Energy Minimizing Configurations for Single-Director Cosserat Shells

Abstract

We propose a model for nonlinearly elastic membranes and consider deformations confined to a regular frictionless surface in R 3 . This is a physically correct model of the sometimes crudely stated motivation for harmonic maps between manifolds. The membrane energy function is convex in the strain pair comprising the deformation gradient and the local area ratio. If the target surface is a plane, the problem reduces to 2-dimensional, polyconvex nonlinear elasticity. On the other hand, the energy density is not rank-one convex for unconstrained deformations into R 3 . We show that the problem admits an energy-minimizing configuration when constrained to lie on the given surface. For a class of Dirichlet-placement problems, we demonstrate that the minimizing deformation is a homeomorphism onto its image and establish the weak Eulerian form of the equilibrium equations.