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 |  موضوع: كتاب Introduction to Nonlinear Finite Element Analysis الجمعة 05 يوليو 2024, 3:33 pm | |
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Introduction to Nonlinear Finite Element Analysis
Nam-Ho Kim
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Contents
1 Preliminary Concepts 1
1.1 Introduction . 1
1.2 Vector and Tensor Calculus . 3
1.2.1 Vector and Tensor . 3
1.2.2 Vector and Tensor Calculus 11
1.2.3 Integral Theorems . 12
1.3 Stress and Strain 14
1.3.1 Stress 15
1.3.2 Strain 26
1.3.3 Stress–Strain Relationship . 31
1.4 Mechanics of Continuous Bodies 36
1.4.1 Boundary-Valued Problem . 37
1.4.2 Principle of Minimum Potential Energy . 38
1.4.3 Principle of Virtual Work 46
1.5 Finite Element Method . 50
1.5.1 Finite Element Approximation 50
1.5.2 Finite Element Equations for a One-Dimensional
Problem 54
1.5.3 Finite Element Equations for 3D Solid Element . 61
1.5.4 A MATLAB Code for Finite Element Analysis 67
1.6 Exercises . 73
References . 79
2 Nonlinear Finite Element Analysis Procedure . 81
2.1 Introduction to Nonlinear Systems in Solid Mechanics 81
2.1.1 Geometric Nonlinearity . 85
2.1.2 Material Nonlinearity . 87
2.1.3 Kinematic Nonlinearity . 89
2.1.4 Force Nonlinearity . 90
xi2.2 Solution Procedures for Nonlinear Algebraic Equations . 91
2.2.1 Newton–Raphson Method . 93
2.2.2 Modified Newton–Raphson Method 101
2.2.3 Incremental Secant Method 103
2.2.4 Incremental Force Method . 109
2.3 Steps in the Solution of Nonlinear Finite Element Analysis . 114
2.3.1 State Determination 114
2.3.2 Residual Calculation . 115
2.3.3 Convergence Check 116
2.3.4 Linearization . 116
2.3.5 Solution 117
2.4 MATLAB Code for a Nonlinear Finite Element
Analysis Procedure 119
2.5 Nonlinear Solution Controls Using Commercial Finite
Element Programs 132
2.5.1 Abaqus . 133
2.5.2 ANSYS . 134
2.5.3 NEiNastran 136
2.6 Summary . 137
2.7 Exercises . 138
References . 140
3 Finite Element Analysis for Nonlinear Elastic Systems . 141
3.1 Introduction . 141
3.2 Stress and Strain Measures in Large Deformation 142
3.2.1 Deformation Gradient 142
3.2.2 Lagrangian and Eulerian Strains . 145
3.2.3 Polar Decomposition . 150
3.2.4 Deformation of Surface and Volume 154
3.2.5 Cauchy and Piola-Kirchhoff Stresses . 158
3.3 Nonlinear Elastic Analysis 161
3.3.1 Nonlinear Static Analysis: Total Lagrangian
Formulation . 162
3.3.2 Nonlinear Static Analysis: Updated Lagrangian
Formulation . 174
3.4 Critical Load Analysis . 179
3.4.1 One-Point Approach . 181
3.4.2 Two-Point Approach . 181
3.4.3 Stability Equation with Actual Critical Load Factor . 182
3.5 Hyperelastic Materials . 183
3.5.1 Strain Energy Density 184
3.5.2 Nearly Incompressible Hyperelasticity 190
3.5.3 Variational Equation and Linearization 197
3.6 Finite Element Formulation for Nonlinear Elasticity 200
3.7 MATLAB Code for Hyperelastic Material Model 205
xii Contents3.8 Nonlinear Elastic Analysis Using Commercial Finite
Element Programs 211
3.8.1 Usage of Commercial Programs . 211
3.8.2 Modeling Examples of Nonlinear Elastic Materials . 214
