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�CBrittle 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�CStatistically 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�CStrain 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 Clo