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The Finite Element Method (FEM) is a powerful technique originally developed for numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems.

Taal / Language : English

Inhoudsopgave:
Part One Basic Concepts and Solution Techniques 21

1 Preliminaries 23

1.1 A simple example of non-linear behaviour 23

1.2 A review of concepts from linear algebra 25

1.3 Vectors and tensors 32

1.4 Stress and strain tensors 36

1.5 Elasticity 42

1.6 The PYFEM finite element library 44

Bibliography 48

2 Non-linear finite element analysis 49

2.1 Equilibrium and virtual work 49

2.2 Spatial discretisation by finite elements 51

2.3 PyFEM: Shape function utilities 55

2.4 Incremental-iterative analysis 59

2.5 Load vs displacement control 68

2.6 PyFEM: A linear finite element code with displacement control 71

Bibliography 80

3 Geometrically non-linear analysis 81

3.1 Truss elements 82

3.1.1 Total Lagrange formulation 85

3.1.2 Updated Lagrange formulation 88

3.1.3 Corotational formulation 89

3.2 PyFEM: The shallow truss problem 94

3.3 Stress and deformation measures in continua 103

3.4 Geometrically non-linear formulation of continuum elements 109

3.4.1 Total and Updated Lagrange formulations 109

3.4.2 Corotational formulation 113

3.5 Linear buckling analysis 117

3.6 PyFEM: A geometrically non-linear continuum element 119

Bibliography 127

4 Solution techniques in quasi-static analysis 129

4.1 Line searches 129

2 Contents

4.2 Path-following or arc-length methods 132

4.3 PYFEM: Implementation of Riks` arc-length solver 140

4.4 Stability and uniqueness in discretised systems 145

4.4.1 Stability of a discrete system 145

4.4.2 Uniqueness and bifurcation in a discrete system 146

4.4.3 Branch switching 150

4.5 Load stepping and convergence criteria 150

4.6 Quasi-Newton methods 153

Bibliography 156

5 Solution techniques for non-linear dynamics 157

5.1 The semi-discrete equations 157

5.2 Explicit time integration 159

5.3 PYFEM: Implementation of an explicit solver 162

5.4 Implicit time integration 167

5.4.1 The Newmark family 167

5.4.2 The HHT --method 168

5.4.3 Alternative implicit methods for time integration 169

5.5 Stability and accuracy in the presence of non-linearities 171

5.6 Energy-conserving algorithms 175

5.7 Time step size control and element technology 178

Bibliography 179

Part Two Material Non-linearities 181

6 Damage mechanics 183

6.1 The concept of damage 183

6.2 Isotropic elasticity-based damage 185

6.3 PYFEM: A plane-strain damage model 188

6.4 Stability, ellipticity, and mesh sensitivity 192

6.4.1 Stability, ellipticity, and mesh sensitivity 192

6.4.2 Mesh sensitivity 195

6.5 Cohesive-zone models 198

6.6 Element technology: Embedded discontinuities 203

6.7 Complex damage models 211

6.7.1 Anisotropic damage models 211

6.7.2 Microplane models 212

6.8 Crack models for concrete and other quasi-brittle materials 214

6.8.1 Elasticity-based smeared crack models 214

6.8.2 Reinforcement and tension stiffening 219

6.9 Regularised damage models 223

6.9.1 Non-local damage models 223

Bibliography 227

7 Plasticity 231

Contents 3

7.1 A simple slip model 231

7.2 Flow theory of plasticity 235

7.2.1 Yield function 235

7.2.2 Flow rule 240

7.2.3 Hardening behaviour 244

7.3 Integration of the stress-strain relation 250

7.4 Tangent stiffness operators 261

7.5 Multi-surface plasticity 264

7.5.1 Koiter`s generalisation 264

7.5.2 Rankine plasticity for concrete 267

7.5.3 Tresca and Mohr-Coulomb plasticity 272

7.6 Soil plasticity: Cam-clay model 279

7.7 Coupled damage-plasticity models 282

