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

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 $W(F)=\hat{W}(F^{T}F)=\hat{W}(C)$ which are convex with respect to the right Cauchy-Green tensor $C=F^{T}F$, where $F$ denotes the gradient of deformation. Examples of such energies exhibiting a blow up for $\det F\to 0$ are given.