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Preface xvii
1 Introduction 1
1.1 Introduction 1
1.2 An Enriched Finite Element Method 3
1.3 A Review on X-FEM: Development and Applications 5
1.3.1 Coupling X-FEM with the Level-Set Method 6
1.3.2 Linear Elastic Fracture Mechanics (LEFM) 7
1.3.3 Cohesive Fracture Mechanics 11
1.3.4 Composite Materials and Material Inhomogeneities 14
1.3.5 Plasticity, Damage, and Fatigue Problems 16
1.3.6 Shear Band Localization 19
1.3.7 Fluid-Structure Interaction 19
1.3.8 Fluid Flow in Fractured Porous Media 20
1.3.9 Fluid Flow and Fluid Mechanics Problems 22
1.3.10 Phase Transition and Solidification 23
1.3.11 Thermal and Thermo-Mechanical Problems 24
1.3.12 Plates and Shells 24
1.3.13 Contact Problems 26
1.3.14 Topology Optimization 28
1.3.15 Piezoelectric and Magneto-Electroelastic Problems 28
1.3.16 Multi-Scale Modeling 29
2 Extended Finite Element Formulation 31
2.1 Introduction 31
2.2 The Partition of Unity Finite Element Method 33
2.3 The Enrichment of Approximation Space 35
2.3.1 Intrinsic Enrichment 35
2.3.2 Extrinsic Enrichment 36
2.4 The Basis of X-FEM Approximation 37
2.4.1 The Signed Distance Function 39
2.4.2 The Heaviside Function 43
2.5 Blending Elements 46
2.6 Governing Equation of a Body with Discontinuity 49
2.6.1 The Divergence Theorem for Discontinuous Problems 50
2.6.2 The Weak form of Governing Equation 51
2.7 The X-FEM Discretization of Governing Equation 53
2.7.1 Numerical Implementation of X-FEM Formulation 55
2.7.2 Numerical Integration Algorithm 57
2.8 Application of X-FEM in Weak and Strong Discontinuities 60
2.8.1 Modeling an Elastic Bar with a Strong Discontinuity 61
2.8.2 Modeling an Elastic Bar with a Weak Discontinuity 63
2.8.3 Modeling an Elastic Plate with a Crack Interface at its Center 66
2.8.4 Modeling an Elastic Plate with a Material Interface at its Center 68
2.9 Higher Order X-FEM 70
2.10 Implementation of X-FEM with Higher Order Elements 73
2.10.1 Higher Order X-FEM Modeling of a Plate with a Material Interface 73
2.10.2 Higher Order X-FEM Modeling of a Plate with a Curved Crack Interface 75
3 Enrichment Elements 77
3.1 Introduction 77
3.2 Tracking Moving Boundaries 78
3.3 Level Set Method 81
3.3.1 Numerical Implementation of LSM 82
3.3.2 Coupling the LSM with X-FEM 83
3.4 Fast Marching Method 85
3.4.1 Coupling the FMM with X-FEM 87
3.5 X-FEM Enrichment Functions 88
3.5.1 Bimaterials, Voids, and Inclusions 88
3.5.2 Strong Discontinuities and Crack Interfaces 91
3.5.3 Brittle Cracks 93
3.5.4 Cohesive Cracks 97
3.5.5 Plastic Fracture Mechanics 99
3.5.6 Multiple Cracks 101
3.5.7 Fracture in Bimaterial Problems 102
3.5.8 Polycrystalline Microstructure 106
3.5.9 Dislocations 111
3.5.10 Shear Band Localization 113
4 Blending Elements 119
4.1 Introduction 119
4.2 Convergence Analysis in the X-FEM 120
4.3 Ill-Conditioning in the X-FEM Method 124
4.3.1 One-Dimensional Problem with Material Interface 126
4.4 Blending Strategies in X-FEM 128
4.5 Enhanced Strain Method 130
4.5.1 An Enhanced Strain Blending Element for the Ramp Enrichment Function 132
4.5.2 An Enhanced Strain Blending Element for Asymptotic Enrichment Functions 134
4.6 The Hierarchical Method 135
4.6.1 A Hierarchical Blending Element for Discontinuous Gradient Enrichment 135
4.6.2 A Hierarchical Blending Element for Crack Tip Asymptotic Enrichments 137
4.7 The Cutoff Function Method 138
4.7.1 The Weighted Function Blending Method 140
4.7.2 A Variant of the Cutoff Function Method 142
4.8 A DG X-FEM Method 143
4.9 Implementation of Some Optimal X-FEM Type Methods 147
4.9.1 A Plate with a Circular Hole at Its Centre 148
4.9.2 A Plate with a Horizontal Material Interface 149
