Non-linear finite element analysis of solids and structures

Non-linear Finite Element Analysis of Solids and Structures......Page 1
Contents......Page 7
Preface......Page 13
Series Preface......Page 15
Notation......Page 17
About the Code......Page 23
PART I: BASIC CONCEPTS AND SOLUTION TECHNIQUES......Page 27
1.1 A Simple Example of Non-linear Behaviour......Page 29
1.2 A Review of Concepts from Linear Algebra......Page 31
1.3 Vectors and Tensors......Page 38
1.4 Stress and Strain Tensors......Page 43
1.5 Elasticity......Page 49
1.6 The PyFEM Finite Element Library......Page 51
References......Page 55
2.1 Equilibrium and Virtual Work......Page 57
2.2 Spatial Discretisation by Finite Elements......Page 59
2.3 PyFEM: Shape Function Utilities......Page 64
2.4 Incremental-iterative Analysis......Page 67
2.5 Load versus Displacement Control......Page 76
2.6 PyFEM: A Linear Finite Element Code with Displacement Control......Page 79
References......Page 88
3 Geometrically Non-linear Analysis......Page 89
3.1 Truss Elements......Page 90
3.1.1 Total Lagrange Formulation......Page 93
3.1.2 Updated Lagrange Formulation......Page 96
3.1.3 Corotational Formulation......Page 98
3.2 PyFEM: The Shallow Truss Problem......Page 102
3.3 Stress and Deformation Measures in Continua......Page 111
3.4.1 Total and Updated Lagrange Formulations......Page 117
3.4.2 Corotational Formulation......Page 122
3.5 Linear Buckling Analysis......Page 126
3.6 PyFEM: A Geometrically Non-linear Continuum Element......Page 129
References......Page 136
4.1 Line Searches......Page 139
4.2 Path-following or Arc-length Methods......Page 142
4.3 PyFEM: Implementation of Riks’ Arc-length Solver......Page 150
4.4.1 Stability of a Discrete System......Page 155
4.4.2 Uniqueness and Bifurcation in a Discrete System......Page 156
4.5 Load Stepping and Convergence Criteria......Page 160
4.6 Quasi-Newton Methods......Page 164
References......Page 167
5.1 The Semi-discrete Equations......Page 169
5.2 Explicit Time Integration......Page 170
5.3 PyFEM: Implementation of an Explicit Solver......Page 175
5.4 Implicit Time Integration......Page 178
5.4.1 The Newmark Family......Page 179
5.4.2 The HHT α-method......Page 180
5.4.3 Alternative Implicit Methods for Time Integration......Page 181
5.5 Stability and Accuracy in the Presence of Non-linearities......Page 182
5.6 Energy-conserving Algorithms......Page 187
5.7 Time Step Size Control and Element Technology......Page 190
References......Page 191
PART II: MATERIAL NON-LINEARITIES......Page 193
6.1 The Concept of Damage......Page 195
6.2 Isotropic Elasticity-based Damage......Page 197
6.3 PyFEM: A Plane-strain Damage Model......Page 201
6.4.1 Stability and Ellipticity......Page 205
6.4.2 Mesh Sensitivity......Page 208
6.5 Cohesive-zone Models......Page 211
6.6 Element Technology: Embedded Discontinuities......Page 216
6.7.1 Anisotropic Damage Models......Page 224
6.7.2 Microplane Models......Page 225
6.8.1 Elasticity-based Smeared Crack Models......Page 227
6.8.2 Reinforcement and Tension Stiffening......Page 232
6.9.1 Non-local Damage Models......Page 236
6.9.2 Gradient Damage Models......Page 237
References......Page 241
7.1 A Simple Slip Model......Page 245
7.2.1 Yield Function......Page 249
7.2.2 Flow Rule......Page 254
7.2.3 Hardening Behaviour......Page 258
7.3 Integration of the Stress–strain Relation......Page 265
7.4 Tangent Stiffness Operators......Page 275
7.5.1 Koiter’s Generalisation......Page 278
7.5.2 Rankine Plasticity for Concrete......Page 280
7.5.3 Tresca and Mohr–Coulomb Plasticity......Page 286
7.6 Soil Plasticity: Cam-clay Model......Page 293
7.7 Coupled Damage–Plasticity Models......Page 296
7.8 Element Technology: Volumetric Locking......Page 297
References......Page 303
8.1 Linear Visco-elasticity......Page 307
