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 |  موضوع: كتاب Large Strain Finite Element Method - A Practical Course الجمعة 17 سبتمبر 2021, 2:16 pm | |
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أخواني في الله
أحضرت لكم كتاب
Large Strain Finite Element Method
A Practical Course
Antonio Munjiza
Queen Mary, University of London
Esteban Rougier
Los Alamos National Laboratory, US
Earl E. Knight
Los Alamos National Laboratory, US
و المحتوى كما يلي :
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 323 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
viii Contents9.3 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
Contents ix14.4 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
x Contents18.4 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
Contents xi24.3 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 46
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 323 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
viii Contents9.3 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
Contents ix14.4 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
x Contents18.4 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
Contents xi24.3 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 46
Index
acceleration, 37, 38, 42, 43, 85, 90, 94, 343
a-conjugate, 51
addition of matrices, 16
additive decomposition, 203, 259, 418
angle stretch, 269, 270, 289, 290, 297–9, 301,
307, 450
anisotropic, 8, 10, 13, 203, 218, 220–222,
238, 246, 255, 259, 260, 263–5,
267, 272, 280, 293, 299, 307, 312,
313, 321, 375, 419, 438–40, 458,
460, 462
anisotropic constitutive law, 313
anisotropic elastic material, 307
anisotropic material element, 220, 264, 265,
458, 460
anisotropic material nonlinearity, 10
anisotropic materials, 10, 13, 203, 218,
220–222, 246, 255, 259, 260, 263–5, 272,
312, 321, 375, 419, 439, 458, 460
a-orthogonal, 49, 53, 55, 57
a-orthonormal column matrices, 32
associativity of the summation, 87
assumption of continuum, 209, 214, 261, 263,
264, 321
axial stretches, 297
base vectors, 91–3, 95, 96, 98, 99, 102,
111–14, 117, 122, 123, 130, 136, 141, 144,
147, 152, 170, 179, 182, 191, 192, 228,
231, 243, 244, 255, 260, 321, 326, 332,
333, 361, 375, 376, 384, 389, 405, 423,
427, 428, 430, 439, 445, 447, 449
bending moment, 77, 78
cartesian coordinate system, 44, 227, 241,
242, 318, 320, 321, 330, 368, 382
Cauchy material element, 223, 225–7, 231,
232, 234, 238, 249–55, 302, 303, 306, 313,
343, 409, 430, 449
Cauchy stress, 167, 168, 173, 186, 188,
203, 225–7, 231, 235–8, 250, 252,
255, 258, 266, 267, 274, 278–80,
299, 302, 313, 326, 343, 349, 353,
355, 363, 377, 378, 392, 396, 399,
406, 407, 409, 412, 429–31, 435,
438, 449, 455, 462
central difference, 35, 37, 38, 40–42, 62
change of angle, 257
change of volume, 258, 357
commutativity, 89, 90
components of stress tensor matrix, 185components of the Cauchy stress, 226, 227,
299, 407, 409, 449, 455
composite elements, 361, 362, 403
composite triangle, 359, 360, 403, 404, 407
computational mechanics, 14, 168, 174, 209,
215, 256, 259, 294, 314
conditional stability, 41
conjugate directions method, 13, 50, 51, 53,
60, 63
conjugate gradient method, 62, 63
constant strain finite element, 325
constitutive law, 13, 213, 231, 258–60,
264–7, 279, 292, 293, 295, 296, 299, 312,
313, 340, 343, 363, 364, 377, 392, 435,
438, 449
contact, 14, 204, 362, 367, 403
continuity conditions, 327
continuum mechanics, 174, 209, 214, 238,
