HANDBOOK OF DAMAGE MECHANICS nano to macro scale for materials and.

Preface to the Second Edition
Preface to the First Edition
Contents
About the Editor
Section Editors
Contributors
Part I: Fundamentals in Damage
	1 The True Nature of the Decomposition of the Damage Variable
		Introduction
		Decomposition for the Scalar Case
			How the Damage Variable Is Decomposed in One Dimension
			Decomposition of Two Defects Only
			Decomposition Due to Three Types of Defects
		Decomposition for the Tensor Case
			Decomposition Due to Two Defect Types Only
			Decomposition Due to Three Defect Types
		The Case of Plane Stress
		Conclusion
		References
	2 Damageability and Integrity of Materials: New Concepts of the Damage and Healing Fields
		Introduction
		A New Damage Variable
			Scalar Formulation
			Tensorial Formulation
			Additional Damage Variables
			Another New Damage Variable
		Integrity and Damageability of Materials
			Scalar Formulation
			Damageability Variable
			Integrity Versus Damageability
		The Integrity Field
			Scalar Formulation
			Illustrative Example
			Approximation of the Integrity Field
		The Healing Field
		Summary/Conclusions
		Appendix
			Appendix I: Are There Any Limits to the Damage Variable
			Appendix II: How to Compose Damage Variables
		References
	3 Damageability and Integrity of Materials: Unrecoverable Damage and Generalized Healing Model
		Introduction
		Unhealable Damage and Nondamageable Integrity
		Generalized Nonlinear Healing
			Scalar Formulation
			Quadratic Healing
			Tensorial Formulation
			Concept of Unhealable Damage
			Damage and Healing Models Compared
		The Healing Process Dissected
		Summary/Conclusions
		References
	4 Partial Damage Mechanics: Introduction
		Introduction
		Review of Continuum Damage Mechanics
		Introduction to Partial Damage Mechanics
		Conclusion
		References
	5 Mechanics of Self-Regenerating Materials
		Introduction
		Basics of Continuum Damage Mechanics
		The Scalar Theory
		The Elastic Stiffness
		The Concept of a Magical Material
		The Tensor Theory
		The Case of Plane Stress
		Conclusion and Discussion
		References
	6 Damage and Nonlinear Super Healing with Application to the Design of New Strengthening Theory
		Introduction
		Damage Healing and Super Healing Mechanics
		Healing and Super Healing in Anisotropic Formulation
		Linear Refined Super Theory
		Anisotropic Formulation of Linear Super Healing Mechanics
		Plane Stress Example of the Refined Super Healing
		One-Dimensional Example
		Generalized Nonlinear Super Healing
		Quadratic Super Healing
		Anisotropic Formulation of Nonlinear Super Healing
		Plane Stress Example of the Generalized Nonlinear Super Healing Theory
		Plane Stress Example of the Quadratic Super Healing Theory
		Non-super-Healed Damage Concept
		Comparative Analysis of Super Healing Models
		Super Healing Efficiency
		Theory of Undamageable Materials
		Conclusions
		References
	7 Vibration Analysis of Cracked Microbeams by Using Finite Element Method
		Introduction
		Modified Couple Stress Theory
		Finite Element Formulation
		Numerical Results
		Summary/Conclusions
		References
	8 Size Effect on Damage Response of Triangular Flexural Test Method
		Introduction
		Experimental Program
			Experimental Setup
			Experimental Results
		Determining Size Effect on Biaxial Tensile Strength
			Biaxial Tensile Strength
			Regression Analysis to Determine Size Effect Parameters
		Numerical Analysis of the Triangular Plate Test (TPT) Specimens
			Microplane Model (M7)
			Validation of the Numerical Procedure
			Finite Element Model for the Triangular Plate Test
				Numerical Analysis of the Triangular Plate Test
				Fracture Pattern of the Triangular Plate Tests
				Numerical Analysis of the Fibrous Triangular Plate Tests
				Fracture Pattern of the Fibrous Triangular Plate Tests
		Summary/Conclusions
		References
