Contents 1 Introduction 1 1.1 Damage and Failure of Heterogeneous Media: Basic Features and Common Characteristics 1 1.1.1 Basic Features 3 1.1.2 Scientific Characteristics 4 1.1.3 Demands for Economic Mechanics 8 1.2 Framework of Statistical Meso-mechanics: Why and How Statistical Meso-mechanics Is 10 1.2.1 Remarks on Multi-scale Approaches 10 1.2.2 Why Statistical Meso-mechanics 12 1.2.3 How Statistical Meso-mechanics Works 14 1.2.4 What the Present Book Deals with 17 1.3 Mathematical Essentials in Statistical Meso-mechanics 21 1.3.1 Statistical 2D-3D Conversion 21 1.3.2 Statistical Differentiation and Correlation of Patterns 40 1.3.3 Ensemble Statistics 53 1.3.4 Weibull Distribution, Heterogeneity Index, and Its Transfer 63 2 Quasi-static Evolution of Deformation and Damage in Meso-heterogeneous Media 71 2.1 Average and Mean Field Approximation (MF) 72 2.1.1 Conventional Averaging 73 2.1.2 Mean Field (MF) Method 74 2.1.3 Mean Field Approximation and Strain Equivalence 76 2.1.4 Coupled Averaging (CA) 77 2.1.5 Two PDF Operations Related to Coupled Averaging (CA) 78 2.2 Elastic and Statistically Brittle (ESB) Model and Its Distinct Features—Global Mean Field (GMF) Approximation 80 2.2.1 Elastic–Brittle Meso-elements and Its Implication 80 2.2.2 Elastic and Statistically Brittle (ESB) Model 81 2.2.3 Full Formulation of Elastic and Statistically Brittle (ESB) Model 84 2.2.4 Energy Variations in ESB Model 89 2.2.5 Stable or not Beyond Peak Load in ESB Model 91 2.2.6 Experimental Extraction of Constitutive Parameters in ESB Model 95 2.3 Continuous Bifurcation and Emergence of Localized Deformation and Damage—Regional Mean Field (RMF) Approximation 97 2.3.1 Experimental Observations and Data Processing of Localization 97 2.3.2 When Localization Emerges 101 2.3.3 Comparison of Experimental and Calculated Results of Localization 106 2.3.4 Continuous Bifurcation with Simultaneous Elastic Unloading and Continuing Damage 108 2.3.5 Constitutive Relation with Localization Resulting from Continuous Bifurcation 111 2.3.6 A Phenomenological Model of Localized Zone c 116 2.3.7 Energy Variation with Localization and Critical State of Stable Deformation Under RMF Approximation 120 2.3.8 Evolution of Statistical Distribution and How GMF Approximation Fails 126 2.4 Size Effect Resulting from Meso-heterogeneity and Its Statistical Understanding 129 2.4.1 Weibull Model—The Weakest Link Model 129 2.4.2 Ba？ant's Theory on Size Effect 130 2.4.3 Size Effect Governed by Elastic Energy Release on Catastrophic Rupture 132 2.4.4 Size Effects Resulting from Finite Meso-elements 133 2.5 Special Experimental Issues in Statistical Meso-mechanics of Damage 158 2.5.1 General View of Experimental Setup Related to Statistical Meso-mechanics 158 2.5.2 Measurement of Surface Deformation 160 2.5.3 Acoustic Inspection 165 2.5.4 X-Ray Computerized Tomography (CT) 170 2.6 Special Issues of Numerical Simulations in Statistical Meso-mechanics of Damage 174 2.6.1 Cellular Automata (CA) with Non-local Interactions 175 2.6.2 Multi-scale Finite Element Methods 179 2.7 Application to Failure Wave Under One-Dimensional Strain Condition—A Moving Front of Expanding Contact Region 186 2.7.1 Fundamentals of Failure Wave 186 2.7.2 Illustrative Problems—Rigid Projectile Against Rigid but Crushable Sample 189 2.7.3 Constitutive Relation Under One-Dimensional Strain State Based on Elastic–Statistically Brittle (ESB) Model 195 2.7.4 Failure Wave—A Moving Front of Expanding Contact Region Due to Heterogeneous Meso-scopic Shear Failure 199 2.8 Application to Metal Foams 207 2.8.1 General Features of Metal Foam 207 2.8.2 Phenomenological and Statistical Formulation of Stress–Strain Relation 209 2.8.3 Cell Model 212 2.8.4 Statistical Formulation of Foam Based on Cell Models 217 2.9 Application to Concrete Under Biaxial Compression 223 2.9.1 General Features of Concrete Under Biaxial Compression 223 2.9.2 ESB Model Under Biaxial Compression and Plane Stress State with GMF Approximation 227 2.9.3 Localization, Catastrophic Rupture, and Gradual Failure 234 3 Time-Dependent Population of Microdamage 239 3.1 Background and Methodology 239 3.1.1 Effects of Microdamage Evolution 240 3.1.2 Methodology 240 3.1.3 Definition of Number Density of Microdamage 243 3.2 Fundamental Equations of Microdamage Evolution 246 3.2.1 Brief Review of the Study on Microdamage Evolution 247 3.2.2 General Equation of Microdamage Evolution 248 3.2.3 Fundamental