A geometrically nonlinear Cosserat shell model for orientable and non-orientable surfaces: Discretization with geometric finite elements

• Published in: arXiv.org
• Main Authors: Nebel, Lisa Julia, Sander, Oliver, Bîrsan, Mircea, Neff, Patrizio
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 01.08.2023
• Subjects: Coordinates; Curvature; Discretization; Finite element method
• ISSN: 2331-8422

We investigate discretizations of a geometrically nonlinear elastic Cosserat shell with nonplanar reference configuration originally introduced by Bîrsan, Ghiba, Martin, and Neff in 2019. The shell model includes curvature terms up to order 5 in the shell thickness, which are crucial to reliably simulate high-curvature deformations such as near-folds or creases. The original model is generalized to shells that are not homeomorphic to a subset of \(\mathbb{R}^2\). For this, we replace the originally planar parameter domain by an abstract two-dimensional manifold, and verify that the hyperelastic shell energy and three-dimensional reconstruction are invariant under changes of the local coordinate systems. This general approach allows to determine the elastic response for even non-orientable surfaces like the M\"obius strip and the Klein bottle. We discretize the model with a geometric finite element method and, using that geometric finite elements are \(H^1\)-conforming, prove that the discrete shell model has a solution. Numerical tests then show the general performance and versatility of the model and discretization method.