{"product_id":"extended-finite-element-method-9781118457689","title":"Extended Finite Element Method","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\u003cb\u003eIntroduces the theory and applications of the extended finite element method (XFEM) in the linear and nonlinear problems of continua, structures and geomechanics\u003c\/b\u003e\u003c\/p\u003e \u003cul\u003e \u003cli\u003eExplores the concept of partition of unity, various enrichment functions, and fundamentals of XFEM formulation.\u003c\/li\u003e \u003cli\u003eCovers numerous applications of XFEM including fracture mechanics, large deformation, plasticity, multiphase flow, hydraulic fracturing and contact problems\u003c\/li\u003e \u003cli\u003eAccompanied by a website hosting source code and examples\u003c\/li\u003e \u003c\/ul\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003eSeries Preface xv \u003cp\u003ePreface xvii\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Introduction 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Introduction 1\u003c\/p\u003e \u003cp\u003e1.2 An Enriched Finite Element Method 3\u003c\/p\u003e \u003cp\u003e1.3 A Review on X-FEM: Development and Applications 5\u003c\/p\u003e \u003cp\u003e1.3.1 Coupling X-FEM with the Level-Set Method 6\u003c\/p\u003e \u003cp\u003e1.3.2 Linear Elastic Fracture Mechanics (LEFM) 7\u003c\/p\u003e \u003cp\u003e1.3.3 Cohesive Fracture Mechanics 11\u003c\/p\u003e \u003cp\u003e1.3.4 Composite Materials and Material Inhomogeneities 14\u003c\/p\u003e \u003cp\u003e1.3.5 Plasticity, Damage, and Fatigue Problems 16\u003c\/p\u003e \u003cp\u003e1.3.6 Shear Band Localization 19\u003c\/p\u003e \u003cp\u003e1.3.7 Fluid–Structure Interaction 19\u003c\/p\u003e \u003cp\u003e1.3.8 Fluid Flow in Fractured Porous Media 20\u003c\/p\u003e \u003cp\u003e1.3.9 Fluid Flow and Fluid Mechanics Problems 22\u003c\/p\u003e \u003cp\u003e1.3.10 Phase Transition and Solidification 23\u003c\/p\u003e \u003cp\u003e1.3.11 Thermal and Thermo-Mechanical Problems 24\u003c\/p\u003e \u003cp\u003e1.3.12 Plates and Shells 24\u003c\/p\u003e \u003cp\u003e1.3.13 Contact Problems 26\u003c\/p\u003e \u003cp\u003e1.3.14 Topology Optimization 28\u003c\/p\u003e \u003cp\u003e1.3.15 Piezoelectric and Magneto-Electroelastic Problems 28\u003c\/p\u003e \u003cp\u003e1.3.16 Multi-Scale Modeling 29\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Extended Finite Element Formulation 31\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Introduction 31\u003c\/p\u003e \u003cp\u003e2.2 The Partition of Unity Finite Element Method 33\u003c\/p\u003e \u003cp\u003e2.3 The Enrichment of Approximation Space 35\u003c\/p\u003e \u003cp\u003e2.3.1 Intrinsic Enrichment 35\u003c\/p\u003e \u003cp\u003e2.3.2 Extrinsic Enrichment 36\u003c\/p\u003e \u003cp\u003e2.4 The Basis of X-FEM Approximation 37\u003c\/p\u003e \u003cp\u003e2.4.1 The Signed Distance Function 39\u003c\/p\u003e \u003cp\u003e2.4.2 The Heaviside Function 43\u003c\/p\u003e \u003cp\u003e2.5 Blending Elements 46\u003c\/p\u003e \u003cp\u003e2.6 Governing Equation of a Body with Discontinuity 49\u003c\/p\u003e \u003cp\u003e2.6.1 The Divergence Theorem for Discontinuous Problems 50\u003c\/p\u003e \u003cp\u003e2.6.2 The Weak form of Governing Equation 51\u003c\/p\u003e \u003cp\u003e2.7 The X-FEM Discretization of Governing Equation 53\u003c\/p\u003e \u003cp\u003e2.7.1 Numerical Implementation of X-FEM Formulation 55\u003c\/p\u003e \u003cp\u003e2.7.2 Numerical Integration Algorithm 57\u003c\/p\u003e \u003cp\u003e2.8 Application of X-FEM in Weak and Strong Discontinuities 60\u003c\/p\u003e \u003cp\u003e2.8.1 Modeling an Elastic Bar with a Strong Discontinuity 61\u003c\/p\u003e \u003cp\u003e2.8.2 Modeling an Elastic Bar with a Weak Discontinuity 63\u003c\/p\u003e \u003cp\u003e2.8.3 Modeling an Elastic Plate with a Crack Interface at its Center 66\u003c\/p\u003e \u003cp\u003e2.8.4 Modeling an Elastic Plate with a Material Interface at its Center 68\u003c\/p\u003e \u003cp\u003e2.9 Higher Order X-FEM 70\u003c\/p\u003e \u003cp\u003e2.10 Implementation of X-FEM with Higher Order Elements 73\u003c\/p\u003e \u003cp\u003e2.10.1 Higher Order X-FEM Modeling of a Plate with a Material Interface 73\u003c\/p\u003e \u003cp\u003e2.10.2 Higher Order X-FEM Modeling of a Plate with a Curved Crack Interface 75\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Enrichment Elements 77\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Introduction 77\u003c\/p\u003e \u003cp\u003e3.2 Tracking Moving Boundaries 78\u003c\/p\u003e \u003cp\u003e3.3 Level Set Method 81\u003c\/p\u003e \u003cp\u003e3.3.1 Numerical Implementation of LSM 82\u003c\/p\u003e \u003cp\u003e3.3.2 Coupling the LSM with X-FEM 83\u003c\/p\u003e \u003cp\u003e3.4 Fast Marching Method 85\u003c\/p\u003e \u003cp\u003e3.4.1 Coupling the FMM with X-FEM 87\u003c\/p\u003e \u003cp\u003e3.5 X-FEM Enrichment Functions 88\u003c\/p\u003e \u003cp\u003e3.5.1 Bimaterials, Voids, and Inclusions 88\u003c\/p\u003e \u003cp\u003e3.5.2 Strong Discontinuities and Crack Interfaces 91\u003c\/p\u003e \u003cp\u003e3.5.3 Brittle Cracks 93\u003c\/p\u003e \u003cp\u003e3.5.4 Cohesive Cracks 97\u003c\/p\u003e \u003cp\u003e3.5.5 Plastic Fracture Mechanics 99\u003c\/p\u003e \u003cp\u003e3.5.6 Multiple Cracks 101\u003c\/p\u003e \u003cp\u003e3.5.7 Fracture in Bimaterial Problems 102\u003c\/p\u003e \u003cp\u003e3.5.8 Polycrystalline Microstructure 106\u003c\/p\u003e \u003cp\u003e3.5.9 Dislocations 111\u003c\/p\u003e \u003cp\u003e3.5.10 Shear Band Localization 113\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Blending Elements 119\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Introduction 119\u003c\/p\u003e \u003cp\u003e4.2 Convergence Analysis in the X-FEM 120\u003c\/p\u003e \u003cp\u003e4.3 Ill-Conditioning in the X-FEM Method 124\u003c\/p\u003e \u003cp\u003e4.3.1 One-Dimensional Problem with Material Interface 126\u003c\/p\u003e \u003cp\u003e4.4 Blending Strategies in