	9 Predicting Damage Behavior of Self-Healing Sandwich Panels: Computational Modeling
		Introduction
		Self-Healing Mechanisms and Syntactic Foams
		Self-Healing Syntactic Foam-Filled Sandwich Structures
		Finite Element Modeling for Self-Healing Materials
		Summary/Conclusions
		References
Part II: Damage and Failure of the Ductile Metals
	10 Problems of Deformation and Damage Studies of Additively Manufactured Regular Cellular Structures
		Introduction
		Experimental Investigations of the Mechanical Response of Regular Cellular Materials
			Quasi-Static Strength Tests
			Medium Strain Rate Tests
			High Strain Rate Tests
		Numerical Investigations of the Mechanical Response of Regular Cellular Structural Materials
			Numerical Erosion
			Constitutive Modelling
			Special Numerical Techniques
		Summary
		References
	11 Thermo-mechanics of Polymers at Extreme and Failure Conditions: Influence of Strain Rate and Temperature
		Current State of Polymers and Future Perspectives
		Mechanical Deformation of Polymers and Material Dependences
			Deformation Mechanisms of Thermoplastic Polymers
			Temperature and Strain Rate Dependences on Polymeric Deformation
			Mechanical Behavior of 3D Printed Polymers by FDM
			Constitutive Modelling of Mechanical Deformation in Polymers
		Damage and Failure Behavior of Polymers
			Failure Mechanisms of Thermoplastic Polymers
			Temperature and Strain Rate Dependences of Polymeric Failure
			Failure Behavior of 3D Printed Polymers by FDM
			Constitutive Modelling of Damage and Failure in Polymers
		Summary/Conclusions
		References
	12 Failure Behavior of Aluminum Alloys Under Different Stress States
		Introduction
		Damage Models
		Experimental and Numerical Analysis of Ductile Failure
		Summary/Conclusions
		References
	13 Modeling of Failure Resulting from High-Velocity Ballistic Impact
		Introduction
		Ballistic Impact Experiment
			Ballistic Impact Testing: Measurement and Observation Techniques
			Common Kinetic Threats
			Failure Modes in Metallic Targets
		Example: Failure Modes in Thick AA7020-T651 Plates
			Ballistic Limit Curve: The Recht-Ipson and Lambert Approaches
			Failure Modes Transition
			Microscopic Observations
		Modeling Approach to Analyze the Observed Failure Mechanism
			The Johnson-Cook Flow and Fracture Model
			Material Characterization
			Numerical Visualization of the Failure Modes Observed Experimentally
		Conclusions
		References
	14 Ductile Crack Growth Using Cohesive GTN Model
		Introduction
		Cohesive Zone Model for Ductile Failure
			Cohesive Interface Model
			GTN Model
			Cohesive GTN Traction-Separation Law
			Estimate of the In-Plane Strain
			Unloading Rule and Initial Stiffness
			Implementation into XPER Computer Code
		Application: 3D Numerical Simulation of a Compact Tension Fracture Specimen
			Modeling
			Results
		Conclusion
		References
	15 Auxetic Damping Systems for Blast Vulnerable Structures
		Introduction
		Blast-Induced Reaction Forces
		Uniaxial Graded Auxetic Damper (UGAD)
		Blast-Vulnerable Steel Gate
		Designing a Reinforced Concrete Supporting Structure
		Conclusions
		References
Part III: Damage in Brittle Materials
	16 Dynamic Deformation, Damage, and Fracture in Geomaterials
		Introduction
		Dynamic Loading Techniques
			Split Hopkinson Bar System
			Traditional Triaxial Hopkinson Bar
			Triaxial Hopkinson Bar System
		Dynamic Testing Methods
			Uniaxial Compression Tests
			Multiaxial Compression Tests
			Dynamic Indirect Tension and Shear Tests under Confinement
			Dynamic Indentation Tests
			Multiple Impact Tests
		Optical Measurement Techniques
			High-Speed Photography
			Digital Image Correlation (DIC)
			X-Ray Imaging and Computed Tomography (CT)
		Dynamic Deformation, Damage, and Fracture Behaviors
			Full-Field Deformation Fields and Stress-Strain Curves
			Dynamic Damage and Fracture
			Modeling Damage Ratio with Typical Experimental Results
