Sharp rank-one convexity conditions in planar isotropic elasticity for the additive volumetric-isochoric split

TL;DR

This paper analyzes rank-one convexity conditions in planar isotropic hyperelasticity with volumetric-isochoric split, reducing complex criteria to simpler differential inequalities and classifying energies based on convexity of a scalar function.

Contribution

It provides a simplified classification of rank-one convexity for planar isotropic hyperelastic energies with volumetric-isochoric split, extending classical criteria.

Findings

Rank-one convexity reduces to one-dimensional differential inequalities.

Energy of the form W(F)=μ/2 * ||F||^2 / det F + f(det F) is rank-one convex iff f is convex.

Legendre-Hadamard ellipticity simplifies in the volumetric-isochoric split context.

Abstract

We consider the volumetric-isochoric split in planar isotropic hyperelasticity and give a precise analysis of rank-one convexity criteria for this case, showing that the Legendre-Hadamard ellipticity condition separates and simplifies in a suitable sense. Starting from the classical two-dimensional criterion by Knowles and Sternberg, we can reduce the conditions for rank-one convexity to a family of one-dimensional coupled differential inequalities. In particular, this allows us to derive a simple rank-one convexity classification for generalized Hadamard energies of the type $W(F)=\frac{\mu}{2}\frac{\lVert F\rVert^2}{\det F}+f(\det F)$; such an energy is rank-one convex if and only if the function $f$ is convex.