Mission $p<n-1$: Possible -- Nonlinear Elasticity Beyond Conventional Limits

TL;DR

This paper establishes the mathematical foundation for nonlinear elasticity models allowing for lower regularity deformations, including cavitations, by proving lower semicontinuity of associated energies.

Contribution

It proves lower semicontinuity of a Neohookean-type energy for models with $p<n-1$, extending elasticity theory beyond conventional limits.

Findings

Proved lower semicontinuity of the energy functional for $p<n-1$.

Developed a model incorporating cavitations with surface measure.

Extended mathematical understanding of nonlinear elasticity with weak limits.

Abstract

In this paper we prove the lower semicontinuity of a Neohookean-type energy for a model of Nonlinear Elasticity that allows, for the first time, for $p<n-1$. Our class of admissible deformations consists of weak limits of Sobolev $W^{1,p}$ homeomorphisms. We also introduce a model that allows for cavitations by studying weak limits of homeomorphisms that can open cavities at some points. In this model we add the measure of the created surface to the energy functional and for this functional we again prove lower semicontinuity.