		Summary
		References
	17 High-Speed Real-Time X-Ray Visualization of Impact Damage Inside Geomaterials
		Introduction
		Materials Investigated
		Experimental Method
			Kolsky (Split-Hopkinson) Bar Setup
			High-Speed Synchrotron X-Ray PCI
			Particle Arrangement
		Damage Mechanisms Under Boundary-Particle Contact Condition
			Soda Lime Glass
			Polycrystalline Silica
			Polycrystalline Silicon
		Damage Mechanisms Under Particle-Particle Contact Condition
			Soda Lime Glass
			Polycrystalline Silica
			Polycrystalline Silicon
			Yttria-Stabilized Zirconia (YSZ)
			Ottawa Sand
			Ohio Gold Sand
			Q-Rock Sand
		Summary of the Visualized Damage Mechanisms
		Conclusions and Prospective
		References
	18 Tensile Damage Mechanisms of Concrete Using X-Ray: In Situ Experiments and Mesoscopic Modeling
		Introduction
		Studied Material: Characteristic Scales and Composition
			Characteristic Sizes of the Specimens
			Specimens Preparation
		Phase Segmentation Procedure of Concrete from X-Ray Tomographic Images
			Scanning of the Specimen
			Segmentation Procedure
			Validation with Neutron Tomography
		FE Meso-model Description
			Local Kinematics Enhancement
				Accounting for the Morphology
				Accounting for the Local Failure Mechanisms
				Double Kinematics Enhancement
			Phenomenology and Resolution Scheme
		In Situ Tensile Test
			In Situ Experimental Setup
			Experimental Stress-Strain Curve and Macrocrack Identification
			Following the Microstructural Evolution
		Numerical Simulations of Tensile Tests
			Identification of Numerical Parameters
			Numerical Prediction
		Conclusions and Perspectives
		References
	19 Damage Model Strategies to Forecast Concrete Structure Behaviors Under Static and Dynamic Loadings
		Introduction
		Modelling Aspects
			Constitutive Equations
			Damage Evolutions
			Model Responses
			1D Version of the Model
			Strain Rate Effects (Mazars and Grange 2017)
		2D FE Description for Structural Applications
			Experimental Program on a Beam Under Cyclic Loading
			Finite Element Description
				Results at the Global Level
				Local Results
		Simplified Modelling for Structural Applications
			Cyclic Behavior
				Hysteretic Loop
				Permanent Strain
				Multifiber Beams and the Steel-Concrete Bond
			Results Obtained with These Enhancements
		Low- and Medium-Velocity Loading on Reinforced Concrete Structures
			Impact on a RC Beam (Mazars and Grange 2017)
			2D-1D Coupling for RC Beam-Column Substructure
				Discretization, 2D-1D Coupling
				Test Results
		Summary/Conclusions
		References
	20 Discrete Element Approach to Model Advanced Damage in Concrete Structures Under Impact
		Introduction
		Discrete Element Model
			Definition of Interactions
			Linar Elastic Constitutive Behavior
			Moment Transfer Law
			Failure Criterion
			Nonlinear Elastic with Damage Tensile Constitutive Behavior
			Constitutive Behavior in Compression - Compaction
			Strain Rate Dependency
		Identification of Constitutive Parameters of the Discrete Element Model for Concrete
			Discrete Element Modeling of Samples
			Identification of Constitutive Parameters by Means of Simulations of Quasi-Static Tests
				Simulation of Quasi-Static Uniaxial Tests
				Simulation of Quasi-Static Triaxial Confined Compression Tests
			Identification of Constitutive Parameters by Means of Simulations of Spalling Tests Using a Hopkinson Bar
		Validation of the Model by the Simulation of Hard Impact Tests
			Edge-On Impact Tests
			Perforation and Penetration Tests Performed by the CEA-Gramat (CEG)
			Simulation of a Drop-Weight Impact on a Reinforced Concrete (RC) Beam
				Test Description
				DE/FE Modeling of the Drop-Weight Test
		Conclusions
		References
	21 Damage in Concrete Subjected to Impact Loading
		Why Damage Modes in Concrete Targets Subjected to Impact Loading Need to be Investigated?
