An introductory approach to the subject of large strains and large displacements in finite elements.
Large Strain Finite Element Method: A Practical Course, takes an introductory approach to the subject of large strains and large displacements in finite elements and starts from the basic concepts of finite strain deformability, including finite rotations and finite displacements. The necessary elements of vector analysis and tensorial calculus on the lines of modern understanding of the concept of tensor will also be introduced.
This book explains how tensors and vectors can be described using matrices and also introduces different stress and strain tensors. Building on these, step by step finite element techniques for both hyper and hypo-elastic approach will be considered.
Material models including isotropic, unisotropic, plastic and viscoplastic materials will be independently discussed to facilitate clarity and ease of learning. Elements of transient dynamics will also be covered and key explicit and iterative solvers including the direct numerical integration, relaxation techniques and conjugate gradient method will also be explored.
This book contains a large number of easy to follow illustrations, examples and source code details that facilitate both reading and understanding.
- Takes an introductory approach to the subject of large strains and large displacements in finite elements. No prior knowledge of the subject is required.
- Discusses computational methods and algorithms to tackle large strains and teaches the basic knowledge required to be able to critically gauge the results of computational models.
- Contains a large number of easy to follow illustrations, examples and source code details.
- Accompanied by a website hosting code examples.
Table of Contents:
Preface xiii
Acknowledgements xv
PART ONE FUNDAMENTALS 1
1 Introduction 3
1.1 Assumption of Small Displacements 3
1.2 Assumption of Small Strains 6
1.3 Geometric Nonlinearity 6
1.4 Stretches 8
1.5 Some Examples of Large Displacement Large Strain Finite Element Formulation 8
1.6 The Scope and Layout of the Book 13
1.7 Summary 13
2 Matrices 15
2.1 Matrices in General 15
2.2 Matrix Algebra 16
2.3 Special Types of Matrices 21
2.4 Determinant of a Square Matrix 22
2.5 Quadratic Form 24
2.6 Eigenvalues and Eigenvectors 24
2.7 Positive Definite Matrix 26
2.8 Gaussian Elimination 26
2.9 Inverse of a Square Matrix 28
2.10 Column Matrices 30
2.11 Summary 32
3 Some Explicit and Iterative Solvers 35
3.1 The Central Difference Solver 35
3.2 Generalized Direction Methods 43
3.3 The Method of Conjugate Directions 50
3.4 Summary 63
4 Numerical Integration 65
4.1 Newton-Cotes Numerical Integration 65
4.2 Gaussian Numerical Integration 67
4.3 Gaussian Integration in 2D 70
4.4 Gaussian Integration in 3D 71
4.5 Summary 72
5 Work of Internal Forces on Virtual Displacements 75
5.1 The Principle of Virtual Work 75
5.2 Summary 78
PART TWO PHYSICAL QUANTITIES 79
6 Scalars 81
6.1 Scalars in General 81
6.2 Scalar Functions 81
6.3 Scalar Graphs 82
6.4 Empirical Formulas 82
6.5 Fonts 83
6.6 Units 83
6.7 Base and Derived Scalar Variables 85
6.8 Summary 85
7 Vectors in 2D 87
7.1 Vectors in General 87
7.2 Vector Notation 91
7.3 Matrix Representation of Vectors 91
7.4 Scalar Product 92
7.5 General Vector Base in 2D 93
7.6 Dual Base 94
7.7 Changing Vector Base 95
7.8 Self-duality of the Orthonormal Base 97
7.9 Combining Bases 98
7.10 Examples 104
7.11 Summary 108
8 Vectors in 3D 109
8.1 Vectors in 3D 109
8.2 Vector Bases 111
8.3 Summary 114
9 Vectors in n-Dimensional Space 117
