目錄
1 Introduction
1.1 Damage and Failure of Heterogeneous Media: Basic Features and Common Characteristics
1.1.1 Basic Features
1.1.2 Scientific Characteristics
1.1.3 Demands for Economic Mechanics
1.2 Framework of Statistical Meso-mechanics: Why and How Statistical Meso-mechanics Is
1.2.1 Remarks on Multi-scale Approaches
1.2.2 Why Statistical Meso-mechanics
1.2.3 How Statistical Meso-mechanics Works
1.2.4 What the Present Book Deals with
1.3 Mathematical Essentials in Statistical Meso-mechanics
1.3.1 Statistical 2D-3D Conversion
1.3.2 Statistical Differentiation and Correlation of Patterns
1.3.3 Ensemble Statistics
1.3.4 Weibull Distribution, Heterogeneity Index, and Its Transfer
2 Quasi-static Evolution of Deformation and Damage in Meso-heterogeneous Media
2.1 Average and Mean Field Approximation (MF)
2.1.1 Conventional Averaging
2.1.2 Mean Field (MF) Method
2.1.3 Mean Field Approximation and Strain Equivalence
2.1.4 Coupled Averaging (CA)
2.1.5 Two PDF Operations Related to Coupled Averaging (CA)
2.2 Elastic and Statistically Brittle (ESB) Model and Its Distinct Features—Global Mean Field (GMF) Approximation
2.2.1 Elastic–Brittle Meso-elements and Its Implication
2.2.2 Elastic and Statistically Brittle (ESB) Model
2.2.3 Full Formulation of Elastic and Statistically Brittle (ESB) Model
2.2.4 Energy Variations in ESB Model
2.2.5 Stable or not Beyond Peak Load in ESB Model
2.2.6 Experimental Extraction of Constitutive Parameters in ESB Model
2.3 Continuous Bifurcation and Emergence of Localized Deformation and Damage—Regional Mean Field (RMF) Approximation
2.3.1 Experimental Observations and Data Processing of Localization
2.3.2 When Localization Emerges
2.3.3 Comparison of Experimental and Calculated Results of Localization
2.3.4 Continuous Bifurcation with Simultaneous Elastic Unloading and Continuing Damage
2.3.5 Constitutive Relation with Localization Resulting from Continuous Bifurcation
2.3.6 A Phenomenological Model of Localized Zone c
2.3.7 Energy Variation with Localization and Critical State of Stable Deformation Under RMF Approximation
2.3.8 Evolution of Statistical Distribution and How GMF Approximation Fails
2.4 Size Effect Resulting from Meso-heterogeneity and Its Statistical Understanding
2.4.1 Weibull Model—The Weakest Link Model
2.4.2 Ba？ant's Theory on Size Effect
2.4.3 Size Effect Governed by Elastic Energy Release on Catastrophic Rupture
2.4.4 Size Effects Resulting from Finite Meso-elements
2.5 Special Experimental Issues in Stat
2.6.2 Multi-scale Finite Element Methods
2.7 Application to Failure Wave Under One-Dimensional Strain Condition—A Moving Front of Expanding Contact Region
2.7.1 Fundamentals of Failure Wave
2.7.2 Illustrative Problems—Rigid Projectile Against Rigid but Crushable Sample
2.7.3 Constitutive Relation Under One-Dimensional Strain State Based on Elastic–Statistically Brittle (ESB) Model
2.7.4 Failure Wave—A Moving Front of Expanding Contact Region Due to Heterogeneous Meso-scopic Shear Failure
2.8 Application to Metal Foams
2.8.1 General Features of Metal Foam
2.8.2 Phenomenological and Statistical Formulation of Stress–Strain Relation
2.8.3 Cell Model
2.8.4 Statistical Formulation of Foam Based on Cell Models
2.9 Application to Concrete Under Biaxial Compression
2.9.1 General Features of Concrete Under Biaxial Compression
2.9.2 ESB Model Under Biaxial Compression and Plane Stress State with GMF Approximation
2.9.3 Localization, Catastrophic Rupture, and Gradual Failure
3 Time-Dependent Population of Microdamage
3.1 Background and Methodology
3.1.1 Effects of Microdamage Evolution
3.1.2 Methodology
3.1.3 Definition of Number Density of Microdamage
3.2 Fundamental Equations of Microdamage Evolution
3.2.1 Brief Review of the Study on Microdamage Evolution