X-FEM 128\u003c\/p\u003e \u003cp\u003e4.5 Enhanced Strain Method 130\u003c\/p\u003e \u003cp\u003e4.5.1 An Enhanced Strain Blending Element for the Ramp Enrichment Function 132\u003c\/p\u003e \u003cp\u003e4.5.2 An Enhanced Strain Blending Element for Asymptotic Enrichment Functions 134\u003c\/p\u003e \u003cp\u003e4.6 The Hierarchical Method 135\u003c\/p\u003e \u003cp\u003e4.6.1 A Hierarchical Blending Element for Discontinuous Gradient Enrichment 135\u003c\/p\u003e \u003cp\u003e4.6.2 A Hierarchical Blending Element for Crack Tip Asymptotic Enrichments 137\u003c\/p\u003e \u003cp\u003e4.7 The Cutoff Function Method 138\u003c\/p\u003e \u003cp\u003e4.7.1 The Weighted Function Blending Method 140\u003c\/p\u003e \u003cp\u003e4.7.2 A Variant of the Cutoff Function Method 142\u003c\/p\u003e \u003cp\u003e4.8 A DG X-FEM Method 143\u003c\/p\u003e \u003cp\u003e4.9 Implementation of Some Optimal X-FEM Type Methods 147\u003c\/p\u003e \u003cp\u003e4.9.1 A Plate with a Circular Hole at Its Centre 148\u003c\/p\u003e \u003cp\u003e4.9.2 A Plate with a Horizontal Material Interface 149\u003c\/p\u003e \u003cp\u003e4.9.3 The Fiber Reinforced Concrete in Uniaxial Tension 151\u003c\/p\u003e \u003cp\u003e4.10 Pre-Conditioning Strategies in X-FEM 154\u003c\/p\u003e \u003cp\u003e4.10.1 Béchet’s Pre-Conditioning Scheme 155\u003c\/p\u003e \u003cp\u003e4.10.2 Menk–Bordas Pre-Conditioning Scheme 156\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Large X-FEM Deformation 161\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Introduction 161\u003c\/p\u003e \u003cp\u003e5.2 Large FE Deformation 163\u003c\/p\u003e \u003cp\u003e5.3 The Lagrangian Large X-FEM Deformation Method 167\u003c\/p\u003e \u003cp\u003e5.3.1 The Enrichment of Displacement Field 167\u003c\/p\u003e \u003cp\u003e5.3.2 The Large X-FEM Deformation Formulation 170\u003c\/p\u003e \u003cp\u003e5.3.3 Numerical Integration Scheme 172\u003c\/p\u003e \u003cp\u003e5.4 Numerical Modeling of Large X-FEM Deformations 173\u003c\/p\u003e \u003cp\u003e5.4.1 Modeling an Axial Bar with a Weak Discontinuity 173\u003c\/p\u003e \u003cp\u003e5.4.2 Modeling a Plate with the Material Interface 177\u003c\/p\u003e \u003cp\u003e5.5 Application of X-FEM in Large Deformation Problems 181\u003c\/p\u003e \u003cp\u003e5.5.1 Die-Pressing with a Horizontal Material Interface 182\u003c\/p\u003e \u003cp\u003e5.5.2 Die-Pressing with a Rigid Central Core 186\u003c\/p\u003e \u003cp\u003e5.5.3 Closed-Die Pressing of a Shaped-Tablet Component 188\u003c\/p\u003e \u003cp\u003e5.6 The Extended Arbitrary Lagrangian–Eulerian FEM 192\u003c\/p\u003e \u003cp\u003e5.6.1 ALE Formulation 192\u003c\/p\u003e \u003cp\u003e5.6.1.1 Kinematics 193\u003c\/p\u003e \u003cp\u003e5.6.1.2 ALE Governing Equations 194\u003c\/p\u003e \u003cp\u003e5.6.2 The Weak Form of ALE Formulation 195\u003c\/p\u003e \u003cp\u003e5.6.3 The ALE FE Discretization 196\u003c\/p\u003e \u003cp\u003e5.6.4 The Uncoupled ALE Solution 198\u003c\/p\u003e \u003cp\u003e5.6.4.1 Material (Lagrangian) Phase 199\u003c\/p\u003e \u003cp\u003e5.6.4.2 Smoothing Phase 199\u003c\/p\u003e \u003cp\u003e5.6.4.3 Convection (Eulerian) Phase 200\u003c\/p\u003e \u003cp\u003e5.6.5 The X-ALE-FEM Computational Algorithm 202\u003c\/p\u003e \u003cp\u003e5.6.5.1 Level Set Update 203\u003c\/p\u003e \u003cp\u003e5.6.5.2 Stress Update with Sub-Triangular Numerical Integration 204\u003c\/p\u003e \u003cp\u003e5.6.5.3 Stress Update with Sub-Quadrilateral Numerical Integration 205\u003c\/p\u003e \u003cp\u003e5.7 Application of the X-ALE-FEM Model 208\u003c\/p\u003e \u003cp\u003e5.7.1 The Coining Test 208\u003c\/p\u003e \u003cp\u003e5.7.2 A Plate in Tension 209\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Contact Friction Modeling with X-FEM 215\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Introduction 215\u003c\/p\u003e \u003cp\u003e6.2 Continuum Model of Contact Friction 216\u003c\/p\u003e \u003cp\u003e6.2.1 Contact Conditions: The Kuhn–Tucker Rule 217\u003c\/p\u003e \u003cp\u003e6.2.2 Plasticity Theory of Friction 218\u003c\/p\u003e \u003cp\u003e6.2.3 Continuum Tangent Matrix of Contact Problem 221\u003c\/p\u003e \u003cp\u003e6.3 X-FEM Modeling of the Contact Problem 223\u003c\/p\u003e \u003cp\u003e6.3.1 The Gauss–Green Theorem for Discontinuous Problems 223\u003c\/p\u003e \u003cp\u003e6.3.2 The Weak Form of Governing Equation for a Contact Problem 224\u003c\/p\u003e \u003cp\u003e6.3.3 The Enrichment of Displacement Field 226\u003c\/p\u003e \u003cp\u003e6.4 Modeling of Contact Constraints via the Penalty Method 227\u003c\/p\u003e \u003cp\u003e6.4.1 Modeling of an Elastic Bar with a Discontinuity at Its Center 231\u003c\/p\u003e \u003cp\u003e6.4.2 Modeling of an Elastic Plate with a Discontinuity at Its Center 233\u003c\/p\u003e \u003cp\u003e6.5 Modeling of Contact Constraints via the Lagrange Multipliers Method 235\u003c\/p\u003e \u003cp\u003e6.5.1 Modeling the Discontinuity in an Elastic Bar 239\u003c\/p\u003e \u003cp\u003e6.5.2 Modeling the Discontinuity in an Elastic Plate 240\u003c\/p\u003e \u003cp\u003e6.6 Modeling of Contact Constraints via the Augmented-Lagrange Multipliers Method 241\u003c\/p\u003e \u003cp\u003e6.6.1 Modeling an Elastic Bar with a Discontinuity 244\u003c\/p\u003e \u003cp\u003e6.6.2 Modeling an Elastic Plate with a Discontinuity 245\u003c\/p\u003e \u003cp\u003e6.7 X-FEM Modeling of Large Sliding Contact Problems 246\u003c\/p\u003e \u003cp\u003e6.7.1 Large Sliding with Horizontal Material Interfaces 249\u003c\/p\u003e \u003cp\u003e6.8 Application of X-FEM Method in Frictional Contact Problems 251\u003c\/p\u003e \u003cp\u003e6.8.1 An Elastic Square Plate with Horizontal Interface 251\u003c\/p\u003e \u003cp\u003e6.8.1.1 Imposing the Unilateral Contact Constraint 252\u003c\/p\u003e \u003cp\u003e6.8.1.2 Modeling the Frictional Stick–Slip Behavior 255\u003c\/p\u003e \u003cp\u003e6.8.2 A Square Plate with an Inclined Crack 256\u003c\/p\u003e \u003cp\u003e6.8.3 A