9.1 Extension from 3D to 4-Dimensional Space 117
9.2 The Dual Base in 4D 118
9.3 Changing the Base in 4D 120
9.4 Generalization to n-Dimensional Space 121
9.5 Changing the Base in n-Dimensional Space 124
9.6 Summary 127
10 First Order Tensors 129
10.1 The Slope Tensor 129
10.2 First Order Tensors in 2D 131
10.3 Using First Order Tensors 132
10.4 Using Different Vector Bases in 2D 134
10.5 Differential of a 2D Scalar Field as the First Order Tensor 137
10.6 First Order Tensors in 3D 141
10.7 Changing the Vector Base in 3D 142
10.8 First Order Tensor in 4D 143
10.9 First Order Tensor in n-Dimensions 147
10.10 Differential of a 3D Scalar Field as the First Order Tensor 149
10.11 Scalar Field in n-Dimensional Space 152
10.12 Summary 153
11 Second Order Tensors in 2D 155
11.1 Stress Tensor in 2D 155
11.2 Second Order Tensor in 2D 158
11.3 Physical Meaning of Tensor Matrix in 2D 159
11.4 Changing the Base 161
11.5 Using Two Different Bases in 2D 163
11.6 Some Special Cases of Stress Tensor Matrices in 2D 167
11.7 The First Piola-Kirchhoff Stress Tensor Matrix 168
11.8 The Second Piola-Kirchhoff Stress Tensor Matrix 169
11.9 Summary 174
12 Second Order Tensors in 3D 175
12.1 Stress Tensor in 3D 175
12.2 General Base for Surfaces 179
12.3 General Base for Forces 182
12.4 General Base for Forces and Surfaces 184
12.5 The Cauchy Stress Tensor Matrix in 3D 186
12.6 The First Piola-Kirchhoff Stress Tensor Matrix in 3D 186
12.7 The Second Piola-Kirchhoff Stress Tensor Matrix in 3D 188
12.8 Summary 189
13 Second Order Tensors in nD 191
13.1 Second Order Tensor in n-Dimensions 191
13.2 Summary 200
PART THREE DEFORMABILITY AND MATERIAL MODELING 201
14 Kinematics of Deformation in 1D 203
14.1 Geometric Nonlinearity in General 203
14.2 Stretch 205
14.3 Material Element and Continuum Assumption 208
14.4 Strain 209
14.5 Stress 213
14.6 Summary 214
15 Kinematics of Deformation in 2D 217
15.1 Isotropic Solids 217
15.2 Homogeneous Solids 217
15.3 Homogeneous and Isotropic Solids 217
15.4 Nonhomogeneous and Anisotropic Solids 218
15.5 Material Element Deformation 221
15.6 Cauchy Stress Matrix for the Solid Element 225
15.7 Coordinate Systems in 2D 227
15.8 The Solid- and the Material-Embedded Vector Bases 228
15.9 Kinematics of 2D Deformation 229
15.10 2D Equilibrium Using the Virtual Work of Internal Forces 231
15.11 Examples 235
15.12 Summary 238
16 Kinematics of Deformation in 3D 241
16.1 The Cartesian Coordinate System in 3D 241
16.2 The Solid-Embedded Coordinate System 241
16.3 The Global and the Solid-Embedded Vector Bases 243
16.4 Deformation of the Solid 244
16.5 Generalized Material Element 246
16.6 Kinematic of Deformation in 3D 247
16.7 The Virtual Work of Internal Forces 249
16.8 Summary 255
17 The Unified Constitutive Approach in 2D 257
17.1 Introduction 257
17.2 Material Axes 259
17.3 Micromechanical Aspects and Homogenization 260
17.4 Generalized Homogenization 263
17.5 The Material Package 264
17.6 Hyper-Elastic Constitutive Law 265
17.7 Hypo-Elastic Constitutive Law 266
17.8 A Unified Framework for Developing Anisotropic Material Models in 2D 267
17.9 Generalized Hyper-Elastic Material 267
17.10 Converting the Munjiza Stress Matrix to the Cauchy Stress Matrix 274
17.11 Developing Constitutive Laws 279
17.12 Generalized Hypo-Elastic Material 288
17.13 Unified Constitutive Approach for Strain Rate and Viscosity 292
17.14 Summary 293
18 The Unified Constitutive Approach in 3D 295
18.1 Material Package Framework 295
18.2 Generalized Hyper-Elastic Material 295
18.3 Generalized Hypo-Elastic Material 299
18.4 Developing Material Models 302
