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

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