Quasiconvex relaxation of planar Biot-type energies and the role of determinant constraints

We derive the quasiconvex relaxation of the Biot-type energy density $\lVert\sqrt{\operatorname{D}\varphi^T \operatorname{D}\varphi}-I_2\rVert^2$ for planar mappings $\varphi\colon\mathbb{R}^2\to \mathbb{R}^2$ in two different scenarios. First, we consider the case $\operatorname{D}\varphi\in\textrm{GL}^+(2)$, in which the energy can be expressed as the squared Euclidean distance $\operatorname{dist}^2(\operatorname{D}\varphi,\textrm{SO}(2))$ to the special orthogonal group $\textrm{SO}(2)$. We then allow for planar mappings with arbitrary $\operatorname{D}\varphi\in\mathbb{R}^{2\times 2}$; in the context of solid mechanics, this lack of determinant constraints on the deformation gradient would allow for self-interpenetration of matter. We demonstrate that the two resulting relaxations do not coincide and compare the analytical findings to numerical results for different relaxation approaches, including a rank-one sequential lamination algorithm, trust-region FEM calculations of representative microstructures and physics-informed neural networks.