3.9 Fitting Hyperelastic Material Parameters from Test Data 221
3.9.1 Elastomer Test Procedures . 222
3.9.2 Data Preparation 224
3.9.3 Curve Fitting . 227
3.9.4 Stability of Constitutive Model 229
3.10 Summary . 231
3.11 Exercises . 232
References . 239
4 Finite Element Analysis for Elastoplastic Problems . 241
4.1 Introduction . 241
4.2 One-Dimensional Elastoplasticity . 242
4.2.1 Elastoplastic Material Behavior . 242
4.2.2 Finite Element Formulation for Elastoplasticity 247
4.2.3 Determination of Stress State . 250
4.3 Multidimensional Elastoplasticity . 265
4.3.1 Yield Functions and Yield Criteria . 266
4.3.2 Von Mises Yield Criterion . 272
4.3.3 Hardening Models . 275
4.3.4 Classical Elastoplasticity Model . 280
4.3.5 Numerical Integration 290
4.3.6 Computational Implementation of Elastoplasticity 299
4.4 Finite Rotation with Objective Integration . 308
4.4.1 Objective Tensor and Objective Rate . 309
4.4.2 Finite Rotation and Objective Rate . 314
4.4.3 Incremental Equation for Finite Rotation
Elastoplasticity . 317
4.4.4 Computational Implementation of Finite Rotation 321
4.5 Finite Deformation Elastoplasticity with Hyperelasticity . 325
4.5.1 Multiplicative Decomposition 325
4.5.2 Finite Deformation Elastoplasticity . 326
4.5.3 Time Integration 330
4.5.4 Return-Mapping Algorithm 333
4.5.5 Consistent Algorithmic Tangent Operator 336
4.5.6 Variational Principles for Finite Deformation . 337
4.5.7 Computer Implementation of Finite Deformation
Elastoplasticity . 338
4.6 Mathematical Formulas from Finite Elasticity 343
4.6.1 Linearization of Principal Logarithmic Stretches . 343
4.6.2 Linearization of the Eigenvector of the Elastic
Trial Left Cauchy-Green Tensor . 344
Contents xiii4.7 MATLAB Code for Elastoplastic Material Model . 345
4.8 Elastoplasticity Analysis of Using Commercial Finite
Element Programs 350
4.8.1 Usage of Commercial Programs . 350
4.8.2 Modeling Examples of Elastoplastic Materials 355
4.9 Summary . 359
4.10 Exercises . 360
References . 365
5 Finite Element Analysis for Contact Problems . 367
5.1 Introduction . 367
5.2 Examples of Simple One-Point Contact . 369
5.2.1 Contact of a Cantilever Beam with a Rigid Block 369
5.2.2 Contact of a Cantilever Beam with Friction . 374
5.3 General Formulation for Contact Problems 378
5.3.1 Contact Condition with Rigid Surface . 379
5.3.2 Variational Inequality in Contact Problems . 382
5.3.3 Penalty Regularization 385
5.3.4 Frictionless Contact Formulation 389
5.3.5 Frictional Contact Formulation 393
5.4 Finite Element Formulation of Contact Problems 398
5.4.1 Contact Between a Flexible Body and a Rigid Body 398
5.4.2 Contact Between Two Flexible Bodies 404
5.4.3 MATLAB Code for Contact Analysis . 406
5.5 Three-Dimensional Contact Analysis . 408
5.6 Contact Analysis Procedure and Modeling Issues 412
5.6.1 Contact Analysis Procedure 413
5.6.2 Contact Modeling Issues 417
5.7 Exercises . 423
References . 426
Index . 427
xiv Contents
Index
A
Assembly, 57
Associative plasticity, 291
B
Back stress, 282, 334
Backward Euler method, 291
Balance of momentum, 37
Basis vectors, 4
Baushinger effect, 278
Broyden, Fletcher, Goldfarb, and
