Nonlinear elasticity with vanishing nonlocal self-repulsion

TL;DR

This paper demonstrates that nonlinear elastic energies with a vanishing nonlocal self-repulsion term can enforce almost everywhere invertibility of deformations in the limit, with implications for elasticity theory.

Contribution

It introduces a novel approach using a nonlocal self-repulsion term with vanishing coefficient to ensure invertibility in nonlinear elasticity models.

Findings

Almost everywhere invertibility achieved in the Gamma-limit

Nonlocal self-repulsion term coincides with a Sobolev-Slobodecki
21 seminorm

Variants near boundary and on surface analyzed

Abstract

We prove that that for nonlinear elastic energies with strong enough energetic control of the outer distortion of admissible deformations, almost everywhere global invertibility as constraint can be obtained in the $\Gamma$-limit of the elastic energy with an added nonlocal self-repulsion term with asymptocially vanishing coefficient. The self-repulsion term considered here formally coincides with a Sobolev-Slobodecki\u{i} seminorm of the inverse deformation. Variants near the boundary or on the surface of the domain are also studied.