3.2.2 General Equation of Microdamage Evolution
3.2.3 Fundamental Equations in Phase Space of Microdamage Sizes {c, c0
3.2.4 Some Other Formulations
3.3 General Solution to Evolution of Microdamage Number Density
3.3.1 Solution to Evolution of Microdamage Number Density n0(c, c0; r)
3.3.2 Evolution of Current Microdamage Number Density n(t, c; r)
3.4 Closed Formulation of Continuum Damage Based on Microdamage Evolution
3.4.1 Continuum Damage Based on Microdamage Number Density
3.4.2 Trans-Scale Field Equations Governing Damage Evolution
3.4.3 Closed One-Dimensional Formulation of Damage Evolution
3.4.4 Dynamic Function of Damage (DFD) and Its Significance
3.4.5 Damage Localization
3.5 Deborah Number and Its Significance in the Evolution of Microdamage
3.5.1 Deborah Number
3.5.2 Competition of Macro- and Mesoscopic Time Scales: Trans-scale Deborah Numbers
3.5.3 Implication of Intrinsic Deborah Number D
3.6 Spallation—Tensile Failure Resulting from Microdamage Under Stress Waves
3.6.1 Historical Remarks and Basic Features
3.6.2 Experimental Study of Mesoscopic Kinetics in Spallation with Sub-microsecond and Multi-stress Pulses Techniques
3.6.3 Distinct Aspects of Spallation Due to Mesoscopic Kinetics of Microcracks
3.7 Short Fatigue
4 Critical Catastrophe in Disordered Heterogeneous Brittle Media
4.1 Evolution Induced Catastrophe (EIC)
4.1.1 What Evolution Induced Catastrophe (EIC) Is
4.1.2 Macroscopic Description of Evolution Induced Catastrophe
4.1.3 Evolution Induced Catastrophe Based on Statistical Driven Nonlinear Threshold Model Under Global Mean Field (GMF) Approximation
4.1.4 Characteristics of Catastrophic Rupture in Simulations
4.2 Catastrophic Rupture and Its Relation to Energy Transfer and Damage Localization
4.2.1 Condition for Catastrophic Rupture in Accord with Energy Transfer Under Global Mean Field (GMF) Approximation
4.2.2 Margining Catastrophic Rupture Under GMF Approximation
4.2.3 Size Effect Governed by Elastic Energy Release on Catastrophic Rupture
4.2.4 Catastrophic Rupture Induced by Localization Under Regional Mean Field (RMF) Approximation
4.3 Sample-Specificity and Trans-Scale Sensitivity
4.3.1 Sample-Specificity of Catastrophic Rupture
4.3.2 Uncertainty Relation in Catastrophe Induced by Damage Localization
4.3.3 Physical Understanding of Sample-Specificity with Load-Sharing Model
4.3.4 Trans-Scale Sensitivity
4.4 Critical Sensitivity and Power Law Singularity of Catastrophe
4.4.1 Critical Sensitivity and Power Law Singularity Based on ESB Model
4.4.2 Loading Rate Effect on Critical Sensitivity
4.4.3 Effect of Discreteness on Critical Sensitivity
4.5 Great Earthquake—The Catastrophic Rupture in Earth's Crust
4.5.1 Great Earthquake and Power Law Singularity
4.5.2 Strain Field Evolution on the Earth's Surface and Its Correlation to Great Earthquake
4.5.3 Relationship Between Critical Sensitivity and Load–Unload Response Ratio (LURR) Before an Earthquake
4.5.4 Lower and Upper Bounds of Catastrophe Occurrence and Earthquake Prediction
4.6 Perspective
Appendices
A.1 : Nomenclature
A.2 : Summary of the Book
A.3 : Statistics—Variation and Correlation
A.3.1 : Variance and Covariance
A.3.2 : Correlations
A.4 : Probability Distribution
A.4.1 : Probability
A.4.2 : Single Continuous Random Variable and Related PDF
A.4.3 : Double Continuous Random Variables, Related PDF and Correlation
A.5 : Basic Combinatorics
A.5.1 : Two Fundamental Principles of Counting
A.5.2 : Basic Permutation and Combination
A.5.3 : Variations of Permutations and Combinations
A.5.4 : Stirling's Formula
A.6 : Weibull Distribution and Weibull Modulus
A.6.1 : Basic of Weibull Distribution
A.6.2 : Fitting of Weibull Modulus M of Strength
A.6.3 : Examples of Weibull Modulus
References
Author Index
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