Some observations on variational elasticity and its application to plates and membranes

TL;DR

This paper examines variational elasticity in thin plates and membranes, highlighting issues with common energy formulations, and proposes a new approach based on Biot strains to improve modeling accuracy.

Contribution

It identifies ambiguities in traditional energy definitions and introduces a novel theory using Biot strains for better elastic modeling of plates.

Findings

Common bending energies have undesirable features

A new theory based on Biot strains is proposed

Divergence form of equations is emphasized

Abstract

Energies and equilibrium equations for thin elastic plates are discussed, with emphasis on several issues pertinent to recent approaches in soft condensed matter. Consequences of choice of basis, choice of invariant strain measures, and of approximating material energies as purely geometric in nature, are detailed. Ambiguities in the definitions of energies based on small-strain expansions, and in a typical informal process of dimensional reduction, are noted. A simple example serves to demonstrate that a commonly used bending energy has undesirable features, and it is suggested that a new theory based on Biot strains be developed. A compact form of the variation of a plate energy is presented. Throughout, the divergence form of equations is emphasized. An appendix relates the naive approach adopted in the main text with standard quantities in continuum mechanics.