Title Page -- Copyright -- Contents -- Series Preface -- Preface -- Chapter 1 Introduction -- 1.1 Introduction -- 1.2 An Enriched Finite Element Method -- 1.3 A Review onX-FEM: Development and Applications -- 1.3.1 CouplingX-FEMwith the Level-Set Method -- 1.3.2 Linear Elastic Fracture Mechanics (LEFM) -- 1.3.3 Cohesive Fracture Mechanics -- 1.3.4 Composite Materials and Material Inhomogeneities -- 1.3.5 Plasticity, Damage, and Fatigue Problems -- 1.3.6 Shear Band Localization -- 1.3.7 Fluid-Structure Interaction -- 1.3.8 Fluid Flow in Fractured Porous Media -- 1.3.9 Fluid Flow and Fluid Mechanics Problems -- 1.3.10 Phase Transition and Solidification -- 1.3.11 Thermal and Thermo-Mechanical Problems -- 1.3.12 Plates and Shells -- 1.3.13 Contact Problems -- 1.3.14 Topology Optimization -- 1.3.15 Piezoelectric and Magneto-Electroelastic Problems -- 1.3.16 Multi-Scale Modeling -- Chapter 2 Extended Finite Element Formulation -- 2.1 Introduction -- 2.2 The Partition of Unity Finite Element Method -- 2.3 The Enrichment of Approximation Space -- 2.3.1 Intrinsic Enrichment -- 2.3.2 Extrinsic Enrichment -- 2.4 The Basis of X-FEM Approximation -- 2.4.1 The Signed Distance Function -- 2.4.2 The Heaviside Function -- 2.5 Blending Elements -- 2.6 Governing Equation of a Body with Discontinuity -- 2.6.1 The Divergence Theorem for Discontinuous Problems -- 2.6.2 The Weak form of Governing Equation -- 2.7 The X-FEM Discretization of Governing Equation -- 2.7.1 Numerical Implementation of X-FEM Formulation -- 2.7.2 Numerical Integration Algorithm -- 2.8 Application of X-FEM in Weak and Strong Discontinuities -- 2.8.1 Modeling an Elastic Bar with a Strong Discontinuity -- 2.8.2 Modeling an Elastic Bar with a Weak Discontinuity -- 2.8.3 Modeling an Elastic Plate with a Crack Interface at its Center.

2.8.4 Modeling an Elastic Plate with a Material Interface at its Center -- 2.9 Higher Order X-FEM -- 2.10 Implementation of X-FEM with Higher Order Elements -- 2.10.1 Higher Order X-FEM Modeling of a Plate with a Material Interface -- 2.10.2 Higher Order X-FEM Modeling of a Plate with a Curved Crack Interface -- Chapter 3 Enrichment Elements -- 3.1 Introduction -- 3.2 Tracking Moving Boundaries -- 3.3 Level Set Method -- 3.3.1 Numerical Implementation of LSM -- 3.3.2 Coupling the LSM with X-FEM -- 3.4 Fast Marching Method -- 3.4.1 Coupling the FMM with X-FEM -- 3.5 X-FEM Enrichment Functions -- 3.5.1 Bimaterials, Voids, and Inclusions -- 3.5.2 Strong Discontinuities and Crack Interfaces -- 3.5.3 Brittle Cracks -- 3.5.4 Cohesive Cracks -- 3.5.5 Plastic Fracture Mechanics -- 3.5.6 Multiple Cracks -- 3.5.7 Fracture in Bimaterial Problems -- 3.5.8 Polycrystalline Microstructure -- 3.5.9 Dislocations -- 3.5.10 Shear Band Localization -- Chapter 4 Blending Elements -- 4.1 Introduction -- 4.2 Convergence Analysis in the X-FEM -- 4.3 Ill-Conditioning in theX-FEM Method -- 4.3.1 One-Dimensional Problem with Material Interface -- 4.4 Blending Strategies in X-FEM -- 4.5 Enhanced Strain Method -- 4.5.1 An Enhanced Strain Blending Element for the Ramp Enrichment Function -- 4.5.2 An Enhanced Strain Blending Element for Asymptotic Enrichment Functions -- 4.6 The Hierarchical Method -- 4.6.1 A Hierarchical Blending Element for Discontinuous Gradient Enrichment -- 4.6.2 A Hierarchical Blending Element for Crack Tip Asymptotic Enrichments -- 4.7 The Cutoff Function Method -- 4.7.1 The Weighted Function Blending Method -- 4.7.2 A Variant of the Cutoff Function Method -- 4.8 A DG X-FEM Method -- 4.9 Implementation of Some OptimalX-FEM Type Methods -- 4.9.1 A Plate with a Circular Hole at Its Centre -- 4.9.2 A Plate with a Horizontal Material Interface.