4.9.3 The Fiber Reinforced Concrete in Uniaxial Tension 151
4.10 Pre-Conditioning Strategies in X-FEM 154
4.10.1 Béchet's Pre-Conditioning Scheme 155
4.10.2 Menk-Bordas Pre-Conditioning Scheme 156
5 Large X-FEM Deformation 161
5.1 Introduction 161
5.2 Large FE Deformation 163
5.3 The Lagrangian Large X-FEM Deformation Method 167
5.3.1 The Enrichment of Displacement Field 167
5.3.2 The Large X-FEM Deformation Formulation 170
5.3.3 Numerical Integration Scheme 172
5.4 Numerical Modeling of Large X-FEM Deformations 173
5.4.1 Modeling an Axial Bar with a Weak Discontinuity 173
5.4.2 Modeling a Plate with the Material Interface 177
5.5 Application of X-FEM in Large Deformation Problems 181
5.5.1 Die-Pressing with a Horizontal Material Interface 182
5.5.2 Die-Pressing with a Rigid Central Core 186
5.5.3 Closed-Die Pressing of a Shaped-Tablet Component 188
5.6 The Extended Arbitrary Lagrangian-Eulerian FEM 192
5.6.1 ALE Formulation 192
5.6.1.1 Kinematics 193
5.6.1.2 ALE Governing Equations 194
5.6.2 The Weak Form of ALE Formulation 195
5.6.3 The ALE FE Discretization 196
5.6.4 The Uncoupled ALE Solution 198
5.6.4.1 Material (Lagrangian) Phase 199
5.6.4.2 Smoothing Phase 199
5.6.4.3 Convection (Eulerian) Phase 200
5.6.5 The X-ALE-FEM Computational Algorithm 202
5.6.5.1 Level Set Update 203
5.6.5.2 Stress Update with Sub-Triangular Numerical Integration 204
5.6.5.3 Stress Update with Sub-Quadrilateral Numerical Integration 205
5.7 Application of the X-ALE-FEM Model 208
5.7.1 The Coining Test 208
5.7.2 A Plate in Tension 209
6 Contact Friction Modeling with X-FEM 215
6.1 Introduction 215
6.2 Continuum Model of Contact Friction 216
6.2.1 Contact Conditions: The Kuhn-Tucker Rule 217
6.2.2 Plasticity Theory of Friction 218
6.2.3 Continuum Tangent Matrix of Contact Problem 221
6.3 X-FEM Modeling of the Contact Problem 223
6.3.1 The Gauss-Green Theorem for Discontinuous Problems 223
6.3.2 The Weak Form of Governing Equation for a Contact Problem 224
6.3.3 The Enrichment of Displacement Field 226
6.4 Modeling of Contact Constraints via the Penalty Method 227
6.4.1 Modeling of an Elastic Bar with a Discontinuity at Its Center 231
6.4.2 Modeling of an Elastic Plate with a Discontinuity at Its Center 233
6.5 Modeling of Contact Constraints via the Lagrange Multipliers Method 235
6.5.1 Modeling the Discontinuity in an Elastic Bar 239
6.5.2 Modeling the Discontinuity in an Elastic Plate 240
6.6 Modeling of Contact Constraints via the Augmented-Lagrange Multipliers Method 241
6.6.1 Modeling an Elastic Bar with a Discontinuity 244
6.6.2 Modeling an Elastic Plate with a Discontinuity 245
6.7 X-FEM Modeling of Large Sliding Contact Problems 246
6.7.1 Large Sliding with Horizontal Material Interfaces 249
6.8 Application of X-FEM Method in Frictional Contact Problems 251
6.8.1 An Elastic Square Plate with Horizontal Interface 251
6.8.1.1 Imposing the Unilateral Contact Constraint 252
6.8.1.2 Modeling the Frictional Stick-Slip Behavior 255
6.8.2 A Square Plate with an Inclined Crack 256
6.8.3 A Double-Clamped Beam with a Central Crack 259
6.8.4 A Rectangular Block with an S-Shaped Frictional Contact Interface 261
7 Linear Fracture Mechanics with the X-FEM Technique 267
7.1 Introduction 267
7.2 The Basis of LEFM 269
7.2.1 Energy Balance in Crack Propagation 270
7.2.2 Displacement and Stress Fields at the Crack Tip Area 271
7.2.3 The SIFs 273
7.3 Governing Equations of a Cracked Body 276
7.3.1 The Enrichment of Displacement Field 277
7.3.2 Discretization of Governing Equations 280
7.4 Mixed-Mode Crack Propagation Criteria 283
7.4.1 The Maximum Circumferential Tensile Stress Criterion 283
7.4.2 The Minimum Strain Energy Density Criterion 284