256, 259, 261, 294, 313, 341, 355, 378,
392, 401, 413, 438, 462
convergence, 50, 357, 363
co-rotational formulation, 203, 293, 302, 313,
418, 438
crack, 14, 157, 175, 176, 209, 214
cross method, 48
current configuration, 204, 265, 324
current position, 204, 209, 214, 225, 227,
229–32, 235, 236, 238, 244, 246, 249, 250,
255, 259, 266–8, 275, 288, 292, 296, 319,
325, 331, 334, 340, 343, 344, 371, 374,
376–8, 382, 387, 389, 391, 392, 404, 409,
421, 424–7, 430, 431, 443, 446–50,
458, 461
current volume, 268, 275, 288, 296, 345, 351,
397, 411, 431, 453
deformation-dependent, 275
deformation independent, 228
deformation-independent matrices, 267
deformation invariant, 293, 312, 313, 378
deformation kinematics, 13, 249, 250, 255,
260, 264, 267, 279, 288, 292, 295, 319,
321, 323, 325, 329, 331, 333, 335, 337,
339–41, 367, 369, 371, 373, 375, 377–9,
381, 383, 385, 387, 389, 391, 392, 438
deformation-objective, 312
deformed solid element, 223, 302
derivation of a scalar field, 372
derivation of the shape functions, 368
determinant, 22–4, 28, 428, 429
diagonal matrix, 21, 60
differential, 35, 63, 73, 78, 86, 108, 115, 127,
137, 139, 149, 150, 152, 154, 174, 189,
200, 209, 214
calculus, 108, 115, 127, 154, 174, 189,
200, 209
equations, 35, 63, 73, 78, 86, 209
differentiating, 137
differentiation, 373, 374, 426, 431, 433, 446
direction of the force, 178
direction of the surface 168, 178
discretization, 37, 317, 318
displacement vector, 104, 130, 131
divergence, 47
dot product, 30, 44, 51, 52, 55, 92, 93, 436
dual base vectors, 98, 99, 170, 405
dual bases, 949, 105, 111, 112, 118, 123, 127,
136, 170, 405, 410, 411, 435
duality condition, 99, 119, 123
dynamic finite element analysis, 35, 256, 294,
313, 326, 341, 355, 392, 401, 413,
438, 462
dynamic relaxation, 13, 37, 43, 367
edge stretches, 269, 272, 277, 288, 289, 297,
300, 307
eigenmatrix, 25, 26
eigenvalues, 24–6, 39, 40, 61, 364
eight-noded quadrilateral finite element, 329
eight-noded solid finite element, 381
elastic constants, 280, 284, 287, 288, 302,
308, 309
engineering definition of stress, 174
engineering strain, 6, 209–12
equilibrium, 3–6, 36, 43, 75, 76, 78, 231, 232,
235, 238, 250, 255, 256, 273, 343–5,
357, 417
equilibrium equations, 343, 357, 417
equilibrium of forces, 3, 4
equilibrium of moments, 273
equivalent nodal forces, 13, 343, 344, 346,
347, 350, 353, 397, 398, 403, 404, 406,
408–10, 412, 430, 434, 435, 449, 452,
453, 462
Euler-Almansi strain, 210–212
exact solution, 6, 45, 47, 52
explicit, 13, 35, 37, 39, 41, 43, 45, 47, 49, 51,
53, 55, 57, 59, 61, 63, 203, 204, 326, 341,
355, 364, 367, 378, 392, 401, 413, 421,
438, 443, 462
464 Indexexplicit dynamic, 367, 378, 392, 401, 413
explicit integration, 37
explicit iterative static formulation, 367
external damping, 42
finite element library, 327, 379
finite strain deformability theory, 203
first derivative, 140, 151
first order tensor, 13, 129, 131–5, 137–41,
143, 145, 147, 149–51, 153, 154, 158,
159, 177
first order tensor to an n-dimensional
space, 147
first Piola-Kirchhoff, 168, 169, 172, 181,
186–8, 258, 275–80, 305, 306
force base, 179, 182, 196, 198
force components, 106, 162, 250
four gauss points integration, 71
four-noded quadrilateral finite elements, 328,
358, 359, 362
fourth dimension, 117, 191
Gaussian elimination, 26, 28, 29, 62, 367
Gaussian integration, 65
Gaussian numerical integration, 67, 352, 355
Gauss integration point, 353–5, 358, 399,
400, 404, 408, 434, 452
Gauss point, 68, 70–72, 348, 352, 355,