4.9.3 The Fiber Reinforced Concrete in Uniaxial Tension -- 4.10 Pre-Conditioning Strategies inX-FEM -- 4.10.1 Béchetś Pre-Conditioning Scheme -- 4.10.2 Menk-Bordas Pre-Conditioning Scheme -- Chapter 5 Large X-FEM Deformation -- 5.1 Introduction -- 5.2 Large FE Deformation -- 5.3 The Lagrangian Large X-FEM Deformation Method -- 5.3.1 The Enrichment of Displacement Field -- 5.3.2 The Large X-FEM Deformation Formulation -- 5.3.3 Numerical Integration Scheme -- 5.4 Numerical Modeling of Large X-FEM Deformations -- 5.4.1 Modeling an Axial Bar with a Weak Discontinuity -- 5.4.2 Modeling a Plate with the Material Interface -- 5.5 Application of X-FEM in Large Deformation Problems -- 5.5.1 Die-Pressing with a Horizontal Material Interface -- 5.5.2 Die-Pressing with a Rigid Central Core -- 5.5.3 Closed-Die Pressing of a Shaped-Tablet Component -- 5.6 The Extended Arbitrary Lagrangian-Eulerian FEM -- 5.6.1 ALE Formulation -- 5.6.1.1 Kinematics -- 5.6.1.2 ALE Governing Equations -- 5.6.2 The Weak Form of ALE Formulation -- 5.6.3 The ALE FE Discretization -- 5.6.4 The Uncoupled ALE Solution -- 5.6.4.1 Material (Lagrangian) Phase -- 5.6.4.2 Smoothing Phase -- 5.6.4.3 Convection (Eulerian) Phase -- 5.6.5 The X-ALE-FEM Computational Algorithm -- 5.6.5.1 Level Set Update -- 5.6.5.2 Stress Update with Sub-Triangular Numerical Integration -- 5.6.5.3 Stress Update with Sub-Quadrilateral Numerical Integration -- 5.7 Application of the X-ALE-FEM Model -- 5.7.1 The Coining Test -- 5.7.2 A Plate in Tension -- Chapter 6 Contact Friction Modeling with X-FEM -- 6.1 Introduction -- 6.2 Continuum Model of Contact Friction -- 6.2.1 Contact Conditions: The Kuhn-Tucker Rule -- 6.2.2 Plasticity Theory of Friction -- 6.2.3 Continuum Tangent Matrix of Contact Problem -- 6.3 X-FEM Modeling of the Contact Problem -- 6.3.1 The Gauss-Green Theorem for Discontinuous Problems.