7.4.3 The Maximum Energy Release Rate 284
7.5 Crack Growth Simulation with X-FEM 285
7.5.1 Numerical Integration Scheme 287
7.5.2 Numerical Integration of Contour J-Integral 289
7.6 Application of X-FEM in Linear Fracture Mechanics 290
7.6.1 X-FEM Modeling of a DCB 290
7.6.2 An Infinite Plate with a Finite Crack in Tension 294
7.6.3 An Infinite Plate with an Inclined Crack 298
7.6.4 A Plate with Two Holes and Multiple Cracks 300
7.7 Curved Crack Modeling with X-FEM 304
7.7.1 Modeling a Curved Center Crack in an Infinite Plate 307
7.8 X-FEM Modeling of a Bimaterial Interface Crack 309
7.8.1 The Interfacial Fracture Mechanics 310
7.8.2 The Enrichment of the Displacement Field 311
7.8.3 Modeling of a Center Crack in an Infinite Bimaterial Plate 314
8 Cohesive Crack Growth with the X-FEM Technique 317
8.1 Introduction 317
8.2 Governing Equations of a Cracked Body 320
8.2.1 The Enrichment of Displacement Field 322
8.2.2 Discretization of Governing Equations 323
8.3 Cohesive Crack Growth Based on the Stress Criterion 325
8.3.1 Cohesive Constitutive Law 325
8.3.2 Crack Growth Criterion and Crack Growth Direction 326
8.3.3 Numerical Integration Scheme 328
8.4 Cohesive Crack Growth Based on the SIF Criterion 328
8.4.1 The Enrichment of Displacement Field 329
8.4.2 The Condition for Smooth Crack Closing 332
8.4.3 Crack Growth Criterion and Crack Growth Direction 332
8.5 Cohesive Crack Growth Based on the Cohesive Segments Method 334
8.5.1 The Enrichment of Displacement Field 334
8.5.2 Cohesive Constitutive Law 335
8.5.3 Crack Growth Criterion and Its Direction for Continuous Crack Propagation 336
8.5.4 Crack Growth Criterion and Its Direction for Discontinuous Crack Propagation 339
8.5.5 Numerical Integration Scheme 341
8.6 Application of X-FEM Method in Cohesive Crack Growth 341
8.6.1 A Three-Point Bending Beam with Symmetric Edge Crack 341
8.6.2 A Plate with an Edge Crack under Impact Velocity 343
8.6.3 A Three-Point Bending Beam with an Eccentric Crack 346
9 Ductile Fracture Mechanics with a Damage-Plasticity Model in X-FEM 351
9.1 Introduction 351
9.2 Large FE Deformation Formulation 353
9.3 Modified X-FEM Formulation 356
9.4 Large X-FEM Deformation Formulation 359
9.5 The Damage-Plasticity Model 364
9.6 The Nonlocal Gradient Damage Plasticity 368
9.7 Ductile Fracture with X-FEM Plasticity Model 369
9.8 Ductile Fracture with X-FEM Non-Local Damage-Plasticity Model 372
9.8.1 Crack Initiation and Crack Growth Direction 372
9.8.2 Crack Growth with a Null Step Analysis 375
9.8.3 Crack Growth with a Relaxation Phase Analysis 377
9.8.4 Locking Issues in Crack Growth Modeling 379
9.9 Application of X-FEM Damage-Plasticity Model 380
9.9.1 The Necking Problem 380
9.9.2 The CT Test 383
9.9.3 The Double-Notched Specimen 385
9.10 Dynamic Large X-FEM Deformation Formulation 387
9.10.1 The Dynamic X-FEM Discretization 388
9.10.2 The Large Strain Model 390
9.10.3 The Contact Friction Model 391
9.11 The Time Domain Discretization: The Dynamic Explicit Central Difference Method 393
9.12 Implementation of Dynamic X-FEM Damage-Plasticity Model 396
9.12.1 A Plate with an Inclined Crack 398
9.12.2 The Low Cycle Fatigue Test 400
9.12.3 The Cyclic CT Test 401
9.12.4 The Double Notched Specimen in Cyclic Loading 405
10 X-FEM Modeling of Saturated/Semi-Saturated Porous Media 409
10.1 Introduction 409
10.1.1 Governing Equations of Deformable Porous Media 411
10.2 The X-FEM Formulation of Deformable Porous Media with Weak Discontinuities 414
10.2.1 Approximation of Displacement and Pressure Fields 415
10.2.2 The X-FEM Spatial Discretization 418
10.2.3 The Time Domain Discretization and Solution Procedure 419
