On the convexity of nonlinear elastic energies in the right Cauchy-Green tensor

TL;DR

This paper establishes a sufficient condition for weak solutions in nonlinear elasticity to be global minimizers, focusing on energies convex in the right Cauchy-Green tensor, with examples including energies that blow up as the determinant approaches zero.

Contribution

It provides a new criterion ensuring weak solutions are global minimizers for energies convex in the right Cauchy-Green tensor, expanding understanding of energy minimization in nonlinear elasticity.

Findings

Criterion for global minimality of weak solutions.

Applicability to energies convex in the right Cauchy-Green tensor.

Examples with blow-up behavior as det F approaches zero.

Abstract

We present a sufficient condition under which a weak solution of the Euler-Lagrange equations in nonlinear elasticity is already a global minimizer of the corresponding elastic energy functional. This criterion is applicable to energies $W(F)=\widehat{W}(F^TF)=\widehat{W}(C)$ which are convex with respect to the right Cauchy-Green tensor $C=F^TF$, where $F$ denotes the gradient of deformation. Examples of such energies exhibiting a blow up for $\det F\to0$ are given.