The aim of this thesis is the fast and exact simulation of modern materials like fibre reinforced thermoplastics and fibre reinforced elastomers. These simulations are in the scope of large strain deformations and contain anisotropic and viscoelastic behaviour. The chapter Differential geometry outlines the necessary tensor analysis and differential geometry. We present the weak formulation in the undeformed domain and use Newton’s method to approximate the solution of this formulation, cf. Section 3.1 and Chapter 4, respectively. For the viscoelasticity we use a special ansatz for the internal variable. Next, we compute all necessary derivations for the Newton system, cf. Sections 4.2 and 4.3. We also investigate the symmetry of the material tensors in Section 4.4. Further, we present three methods to improve the convergence of Newton’s method, cf. Section 4.5. With these three methods we are able to consider more problems, compute them faster and in a more robust way. In Chapter 5 we concisely discuss the FEM and show the appearing matrices in detail. The aim of Chapter 6 is the application of the a posteriori error estimator to this complex material behaviour. We present some numerical examples in Chapter 7. In Chapter 8 the problems that arise in the simulation of fibre-reinforced elastomers are analysed and tackled with help of mixed formulations. We derive a symmetric mixed formulation from a reduced form of the energy density. Also, we reformulate the mixed variable for inextensibility to avoid the numerical cancellation in Section 8.3. The Section 8.4 is about a joined mixed formulation to solve problems with inextensible fibres in an incompressible matrix, like fibre-reinforced rubber. The succeeding section Section 8.5 deals with the arising indefinite block matrix system.:Contents
Glossary 5
1 Introduction – motivation 13
2 Differential geometry 15
2.1 From parametrisations to the Lagrangian strain 15
2.2 Derivatives of tensors 20
3 Physical foundations 25
3.1 Large Deformation 25
3.1.1 Balance of forces 25
3.1.2 Energy minimisation 28
3.2 Anisotropic energy density 29
3.3 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Newton’s method 37
4.1 Newton system 37
4.2 Anisotropic material tensor 40
4.3 Viscoelastic material tensor 41
4.4 Symmetry of the material tensor 44
4.5 Load steps and line-search 47
4.5.1 Load steps – time steps 47
4.5.2 Backtracking for det ℱ > 0 48
4.5.3 Line search for energy minimisation 49
5 Implementation 53
5.1 Numerical Integration 53
5.2 Finite element discretisation 54
5.3 Voigt notation 56
6 Mesh control 65
7 Numerical results 69
7.1 Semi-analytical example 69
7.2 Cook’s membrane 71
7.2.1 Viscoelastic example 72
7.3 Chemnitz hook – Chemnitzer Haken 72
8 Mixed formulation 75
8.1 Motivation 75
8.2 General considerations 78
8.3 Smooth square root 81
8.4 Joined mixed formulation 84
8.5 Matrix representation 86
9 Conclusion 91
10 Theses 93
11 Appendix 95
11.1 Derivatives of the distortion-invariants with respect to the pseudo invariants 95
Bibliography 101
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:23546 |
| Date | 14 August 2018 |
| Creators | Schmidt, Hansjörg |
| Contributors | Meyer, Arnd, Tobiska, Lutz, Apel, Thomas, Technische Universität Chemnitz |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/acceptedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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