		Examples of Damage Processes Involved in Concrete Targets Subjected to Small Caliber Projectile
		Investigation of Damage Processes in Concrete Under Edge-On Impact Test
			Presentation of the EOI Testing Technique Applied to Concrete
			EOI Test Applied to Ductal UHPC Concrete
			EOI Test Applied to Dry and Wet MB50 Microconcrete and R30A7 Common Concrete
		Summary of Damage Modes Expected in Concrete Under Various Loading Conditions
		Numerical Simulation of a Ballistic Impact Against a Concrete Target
			Presentation of the KST-DFH Coupled Plasticity-Anisotropic Damage Model
			Numerical Simulation of the Impact of a Striker Against a Common Concrete Slab
		Conclusion
		References
	22 Failure Mechanisms of Ceramics Under Quasi-static and Dynamic Loads: Overview
		Main Body Text
			Introduction
			Failure Mechanisms Based on Microstructural Defects
				Effect of Lateral Confinement
				Effect of Strain Rate
			Failure Mechanisms Under Dynamic Loads
				Fragmentation and Spall
				Cavity Expansion
				Phase Transformation
				Amorphization
			Constitutive Models and Damage Formulations
				Micromechanical Models
				Phenomenological Models
				Damage Formulations
			Summary
		References
	23 Damage in Armor Ceramics Subjected to High-Strain-Rate Dynamic Loadings: The Spherical Expansion Shock Wave Pyrotechnic Test
		Introduction
		Description of the Problem
		State of the Art and Position of the Problem for the Divergent Spherical Wave Test
		Numerical Investigation of the Loading History
			Elastic Behavior Hypothesis
			Elastic Perfectly Plastic Behavior Hypothesis
		Experiments Performed on Two Alumina Ceramics
			Presentation of the Two Alumina Ceramics
			Experimental Setup and Instrumentation
			Experimental Results Obtained with Coarse-Grained Alumina AL23
			Experimental Results Obtained with Fine-Grained Alumina T299
			Discussion on the Experimental Results
		Postmortem Macroscopic Observations After Recovery Tests
			Macroscopic Observations for Coarse-Grained Alumina
			Macroscopic Observations for Fine-Grained Alumina
		Postmortem Analysis with SEM Microscopy
			SEM Observations for Coarse-Grained Alumina
			SEM Observations for Fine-Grained Alumina
		Postmortem Analysis with TEM Microscopy
			TEM Observations for Coarse-Grained Alumina
			TEM Observations for Fine-Grained Alumina
		Discussion About the Physical Phenomena
			Analysis of Microscopic Observations
			Influence of Microstructural Parameters
		Conclusion
		References
	24 Damage in Armor Ceramics Subjected to High-Strain-Rate Dynamic Loadings: The Edge-On Impact Test
		Introduction: Why Tensile Damage in Armor Ceramics Needs to Be Investigated?
		The Edge-On Impact Testing Technique: A Short Review
		Numerical Investigation of an Edge-On Impact Test
		Experiments Conducted on Four Silicon Carbide Ceramics
			Presentation of the SiC Ceramics
			Experimental Setup and Instrumentation
			Experimental Results Obtained in Open Configuration
			Experimental Results Obtained in Sarcophagus Configuration
			Tomographic Analysis of Fragments
		Discussion: Role of the Flaws Population on the Fragmentation Process in Ceramics
		Conclusions
		References
Part IV: Composite Damage Mechanics
	25 Effective Modeling of Interlaminar Damage in Multilayered Composite Structures Using Zigzag Kinematic Approximations
		Introduction
		Zigzag Theories for Structures with Perfect Bonding Between Adjacent Layers
			Original Zigzag Theories
			Refined Zigzag Theories
		Zigzag Theories for Structures with Continuous Interfacial Imperfections, Damaged Layers, and Stationary Delaminations
			Theories Using the Compliant Layer Concept
			Theories Using Imperfect Interfaces and the Spring Layer Model
				Interfaces and the Spring Layer Model
				Original Zigzag Theory with Imperfect and Smooth Interfaces
				Refined Zigzag Theory with Imperfect and Smooth Interfaces
			Theories Using the Sublaminate Approach or Additional Degrees of Freedom
		Zigzag Theories for Effective Modeling of Progressive Delamination Damage
			The Cohesive Crack Model, Cohesive Interfaces, and Brittle Fracture
			Zigzag Theories Using Compliant Layers and Continuum Damage Mechanics
			Zigzag Theories Using Cohesive Interfaces and Fracture Mechanics
		Conclusions
		References
Part V: Rock Damage Mechanics
	26 Numerical Analysis of Damage by Phase-Field Method