Double-Clamped Beam with a Central Crack 259\u003c\/p\u003e \u003cp\u003e6.8.4 A Rectangular Block with an S–Shaped Frictional Contact Interface 261\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Linear Fracture Mechanics with the X-FEM Technique 267\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Introduction 267\u003c\/p\u003e \u003cp\u003e7.2 The Basis of LEFM 269\u003c\/p\u003e \u003cp\u003e7.2.1 Energy Balance in Crack Propagation 270\u003c\/p\u003e \u003cp\u003e7.2.2 Displacement and Stress Fields at the Crack Tip Area 271\u003c\/p\u003e \u003cp\u003e7.2.3 The SIFs 273\u003c\/p\u003e \u003cp\u003e7.3 Governing Equations of a Cracked Body 276\u003c\/p\u003e \u003cp\u003e7.3.1 The Enrichment of Displacement Field 277\u003c\/p\u003e \u003cp\u003e7.3.2 Discretization of Governing Equations 280\u003c\/p\u003e \u003cp\u003e7.4 Mixed-Mode Crack Propagation Criteria 283\u003c\/p\u003e \u003cp\u003e7.4.1 The Maximum Circumferential Tensile Stress Criterion 283\u003c\/p\u003e \u003cp\u003e7.4.2 The Minimum Strain Energy Density Criterion 284\u003c\/p\u003e \u003cp\u003e7.4.3 The Maximum Energy Release Rate 284\u003c\/p\u003e \u003cp\u003e7.5 Crack Growth Simulation with X-FEM 285\u003c\/p\u003e \u003cp\u003e7.5.1 Numerical Integration Scheme 287\u003c\/p\u003e \u003cp\u003e7.5.2 Numerical Integration of Contour J–Integral 289\u003c\/p\u003e \u003cp\u003e7.6 Application of X-FEM in Linear Fracture Mechanics 290\u003c\/p\u003e \u003cp\u003e7.6.1 X-FEM Modeling of a DCB 290\u003c\/p\u003e \u003cp\u003e7.6.2 An Infinite Plate with a Finite Crack in Tension 294\u003c\/p\u003e \u003cp\u003e7.6.3 An Infinite Plate with an Inclined Crack 298\u003c\/p\u003e \u003cp\u003e7.6.4 A Plate with Two Holes and Multiple Cracks 300\u003c\/p\u003e \u003cp\u003e7.7 Curved Crack Modeling with X-FEM 304\u003c\/p\u003e \u003cp\u003e7.7.1 Modeling a Curved Center Crack in an Infinite Plate 307\u003c\/p\u003e \u003cp\u003e7.8 X-FEM Modeling of a Bimaterial Interface Crack 309\u003c\/p\u003e \u003cp\u003e7.8.1 The Interfacial Fracture Mechanics 310\u003c\/p\u003e \u003cp\u003e7.8.2 The Enrichment of the Displacement Field 311\u003c\/p\u003e \u003cp\u003e7.8.3 Modeling of a Center Crack in an Infinite Bimaterial Plate 314\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Cohesive Crack Growth with the X-FEM Technique 317\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Introduction 317\u003c\/p\u003e \u003cp\u003e8.2 Governing Equations of a Cracked Body 320\u003c\/p\u003e \u003cp\u003e8.2.1 The Enrichment of Displacement Field 322\u003c\/p\u003e \u003cp\u003e8.2.2 Discretization of Governing Equations 323\u003c\/p\u003e \u003cp\u003e8.3 Cohesive Crack Growth Based on the Stress Criterion 325\u003c\/p\u003e \u003cp\u003e8.3.1 Cohesive Constitutive Law 325\u003c\/p\u003e \u003cp\u003e8.3.2 Crack Growth Criterion and Crack Growth Direction 326\u003c\/p\u003e \u003cp\u003e8.3.3 Numerical Integration Scheme 328\u003c\/p\u003e \u003cp\u003e8.4 Cohesive Crack Growth Based on the SIF Criterion 328\u003c\/p\u003e \u003cp\u003e8.4.1 The Enrichment of Displacement Field 329\u003c\/p\u003e \u003cp\u003e8.4.2 The Condition for Smooth Crack Closing 332\u003c\/p\u003e \u003cp\u003e8.4.3 Crack Growth Criterion and Crack Growth Direction 332\u003c\/p\u003e \u003cp\u003e8.5 Cohesive Crack Growth Based on the Cohesive Segments Method 334\u003c\/p\u003e \u003cp\u003e8.5.1 The Enrichment of Displacement Field 334\u003c\/p\u003e \u003cp\u003e8.5.2 Cohesive Constitutive Law 335\u003c\/p\u003e \u003cp\u003e8.5.3 Crack Growth Criterion and Its Direction for Continuous Crack Propagation 336\u003c\/p\u003e \u003cp\u003e8.5.4 Crack Growth Criterion and Its Direction for Discontinuous Crack Propagation 339\u003c\/p\u003e \u003cp\u003e8.5.5 Numerical Integration Scheme 341\u003c\/p\u003e \u003cp\u003e8.6 Application of X-FEM Method in Cohesive Crack Growth 341\u003c\/p\u003e \u003cp\u003e8.6.1 A Three-Point Bending Beam with Symmetric Edge Crack 341\u003c\/p\u003e \u003cp\u003e8.6.2 A Plate with an Edge Crack under Impact Velocity 343\u003c\/p\u003e \u003cp\u003e8.6.3 A Three-Point Bending Beam with an Eccentric Crack 346\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Ductile Fracture Mechanics with a Damage-Plasticity Model in X-FEM 351\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Introduction 351\u003c\/p\u003e \u003cp\u003e9.2 Large FE Deformation Formulation 353\u003c\/p\u003e \u003cp\u003e9.3 Modified X-FEM Formulation 356\u003c\/p\u003e \u003cp\u003e9.4 Large X-FEM Deformation Formulation 359\u003c\/p\u003e \u003cp\u003e9.5 The Damage–Plasticity Model 364\u003c\/p\u003e \u003cp\u003e9.6 The Nonlocal Gradient Damage Plasticity 368\u003c\/p\u003e \u003cp\u003e9.7 Ductile Fracture with X-FEM Plasticity Model 369\u003c\/p\u003e \u003cp\u003e9.8 Ductile Fracture with X-FEM Non-Local Damage-Plasticity Model 372\u003c\/p\u003e \u003cp\u003e9.8.1 Crack Initiation and Crack Growth Direction 372\u003c\/p\u003e \u003cp\u003e9.8.2 Crack Growth with a Null Step Analysis 375\u003c\/p\u003e \u003cp\u003e9.8.3 Crack Growth with a Relaxation Phase Analysis 377\u003c\/p\u003e \u003cp\u003e9.8.4 Locking Issues in Crack Growth Modeling 379\u003c\/p\u003e \u003cp\u003e9.9 Application of X-FEM Damage-Plasticity Model 380\u003c\/p\u003e \u003cp\u003e9.9.1 The Necking Problem 380\u003c\/p\u003e \u003cp\u003e9.9.2 The CT Test 383\u003c\/p\u003e \u003cp\u003e9.9.3 The Double-Notched Specimen 385\u003c\/p\u003e \u003cp\u003e9.10 Dynamic Large X-FEM Deformation Formulation 387\u003c\/p\u003e \u003cp\u003e9.10.1 The Dynamic X-FEM Discretization 388\u003c\/p\u003e \u003cp\u003e9.10.2 The Large Strain Model 390\u003c\/p\u003e \u003cp\u003e9.10.3 The Contact Friction Model 391\u003c\/p\u003e \u003cp\u003e9.11 The Time Domain Discretization: The Dynamic Explicit Central Difference Method 393\u003c\/p\u003e \u003cp\u003e9.12 Implementation of Dynamic X-FEM Damage-Plasticity Model 