The Catalogue of Computational Material Models: Basic Geometrically Linear Models in 1d

ISBN-13: 9783030636838 / Angielski / Twarda / 2021 / 402 str.

Paul Steinmann; Kenneth Runesson

Spis treści

Contents

1 Introduction 9

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Compositions of RheologicalModels . . . . . . . . . . . . . . . 13

1.2.1 Serial Arrangement of ERMs . . . . . . . . . . . . . . . 14

1.2.2 Parallel Arrangement of ERMs . . . . . . . . . . . . . 14

1.2.3 Arrangement of Serial-CRMs and Parallel-CRMs . . . 16

1.2.4 Serial Arrangement of ERMs and Parallel-CRMs . . . 18

1.2.5 Parallel Arrangement of ERMs and Serial-CRMs . . . 20

1.3 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 Preliminaries 35

2.1 Modelling Tools . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.1.1 ContinuumMechanics . . . . . . . . . . . . . . . . . . 36

2.1.2 Dissipation Consistent Material Modelling . . . . . . . 38

2.2 Computational Tools . . . . . . . . . . . . . . . . . . . . . . . 47

2.2.1 Constitutive Integrator . . . . . . . . . . . . . . . . . . 47

2.2.2 Finite ElementMethod . . . . . . . . . . . . . . . . . . 57

2.3 Mathematical Tools . . . . . . . . . . . . . . . . . . . . . . . . 65

2.3.1 Heaviside Function and Causal Signals . . . . . . . . . 65

2.3.2 Laplace Transformation . . . . . . . . . . . . . . . . . 66

2.3.3 Complex Representations . . . . . . . . . . . . . . . . 69

2.3.4 Legendre Transformation . . . . . . . . . . . . . . . . . 77

2.3.5 Constrained Optimization . . . . . . . . . . . . . . . . 79

3 Elasticity 85

3.1 HookeModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.1.1 Specific HookeModel: Formulation . . . . . . . . . . . 86

3.1.2 Specific Hooke Model: Algorithmic Update . . . . . . . 94

3

4 CONTENTS

3.1.3 Specific HookeModel: Response Analysis . . . . . . . . 94

3.1.4 Generic HookeModel: Formulation . . . . . . . . . . . 98

4 Visco-Elasticity 101

4.1 NewtonModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.1.1 Specific NewtonModel: Formulation . . . . . . . . . . 102

4.1.2 Specific Newton Model: Algorithmic Update . . . . . . 111

4.1.3 Specific NewtonModel: Response Analysis . . . . . . . 113

4.1.4 Generic NewtonModel: Formulation . . . . . . . . . . 123

4.2 KelvinModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.2.1 Specific KelvinModel: Formulation . . . . . . . . . . . 126

