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