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We consider conformally invariant energies $W$ on the group$\operatorname{GL}^+(2)$ of $2\times2$-matrices with positive determinant, i.e.$W\colon\operatorname{GL}^+(2)\to\mathbb{R}$ 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$\operatorname{SO}(2)$ denotes the special orthogonal group, and provide anexplicit formula for the (notoriously difficult to compute) quasiconvexenvelope 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 inorder to demonstrate the convenience of the eigenvalue-based expressioncompared to the more common representation in terms of the distortion$\mathbb{K}:=\frac12\frac{\lVert F\rVert^2}{\det F}$. Special cases of ourresults can be obtained from earlier works by Astala et al. and Yan.

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