The quasiconvex envelope of conformally invariant planar energy functions in isotropic hyperelasticity

Abstract

We consider conformally invariant energies $W$ on the group ${\mathrm{GL}}^{+}(2)$ of $2\times 2$-matrices with positive determinant, i.e. $W:{\mathrm{GL}}^{+}(2)\to \mathbb{R}$ such that \[W(AFB) = W(F) \qquad\text{for all }\; A,B\in\{aR\in\operatorname{GL}^+(2) \,|\, a\in(0,\infty)\,,\; R\in\operatorname{SO}(2)\}\,,\] where $\mathrm{SO}(2)$ denotes the special orthogonal group, and provide an explicit formula for the (notoriously difficult to compute) quasiconvex envelope of these functions. Our results, which are based on the representation $W(F)=h(\frac{{\lambda }_1}{{\lambda }_2})$ of $W$ in terms of the singular values ${\lambda }_1,{\lambda }_2$ of $F$, are applied to a number of example energies in order to demonstrate the convenience of the eigenvalue-based expression compared to the more common representation in terms of the distortion $\mathbb{K}:=\frac{1}{2}\frac{\Vert F{\Vert }^2}{\det F}$. Special cases of our results can be obtained from earlier works by Astala et al. and Yan.