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).