Equations in Phase Space of Microdamage Sizes {c, c0} 251 3.2.4 Some Other Formulations 253 3.3 General Solution to Evolution of Microdamage Number Density 253 3.3.1 Solution to Evolution of Microdamage Number Density n0(c, c0; r) 253 3.3.2 Evolution of Current Microdamage Number Density n(t, c; r) 257 3.4 Closed Formulation of Continuum Damage Based on Microdamage Evolution 264 3.4.1 Continuum Damage Based on Microdamage Number Density 265 3.4.2 Trans-Scale Field Equations Governing Damage Evolution 267 3.4.3 Closed One-Dimensional Formulation of Damage Evolution 269 3.4.4 Dynamic Function of Damage (DFD) and Its Significance 271 3.4.5 Damage Localization 276 3.5 Deborah Number and Its Significance in the Evolution of Microdamage 280 3.5.1 Deborah Number 281 3.5.2 Competition of Macro- and Mesoscopic Time Scales: Trans-scale Deborah Numbers 283 3.5.3 Implication of Intrinsic Deborah Number D* 286 3.6 Spallation—Tensile Failure Resulting from Microdamage Under Stress Waves 290 3.6.1 Historical Remarks and Basic Features 292 3.6.2 Experimental Study of Mesoscopic Kinetics in Spallation with Sub-microsecond and Multi-stress Pulses Techniques 295 3.6.3 Distinct Aspects of Spallation Due to Mesoscopic Kinetics of Microcracks 304 3.7 Short Fatigue Cracks 314 3.7.1 Special Features of Short Fatigue Cracks 314 3.7.2 Statistical Aspects of Short Fatigue Cracks in Experimental Observations 316 3.7.3 Statistical Formulation of Collective Evolution of Short Fatigue Cracks 320 3.8 More Cases of Time-Dependent Processes Related to Microdamage Evolution 328 3.8.1 Damage Evolution in Polymer Owing to Crazing 329 3.8.2 Superdeep Penetration of Particles into a Metal Target 333 3.9 Brief Summary 334 4 Critical Catastrophe in Disordered Heterogeneous Brittle Media 337 4.1 Evolution Induced Catastrophe (EIC) 337 4.1.1 What Evolution Induced Catastrophe (EIC) Is 337 4.1.2 Macroscopic Description of Evolution Induced Catastrophe 340 4.1.3 Evolution Induced Catastrophe Based on Statistical Driven Nonlinear Threshold Model Under Global Mean Field (GMF) Approximation 347 4.1.4 Characteristics of Catastrophic Rupture in Simulations 353 4.2 Catastrophic Rupture and Its Relation to Energy Transfer and Damage Localization 360 4.2.1 Condition for Catastrophic Rupture in Accord with Energy Transfer Under Global Mean Field (GMF) Approximation 360 4.2.2 Margining Catastrophic Rupture Under GMF Approximation 372 4.2.3 Size Effect Governed by Elastic Energy Release on Catastrophic Rupture 377 4.2.4 Catastrophic Rupture Induced by Localization Under Regional Mean Field (RMF) Approximation 380 4.3 Sample-Specificity and Trans-Scale Sensitivity 392 4.3.1 Sample-Specificity of Catastrophic Rupture 392 4.3.2 Uncertainty Relation in Catastrophe Induced by Damage Localization 394 4.3.3 Physical Understanding of Sample-Specificity with Load-Sharing Model 398 4.3.4 Trans-Scale Sensitivity 407 4.4 Critical Sensitivity and Power Law Singularity of Catastrophe 410 4.4.1 Critical Sensitivity and Power Law Singularity Based on ESB Model 410 4.4.2 Loading Rate Effect on Critical Sensitivity 421 4.4.3 Effect of Discreteness on Critical Sensitivity 427 4.5 Great Earthquake—The Catastrophic Rupture in Earth's Crust 432 4.5.1 Great Earthquake and Power Law Singularity 432 4.5.2 Strain Field Evolution on the Earth's Surface and Its Correlation to Great Earthquake 434 4.5.3 Relationship Between Critical Sensitivity and Load–Unload Response Ratio (LURR) Before an Earthquake 439 4.5.4 Lower and Upper Bounds of Catastrophe Occurrence and Earthquake Prediction 442 4.6 Perspective 450 Appendices 451 A.1: Nomenclature 451 A.2: Summary of the Book 454 A.3: Statistics—Variation and Correlation 461 A.3.1: Variance and Covariance 461 A.3.2: Correlations 462 A.4: Probability Distribution 463 A.4.1: Probability 463 A.4.2: Single Continuous Random Variable and Related PDF 465 A.4.3: Double Continuous Random Variables, Related PDF and Correlation 467 A.5: Basic Combinatorics 471 A.5.1: Two Fundamental Principles of Counting 472 A.5.2: Basic Permutation and Combination 472 A.5.3: Variations of Permutations and Combinations 473 A.5.4: Stirling's Formula 475 A.6: Weibull Distribution and Weibull Modulus 475 A.6.1: Basic of Weibull Distribution 475 A.6.2: Fitting of Weibull Modulus M of Strength 476 A.6.3: Examples of Weibull Modulus 477 References 479 Author Index 489 Subject Index 495
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