Energy Minimizing Configurations for Single-Director Cosserat Shells

TL;DR

This paper develops a mathematical framework for energy minimization in single-director Cosserat shell models, incorporating curvature and strains, and proves the existence of minimizers under certain conditions.

Contribution

It introduces a new existence proof for minimizers in Cosserat shell models with polyconvex energy functions, including constrained variants.

Findings

Existence of energy minimizers established under growth conditions.

Energy blowup ensures local orientation stability.

Results applicable to common constrained shell models.

Abstract

We consider a class of single-director Cosserat shell models accounting for both curvature and finite mid-plane strains. We assume a polyconvexity condition for the stored-energy function that reduces to a physically correct membrane model in the absence of bending. With appropriate growth conditions, we establish the existence of energy minimizers. The local orientation of a minimizing configuration is maintained via the blowup of the stored energy as a version of the local volume ratio approaches zero. Finally, we specialize our results to three constrained versions of the theory commonly employed in the subject.