358–64, 397, 400, 401, 404, 405, 408, 409,
452, 453
Gauss-seidel method, 46–8, 53
general base, 12, 98, 100, 102–4, 106, 108,
111, 114, 117, 120, 124, 136, 140, 142,
151, 167, 179, 182, 184
general base for forces and surfaces, 184
general force base, 182, 196
generalized material element, 222–5, 231, 238,
246, 247, 249, 255, 257–60, 264, 265,
267–9, 271, 273–7, 279–81, 288–90, 293,
295–300, 302, 303, 307, 308, 312, 321,
324–6, 334, 343, 361, 363, 364, 386, 389,
392, 419, 439, 440, 449
generalized material-embedded base vectors,
247–9, 260
generalized stress matrix components, 275
generalized stretch, 280, 292
general virtual displacement, 235, 254
global base, 98–100, 109, 111, 113, 161, 164,
166, 167, 177–82, 192, 196, 324, 373,
425, 430
global base vector, 111, 430
global coordinates, 229, 232, 244, 320, 330,
331, 336, 369, 373, 374, 382, 385,
420–423, 432, 443, 458
global derivatives, 349, 431
global orthonormal base, 98, 101, 111, 134,
141, 168, 170, 176, 186, 187, 191, 196,
219, 226, 299, 302, 334, 343, 386
global orthonormal vector base, 144
global vector base, 165, 260, 321
Green-Lagrange strain, 210–212
Green-Naghdi rate, 259, 267
heterogeneous solid, 217
homogeneous and anisotropic solids,
218, 238
homogeneous and isotropic solid, 219
homogeneous deformation triangle, 317
homogeneous deformation finite element,
317, 438, 453
homogeneous internal forces, 175, 176
homogenization, 260, 262–4, 272, 273
homogenized material element, 321
Hooke’ s law, 213, 281, 282, 284–7,
309–11, 313
hyper-elastic material, 13, 258, 259, 264, 265,
267, 280, 295
hyper-lines, 48, 49
hyper-point, 44, 45
hyper-space, 44
hyper-surfaces, 45
hypo-elastic formulation, 13, 258, 259, 264,
266, 288, 299
identity matrix, 21
implicit equilibrium formulation, 4
implicit formulation, 42
inertia forces, 36, 43, 250, 256, 343
infi-meter, 77, 207, 208, 221, 243, 259, 373,
393, 439
infinitesimal base, 321, 323, 324, 445, 449
infinitesimally small, 5–7, 75, 77, 139, 150,
176, 206–8, 221, 222, 224, 227, 228,
232, 233, 236, 243, 246, 249, 251, 252,
259, 284, 321, 373, 393, 419, 423,
430, 439
infinitesimally small vector, 150
infinitesimal vicinity, 7, 176, 210, 231, 238,
243, 247, 255, 333, 372, 375, 376,
458, 460
Index 465initial geometry, 3, 330, 382
initial position, 204, 206, 209, 223, 224, 227,
236, 245, 248, 249, 260, 266, 267, 269,
295, 297, 321, 322, 330, 333, 334, 339–41,
371, 373, 376, 377, 382, 386, 388–90, 392,
421, 423, 425, 427, 428, 442, 446, 448,
456, 459
inner product, 30, 110
instability, 5, 6, 60–62
integrating over the volume, 394
integrating the stress over boundaries, 403
integration scheme, 37, 38, 40–42, 62
internal damping, 42, 43
internal forces, 3–5, 7, 13, 35, 36, 38, 59,
75, 77, 155–8, 160, 162, 164, 168, 173,
175–7, 184, 185, 187, 188, 225–7,
231–4, 236–8, 249–55, 258, 261, 267,
271–4, 276, 277, 279, 292, 293, 299–302,
304–8, 312, 313, 343, 344, 358, 378,
392–4, 406–9, 412, 417, 430, 436, 449,
452, 462
inverse transpose matrix, 100
iso-parametric finite elements, 329, 331, 333,
335, 337–9, 341
isotropic, 154, 217–19, 238, 256, 259, 263,
280, 281, 284, 285, 292–4, 307–9, 313
isotropic plane strain, 293
isotropic plane stress, 292
isotropic solid, 217–20, 238, 280, 299, 307,
308, 313
Jaumann rate, 203, 259, 267, 275
kinematic hardening, 259
kinematics of deformation, 203, 205, 207,
209, 211, 213, 214, 217, 219, 221, 223,
225, 227, 229, 231, 233, 235, 237, 239,
241, 243, 245, 247, 249, 251, 253, 255,