7.8 Element technology: volumetric locking 283

Bibliography 289

8 Time-dependent material models 293

8.1 Linear visco-elasticity 293

8.1.1 One-dimensional linear visco-elasticity 293

8.1.2 Three-dimensional visco-elasticity 296

8.1.3 Algorithmic aspects 297

8.2 Creep models 299

8.3 Visco-plasticity 301

8.3.1 One-dimensional visco-plasticity 302

8.3.2 Integration of the rate equations 303

8.3.3 Perzyna visco-plasticity 304

8.3.4 Duvaut-Lions visco-plasticity 306

8.3.5 Consistency model 308

8.3.6 Propagative or dynamic instabilities 310

Bibliography 315

Part Three Structural Elements 317

9 Beams and arches 319

9.1 A shallow arch 319

9.1.1 Kirchhoff formulation 319

9.1.2 Including shear deformation: Timoshenko beam 326

9.2 PYFEM: A Kirchhoff beam element 329

9.3 Corotational elements 333

9.3.1 Kirchhoff theory 333

9.3.2 Timoshenko beam theory 337

9.4 An isoparametric degenerate continuum 2D beam element 339

9.5 An isoparametric degenerate continuum 3D beam element 344

Bibliography 352

10 Plates and shells 355

4 Contents

10.1 Shallow-shell formulations 356

10.2 An isoparametric degenerate continuum shell element 363

10.3 Solid-like shell elements 368

10.4 Shell plasticity: Ilyushin`s criterion 369

Bibliography 373

Part Four Large Strains 375

11 Hyperelasticity 377

11.1 More continuum mechanics 377

11.1.1 Momentum balance and stress tensors 377

11.1.2 Objective stress rates 380

11.1.3 Principal stretches and invariants 384

11.2 Strain-energy functions 386

11.2.1 Incompressibility and near-incompressibility 388

11.2.2 Strain energy as a function of stretch invariants 390

11.2.3 Strain energy as a function of principal stretches 394

11.2.4 Logarithmic extension of linear elasticity: Hencky model 398

11.3 Element technology 400

11.3.1 u/p formulation 401

11.3.2 Enhanced Assumed Strain elements 404

11.3.3 F-bar approach 406

11.3.4 Corotational approach 407

Bibliography 409

12 Large-strain elastoplasticity 411

12.1 Eulerian formulations 412

12.2 Multiplicative elastoplasticity 417

12.3 Multiplicative elastoplasticity vs rate formulations 421

12.4 Integration of the rate equations 424

12.5 Exponential return-mapping algorithms 428

Bibliography 432

Part Five Advanced Discretisation Concepts 435

13 Interfaces and discontinuities 437

13.1 Interface elements 437

13.2 Discontinuous Galerkin methods 446

Bibliography 450

14 Meshless and partition-of-unity methods 451

14.1 Meshless methods 452

14.1.1 The element-free Galerkin method 452

14.1.2 Application to fracture 456

14.1.3 Higher-order damage mechanics 458

Contents 5

14.1.4 Volumetric locking 460

14.2 Partition-of-unity approaches 461

14.2.1 Application to fracture 465

14.2.2 Extension to large deformations 470

14.2.3 Dynamic fracture 476

14.2.4 Weak discontinuities 479

Bibliography 480

15 Isogeometric finite element analysis 483

15.1 Basis functions in computer aided geometric design 483

15.1.1 Univariate B-splines 484

15.1.2 Univariate non-uniform rational B-splines 487

15.1.3 Multivariate B-splines and NURBS patches 488

15.1.4 T-splines 490

15.2 Isogeometric finite elements 493

15.2.1 B´ezier element representation 493

15.2.2 B´ezier extraction 495

15.3 PYFEM: Shape functions for isogeometric analysis 497

15.4 Isogeometric analysis in non-linear solid mechanics 500

15.4.1 Design-through-analysis of shell structures 500

15.4.2 Higher-order damage models 505

15.4.3 Cohesive-zone models 510

Bibliography 518
Extra informatie:
Hardback
540 pagina's
Januari 2012
948 gram
249 x 174 x 30 mm
Wiley-Blackwell

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