Surface penalization of self-interpenetration in linear and nonlinear elasticity

TL;DR

This paper introduces a surface penalization method for hyperelastic models to prevent self-interpenetration, ensuring deformations are almost everywhere invertible, with convergence proofs and numerical validation in 3D contact scenarios.

Contribution

It proposes a novel surface penalization approach that approximates the Ciarlet-Nečas condition in both linear and nonlinear elasticity models, including non-simple materials.

Findings

Penalized functionals converge to the original constrained functional.

Method effectively prevents self-interpenetration in numerical simulations.

Applicable to complex 3D self-contact problems.

Abstract

We analyze a term penalizing surface self-penetration, as a soft constraint for models of hyperelastic materials to approximate the Ciarlet-Ne\v{c}as condition (almost everywhere global invertibility of deformations). For a linear elastic energy subject to an additional local invertibility constraint, we prove that the penalized elastic functionals converge to the original functional subject to the Ciarlet-Ne\v{c}as condition. The approach also works for nonlinear models of non-simple materials including a suitable higher order term in the elastic energy, without artificial local constraints. Numerical experiments illustrate our results for a self-contact problem in 3d.