The Modeling of Deformable Bodies with Frictionless (Self-)Contacts

Abstract

We propose a mathematical model for m-dimensional deformable bodies moving in $\mathbb {R}^n$ , that allows for frictionless contacts or self-contacts while forbidding transversal (self-)intersection. To this end, a topological constraint is imposed to the set of admissible deformations. We restrict our analysis to the static case (although the dynamic case is briefly addressed at the end of the article). In this case, no transversal self-intersection can occur as long as 2m < n, so our modeling is mainly designed to handle the case $2m \geqq n$ . For nonlinear hyperelastic bodies, we prove the existence of at least one minimizer of the energy on the set of admissible deformations, under suitable assumptions on the stored energy function. Moreover, for certain choices of m and n, under regularity assumptions on the minimizers, the solutions of the minimization problem satisfy Euler–Lagrange equations.