In the Self-Contact Problem in Nonlinear Elasticity

Abstract

In this paper, we consider the minimization problem of $3$-dimensional nonlinear hyperelastic bodies moving in $\mathbb{R}^{3}$, which enables frictionless self-contact and forbids self-intersection. For this, we define a new class of admissible deformations based on a natural homotopy constraint. We study strictly orientation-preserving Sobolev maps in this new class and their global invertibility properties from a topological point of view. In this fashion, we prove that, under suitable hypotheses, such maps are actually homeomorphisms. Applying this result to the mixed displacement-traction problem, the existence of homeomorphic minimizers is shown for nonlinear stored energy functions with suitable properties.