# Quasiconvex relaxation of isotropic functions in incompressible planar hyperelasticity

## TL;DR

This paper derives an explicit formula for the quasiconvex envelope of isotropic functions on SL(2), aiding the analysis of incompressible planar hyperelastic materials by connecting relaxation techniques with isotropic energy properties.

## Contribution

It provides a novel explicit formula for quasiconvex envelopes of isotropic functions on SL(2), combining previous relaxation results with recent equivalence findings in planar elasticity.

## Key findings

- Explicit quasiconvex envelope formula for isotropic functions on SL(2)
- Connection between polyconvexity and rank-one convexity in planar elasticity
- Enhanced understanding of energy relaxation in incompressible hyperelasticity

## Abstract

In this note, we provide an explicit formula for computing the quasiconvex envelope 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 special orthogonal group, respectively. In order to obtain our result, we combine earlier work by Dacorogna and Koshigoe on the relaxation of certain conformal planar energy functions with a recent result on the equivalence between polyconvexity and rank-one convexity for objective and isotropic energies in planar incompressible nonlinear elasticity.