Rank-one convexity vs. ellipticity for isotropic functions

Abstract

It is well known that a twice-differentiable real-valued function $W:{\mathrm{GL}}^{+}(n)\to \mathbb{R}$ on the group ${\mathrm{GL}}^{+}(n)$ of invertible $n\times n-$matrices with positive determinant is rank-one convex if and only if it is Legendre-Hadamard elliptic. Many energy functions arising from interesting applications in isotropic nonlinear elasticity, however, are not necessarily twice differentiable everywhere on ${\mathrm{GL}}^{+}(n)$, especially at points with non-simple singular values. Here, we show that if an isotropic function $W$ on ${\mathrm{GL}}^{+}(n)$ is twice differentiable at each $F\in {\mathrm{GL}}^{+}(n)$ with simple singular values and Legendre-Hadamard elliptic at each such $F$, then $W$ is already rank-one convex under strongly reduced regularity assumptions. In particular, this generalization makes (local) ellipticity criteria accessible as criteria for (global) rank-one convexity to a wider class of elastic energy potentials expressed in terms of ordered singular values. Our results are also directly applicable to so-called conformally invariant energy functions. We also discuss a classical ellipticity criterion for the planar case by Knowles and Sternberg which has often been used in the literature as a criterion for global rank-one convexity and show that for this purpose, it is still applicable under weakened regularity assumptions.

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Cited by in corpus (2)

• Polyconvex anisotropic hyperelasticity with neural networks
• Morrey's conjecture for the planar volumetric-isochoric split. Part I: least convex energy functions