10.2.4 Numerical Integration Scheme 421
10.3 Application of the X-FEM Method in Deformable Porous Media with Arbitrary Interfaces 422
10.3.1 An Elastic Soil Column 422
10.3.2 An Elastic Foundation 424
10.4 Modeling Hydraulic Fracture Propagation in Deformable Porous Media 427
10.4.1 Governing Equations of a Fractured Porous Medium 428
10.4.2 The Weak Formulation of a Fractured Porous Medium 430
10.5 The X-FEM Formulation of Deformable Porous Media with Strong Discontinuities 434
10.5.1 Approximation of the Displacement and Pressure Fields 434
10.5.2 The X-FEM Spatial Discretization 437
10.5.3 The Time Domain Discretization and Solution Procedure 438
10.6 Alternative Approaches to Fluid Flow Simulation within the Fracture 442
10.6.1 A Partitioned Solution Algorithm for Interfacial Pressure 442
10.6.2 A Time-Dependent Constant Pressure Algorithm 444
10.7 Application of the X-FEM Method in Hydraulic Fracture Propagation of Saturated Porous Media 445
10.7.1 An Infinite Saturated Porous Medium with an Inclined Crack 446
10.7.2 Hydraulic Fracture Propagation in an Infinite Poroelastic Medium 449
10.7.3 Hydraulic Fracturing in a Concrete Gravity Dam 452
10.8 X-FEM Modeling of Contact Behavior in Fractured Porous Media 455
10.8.1 Contact Behavior in a Fractured Medium 455
10.8.2 X-FEM Formulation of Contact along the Fracture 456
10.8.3 Consolidation of a Porous Block with a Vertical Discontinuity 457
11 Hydraulic Fracturing in Multi-Phase Porous Media with X-FEM 461
11.1 Introduction 461
11.2 The Physical Model of Multi-Phase Porous Media 463
11.3 Governing Equations of Multi-Phase Porous Medium 465
11.4 The X-FEM Formulation of Multi-Phase Porous Media with Weak Discontinuities 467
11.4.1 Approximation of the Primary Variables 469
11.4.2 Discretization of Equilibrium and Flow Continuity Equations 473
11.4.3 Solution Procedure of Discretized Equilibrium Equations 476
11.5 Application of X-FEM Method in Multi-Phase Porous Media with Arbitrary Interfaces 477
11.6 The X-FEM Formulation for Hydraulic Fracturing in Multi-Phase Porous Media 482
11.7 Discretization of Multi-Phase Governing Equations with Strong Discontinuities 487
11.8 Solution Procedure for Fully Coupled Nonlinear Equations 493
11.9 Computational Notes in Hydraulic Fracture Modeling 497
11.10 Application of the X-FEM Method to Hydraulic Fracture Propagation of Multi-Phase Porous Media 499
12 Thermo-Hydro-Mechanical Modeling of Porous Media with X-FEM 509
12.1 Introduction 509
12.2 THM Governing Equations of Saturated Porous Media 511
12.3 Discontinuities in a THM Medium 513
12.4 The X-FEM Formulation of THM Governing Equations 514
12.4.1 Approximation of Displacement, Pressure, and Temperature Fields 515
12.4.2 The X-FEM Spatial Discretization 517
12.4.3 The Time Domain Discretization 520
12.5 Application of the X-FEM Method to THM Behavior of Porous Media 521
12.5.1 A Plate with an Inclined Crack in Thermal Loading 521
12.5.2 A Plate with an Edge Crack in Thermal Loading 522
12.5.3 An Impermeable Discontinuity in Saturated Porous Media 524
12.5.4 An Inclined Fault in Porous Media 527
References 533
Index 557