8.1.1 One-dimensional Linear Visco-elasticity......Page 308
8.1.2 Three-dimensional Visco-elasticity......Page 310
8.1.3 Algorithmic Aspects......Page 311
8.2 Creep Models......Page 313
8.3.1 One-dimensional Visco-plasticity......Page 315
8.3.2 Integration of the Rate Equations......Page 317
8.3.3 Perzyna Visco-plasticity......Page 318
8.3.4 Duvaut–Lions Visco-plasticity......Page 320
8.3.5 Consistency Model......Page 322
8.3.6 Propagative or Dynamic Instabilities......Page 324
References......Page 329
PART III: STRUCTURAL ELEMENTS......Page 331
9.1.1 Kirchhoff Formulation......Page 333
9.1.2 Including Shear Deformation: Timoshenko Beam......Page 340
9.2 PyFEM: A Kirchhoff Beam Element......Page 343
9.3.1 Kirchhoff Theory......Page 347
9.3.2 Timoshenko Beam Theory......Page 352
9.4 A Two-dimensional Isoparametric Degenerate Continuum Beam Element......Page 354
9.5 A Three-dimensional Isoparametric Degenerate Continuum Beam Element......Page 359
References......Page 367
10 Plates and Shells......Page 369
10.1 Shallow-shell Formulations......Page 370
10.2 An Isoparametric Degenerate Continuum Shell Element......Page 377
10.3 Solid-like Shell Elements......Page 382
10.4 Shell Plasticity: Ilyushin’s Criterion......Page 383
References......Page 387
PART IV: LARGE STRAINS......Page 389
11.1.1 Momentum Balance and Stress Tensors......Page 391
11.1.2 Objective Stress Rates......Page 394
11.1.3 Principal Stretches and Invariants......Page 398
11.2 Strain Energy Functions......Page 400
11.2.1 Incompressibility and Near-incompressibility......Page 402
11.2.2 Strain Energy as a Function of Stretch Invariants......Page 404
11.2.3 Strain Energy as a Function of Principal Stretches......Page 408
11.2.4 Logarithmic Extension of Linear Elasticity: Hencky Model......Page 412
11.3.1 u/p Formulation......Page 415
11.3.2 Enhanced Assumed Strain Elements......Page 418
11.3.3 F-bar Approach......Page 421
11.3.4 Corotational Approach......Page 422
References......Page 424
12 Large-strain Elasto-plasticity......Page 427
12.1 Eulerian Formulations......Page 428
12.2 Multiplicative Elasto-plasticity......Page 433
12.3 Multiplicative Elasto-plasticity versus Rate Formulations......Page 437
12.4 Integration of the Rate Equations......Page 440
12.5 Exponential Return-mapping Algorithms......Page 444
References......Page 448
PART V: ADVANCED DISCRETISATION CONCEPTS......Page 451
13 Interfaces and Discontinuities......Page 453
13.1 Interface Elements......Page 454
13.2 Discontinuous Galerkin Methods......Page 462
References......Page 465
14 Meshless and Partition-of-unity Methods......Page 467
14.1.1 The Element-free Galerkin Method......Page 468
14.1.2 Application to Fracture......Page 472
14.1.3 Higher-order Damage Mechanics......Page 474
14.1.4 Volumetric Locking......Page 476
14.2 Partition-of-unity Approaches......Page 477
14.2.1 Application to Fracture......Page 481
14.2.2 Extension to Large Deformations......Page 486
14.2.3 Dynamic Fracture......Page 491
14.2.4 Weak Discontinuities......Page 494
References......Page 496
15.1 Basis Functions in Computer Aided Geometric Design......Page 499
15.1.1 Univariate B-splines......Page 500
15.1.1.4 Partition-of-unity Property......Page 502
15.1.1.6 Refinement......Page 503
15.1.3 Multivariate B-splines and NURBS Patches......Page 504
15.1.4 T-splines......Page 506
15.2.1 B´ezier Element Representation......Page 509
15.2.2 B´ezier Extraction......Page 511
15.3 PyFEM: Shape Functions for Isogeometric Analysis......Page 513
15.4 Isogeometric Analysis in Non-linear Solid Mechanics......Page 516
15.4.1 Design-through-analysis of Shell Structures......Page 517
15.4.2 Higher-order Damage Models......Page 522
15.4.3 Cohesive Zone Models......Page 526
References......Page 532
Index......Page 535