relationship 288C.2 Brittle to Quasi-brittle Transition and Tricritical Point 290C.2.1 Abrupt Failure and Tricritical Point 292D Quantum Breakdown 293E Fractals 295F Two-fractal Overlap Model 297F.1 Renormalization Group Study: Continuum Limit 297F.2 Discrete Limit 299F.2.1 Gutenberg-Richter Law 299F.2.2 Omori Law 300G Microscopic Theories of Friction 303G.1 Frenkel-Kontorova Model 303G.2 Two-chain Model 304G.2.1 Effect of Fractal Disorder 305References 309Index 323