267, 299, 340, 377, 449
Lagrange polynomials, 330
Lagrange shape functions, 330, 381
large displacement formulation, 6, 10
large displacements, 6, 8–10, 12, 13, 203,
343, 417, 418, 438, 440
formulation, 6, 10
and large strains, 440
small strain, 8
large strains, 6, 8, 10, 12, 203, 213, 214, 255,
312, 313, 417, 418, 440, 462
large strains large displacement shells, 10
linear mapping, 131–4, 141, 143, 144, 147,
153, 158, 163, 174, 176, 177, 181, 184,
192, 196, 225, 258
linear mapping of surfaces to forces, 158, 174
load, 3, 5, 43, 50–53, 155, 156, 175, 213, 417,
418, 421
local coordinates, 229, 244, 320, 325, 331,
336, 382, 385, 419–21
local derivatives, 349, 431
logarithmic strains, 210–212, 214, 280, 294,
307, 308, 313
lower triangular matrix, 21
mapping, 130–134, 141, 143, 144, 147, 153,
158, 163, 164, 174, 176, 177, 181, 184,
185, 192, 196, 225, 258
material axes, 218, 219, 259, 321, 326, 335
material element, 208, 214, 218, 221–7, 231,
232, 234, 238, 246, 247, 249–51, 253–5,
257–61, 264, 265, 267–9, 271, 273–7,
280, 281, 288–90, 293, 295–300, 302,
303, 306–8, 312, 313, 321, 324–6,
334, 343, 363, 364, 375, 378, 386, 389,
392, 394, 409, 419, 439, 440, 449,
458, 460
material-embedded infinitesimal base, 321
material-embedded vector base, 228, 231,
238, 247, 249, 323, 325, 375–7, 389, 392,
427, 429, 438, 439, 447, 448, 459
material model, 13, 14, 264, 267, 280, 301,
302, 307
material package, 13, 264–7, 268, 274, 278,
292, 293, 295, 296, 299, 302, 312, 313,
326, 355, 361, 363, 364, 377, 378, 392,
401, 430, 435, 449
material point, 7, 75, 76, 78, 168, 203–5, 217,
219, 221, 224–9, 231, 232, 235, 243, 244,
246, 247, 249–51, 255, 260, 264, 317–24,
326, 328, 329, 331–6, 340, 341, 344–6,
352, 364, 368, 369, 372, 375, 376, 382,
383, 385, 386, 389, 392, 393, 419–21, 423,
424, 427, 439, 440, 443, 444, 447,
449, 457
material properties, 218, 219, 377
matrices, 15–19, 21, 23, 25–7, 29–33, 50, 51,
53, 60, 62, 83, 94, 110, 114, 115, 121,
132–4, 141, 142, 153, 154, 167, 168, 170,
179, 186, 230, 258, 264, 267, 292,
326, 398
466 Indexmatrix algebra, 13, 16, 32, 108, 115, 127
matrix inversions, 367
matrix multiplication, 19, 108
mechanics of discontinua, 209
mesh, 317, 319, 327, 330, 357, 358, 363, 382
method of residuals, 75
microstructure, 209, 210, 261, 262
multiplication of matrices by a scalar, 18
multiplicative decomposition, 7, 8, 13, 203,
204, 214, 215, 239, 256, 259, 266, 267,
288–90, 292, 294, 300, 301, 313, 314, 367,
368, 418, 438, 462
Munjiza generalized material element, 222,
255, 312
n-dimensional space, 44, 117, 119, 121, 123,
125, 127, 147, 152, 200
nd space, 45, 49, 50, 127, 152, 154
Newton-Cotes integration, 65–7
nodal forces, 13, 343–5, 347–53, 355, 357,
359, 361, 363, 377, 378, 393, 395, 397–9,
401, 403–13, 430, 434, 435, 437, 438, 449,
452, 455, 462
non-cartesian coordinate system, 318, 320
non-homogeneous, 247
nonlinear algebraic equations, 13, 58
nonlinear elasticity, 213
nonlinearity, 6, 8, 10, 203, 235, 325, 412
nonlinear material laws, 364, 403
nonlinear materials, 10, 302
non-orthonormal bases, 101, 111, 168
numerical integration, 65, 67–9, 71–3, 352,
354, 355, 401
numerical integration of virtual work, 401
objective, 280, 293, 302, 313, 363
one Gauss point integration, 70, 405
one Gauss point numerical integration, 68
orthonormal base, 91–3, 97, 98, 101, 104,
106, 109, 111, 117, 120–122, 134,
139, 141, 158, 168, 170, 176, 186,
187, 195, 196, 219, 226, 299, 302,
343, 386
orthonormal column matrices, 32