		Introduction
		Presentation of a Double-Phase-Field Method
			Regularized Crack Density Distribution
			Free Energy of Cracked Materials
		Initially Isotropic Materials
		Initially Anisotropic Materials
			Evolution of Damage Fields
			Numerical Implementation in Finite Element Method
		Numerical Assessment
			Tension Test of a Single-Edge Notched Plate
			Shear Test of a Single-Edge Notched Plate
		Analysis of Laboratory Tests
			Sandstone
			Jinping Marble
		Conclusion
		References
	27 Micromechanics-Based Models for Induced Damage in Rock-Like Materials
		Introduction
		Fundamentals of Upscaling Analyses by Homogenization
			Description of the Representative Volume Element
			Principle of the Linear Homogenization Method
				Strain Boundary Condition
				Stress Boundary Condition
				Strain Localization Tensor and Effective Elasticity Tensor
				Dilute Homogenization Scheme
				Eshelby Tensor for Penny-Shaped Microcracks
				Mori-Tanaka Method
				Ponte-Castaneda-Willis (PCW) Homogenization Scheme
			Comparison in the Case of Randomly Distributed Microcracks
		Damage-Friction Coupling Under Compression-Dominated Loading
			Strain Problem Decomposition
			Auxiliary Problem Decomposition
			Full Determination of the System Free Energy
			State Equations
			Friction Criterion and Plastic Flow
			Damage Criterion and Damage Evolution
			Consistency Conditions and Stress-Strain Relation in Rate Form
		Analytical Solution to the Constitutive Equations
			Case of Conventional Triaxial Compression
			Case of Triaxial Proportional Compression
			Basic Features of the Damage Resistance Function R(d)
			Illustration of the Model´s Predictions
			Extension to Take into Account Cracking-Induced Material Anisotropies
		Summary/Conclusions
		References
	28 Application of Continuum Damage Mechanics in Hydraulic Fracturing Simulations
		Introduction
		Simulation Techniques and Hydraulic Fracturing Design
		Thermodynamic Principles and Continuum Damage Mechanics of Porous Rocks
		CDM-Based Fluid-Driven Fracture in Porous Rocks
		Simulation Results
		Concluding Remarks
		References
	29 Damage and Fracture in Brittle Materials with Enriched Finite Element Method: Numerical Study
		Introduction
		Kinematics of Discontinuities in Solids
			Kinematics Description of Weak Discontinuity
			Kinematic Description of Strong Discontinuity
		Finite Element Implementation
			Incompatible Modes
			Finite Element Discretization
		Admissible Discrete Model with Closure Mechanism on the Discontinuity Surface
			Localization Criterion
			Failure Criterion: Traction Separation Law
			Closure Criterion
				Unloading Procedure
				Closure of Cracks
				Reloading Procedure
				Governing Equations
		Numerical Resolution of the Discrete Finite Element System
			Linearization of Equations
			Solving the System
			Resolution of the Cohesive Criterion
		Numerical Application to a Cubic Specimen with Heterogeneous Structure
		Conclusion
		References
	30 Damage and Failure of Hard Rocks Under True Triaxial Compression
		Introduction
		Fracture Evolution of Hard Rock Under True Triaxial Compression
			Experimental Method
			Pre-peak Progressive Cracking Process Induced by Stress
			The Effect of Intermediate Principal Stress on the Crack Stress Thresholds
			Energy Analysis of the Rock Cracking Process
		Damage Evolution of Hard Rock Subjected to Cyclic True Triaxial Loading
			Experimental Method
			Irreversible Strain Characteristics
			Strain Energy Characteristics
			Cohesion and Internal Friction Angle Characteristics
		Conclusions
		References
	31 Plastic Deformation and Damage in Rocks Under Coupled Thermo-hydromechanical Conditions: Numerical Study
		Introduction
		Thermodynamics of Thermoporous Elastoplastic-Damage Materials
			Thermoelastoplastic Material
			Thermoelastoplastic-Damaged Material
		State Equations for Hydraulic and Thermic Behavior
		An Elastoplastic Damage Model for Rocks Used in Case Studies
		Numerical Issues on Porous Elastoplastic-Damage Model Implementation
			Numerical Strategies for Multiphysics Coupling
			Integration of Constitutive Elastoplastic-Damage Equations
		Case Study: Evolution of Rock Mass State Around a Circular Horizontal Drift Under Thermo-Hydromechanical Loads
			Problem Description
			Results and Discussions
		Summary/Conclusions
