A relaxation approach to the minimisation of the neo-Hookean energy in 3D

TL;DR

This paper introduces a relaxation method for the neo-Hookean energy in 3D elasticity, enlarging the minimisation space and adding a penalty term to ensure the existence of minimisers.

Contribution

It proposes a modified energy functional that guarantees minimisers exist by penalising singularities, transforming the original existence problem into a regularity question.

Findings

The modified energy attains a minimum in the larger space.

The approach links minimiser existence to regularity of the new energy.

Provides a framework for analyzing neo-Hookean energy minimisers.

Abstract

Despite its high significance in nonlinear elasticity, the neo-Hookean energy is still not known to admit minimisers in some appropriate admissible class. Using ideas from relaxation theory, we propose a larger minimisation space and a modified functional that coincides with the neo-Hookean energy on the original space. This modified energy is the sum of the neo-Hookean energy and a term penalising the singularities of the inverse deformation. The new functional attains its minimum in the larger space, so the initial question of existence of minimisers of the neo-Hookean energy is thus transformed into a question of regularity of minimisers of this new energy.