orthonormal vector base, 127, 144, 155
parallelepiped, 246, 386
parallelization, 367
parallelogram-shaped infinitesimal material
element, 222
physical meaning, 94, 102, 159, 160, 163,
173, 185, 222, 374
physical reality, 89, 131–3, 168, 185
plane strain, 284, 285, 287, 288, 293, 364
plane strain isotropic material, 284
plane stress, 281, 283, 284, 288, 292,
430, 435
plane stress formulation, 283
plastically, 213, 440
Poisson’s ratio, 288, 309
polygon, 89–91, 117, 118
polygon of vectors, 89, 90, 117, 118
positive definite matrix, 26, 50, 53, 63
potential energy method, 75
preconditioner, 61
preconditioning, 60, 61
previous position, 238, 245, 248, 260, 266,
288, 292, 325, 330, 333, 340, 341, 371,
374, 377, 382, 384, 388, 389, 391, 421,
424, 425, 427, 447, 448, 460, 461
principle of virtual work, 75, 78, 232, 250,
343–5, 352, 353, 355, 393, 412, 430,
438, 452
pure shear, 284, 287, 310
quadratic form, 24, 26, 62
rate of deformation, 330
rate of stretching, 292
reduced integration, 358, 359, 363, 399, 400
reference configuration, 204
representative sample, 262–4
representative volume, 261, 263, 272,
273, 321
residual, 53–7, 59–61, 75
resultant internal forces, 177, 188, 226,
258, 274, 299, 301, 302, 305, 406,
407, 409
rotated, 222, 257
rotation, 7, 8, 77, 222–4, 257, 280, 293, 302,
312, 440, 456
sampling points, 37, 364, 400, 401, 407,
412, 449–51
scalar field, 45, 137–40, 149–52, 154, 372
scalar function, 81, 82, 85
scalar graphs, 82, 85
scalar variable, 81–3, 85, 86, 204
scouting, 45
second order formulation, 6
Index 467second order tensors, 8, 13, 155, 157–61, 163,
165, 167, 169, 171, 173–5, 177, 179, 181,
183, 185–7, 189, 191, 193, 195, 197,
199, 200
second Piola-Kirchhoff stress tensor matrix,
169, 172, 188, 258
selective integration, 13, 357–61, 363, 364,
400, 403, 409
serendipity, 330, 381
serendipity shape functions, 330
shape function, 319, 320, 327, 328, 330, 331,
336, 337, 339–41, 344, 349, 368–70, 379,
381, 385, 387, 389, 393, 401, 412, 419,
420, 422, 431, 433, 438, 443, 446, 449,
452, 457
shear forces, 168
shear locking, 362–4, 399, 412, 449, 450,
460, 462
shear strain, 271, 363
shear stress, 155, 363, 449, 450
shear stretches, 460
shearing, 258, 271, 290, 297, 357
simply supported beam, 76–8
singular square matrix, 28
six-noded shell element, 440, 441, 449,
452, 462
six-noded triangle finite element, 328
slope, 62, 129–34, 137, 139, 140, 149,
151, 154
slope of the tangential line, 140, 151
slope of the tangential plane, 139, 140, 149
slope tensor, 129, 132–4, 154
small strain formulations, 271, 417
small strains, 6, 8–10, 203, 213, 271, 281–4,
286, 287, 309, 310, 313
solid-embedded coordinate lines, 374, 380,
381, 383
solid-embedded coordinate surfaces, 380,
381, 441
solid-embedded coordinate system, 227, 241,
242, 319, 328, 329, 353, 368, 369, 372,
419, 441, 444
solid-embedded infinitesimal vector base,
322, 332, 383
solid-embedded vector base, 228, 231, 243,
245, 247, 248, 322, 323, 325, 326, 332,
333, 339, 340, 372–7, 383, 384, 388–91,
425–7, 438, 445, 447, 458
spectral radius, 25, 40, 41, 367
speed, 50, 67, 81, 82, 85
square matrices, 16, 30
square matrix, 16, 21–3, 24, 26, 28, 30–32,
98, 99
square-shaped material element, 221,
222, 225
square-shaped solid body, 235, 236
stability, 5, 6, 38, 41, 256, 294,
313, 364
static equilibrium, 43
static problems, 43, 204, 256, 293, 421
static solutions, 37
stiffness matrix, 35, 36, 50, 204, 367
strain increments, 301
strain measures, 8, 209–12, 301, 307, 308