		References
	32 Frictional Contact Between a Blunt Tool and Quasi-brittle Rock with Damage: Numerical Modeling
		Introduction
		Problem Definition and Governing Parameters
		Elasto-Plastic-Damage model
			Model Review and Adjustment
			Damage Initiation and Evolution
			Single Element Test
		Frictional Contact Using an Elastoplastic Model
			Finite Element Model
			Verification of the Finite Element Model in Asymptotic Regimes
			Influence of Parameter η
			Mesh Size Sensitivity
		Frictional Contact Using an Elasto-Plastic-Damage model
			Typical Case
			Invariance with Respect to Fixed η and ξ
			Influence of Parameter η
			Influence of Parameter ξ
		Discussion
			Two Governing Dimensionless Parameters
			Comparison with Experimental Results
			Limitation of the Numerical Study
		Conclusions
		References
Part VI: Micromechanical Damage and Healing for Concrete
	33 Stochastic Micromechanics-Based Probabilistic Damage and Repair Models for Cementitious Composites
		Introduction
		Micromechanical Modeling for the Probabilistic Damage Evolution for Cementitious Composites
			Basis of Micromechanical Damage Model
			The RVE Representations of Microcracked Cementitious Composite
			The Cementitious Composite´s Undamaged Compliance Tensor
			The Equivalent Isotropic Matrix
			The Damage-Induced Compliance Tensor of Cementitious Material
				Inelastic Compliance Tensor Induced by the Open Microcrack
				Overall Compliance Tensor Induced by Microcracks
				The Compliance Tensor Caused by the Unstable Microcracks
			Discussions on the Probabilistic Behavior of the Solid Phase
		The Repaired Concrete´s Stochastic Micromechanical Model
			The Repaired Concrete´s Deterministic Micromechanical Models
			Stochastic Descriptions for the Microstructures of the Repaired Concrete
				Uncertainty Quantifications for the Deposition Products
				Approximation for the Gaussian Process
				Uncertainty Quantifications for the Constituent Properties
			Multilevel Predictions for the Repaired Concrete´s Properties
				The First-Level Predictions
				The Second-Level Predictions
				Modifications for the Dry States
			The Composite´s Statistical Behavior
				Univariate Approximation for Multivariate Function
				Newton Interpolations
				Monte Carlo Simulation
			Numerical Examples
				Verifications
				Discussion on the Unsaturated Situation
				Discussion on the Imperfect Bonding
				Discussions on the Other Factors
		Conclusions
		References
	34 Class of Damage-Healing Models for Cementitious Composites at Multi-scales
		Introduction
		The Compliance of Damaged Materials
		The Healing Process
		Verification and Parametric Analysis
		The DEM Self-Healing Model
		Parametric Analysis
		Summary/Conclusions
		References
	35 Influences of Imperfect Interfaces on Effective Elastoplastic Responses of Particulate Composites
		Introduction
		Perfect and Imperfect Interface
		Modified Eshelby Inclusion Problem
			Iterative Method
			Decomposition Method
			Direct Computation Method
			Comparisons of Modified Eshelby Tensors
		Effective Stiffness with Modified Eshelby Tensor
			Mori-Tanaka Method
			Self-Consistent Method
			Degeneration of Effective Elastic Moduli of a Composite
		Equivalent Stiffness Method
		Effective Elastoplastic Responses of Two-Phase Composites
			Effective Elastic Stiffness with Pair-Wise Interactions
			Effective Yield Function
			Elastoplastic Deformation Responses of Ductile Matrix Composites
		Concluding Remarks
		Appendix: Contraction and Inversion of Fourth-Order Tensors
		References
	36 New Chemo-Mechanical Theory of Corrosion Damage in Concrete Under Sulfate Attack
		Introduction
		Corrosion Damage Model of Concrete Under Sulfate Attack
			Experiment for Damage Evolution
			Theoretical Model of Damage Evolution
		Nucleation of Corrosion Damage
		Evolution of Expansive Stress
		Model of Damage Evolution
		Entropy Evolution for Crack Propagation in Concrete Surface Under Sulfate Attack
			Experiment on Surface Cracks Evolution
				Material and Samples
				Methods
			Evolution Model of Entropy
				Evolution of Entropy of Concrete with Different Water-to-Cement Ratio
				Theoretical Model for Entropy Evolution
		Conclusions
		References
	37 Strain Energy-Based Thermo-elastoviscoplastic Two-Parameter Damage Self-Healing Formulations for Bituminous Composite Mater...
