Existence of Weak Solutions for Non-Simple Elastic Surface Models

TL;DR

This paper proves the existence of weak solutions for a class of nonlinear elastic surface models that account for large deformations, bending, and thickness changes, using polyconvex energy assumptions.

Contribution

It establishes the existence of weak solutions for non-simple elastic surface models with polyconvex energy depending on the second gradient of deformation.

Findings

Weak solutions exist under certain growth conditions.

The model accounts for finite strains and thickness changes.

Weak form of equilibrium equations derived rigorously.

Abstract

We consider a class of models for nonlinearly elastic surfaces in this work. We have in mind thin, highly deformable structures modeled directly as two-dimensional nonlinearly elastic continua, accounting for finite membrane and bending strains and thickness change. We assume that the stored-energy density is polyconvex with respect to the second gradient of the deformation, and we require that it grow unboundedly as the local area ratio approaches zero. For sufficiently fast growth, we show that the latter is uniformly bounded away from zero at an energy minimizer. With this in hand, we rigorously derive the weak form of the Euler-Lagrange equilibrium equations.