strain rate, 214, 292
stress distribution, 408, 412, 451
stress increments, 291
stress integration, 13, 408
stress sampling, 401, 407, 409, 451
stress-strain relationship, 213
stress tensor, 13, 155, 158–61, 163, 164,
167–9, 172–8, 180, 181, 184–9, 226, 227,
231, 238, 249, 250, 252, 255, 258, 265,
267, 272, 278, 279, 292, 293, 302, 305,
306, 313, 343, 349, 355, 363, 377, 378,
392, 394, 396, 399, 407, 409, 429, 430,
435, 438, 449, 455, 462
stress transformation, 302
stress updates, 275
stretch 7, 8, 13, 205, 206, 208–12, 214,
223, 224, 257, 259, 268–72, 288–90, 292,
293, 296–302, 359–61, 363, 364, 368,
400, 401, 407, 412, 417, 427, 439,
449–51, 460
stretching, 204, 206, 207, 209, 214, 223, 225,
255, 257, 258, 263, 266, 268–72, 288–92,
297–300, 312, 364, 451
stretching modes, 291, 298–300
structural elements, 42, 249, 261
subtraction of matrices, 17
surface base vectors, 192
surface’s base, 179–81, 186, 187, 192,
195, 198
symmetric matrix, 21, 26, 62
system of equations, 44–6, 62, 63, 367
tangential stiffness matrix, 204, 367
tangential vectors, 323
ten-noded tetrahedron, 380, 392
tensorial calculus, 13
tensor matrix, 159, 160, 163, 164, 167–9,
172, 173, 178, 181, 185–8, 194, 198,
468 Index226, 227, 231, 238, 250, 252, 255, 258,
265, 267, 272, 278, 292, 302, 305, 306,
313, 343, 349, 355, 363, 377, 378, 392,
394, 396, 399, 406, 407, 409, 429, 430,
435, 438, 449, 455, 462
tensor transformation, 133, 185
tetrahedron finite element, 367–71, 373, 375,
377, 379, 380, 392, 396, 409
theoretically exact, 6, 7, 13, 313
three points Newton-Cotes integration, 67
time integration, 38, 40–42, 62, 63
total virtual work, 235, 237, 238, 254, 345,
346, 394
transformation, 95, 133, 185, 302
translating, 225
translation, 7, 8, 203, 222, 257, 456
transposition of a matrix, 18
triangular finite element, 318, 320, 322, 327,
357, 435
Truesdell rate, 259, 267
two Gauss points numerical integration, 68
two points Newton-Cotes integration, 66
under-integrated, 358, 400
unified constitutive approach, 257, 259, 261,
263, 265, 267, 269, 271, 273, 275, 277,
279, 281, 283, 285, 287, 289, 291–3, 295,
297, 299, 301, 303, 305, 307, 309,
311, 313
uniformly distributed load, 155
unit length, 32, 207, 208, 332
unit length column matrix, 32
unit prefixes, 83, 84
unit set, 84, 85
unit-length, 206, 207
upper triangular matrix, 21
vector base, 93–7, 108, 111, 113, 114, 127,
131–4, 142, 144, 147, 153–5, 161, 164,
165, 167, 168, 172, 174, 178, 186, 218,
219, 221, 228, 230, 231, 238, 243, 245–9,
260, 321–3, 325, 326, 332–4, 339–41,
372–7, 383, 384, 386, 388–92, 405, 410,
419, 423, 425–7, 429, 438–40,
444–9, 458–60
vector of the dual base, 112, 123
vector transformation, 95
vector variable equations, 91
vertical displacement, 357
virtual displacement, 75–7, 78, 232–4, 236–8,
250–254, 344–6, 393, 394, 397, 409,
430, 452
virtual work, 75–8, 231–4, 236–8, 249–54,
256, 343–6, 348, 353, 355, 358, 363, 393,
394, 400, 401, 405, 407, 412, 430, 438,
452, 455, 462
virtual work of the internal forces, 232, 234,
236, 237, 394
viscosity, 14, 267, 292, 293
volume change, 269, 274, 286, 287, 357, 359,
362, 363
volumetric integration, 438
volumetric locking, 357, 358, 363, 400
volumetric strain, 281, 283, 286, 287,
309, 311
volumetric stretch, 7, 268, 269, 271, 288, 290,
296, 297, 301, 307, 360, 361, 363, 364,
400, 401, 407
Young’s modulus, 288, 309
zero energy modes, 358, 359, 363, 400
zero matrix,
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