Shanno (BFGS) method, 107
Bisection, 116
Boundary condition, 38, 54
essential, 54
natural, 54
Boundary valued problem, 38, 54
Bulk modulus, 192
C
Cauchy–Green tensor, 145, 147, 176, 191,
327, 331, 343
left, 147, 327, 331, 343
right, 145, 327, 343
Cauchy’s Lemma, 20
Consistency condition, 372, 380
contact, 380, 409
Constitutive relation, 31
Constrained optimization, 384
contact, 384
Contact force, 372, 410, 417
normal, 410
Contact form, 387
normal, 387
tangential, 387
Contact pair, 413
Contact problem, 367
Contact search, 414
Contact stiffness, 410, 416
Contact tolerance,415
Contraction, 8
Convergence, 94, 421
Convex set, 382
Coulomb friction, 375, 393
Critical displacement, 180
Critical load, 179, 181, 183
actual load factor, 183
load factor, 181
one-point, 181
two-point, 181
Cross product. See Vector, product
D
Deformation field, 27
Deformation gradient, 144, 330
relative, 330
Deviator, 274
Directional derivative, 385
Displacement field, 27
Displacement gradient, 144
Dissipation function, 327, 328
Dissipation inequality, 328
Distortion energy theory, 268
Divergence, 11
Divergence theorem, 12
Dual vector, 10
Dyadic product, 5
 Springer Science+Business Media New York 2015
N.-H. Kim, Introduction to Nonlinear Finite Element Analysis,
DOI 10.1007/978-1-4419-1746-1
427E
Effective plastic strain, 282
Eigenvalue, 23, 182
Eigenvector, 23
Elastic domain, 282, 326
Elasticity matrix, 34
Elasticity tensor, 32
Elastic limit, 32
Elastic modulus, 243
Elastic predictor, 291
Elastoplasticity, 241, 273, 308, 325, 360
finite deformation, 360
finite rotation, 308
infinitesimal, 273
multiplicative plasticity, 325
Euclidean norm, 157
F
Failure envelope, 267
Finite element, 50, 51, 62
shape function, 62
Flow potential, 283
Form, 44
energy bilinear, 44
load linear, 44
Frame indifference, 21
Fre´chet differentiable, 43
Free energy, 327, 332
Friction, 374
G
Gap, 370, 390
Gap function, 410
Gauss integration, 65
Gauss’ theorem, 47
Generalized Hooke’s law, 31, 32
Generalized solution, 40
Gradient, 11
Green’s identity, 14
H
Hooke’s law, 15
generalized, 15
Hydrostatic pressure, 192
Hyper-elastic material, 184
I
Impenetrability, 372
Impenetrability condition, 379, 380
Incremental force method, 109
Initial stiffness, 170, 298
Inner product, 4
Integration-by-parts, 13
Interpolation function, 53
Invariant, 185
Isoparametric mapping, 62
Isotropic hardening, 282
J
Jacobian, 94
Jacobian matrix, 116
K
Kinematically admissible displacement, 40
Kinematic hardening, 282, 283
Kronecker delta symbol, 4, 164
Kuhn–Tucker condition, 284, 329
L
Lagrange multiplier, 284, 368, 372, 376
Lagrangian strain, 167
Lame’s constants, 33, 163, 281
Laplace operator, 11
Lie derivative, 327
Load step, 110
Lower and upper (LU) decomposition, 101
M
Master, 371
Master element, 408
Material description, 168
Matrix, 5, 23, 34
determinant, 23
elasticity, 34
Modified Newton–Raphson method, 101–103
Mooney–Rivlin material, 186–187
N
Natural coordinate, 379
contact problem, 379
Necking, 32
Neo–Hookean material, 186
Newton–Raphson method, 93, 168
Nonlinear elastic problem, 162
Nonlinearity, 162, 241, 367
boundary, 367
