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

TL;DR

This paper derives an explicit formula for the quasiconvex envelope of conformally invariant energies in 2D hyperelasticity, simplifying the analysis of such energies using eigenvalues instead of distortion measures.

Contribution

It provides a novel explicit formula for the quasiconvex envelope of conformally invariant energies in 2D, based on singular values, improving upon previous eigenvalue-based representations.

Findings

Explicit quasiconvex envelope formula for conformally invariant energies

Eigenvalue-based expression simplifies analysis of these energies

Connections to earlier work by Astala et al. and Yan

Abstract

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 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 $\mathbb{K}:=\frac12\frac{\lVert F\rVert^2}{\det F}$. Special cases of our results can be obtained from earlier works by Astala et al. and Yan.