We consider thin bodies made from elastomers and nematic liquid crystal elastomers. Starting from a nonlinear 3d hyperelastic model, and using the Gamma-convergence method, we derive lower dimensional models for 2d and 1d. The limit models describe the interplay between free liquid crystal orientations and bending deformations.:1 Introduction
1.1 Main results and structure of the text
1.2 Survey of the literature
1.2.1 Dimension reduction in nonlinear elasticity
1.2.2 Relation to other bending regime results in detail
1.2.3 Relation to other Gamma-convergence results of LCEs
2 Liquid crystal elastomers
2.1 Properties
2.2 Modeling
3 Rods
3.1 Setup and statement of analytical main results
3.1.1 The 3d-model and assumptions
3.1.2 The effective 1d-model
3.1.3 The Gamma-convergence result without boundary conditions
3.1.4 Boundary conditions for y
3.1.5 Weak and strong anchoring of n
3.1.6 Definition and properties of the effective coefficients
3.2 Numerical 1d-model exploration
3.3 Dimensional analysis and scalings
3.3.1 Non-dimensionalization and rescaling
3.3.2 Scaling assumptions
3.3.3 Dimensional analysis and applicability of the 1d-model
3.4 Smooth approximation of framed curves
3.5 Proofs
3.5.1 Compactness: proofs of Theorem 3.1.3 (a) and Proposition 3.1.4 (a)
3.5.2 Lower bound: proof of Theorem 3.1.3 (b) . . . . . . . . . . . . 68
3.5.3 Upper bound: proofs of Theorem 3.1.3 (c) and Proposition 3.1.4 (b)
3.5.4 Anchoring: proof of Proposition 3.1.5
3.5.5 Properties of the effective coefficients
4 Plates
4.1 Setup and statement of analytical main results
4.1.1 The 3d-model and assumptions
4.1.2 The effective 2d-model
4.1.3 The Gamma-convergence result without boundary conditions
4.1.4 Definition and properties of the effective coefficients
4.1.5 Boundary conditions for y
4.1.6 Weak and strong anchoring of n
4.2 Analytical and numerical 2d-model exploration
4.2.1 Analytical 2d-model exploration
4.2.2 Numerical 2d-model exploration
4.3 Dimensional analysis and scalings
4.3.1 Non-dimensionalization and rescaling
4.3.2 Scaling assumptions
4.3.3 Dimensional analysis and applicability
4.4 Geometry and approximation of bending deformations
4.4.1 Proofs of the geometric properties in the smooth case
4.4.2 Proof for the smooth approximations
4.5 Proofs
4.5.1 Compactness: proofs of Theorems 4.1.1 (a) and 4.1.8 (a)
4.5.2 Lower bound: proof of Theorem 4.1.1 (b)
4.5.3 Upper bound: proofs of Theorem 4.1.1 (c) and Theorem 4.1.8 (b)
4.5.4 Properties of the effective coefficients
4.5.5 Anchorings
4.5.6 Approximation of nonlinear strains: proof of Proposition 4.5.4
5 Conclusions and outlooks
Bibliography
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:91561 |
| Date | 22 May 2024 |
| Creators | Griehl, Max |
| Contributors | Neukamm, Stefan Minsu, Bartels, Sören, Schmidt, Bernd, Technische Universität Dresden |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/updatedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | https://doi.org/10.1142/S0218202523500331, https://doi.org/10.48550/arXiv.2205.15174 |
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