396\u003c\/p\u003e \u003cp\u003e9.12.1 A Plate with an Inclined Crack 398\u003c\/p\u003e \u003cp\u003e9.12.2 The Low Cycle Fatigue Test 400\u003c\/p\u003e \u003cp\u003e9.12.3 The Cyclic CT Test 401\u003c\/p\u003e \u003cp\u003e9.12.4 The Double Notched Specimen in Cyclic Loading 405\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 X-FEM Modeling of Saturated\/Semi-Saturated Porous Media 409\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Introduction 409\u003c\/p\u003e \u003cp\u003e10.1.1 Governing Equations of Deformable Porous Media 411\u003c\/p\u003e \u003cp\u003e10.2 The X-FEM Formulation of Deformable Porous Media with Weak Discontinuities 414\u003c\/p\u003e \u003cp\u003e10.2.1 Approximation of Displacement and Pressure Fields 415\u003c\/p\u003e \u003cp\u003e10.2.2 The X-FEM Spatial Discretization 418\u003c\/p\u003e \u003cp\u003e10.2.3 The Time Domain Discretization and Solution Procedure 419\u003c\/p\u003e \u003cp\u003e10.2.4 Numerical Integration Scheme 421\u003c\/p\u003e \u003cp\u003e10.3 Application of the X-FEM Method in Deformable Porous Media with Arbitrary Interfaces 422\u003c\/p\u003e \u003cp\u003e10.3.1 An Elastic Soil Column 422\u003c\/p\u003e \u003cp\u003e10.3.2 An Elastic Foundation 424\u003c\/p\u003e \u003cp\u003e10.4 Modeling Hydraulic Fracture Propagation in Deformable Porous Media 427\u003c\/p\u003e \u003cp\u003e10.4.1 Governing Equations of a Fractured Porous Medium 428\u003c\/p\u003e \u003cp\u003e10.4.2 The Weak Formulation of a Fractured Porous Medium 430\u003c\/p\u003e \u003cp\u003e10.5 The X-FEM Formulation of Deformable Porous Media with Strong Discontinuities 434\u003c\/p\u003e \u003cp\u003e10.5.1 Approximation of the Displacement and Pressure Fields 434\u003c\/p\u003e \u003cp\u003e10.5.2 The X-FEM Spatial Discretization 437\u003c\/p\u003e \u003cp\u003e10.5.3 The Time Domain Discretization and Solution Procedure 438\u003c\/p\u003e \u003cp\u003e10.6 Alternative Approaches to Fluid Flow Simulation within the Fracture 442\u003c\/p\u003e \u003cp\u003e10.6.1 A Partitioned Solution Algorithm for Interfacial Pressure 442\u003c\/p\u003e \u003cp\u003e10.6.2 A Time-Dependent Constant Pressure Algorithm 444\u003c\/p\u003e \u003cp\u003e10.7 Application of the X-FEM Method in Hydraulic Fracture Propagation of Saturated Porous Media 445\u003c\/p\u003e \u003cp\u003e10.7.1 An Infinite Saturated Porous Medium with an Inclined Crack 446\u003c\/p\u003e \u003cp\u003e10.7.2 Hydraulic Fracture Propagation in an Infinite Poroelastic Medium 449\u003c\/p\u003e \u003cp\u003e10.7.3 Hydraulic Fracturing in a Concrete Gravity Dam 452\u003c\/p\u003e \u003cp\u003e10.8 X-FEM Modeling of Contact Behavior in Fractured Porous Media 455\u003c\/p\u003e \u003cp\u003e10.8.1 Contact Behavior in a Fractured Medium 455\u003c\/p\u003e \u003cp\u003e10.8.2 X-FEM Formulation of Contact along the Fracture 456\u003c\/p\u003e \u003cp\u003e10.8.3 Consolidation of a Porous Block with a Vertical Discontinuity 