Graphical Representations of the Regions of Rank-One-Convexity of some Strain Energies

Abstract

Isotropic elastic energies which are quadratic in the strain measures of the Seth family are known not to be rankone-convex in the entire domain of invertible deformation gradients with positive determinant. Therefore, they are in principle capable of displaying a laminated microstructure. Nevertheless, they are commonly used for standard elastic solids. In general one does not observe a microstructure evolution due to the fact that the solution is not sought outside of the region of rank-one-convexity. Consequently, the question for the boundaries of the region of rank-one-convexity arises. We address this question by applying a set of necessary and sufficient conditions for rank-one-convexity to the mentioned elastic energies, and give graphical representations for the regions of rank-one-convexity