Title Page	4
	Copyright Page	5
	PREFACE	6
	Table of Contents	7
	Damage and Smeared Crack Models	8
	1 Isotropic Damage Models	8
	1.1 One-Dimensional DamageModel	8
	1.2 DamageModels with Strain-Based Loading Functions	12
	2 Smeared Crack Models	22
	2.1 One-Dimensional Smeared Crack Model	23
	2.2 Multi-Dimensional Smeared Crack Models	27
	2.3 Fixed Crack Model	28
	2.4 Rotating Crack Model	30
	2.5 Rotating Crack Model with Transition to Scalar Damage	32
	2.6 Examples of Failure Simulations	34
	3 Strain Localization due to Softening	36
	3.1 Strain Localization in One Dimension	36
	3.2 Strain Localization in Multiple Dimensions	39
	3.3 Mesh-Adjusted Softening Modulus (Crack Band Approach)	42
	4 Regularized Softening Models	46
	4.1 Integral-Type Nonlocal Models	46
	4.2 Gradient-Enhanced Models	49
	4.3 Examples of Failure Simulations	51
	Acknowledgment	53
	Bibliography	53
	Cracking and Fracture of Concrete at Meso-levelusing Zero-thickness Interface Elements	57
	1. Introduction	57
	1.1. Historical Aspects, General Considerations	57
	1.2. Meso-structural Geometries, Discretization	58
	1.3. Fracture-based Interface Constitutive Law with Aging Effect	58
	1.4. Aging Viscoelastic Model for the Matrix-phase	66
	2. Original Results in 2D under Mechanical Loading	67
	3. Extension to 3D	70
	3.1. Geometry and Meshes	70
	3.2. Extension of the Constitutive Model to 3D	72
	3.3. Computational and Graphics Aspects	73
	3.4. Results	73
	4. Deterioration due to Diffusion-driven EnvironmentalPhenomena: Drying Shrinkage and External Sulphate Attack	79
	4.1. Hygro-mechanical Modeling of Drying Shrinkage	79
	4.2. Chemo-mechanical Analysis of External Sulfate Attack	88
	Description of the chemo-transport model	89
	Main results of the model	93
	5. Concluding remarks and on-going developments	96
	Acknowledgements	97
	References	97
	Crack Models with Embedded Discontinuities	104
	1 Fracture Problem Approaches Based on ContinuumConstitutive Relations	104
	1.1 Motivation: Idealization of the Fracture Process Zone (FPZ) in Quasi-Brittle Material	104
	1.2 Stability and Uniqueness of the Mechanical Boundary Value Problem(BVP)	108
	1.3 Example of a Material Model Subjected to Stability Loss and Bifurcation:Isotropic continuum damage model for concrete	111
	2 Material Failure Analysis Using the Continuum-StrongDiscontinuity Approach (CSDA)	113
	2.1 Motivation	113
	2.2 The 1D Continuum-Strong Discontinuity Approach (CSDA)	118
	2.3 The Continuum-Strong Discontinuity Approach in 3D Problems	123
	3 Finite Elements with Embedded Discontinuities	125
	3.1 Strong Discontinuities: The Local Form of the BVP Governing Equations	126
	3.2 A Variational Consistent Formulation of the BVP with Strong Discontinuities	127
	3.3 Finite Elements with Embedded Discontinuities	129
	3.4 An Embedded Strong Discontinuity FE not Needing the Crack PathContinuity Enforcement	133
	4 Algorithmic Aspects of the CSDA	136
	4.1 A Global Tracking Algorithm	137
	4.2 Factors Influencing the Stability and Accuracy of the Numerical Method	139
	5 Applications of the CSDA Methodology to ConcreteFracture Problems	143
	5.1 Modeling the Fracture Process Zone	144
	5.1.1 The Three-Point Bending Test	145
	5.1.2 Double Cantilever Beam (DCB Test)	148
	5.2 Size Effect Analysis	149
	5.2.1 The Brazilian Test	150
	5.3 Dynamic Fracture Simulation	151
	6 A Model for Reinforced Concrete Fracture via CSDA andMixture Theory	153
	6.1 A Mixture Theory for Reinforced Concrete	154
	6.2 Constitutive Model for the Composite: Regularization Procedure basedon the CSDA	155
	6.3 Representative numerical simulations	157
	Bibliography	162
	Plasticity based crack models and applications	165
	1 Introduction	165
	2 Plasticity Based 2D Concrete Model with SmearedCracks	166
	2.1 Formulation for plain concrete	166
	2.2 Extension to reinforced concrete	174
	2.3 Validation	178
	2.4 Applications	182
	3 Plasticity Based Crack Model with EmbeddedDiscontinuities	195
	3.1 Strong Discontinuity Kinematics	195
	3.2 Determination of the displacement jump and the stresses	200
	3.3 Finite element discretization	204
	3.4 Determination of the crack direction	206
	3.5 Crack Tracking Algorithm	208
	3.6 Validation	209
	3.7 Application	216
	Bibliography	219
	Crack models based on the extended finiteelement method	224
	1 Introduction	224
	2 Background on discretization methods	225
	2.1 Problem description and notations	225
	2.3 The finite element method	226
	2.4 Meshless methods	228
	2.5 The partition of unity	230
	3 Discontinuity modeling with the X-FEM and levelsets	231
	3.1 A simple 1D problem	231
	3.2 Extension to 2D and 3D	235
	3.3 Cracks located by level sets	239
	4 Technical and mathematical aspects	243
	4.1 Integration of the element stiffness	243
	4.2 Topological and geometrical enrichment strategies	243
	4.3 Solver and condition number	246
	4.4 Inf-sup condition for cracks under contact	248
	5 Configurational analysis of the crack front	256
	5.1 The Eshelby tensor	256
	5.2 Energy integrals	259
	5.3 Energetic information for cohesive cracks	260
	Bibliography	264
	Smeared Crack and X-FEM Modelsin the Context of Poromechanics	268
	1 Elastoplastic-Damage Models for Concrete	268
	1.1 Introductory Remarks	268
	1.2 Evolution Equations	269
	1.3 Damage and Yield Surfaces	272
	1.4 Numerical Analysis of a Notched Concrete Beam	274
	2 Hygro-mechanical Couplings in Concrete	276
	2.1 Concrete Microstructure and Hygral Forces	276
	2.2 Drying Creep	278
	3 Fundamentals of Poromechanics	280
	3.1 Concept of Volume Fractions	282
	3.2 Kinematics	283
	3.3 Balance of Momentum	283
	3.4 Mass Balance Equations	284
	3.5 Constitutive Equations	285
	3.6 Transport of Water and Heat	286
	4 Poroplastic-Damage Model for Concrete	286
	4.1 State Equations	287
	4.2 Coupling Coefficients	289
	4.3 Effective stresses	290
	4.4 Multisurface Poroplastic-Damage Model	292
	4.5 Long-term Creep	293
	4.6 Capillary-Pressure Relation	294
	4.7 Moisture Transport	295
	4.8 Finite Element Formulation	298
	4.9 Application: Drying and Re-wetting of a Base-RestrainedConcrete Wall	300
	5 Hygromechanical Extended Finite Element Modelfor Concrete	302
	5.1 Introduction	302
	5.2 X-FEM Resolution of Cracks	303
	5.3 Weak Form of Balance of Momentum	304
	5.4 Extension to Coupled Hygro-Mechanical Analyses	306
	5.5 Finite Element Implementation	306
	5.6 Crack Tracking Algorithm	307
	5.7 Numerical Solution	307
	5.8 Numerical Applications	312
	Bibliography	323