Large Strain Finite Element Method

Antonio A. Munjiza, Queen Mary College, London, UKAntonio 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.

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.

An introductory approach to the subject of large strains andlarge displacements in finite elements.

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