		Introduction
			Continuum Thermodynamics Framework for Two-Parameter Damage Self-Healing Formulations
				A Coupled Two-Parameter Model
				Definition of Net (Combined) Variables of the Volumetric and Deviatoric Damage and Healing
				Net Stress Concept and Hypothesis of Strain Equivalence
				Characterization of the Initial Elastic Strain Energy-Based Volumetric and Deviatoric Damage
				Characterization of Initial Elastic Strain Energy-Based Volumetric and Deviatoric Healing
				Net (Combined) Effect of the Volumetric and Deviatoric Damage and Healing
		A Conceptual Illustration
		Conclusions
		Appendix
		References
	38 Computational Algorithms for Strain Energy-Based Thermo-elastoviscoplastic Two-Parameter Damage Self-Healing Models for Bit...
		Introduction
			Computational Algorithms for Strain Energy-Based Thermo-elastoviscoplastic Two-Parameter Damage Self-Healing Formulations
				Computational Algorithms Called Two-Step Operator Splitting Methodology
				The Elastic Two-Parameter Damage Self-Healing Predictor
				The Net Viscoplastic Return Mapping Corrector
			Application to Asphalt Concrete and Verification for Thermo-elastoviscoplastic Damage Self-Healing Formulations
				Three-Dimensional Driver Problem
					Thermo-elastoviscoplastic Two-Parameter Damage Self-Healing Formulation
				Comparison Between Experimental Measurements and Predictions
					TAMU Asphalt Concrete Measurement
					NCSU Asphalt Mixture Measurements
		Conclusions
		References
Part VII: Damage for Disordered Materials
	39 Failure Mechanics of Geomaterials
		Introduction: Main Features of Failure in Geomaterials
		A General Criterion for Failure by Divergence Instabilities
			Kinetic Energy and Second-Order Work
			Micromechanically Based Formulation
		The Second-Order Work Criterion, Features, and Illustrative 3D Examples (Multiaxial Loading)
			General Equation of Local Second-Order Work Criterion
			Illustration of Instability Cones Using Darve Model
			Conditions of Effective Failure
		Failure Analysis in Granular Materials by the Discrete Element Method
		Rock Joint Failure Modeling
			Two Rock Joint Constitutive Relations
			The Use of the Second-Order Work Criterion
			Application
		Failure Modeling by Finite Element Method: Homogeneous Cases and Boundary Value Problems
			Material Instabilities in the Triaxial Test
				Finite Element Analysis of Diffuse Failure
				Loading Direction, Control Parameters, and Bifurcation Domain
				Second-Order Work, Relevance to Loss of Uniqueness, and Localization
			FEM Modeling of the Petacciato Landslide
		Conclusion
		References
	40 Fractals and Mechanics of Fracture
		Introduction
		Basic Concepts in Fractal Fracture Mechanics
		Delayed Fracture in Viscoelastic Solids for Euclidian and Fractal Geometries: Motion of a Smooth Crack in a Viscoelastic Medium
		Growth of Fractal Cracks in Viscoelastic Media
		Some Fundamental Concepts
		Conclusions
		References
	41 Lattice and Particle Modeling of Damage Phenomena
		Introduction
		Basic Idea of a Spring Network Representation
			Anti-plane Elasticity on Square Lattice
			In-Plane Elasticity: Triangular Lattice with Central Interactions
			In-Plane Elasticity: Triangular Lattice with Central and Angular Interactions
			Triple Honeycomb Lattice
		Spring Network Models
			Representation by a Fine Mesh
			Damage in Macro-Homogeneous Materials
				Spring Network for Inelastic Materials
				Hill-Mandel Macrohomogeneity Condition
				Modeling Elastic-Brittle Materials
				Modeling Elastic-Plastic Materials
				Modeling Elastic-Plastic-Brittle Materials
			Damage Patterns and Maps of Disordered Elastic-Brittle Composites
		Particle Models
			Governing Equations
				Basic Concepts
				Leapfrog Method
			Examples
			Other Models
		Scaling and Stochastic Evolution in Damage Phenomena
		Concluding Remarks
		References
	42 Toughening and Instability Phenomena in Quantized Fracture Process: Euclidean and Fractal Cracks
		Introduction
		Displacements and Strains Associated with a Discrete Cohesive Crack Model
		Quantization of the Panin Strain and the Criterion for Subcritical Crack Growth
		Stability of Fractal Cracks
		Conclusions
		Appendix A
		Appendix B