The finite element method (FEM) is one of the most common numerical tools for obtaining the approximate solutions of partial differential equations. It has been applied successfully in many areas of engineering sciences to study, model, and predict the behavior of structures. The area ranges across aeronautical and aerospace engineering, the automobile industry, mechanical engineering, civil engineering, biomechanics, geomechanics, material sciences, and many more. The FEM does not operate on differential equations; instead, continuous boundary and initial value problems are reformulated into equivalent variational forms. The FEM requires the domain to be subdivided into non-overlapping regions, called the elements. In the FEM, individual elements are connected together by a topological map, called a mesh, and local polynomial representation is used for the fields within the element. The solution obtained is a function of the quality of mesh and the fundamental requirement is that the mesh has to conform to the geometry. The main advantage of the FEM is that it can handle complex boundaries without much difficulty. Despite its popularity, the FEM suffers from certain drawbacks. There are number of instances where the FEM poses restrictions to an efficient application of the method. The FEM relies on the approximation properties of polynomials; hence, they often require smooth solutions in order to obtain optimal accuracy. However, if the solution contains a non-smooth behavior, like high gradients or singularities in the stress and strain fields, or strong discontinuities in the displacement field as in the case of cracked bodies, then the FEM becomes computationally expensive to get optimal convergence.
One of the most significant interests in solid mechanics problems is the simulation of fracture and damage phenomena (Figure 1.1). Engineering structures, when subjected to high loading, may result in stresses in the body exceeding the material strength and thus, in progressive failure. These material failure processes manifest themselves in various failure mechanisms such as the fracture process zone (FPZ) in rocks and concrete, the shear band localization in ductile metals, or the discrete crack discontinuity in brittle materials. The accurate modeling and evolution of smeared and discrete discontinuities have been a topic of growing interest over the past few decades, with quite a few notable developments in computational techniques over the past few years. Early numerical techniques for modeling discontinuities in finite elements can be seen in the work of Ortiz, Leroy, and Needleman (1987) and Belytschko, Fish, and Englemann (1988). They modeled the shear band localization as a "weak" (strain) discontinuity that could pass through the finite element mesh using a multi-field variational principle. Dvorkin, Cuitiño, and Gioia (1990) considered a "strong" (displacement) discontinuity by modifying the principle of virtual work statement. A unified framework for modeling the strong discontinuity by taking into account the softening constitutive law and the interface traction-displacement relation was proposed by Simo, Oliver, and Armero (1993). In the strong discontinuity approach, the displacement consists of regular and enhanced components, where the enhanced component yields a jump across the discontinuity surface. An assumed enhanced strain variational formulation is used, and the enriched degrees of freedom (DOF) are statically condensed on an element level to obtain the tangent stiffness matrix for the element. An alternative approach for modeling fracture phenomena was introduced by Xu and Needleman (1994) based on the cohesive surface formulation, which was used later by Camacho and Ortiz (1996) to model the damage in brittle materials. The cohesive surface formulation is a phenomenological framework in which the fracture characteristics of the material are embedded in a cohesive surface traction-displacement relation. Based on this approach, an inherent length scale is introduced into the model, and in addition, no fracture criterion is required so the crack growth and the crack path are outcomes of the analysis.
