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

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.