On duality principles for one and three-dimensional non-linear models in elasticity

TL;DR

This paper develops duality principles for non-linear elasticity models in one and three dimensions, demonstrating no duality gap and applicability to broader elasticity problems.

Contribution

It introduces duality principles for non-linear elasticity variational problems and proves the absence of duality gap in these models.

Findings

Duality principles established for 1D and 3D models

Proof of no duality gap in local extremal cases

Applicability to broader elasticity models

Abstract

In this article, we develop duality principles applicable to primal variational formulations found in the non-linear elasticity theory. As a first application, we establish the concerning results in details for one and three-dimensional models. We emphasize such duality principles are applicable to a larger class of variational optimization problems, such as non-linear models of plates and shells and other models in elasticity. Finally, we formally prove there is no duality gap between the primal and dual formulations, in a local extremal context.