Nonlinearly Elastic Maps: Energy Minimizing Configurations of Membranes on Prescribed Surfaces

TL;DR

This paper introduces a physically accurate model for nonlinear elastic membranes constrained to surfaces, proving existence of energy-minimizing configurations and analyzing their properties, including homeomorphism and equilibrium conditions.

Contribution

It develops a new convex energy density model for membranes on surfaces and proves the existence and properties of energy-minimizing configurations under constraints.

Findings

Existence of energy-minimizing configurations on prescribed surfaces.

Minimizers are homeomorphisms onto their images.

Weak Eulerian equilibrium equations are established.

Abstract

We propose a model for nonlinearly elastic membranes undergoing finite deformations while confined to a regular frictionless surface in $\mathbb{R}^3$. This is a physically correct model of the analogy sometimes given to motivate harmonic maps between manifolds. The proposed energy density 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 addressed by J.M. Ball. On the other hand, the energy density is not rank-one convex for unconstrained deformations into $\mathbb{R}^3$. We show that the problem admits an energy-minimizing configuration when constrained to lie on the given surface. For a class of Dirichlet problems, we demonstrate that the minimizing deformation is a homeomorphism onto its image on the given surface and establish the weak Eulerian form of the equilibrium equations.