7. Februar 2022
Philipp Reiter (Technische Universität Chemnitz)
Impermeability in nonlinear elasticity models
Maintaining the topology of objects undergoing deformations is a crucial aspect of elasticity models. In this talk we consider two different settings in which impermeability is implemented via regularization by a suitable nonlocal functional.
The behavior of long slender objects may be characterized by the classic Kirchhoff model of elastic rods. Phenomena like supercoiling which play an essential role in molecular biology can only be observed if self-penetrations are precluded. This can be achieved by adding a self-repulsive functional such as the tangent-point energy. We discuss the discretization of this approach and present some numerical simulations.
In case of elastic solids whose shape is described by the image of a reference domain under a deformation map, self-interpenetrations can be ruled out by claiming global invertibility. Given a suitable stored energy density, the latter is ensured by the Ciarlet–Nečas condition which, however, is difficult to handle numerically in an efficient way. This motivates approximating the latter by adding a self-repulsive functional which formally corresponds to a suitable Sobolev–Slobodeckiĭ seminorm of the inverse deformation.
This is joint work with Sören Bartels (Freiburg) and Stefan Krömer (Prague).