18.5 Calculation of the Cauchy Stress Tensor Matrix 302
18.6 Summary 312
PART FOUR THE FINITE ELEMENT METHOD IN 2D 315
19 2D Finite Element: Deformation Kinematics Using the Homogeneous Deformation Triangle 317
19.1 The Finite Element Mesh 317
19.2 The Homogeneous Deformation Finite Element 317
19.3 Summary 326
20 2D Finite Element: Deformation Kinematics Using Iso-Parametric Finite Elements 327
20.1 The Finite Element Library 327
20.2 The Shape Functions 327
20.3 Nodal Positions 330
20.4 Positions of Material Points inside a Single Finite Element 331
20.5 The Solid-Embedded Vector Base 332
20.6 The Material-Embedded Vector Base 334
20.7 Some Examples of 2D Finite Elements 337
20.8 Summary 340
21 Integration of Nodal Forces over Volume of 2D Finite Elements 343
21.1 The Principle of Virtual Work in the 2D Finite Element Method 343
21.2 Nodal Forces for the Homogeneous Deformation Triangle 348
21.3 Nodal Forces for the Six-Noded Triangle 352
21.4 Nodal Forces for the Four-Noded Quadrilateral 353
21.5 Summary 355
22 Reduced and Selective Integration of Nodal Forces over Volume of 2D Finite Elements 357
22.1 Volumetric Locking 357
22.2 Reduced Integration 358
22.3 Selective Integration 359
22.4 Shear Locking 362
22.5 Summary 364
PART FIVE THE FINITE ELEMENT METHOD IN 3D 365
23 3D Deformation Kinematics Using the Homogeneous Deformation Tetrahedron Finite Element 367
23.1 Introduction 367
23.2 The Homogeneous Deformation Four-Noded Tetrahedron Finite Element 368
23.3 Summary 377
24 3D Deformation Kinematics Using Iso-Parametric Finite Elements 379
24.1 The Finite Element Library 379
24.2 The Shape Functions 379
24.3 Nodal Positions 381
24.4 Positions of Material Points inside a Single Finite Element 382
24.5 The Solid-Embedded Infinitesimal Vector Base 383
24.6 The Material-Embedded Infinitesimal Vector Base 386
24.7 Examples of Deformation Kinematics 387
24.8 Summary 392
25 Integration of Nodal Forces over Volume of 3D Finite Elements 393
25.1 Nodal Forces Using Virtual Work 393
25.2 Four-Noded Tetrahedron Finite Element 396
25.3 Reduce Integration for Eight-Noded 3D Solid 399
25.4 Selective Stretch Sampling-Based Integration for the Eight-Noded Solid Finite Element 400
25.5 Summary 401
26 Integration of Nodal Forces over Boundaries of Finite Elements 403
26.1 Stress at Element Boundaries 403
26.2 Integration of the Equivalent Nodal Forces over the Triangle Finite Element 404
26.3 Integration over the Boundary of the Composite Triangle 407
26.4 Integration over the Boundary of the Six-Noded Triangle 408
26.5 Integration of the Equivalent Internal Nodal Forces over the Tetrahedron Boundaries 409
26.6 Summary 412
PART SIX THE FINITE ELEMENT METHOD IN 2.5D 415
27 Deformation in 2.5D Using Membrane Finite Elements 417
27.1 Solids in 2.5D 417
27.2 The Homogeneous Deformation Three-Noded Triangular Membrane Finite Element 419
27.3 Summary 438
28 Deformation in 2.5D Using Shell Finite Elements 439
28.1 Introduction 439
28.2 The Six-Noded Triangular Shell Finite Element 440
28.3 The Solid-Embedded Coordinate System 441
28.4 Nodal Coordinates 442
28.5 The Coordinates of the Finite Element’s Material Points 443
28.6 The Solid-Embedded Infinitesimal Vector Base 444
28.7 The Solid-Embedded Vector Base versus the Material-Embedded Vector Base 447
28.8 The Constitutive Law 449
28.9 Selective Stretch Sampling Based Integration of the Equivalent Nodal Forces 449
28.10 Multi-Layered Shell as an Assembly of Single Layer Shells 455
28.11 Improving the CPU Performance of the Shell Element 456
28.12 Summary 462
Index 463