4.2.2 Specific Kelvin Model: Algorithmic Update . . . . . . 134

4.2.3 Specific KelvinModel: Response Analysis . . . . . . . 138

4.2.4 Generic KelvinModel: Formulation . . . . . . . . . . . 148

4.3 Generalized-KelvinModel . . . . . . . . . . . . . . . . . . . . 150

4.3.1 Standard-Linear-Solid Kelvin Model:

Formulation . . . . . . . . . . . . . . . . . . . . . . . . 151

4.3.2 Standard-Linear-Solid Kelvin Model:

Algorithmic Update . . . . . . . . . . . . . . . . . . . . 165

4.3.3 Standard-Linear-Solid Kelvin Model:

Response Analysis . . . . . . . . . . . . . . . . . . . . 168

4.3.4 Generic Generalized-Kelvin Model:

Formulation . . . . . . . . . . . . . . . . . . . . . . . . 179

4.4 MaxwellModel . . . . . . . . . . . . . . . . . . . . . . . . . . 183

4.4.1 SpecificMaxwellModel: Formulation . . . . . . . . . . 184

4.4.2 Specific Maxwell Model: Algorithmic Update . . . . . . 192

4.4.3 SpecificMaxwellModel: Response Analysis . . . . . . 197

4.4.4 GenericMaxwellModel: Formulation . . . . . . . . . . 208

4.5 Generalized-MaxwellModel . . . . . . . . . . . . . . . . . . . 210

4.5.1 Standard-Linear-Solid Maxwell Model:

Formulation . . . . . . . . . . . . . . . . . . . . . . . . 211

4.5.2 Standard-Linear-Solid Maxwell Model:

Algorithmic Update . . . . . . . . . . . . . . . . . . . . 225

4.5.3 Standard-Linear-Solid Maxwell Model:

Response Analysis . . . . . . . . . . . . . . . . . . . . 227

4.5.4 Generic Generalized-Maxwell Model:

Formulation . . . . . . . . . . . . . . . . . . . . . . . . 238

CONTENTS 5

5 Plasticity 241

5.1 St. VenantModel . . . . . . . . . . . . . . . . . . . . . . . . . 242

5.1.1 Specific St. VenantModel: Formulation . . . . . . . . . 242

5.1.2 Specific St. Venant Model: Algorithmic Update . . . . 250

5.1.3 Specific St. VenantModel: Response Analysis . . . . . 253

5.1.4 Generic St. VenantModel: Formulation . . . . . . . . . 259

5.2 PrandtlModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

5.2.1 Specific PrandtlModel: Formulation . . . . . . . . . . 263

5.2.2 Specific Prandtl Model: Algorithmic Update . . . . . . 267

5.2.3 Specific PrandtlModel: Response Analysis . . . . . . . 269

5.2.4 Generic PrandtlModel: Formulation . . . . . . . . . . 277

5.3 Prandtl Hardening Model . . . . . . . . . . . . . . . . . . . . 280

5.3.1 Specific Prandtl Isotropic Hardening Model:

Formulation . . . . . . . . . . . . . . . . . . . . . . . . 281

5.3.2 Specific Prandtl Isotropic Hardening Model:

Algorithmic Update . . . . . . . . . . . . . . . . . . . . 286

5.3.3 Specific Prandtl Isotropic Hardening Model:

Response Analysis . . . . . . . . . . . . . . . . . . . . 291

5.3.4 Specific Prandtl Kinematic Hardening Model:

Formulation . . . . . . . . . . . . . . . . . . . . . . . . 303

5.3.5 Specific Prandtl Kinematic Hardening Model:

Algorithmic Update . . . . . . . . . . . . . . . . . . . . 308

5.3.6 Specific Prandtl Kinematic Hardening Model:

Response Analysis . . . . . . . . . . . . . . . . . . . . 313

5.3.7 Specific Prandtl Mixed Hardening Model:

Formulation . . . . . . . . . . . . . . . . . . . . . . . . 325

5.3.8 Specific Prandtl Mixed Hardening Model:

Algorithmic Update . . . . . . . . . . . . . . . . . . . . 331

5.3.9 Specific Prandtl Mixed Hardening Model:

Response Analysis . . . . . . . . . . . . . . . . . . . . 335

5.3.10 Generic Prandtl Hardening Model:

Formulation . . . . . . . . . . . . . . . . . . . . . . . . 349

6 Visco-Plasticity 353

6.1 BinghamModel . . . . . . . . . . . . . . . . . . . . . . . . . . 354

6.1.1 Specific BinghamModel: Formulation . . . . . . . . . 355

6.1.2 Specific Bingham Model: Algorithmic Update . . . . . 366

6.1.3 Specific BinghamModel: Response Analysis . . . . . . 368

6 CONTENTS

6.1.4 Generic BinghamModel: Formulation . . . . . . . . . 380

6.2 PerzynaModel . . . . . . . . . . . . . . . . . . . . . . . . . . 383

6.2.1 Specific PerzynaModel: Formulation . . . . . . . . . . 384

6.2.2 Specific Perzyna Model: Algorithmic Update . . . . . . 391

6.2.3 Specific PerzynaModel: Response Analysis . . . . . . . 396

6.2.4 Generic PerzynaModel: Formulation . . . . . . . . . . 407

6.3 Perzyna Hardening Model . . . . . . . . . . . . . . . . . . . . 410

6.3.1 Specific Perzyna Isotropic Hardening Model:

Formulation . . . . . . . . . . . . . . . . . . . . . . . . 411

6.3.2 Specific Perzyna Isotropic Hardening Model:

Algorithmic Update . . . . . . . . . . . . . . . . . . . . 420

6.3.3 Specific Perzyna Isotropic Hardening Model:

Response Analysis . . . . . . . . . . . . . . . . . . . . 424

6.3.4 Specific Perzyna Kinematic Hardening Model:

Formulation . . . . . . . . . . . . . . . . . . . . . . . . 437

6.3.5 Specific Perzyna Kinematic Hardening Model:

Algorithmic Update . . . . . . . . . . . . . . . . . . . . 448

6.3.6 Specific Perzyna Kinematic Hardening Model:

Response Analysis . . . . . . . . . . . . . . . . . . . . 450

6.3.7 Specific Perzyna Mixed Hardening Model:

Formulation . . . . . . . . . . . . . . . . . . . . . . . . 463

6.3.8 Specific Perzyna Mixed Hardening Model:

Algorithmic Update . . . . . . . . . . . . . . . . . . . . 474

6.3.9 Specific Perzyna Mixed Hardening Model:

Response Analysis . . . . . . . . . . . . . . . . . . . . 478

6.3.10 Generic Perzyna Hardening Model:

Formulation . . . . . . . . . . . . . . . . . . . . . . . . 491

O autorze

Paul Steinmann is Professor for Mechanics at the Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Germany, where he heads the Institute of Applied Mechanics in the Department of Mechanical Engineering within FAU’s Faculty of Engineering. He is also Director of the Glasgow Computational Engineering Centre (GCEC) in the School of Engineering at the University of Glasgow, UK. His research interests include nonlinear continuum mechanics, material modelling, coupled and multiscale problems and corresponding computational methods both for engineering as well as for biomechanical applications.

Kenneth Runesson is Professor Emeritus of Material and Structural Mechanics at Chalmers University of Technology, Gothenburg, Sweden. His research interests include the computational modelling of coupled problems for porous media including inelasticity, damage and fracture. A trademark is control of discretization and modelling errors via space-time adaptivity. Computational homogenization and scale-bridging strategies are another major research field.

Blurb

This book gives a comprehensive account of the formulation and computational treatment of basic geometrically linear models in 1D. To set the stage, it assembles some preliminaries regarding necessary modelling, computational and mathematical tools. Thereafter, the remaining parts are concerned with the actual catalogue of computational material models. To this end, after starting out with elasticity as a reference, further 15 different basic variants of material models (5 x each of {visco-elasticity, plasticity, visco-plasticity}, respectively) are systematically explored. The presentation for each of these basic material models is a stand-alone account and follows in each case the same structure. On the one hand, this allows, in the true sense of a catalogue, to consult each of the basic material models separately without the need to refer to other basic material models. On the other hand, even though this somewhat repetitious concept may seem tedious, it allows to compare the formulation and resulting algorithmic setting of the various basic material models and thereby to uncover, in detail, similarities and differences. In particular, the response of each basic material model is analysed for the identical histories (Zig-Zag, Sine, Ramp) of prescribed strain and stress so as to clearly showcase and to contrast to each other the characteristics of the various modelling options.