6.3.2 The Weak Form of Governing Equation for a Contact Problem -- 6.3.3 The Enrichment of Displacement Field -- 6.4 Modeling of Contact Constraints via the Penalty Method -- 6.4.1 Modeling of an Elastic Bar with a Discontinuity at Its Center -- 6.4.2 Modeling of an Elastic Plate with a Discontinuity at Its Center -- 6.5 Modeling of Contact Constraints via the Lagrange Multipliers Method -- 6.5.1 Modeling the Discontinuity in an Elastic Bar -- 6.5.2 Modeling the Discontinuity in an Elastic Plate -- 6.6 Modeling of Contact Constraints via the Augmented-Lagrange Multipliers Method -- 6.6.1 Modeling an Elastic Bar with a Discontinuity -- 6.6.2 Modeling an Elastic Plate with a Discontinuity -- 6.7 X-FEM Modeling of Large Sliding Contact Problems -- 6.7.1 Large Sliding with Horizontal Material Interfaces -- 6.8 Application of X-FEM Method in Frictional Contact Problems -- 6.8.1 An Elastic Square Plate with Horizontal Interface -- 6.8.1.1 Imposing the Unilateral Contact Constraint -- 6.8.1.2 Modeling the Frictional Stick-Slip Behavior -- 6.8.2 A Square Plate with an Inclined Crack -- 6.8.3 A Double-Clamped Beam with a Central Crack -- 6.8.4 A Rectangular Block with an S-Shaped Frictional Contact Interface -- Chapter 7 Linear Fracture Mechanics with theX-FEMTechnique -- 7.1 Introduction -- 7.2 The Basis of LEFM -- 7.2.1 Energy Balance in Crack Propagation -- 7.2.2 Displacement and Stress Fields at the Crack Tip Area -- 7.2.3 The SIFs -- 7.3 Governing Equations of a Cracked Body -- 7.3.1 The Enrichment of Displacement Field -- 7.3.2 Discretization of Governing Equations -- 7.4 Mixed-Mode Crack Propagation Criteria -- 7.4.1 The Maximum Circumferential Tensile Stress Criterion -- 7.4.2 The Minimum Strain Energy Density Criterion -- 7.4.3 The Maximum Energy Release Rate -- 7.5 Crack Growth Simulation withX-FEM -- 7.5.1 Numerical Integration Scheme.

7.5.2 Numerical Integration of ContourJ-Integral -- 7.6 Application ofX-FEMin Linear Fracture Mechanics -- 7.6.1 X-FEMModeling of aDCB -- 7.6.2 An Infinite Plate with a Finite Crack in Tension -- 7.6.3 An Infinite Plate with an Inclined Crack -- 7.6.4 A Plate with Two Holes and Multiple Cracks -- 7.7 Curved Crack Modeling withX-FEM -- 7.7.1 Modeling a Curved Center Crack in an Infinite Plate -- 7.8 X-FEM Modeling of a Bimaterial Interface Crack -- 7.8.1 The Interfacial Fracture Mechanics -- 7.8.2 The Enrichment of the Displacement Field -- 7.8.3 Modeling of a Center Crack in an Infinite Bimaterial Plate -- Chapter 8 Cohesive Crack Growth with the X-FEM Technique -- 8.1 Introduction -- 8.2 Governing Equations of a Cracked Body -- 8.2.1 The Enrichment of Displacement Field -- 8.2.2 Discretization of Governing Equations -- 8.3 Cohesive Crack Growth Based on the Stress Criterion -- 8.3.1 Cohesive Constitutive Law -- 8.3.2 Crack Growth Criterion and Crack Growth Direction -- 8.3.3 Numerical Integration Scheme -- 8.4 Cohesive Crack Growth Based on the SIF Criterion -- 8.4.1 The Enrichment of Displacement Field -- 8.4.2 The Condition for Smooth Crack Closing -- 8.4.3 Crack Growth Criterion and Crack Growth Direction -- 8.5 Cohesive Crack Growth Based on the Cohesive Segments Method -- 8.5.1 The Enrichment of Displacement Field -- 8.5.2 Cohesive Constitutive Law -- 8.5.3 Crack Growth Criterion and Its Direction for Continuous Crack Propagation -- 8.5.4 Crack Growth Criterion and Its Direction for Discontinuous Crack Propagation -- 8.5.5 Numerical Integration Scheme -- 8.6 Application of X-FEM Method in Cohesive Crack Growth -- 8.6.1 A Three-Point Bending Beam with Symmetric Edge Crack -- 8.6.2 A Plate with an Edge Crack under Impact Velocity -- 8.6.3 A Three-Point Bending Beam with an Eccentric Crack.

Chapter 9 Ductile Fracture Mechanics with a Damage-Plasticity Model in X-FEM.