457\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Hydraulic Fracturing in Multi-Phase Porous Media with X-FEM 461\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Introduction 461\u003c\/p\u003e \u003cp\u003e11.2 The Physical Model of Multi-Phase Porous Media 463\u003c\/p\u003e \u003cp\u003e11.3 Governing Equations of Multi-Phase Porous Medium 465\u003c\/p\u003e \u003cp\u003e11.4 The X-FEM Formulation of Multi-Phase Porous Media with Weak Discontinuities 467\u003c\/p\u003e \u003cp\u003e11.4.1 Approximation of the Primary Variables 469\u003c\/p\u003e \u003cp\u003e11.4.2 Discretization of Equilibrium and Flow Continuity Equations 473\u003c\/p\u003e \u003cp\u003e11.4.3 Solution Procedure of Discretized Equilibrium Equations 476\u003c\/p\u003e \u003cp\u003e11.5 Application of X-FEM Method in Multi-Phase Porous Media with Arbitrary Interfaces 477\u003c\/p\u003e \u003cp\u003e11.6 The X-FEM Formulation for Hydraulic Fracturing in Multi-Phase Porous Media 482\u003c\/p\u003e \u003cp\u003e11.7 Discretization of Multi-Phase Governing Equations with Strong Discontinuities 487\u003c\/p\u003e \u003cp\u003e11.8 Solution Procedure for Fully Coupled Nonlinear Equations 493\u003c\/p\u003e \u003cp\u003e11.9 Computational Notes in Hydraulic Fracture Modeling 497\u003c\/p\u003e \u003cp\u003e11.10 Application of the X-FEM Method to Hydraulic Fracture Propagation of Multi-Phase Porous Media 499\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Thermo-Hydro-Mechanical Modeling of Porous Media with X-FEM 509\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 Introduction 509\u003c\/p\u003e \u003cp\u003e12.2 THM Governing Equations of Saturated Porous Media 511\u003c\/p\u003e \u003cp\u003e12.3 Discontinuities in a THM Medium 513\u003c\/p\u003e \u003cp\u003e12.4 The X-FEM Formulation of THM Governing Equations 514\u003c\/p\u003e \u003cp\u003e12.4.1 Approximation of Displacement, Pressure, and Temperature Fields 515\u003c\/p\u003e \u003cp\u003e12.4.2 The X-FEM Spatial Discretization 517\u003c\/p\u003e \u003cp\u003e12.4.3 The Time Domain Discretization 520\u003c\/p\u003e \u003cp\u003e12.5 Application of the X-FEM Method to THM Behavior of Porous Media 521\u003c\/p\u003e \u003cp\u003e12.5.1 A Plate with an Inclined Crack in Thermal Loading 521\u003c\/p\u003e \u003cp\u003e12.5.2 A Plate with an Edge Crack in Thermal Loading 522\u003c\/p\u003e \u003cp\u003e12.5.3 An Impermeable Discontinuity in Saturated Porous Media 524\u003c\/p\u003e \u003cp\u003e12.5.4 An Inclined Fault in Porous Media 527\u003c\/p\u003e \u003cp\u003eReferences 533\u003c\/p\u003e \u003cp\u003eIndex 557\u003c\/p\u003e","brand":"John Wiley \u0026 Sons Inc","offers":[{"title":"Default Title","offer_id":49406873993559,"sku":"9781118457689","price":93.56,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781118457689.jpg?v=1730497408","url":"https:\/\/bookcurl.com\/products\/extended-finite-element-method-9781118457689","provider":"Book Curl","version":"1.0","type":"link"}