9781118405307

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.

Antonio A. Munjiza, Queen Mary College, London, UK
Antonio Munjiza is a professor of computational mechanics in the Department of Computational Mechanics at Queen Mary College, London. His research interests include finite element methods, discrete element methods, molecular dynamics, structures and solids, structural dynamics, software engineering, blasts, impacts, and nanomaterials. He has authored two books, The Combined Finite-Discrete Element Method (Wiley 2004) and Computational Mechanics of Discontinua (Wiley 2011) and over 110 refereed journal papers. In addition, he is on the editorial board of seven international journals. Dr Munjiza is also an accomplished software engineer with three research codes behind him and one commercial code all based on his technology.

Earl E. Knight, Esteban Rougier and Ted Carney, Los Alamos National Laboratories, USA
Earl Knight is a Team Leader in the Geodynamics Team at Los Alamos National Laboratory. His research interests include geodynamic modeling, rock mechanical modeling for deep water oil reservoirs and ground based nuclear explosion monitoring.

Esteban Rougier is a Post Doctoral Research Associate at LANL. He has received his Ph.D. from Queen Mary, University of London in 2008` on Computational Mechanics of Discontinuum and its Application to the Simulation of Micro-Flows.

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

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Large Strain Finite Element Method A Practical Course

Large Strain Finite Element Method A Practical Course

1118405307

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