The Catalogue of Computational Material Models

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.

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.

Autorentext

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.

Inhalt
Contents1 Introduction 91.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Compositions of RheologicalModels . . . . . . . . . . . . . . . 131.2.1 Serial Arrangement of ERMs . . . . . . . . . . . . . . . 141.2.2 Parallel Arrangement of ERMs . . . . . . . . . . . . . 141.2.3 Arrangement of Serial-CRMs and Parallel-CRMs . . . 161.2.4 Serial Arrangement of ERMs and Parallel-CRMs . . . 181.2.5 Parallel Arrangement of ERMs and Serial-CRMs . . . 201.3 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . 292 Preliminaries 352.1 Modelling Tools . . . . . . . . . . . . . . . . . . . . . . . . . . 362.1.1 ContinuumMechanics . . . . . . . . . . . . . . . . . . 362.1.2 Dissipation Consistent Material Modelling . . . . . . . 382.2 Computational Tools . . . . . . . . . . . . . . . . . . . . . . . 472.2.1 Constitutive Integrator . . . . . . . . . . . . . . . . . . 472.2.2 Finite ElementMethod . . . . . . . . . . . . . . . . . . 572.3 Mathematical Tools . . . . . . . . . . . . . . . . . . . . . . . . 652.3.1 Heaviside Function and Causal Signals . . . . . . . . . 652.3.2 Laplace Transformation . . . . . . . . . . . . . . . . . 662.3.3 Complex Representations . . . . . . . . . . . . . . . . 692.3.4 Legendre Transformation . . . . . . . . . . . . . . . . . 772.3.5 Constrained Optimization . . . . . . . . . . . . . . . . 793 Elasticity 853.1 HookeModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.1.1 Specific HookeModel: Formulation . . . . . . . . . . . 863.1.2 Specific Hooke Model: Algorithmic Update . . . . . . . 9434 CONTENTS3.1.3 Specific HookeModel: Response Analysis . . . . . . . . 943.1.4 Generic HookeModel: Formulation . . . . . . . . . . . 984 Visco-Elasticity 1014.1 NewtonModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.1.1 Specific NewtonModel: Formulation . . . . . . . . . . 1024.1.2 Specific Newton Model: Algorithmic Update . . . . . . 1114.1.3 Specific NewtonModel: Response Analysis . . . . . . . 1134.1.4 Generic NewtonModel: Formulation . . . . . . . . . . 1234.2 KelvinModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.2.1 Specific KelvinModel: Formulation . . . . . . . . . . . 1264.2.2 Specific Kelvin Model: Algorithmic Update . . . . . . 1344.2.3 Specific KelvinModel: Response Analysis . . . . . . . 1384.2.4 Generic KelvinModel: Formulation . . . . . . . . . . . 1484.3 Generalized-KelvinModel . . . . . . . . . . . . . . . . . . . . 1504.3.1 Standard-Linear-Solid Kelvin Model:Formulation . . . . . . . . . . . . . . . . . . . . . . . . 1514.3.2 Standard-Linear-Solid Kelvin Model:Algorithmic Update . . . . . . . . . . . . . . . . . . . . 1654.3.3 Standard-Linear-Solid Kelvin Model:Response Analysis . . . . . . . . . . . . . . . . . . . . 1684.3.4 Generic Generalized-Kelvin Model:Formulation . . . . . . . . . . . . . . . . . . . . . . . . 1794.4 MaxwellModel . . . . . . . . . . . . . . . . . . . . . . . . . . 1834.4.1 SpecificMaxwellModel: Formulation . . . . . . . . . . 1844.4.2 Specific Maxwell Model: Algorithmic Update . . . . . . 1924.4.3 SpecificMaxwellModel: Response Analysis . . . . . . 1974.4.4 GenericMaxwellModel: Formulation . . . . . . . . . . 2084.5 Generalized-MaxwellModel . . . . . . . . . . . . . . . . . . . 2104.5.1 Standard-Linear-Solid Maxwell Model:Formulation . . . . . . . . . . . . . . . . . . . . . . . . 2114.5.2 Standard-Linear-Solid Maxwell Model:Algorithmic Update . . . . . . . . . . . . . . . . . . . . 2254.5.3 Standard-Linear-Solid Maxwell Model:Response Analysis . . . . . . . . . . . . . . . . . . . . 2274.5.4 Generic Generalized-Maxwell Model:Formulation ....