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In this note, we provide an explicit formula for computing the quasiconvexenvelope of any real-valued function $W\colon\operatorname{SL}(2)\to\mathbb{R}$with $W(RF)=W(FR)=W(F)$ for all $F\in\operatorname{SL}(2)$ and all$R\in\operatorname{SO}(2)$, where $\operatorname{SL}(2)$ and$\operatorname{SO}(2)$ denote the special linear group and the specialorthogonal group, respectively. In order to obtain our result, we combineearlier work by Dacorogna and Koshigoe on the relaxation of certain conformalplanar energy functions with a recent result on the equivalence betweenpolyconvexity and rank-one convexity for objective and isotropic energies inplanar incompressible nonlinear elasticity.

Quasiconvex relaxation of isotropic functions in incompressible planar hyperelasticity