force, 90–91
428 Indexgeometric, 85–87, 164
kinematic, 89–90
material, 87–89, 241
Nonlinear solution procedure, 91
Norm, 5, 8
Normal gap, 380
O
Objective rate, 360
Operator, linear, 81
P
Penalty, 368, 372, 377
Penalty method, 386
Penalty parameter, 373, 386
Penetration, 372
Permutation, 10, 156
Plane strain, 34
Plane stress, 34
Plastic consistency
parameter, 283
Plastic corrector, 291
Plastic modulus, 246, 284, 334
Poisson’s ratio, 33
Polar decomposition, 150
Potential energy, 166, 384, 386
Principal stress direction, 22, 24
Principal stretch, 332
Principle of minimum potential
energy, 39
Principle of virtual work, 46
Projection, 4, 290, 380
Proportional limit, 32
R
Reference element, 65
Residual, 94, 116
Residual load, 170, 299
Return mapping, 292, 333, 360
Reynolds transport theorem, 13
Rigid-body motion, 421
Rigid body rotation, 315
Rotation tensor, 150
S
Secant method, 104
Secant stiffness matrix, 107
Shape function, 53
Shear modulus, 33
Slave, 371
Slave-master, 368
Slave node, 399
Slip, 375, 393
Slip condition, 394
Sobolev space, 40
Solution, 44, 52
generalized, 44
trial, 52
Spatial description, 174
Spatial velocity gradient, 327
Spin tensor, 315
Stick, 375
Stick condition, 395
Stiffness matrix, 57, 64, 296
consistent, 296
solid, 64
Strain, 7, 26, 28–30, 145, 147, 167,
170, 174, 268, 332, 334
deviatoric, 30, 268
effective plastic, 334
elastic principal stretch, 332
engineering, 174
engineering shear, 28
Eulerian, 147
infinitesimal, 145, 172
Lagrangian, 145, 167, 170
normal, 28
shear, 28
symmetric, 29
tensorial shear, 28
volumetric, 30
Strain energy, 39, 163, 281
elastic, 281
Strain energy density, 268
distortion, 268
Strain hardening, 32
Stress, 17, 18, 20–22, 24, 31, 32, 159,
160, 174, 268, 291, 314, 316, 326
Cauchy, 159, 174, 314
deviatoric, 21, 268
first Piola–Kirchhoff, 159, 318
invariant, 24
Kirchhoff, 160, 326
mean, 21
normal, 20
principal, 22
second Piola–Kirchhoff, 159
shear, 20
symmetry, 18
tensor, 17
trial, 291
ultimate, 32
Index 429Stress (cont.)
uniaxial, 31
yield, 32
Stress rate, 315
Jaumann, 315
Stress vector, 15
Stretch tensor, 150
Strong form, 38
Structural energy form, 168, 185,
298, 337
elastic, 168
elastoplasticity, 298
finite deformation, 337
nonlinear, 175
St. Venant–Kirchhoff material, 163
Surface traction, 15
T
Tangential slip, 379, 380
Tangential traction force, 387
Tangent modulus, 243
Tangent operator, 297, 336, 337
consistent, 297, 337
material, 336
spatial, 336
Tangent stiffness matrix, 94
Tensor, 5–7, 9, 10, 17, 32
Cartesian, 5
elasticity, 32
identity, 5
orthogonal, 9
skew, 6, 10
spin, 7
stress, 17
symmetric, 6
Tensor product, 269
Time step, 110
Total Lagrangian formulation, 168
Trace, 8, 268
Transpose, 3
Trial function, 50
U
Updated Lagrangian formulation, 174
V
Variational equation, 43, 167
Variational inequality, 383
Vector, 3, 10
dual, 10
product, 10
Virtual displacement, 42
Virtual work, 391
contact, 391
W
Weak form, 44, 115, 166, 249, 298, 383
Work, 39
Y
Yield criterion, 282
von Mises, 282
Yield function, 282, 327
Yield surface, 282
Young’s modulus, 33, 83
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