		References
	43 Two-Dimensional Discrete Damage Models: Lattice and Rational Models
		Introduction
		Lattices with Central Interactions (α-Models)
		Triangular Lattice with Central Interactions
		Triangular Lattice with First and Second Neighbor Central Interactions
		Examples of Applications of α-Model
		Rational Models of Brittle Materials
		Lattices with Central and Angular Interactions (α-β Models)
		Square Lattice with Central and Angular Interactions
		Examples of Applications of Lattices with Central and Angular Interactions
		Lattices with Beam Interactions
		Triangular Bernoulli-Euler Beam Lattice
		Triangular Timoshenko Beam Lattice
		Computer Implementation Procedure for Beam Lattices
		Examples of Applications of Lattices with Beam Interactions
		Conclusion
		References
Part VIII: Damage in Crystalline Metals and Alloys
	44 From Single Crystal to Polycrystal Plasticity: Overview of Main Approaches
		Introduction
		Continuum Discretization of a Boundary Value Problem
		Single Crystal Plasticity
		Local Single Crystal Approaches
		An Application of Crystal Plasticity to the Study of Intergranular Damage in an FCC Alloy
		Single Crystal Formulation for the FCC Material
		Intergranular Crack Observations, Strain Field Measurements, and Predicted Local Stress and Strain Fields
		Nonlocal Single Crystal Approaches
		Nonlocal Models Based on Internal Strain Gradient Variables
		Nonlocal Models Based on the Mechanics of Generalized Continua
		Microcurl Model: Balance and Constitutive Equations
		Application of the Microcurl Model to Study the Deformation Behavior of a Polycrystalline Aggregate
		Concluding Remarks
		References
	45 Micromechanics for Heterogeneous Material Property Estimation
		Introduction
			Overall Property of Heterogeneous Material
			Average Field Theory and Homogenization Theory
			Field Equations
		Average Field Theory
			Averaging Scheme
			Average Field
			Explicit Expression of Overall Elasticity in Terms of Strain Concentration Tensor
			Use of Eshelby´s Tensor for Evaluation of Strain Concentration Tensor
		Homogenization Theory
			Singular Perturbation Expansion
			Use of Periodic Structure as Microstructure Model
			Comparison of Average Field Theory and Homogenization Theory
		Strain Energy Consideration
			Consistency of Overall Elasticity
			Condition for Consistent Overall Elasticity
			Dependence of Overall Elasticity on Loading Condition
		Hashin-Shtrikman Variational Principle
			Fictitious Uniform RVE
			Hashin-Shtrikman Functional for Eigen-stress
			Application of Hashin-Shtrikman Variational Principle to Periodic Structure
		Overall Property at Dynamics State
			Averaging Scheme at Dynamic State
			Fictitious Uniform RVE at Dynamic State
			Application of Singular Perturbation Expansion
		Conclusion
		References
	46 Microstructural Behavior and Fracture in Crystalline Materials: Overview
		Introduction
		Dislocation-Density-Based Multiple Slip Formulation
		Multiple-Slip Crystal Plasticity Formulation
		Mobile and Immobile Dislocation-Density Evolution Equations
		Determination of Dislocation-Density Evolution Coefficients
		Dislocation-Density GB Interaction Scheme
		Martensitic Microstructural Representation
		Computational Representation of Failure Surfaces and Microstructural Failure Criterion
		Results and Discussion
		Martensitic Block Size
		Low-Carbon Steel
		High-Carbon Steel
		Dynamic Behavior
		Martensitic Block Distribution
		Random Variant Distribution
		Optimized Variant Distribution
		Dynamic Behavior
		Conclusion
		References
	47 Molecular Dynamics Simulations of Plastic Damage in Metals
		Introduction and Historical Perspective
		Molecular Dynamics Simulations
		Initial Conditions
		Interatomic Force Expressions
		Classical Equations of Motion, Numerical Integrators, and Thermostats
		Analyzing Atomic Simulations
		Multiscale Modeling
		Available Codes
		Example Simulations of Metal Dynamics
		Simulations of Shock-Loaded Crystals
		Deformed Nanocrystalline Metals
		Simulations of Grain Boundary Migration
		Current Challenges
		Interatomic Forces
		Length Scales
		Timescales
		Quantum Dynamics
		Interpretation of MDS Results
		Conclusion
		References
Index