A Polyconvex Integrand; Euler–Lagrange Equations and Uniqueness of Equilibrium

Abstract

In this manuscript we are interested in stored energy functionals W defined on the set of d × d matrices, which not only fail to be convex but satisfy ${{\rm lim}_{\det \xi \rightarrow 0^+} W(\xi)=\infty.}$ We initiate a study which we hope will lead to a theory for the existence and uniqueness of minimizers of functionals of the form ${E(\mathbf{u})=\int_\Omega (W(\nabla \mathbf{u}) -\mathbf{F} \cdot \mathbf{u}) {\rm d}x}$ , as well as their Euler–Lagrange equations. The techniques developed here can be applied to a class of functionals larger than those considered in this manuscript, although we keep our focus on polyconvex stored energy functionals of the form ${W(\xi)=f(\xi) +h( {\rm det} \xi)}$ – such that ${{\rm lim}_{t \rightarrow 0^+} h(t)=\infty}$ – which appear in the study of Ogden material. We present a collection of perturbed and relaxed problems for which we prove uniqueness results. Then, we characterize these minimizers by their Euler–Lagrange equations.