Figure 1.1 Building destroyed by a 8.8 magnitude earthquake on Saturday, February 27, 2010, with intense shaking lasting for about 3 minutes, which occurred off the coast of central Chile.
(Source: Vladimir Platonow (Agência Brasil) [CC-BY-3.0-br (http://creativecommons.org/licenses/by/3.0/br/deed.en)], via Wikimedia Commons; http://commons.wikimedia.org/wiki/File:Terremoto_no_Chile_2010.JPG)
In the FEM, the non-smooth displacement near the crack tip is basically captured by refining the mesh locally. The number of DOF may drastically increase, especially in three-dimensional applications. Moreover, the incremental computation of a crack growth needs frequent remeshings. Reprojecting the solution on the updated mesh is not only a costly operation but also it may have a troublesome impact on the quality of results. The classical FEM has achieved its limited ability for solving fracture mechanics problems. To avoid these computational difficulties, a new approach to the problem consists in taking into account the a priori knowledge of the exact solution. Applying the asymptotic crack tip displacement solution to the finite element basis seems to have been a somewhat early idea. A significant improvement in crack modeling was presented with the development of a partition of unity (PU) based enrichment method for discontinuous fields in the PhD dissertation by Dolbow (1999), which was referred to as the extended FEM (X-FEM). In the X-FEM, special functions are added to the finite element approximation using the framework of PU. For crack modeling, a discontinuous function such as the Heaviside step function and the two-dimensional linear elastic asymptotic crack tip displacement fields, are used to account for the crack. This enables the domain to be modeled by finite elements without explicitly meshing the crack surfaces. The location of the crack discontinuity can be arbitrary with respect to the underlying finite element mesh, and the crack propagation simulation can be performed without the need to remesh as the crack advances. A particularly appealing feature is that the finite element framework and its properties, such as the sparsity and symmetry, are retained and a single-field (displacement) variational principle is used to obtain the discrete equations. This technique provides an accurate and robust numerical method to model strong (displacement) discontinuities.
The original research articles on the X-FEM were presented by Belytschko and Black (1999) and Moës, Dolbow, and Belytschko (1999) for elastic fracture propagation on the topic of "A FEM for crack growth without remeshing". They presented a minimal remeshing FEM for crack growth by including the discontinuous enrichment functions to the finite element approximation in order to account for the presence of the crack. The essential idea was based on adding enrichment functions to the approximation space that contains a discontinuous displacement field. Hence, the method allows the crack to be arbitrarily aligned within the mesh. The same span of functions was earlier developed by Fleming et al. (1997) for the enrichment of the element-free Galerkin method. The method exploits the PU property of finite elements that was noted by Melenk and Babuska (1996), namely that the sum of the shape functions must be unity. This property has long been known, since it corresponds to the ability of the shape functions to reproduce a constant that represents translation, which is crucial for convergence.
The X-FEM provides a powerful tool for enriching solution spaces with information from asymptotic solutions and other knowledge of the physics of the problem. This has proved very useful for cracks and dislocations where near-field solutions can be embedded by the PU method to tremendously increase the accuracy of relatively coarse meshes. The technique offers possibilities in treating phenomena such as surface effects in nano-mechanics, void growth, subscale models of interface behavior, and so on. Thus, the X-FEM method has greatly enhanced the power of the FEM for many of the problems of interest in mechanics of materials. The aim of this chapter is to provide an overview of the X-FEM with an emphasis on various applications of the technique to materials modeling problems, including linear elastic fracture mechanics ( LEFM); cohesive fracture mechanics; composite materials and material inhomogeneities; plasticity, damage and fatigue problems; shear band localization; fluid-structure interaction; fluid flow in fractured porous media; fluid flow and fluid mechanics problems; phase transition and solidification; thermal and thermo-mechanical problems; plates and shells; contact problems; topology optimization; piezoelectric and magneto-electroelastic problems; and multi-scale modeling.
The FEM is widely used in industrial design applications, and many different software packages based on FEM techniques have been developed. It has undoubtedly become the most popular and powerful analytical tool for studying the behavior of a wide range of engineering and physical problems. Its applications have been developed from basic mechanical problems to fracture mechanics, fluid dynamics,...
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