Some Implications of Empirical Inequalities for the Plane Strain Deformations of Unconstrained,Isotropic,Hyperelastic Solids

Abstract

A strengthened form of one of the two empirical inequalities in plane strain isotropic elasticity is shown to imply that the restriction of the strain energy function to the class of deformation gradients that shares the same average of the squares of the principal stretches is bounded from below by the strain energy corresponding to the conformal deformation in this class. A weakened version of this minimum property (together with a strengthened form of the pressure-compression condition) is then shown to imply (i) that for zero body forces and tensile prescribed boundary displacements, the radial conformal deformations are globally stable; (ii) that the strain energy function satisfies growth conditions of the type employed in the direct calculus of variations; and (iii) that for zero body forces and prescribed boundary displacements that render a given deformation overall expansive, the total energy of this deformation is bounded from below by a computable constant.