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

Abstract

We derive the quasiconvex relaxation of the Biot-type energy density $\Vert \sqrt{D\ddot{I}{†}^{T}D\ddot{I}†}-I_2{\Vert }^2$ for planar mappings $\ddot{I}†:{\mathbb{R}}^2\to {\mathbb{R}}^2$ in two different scenarios. First, we consider the case $D\ddot{I}†\in {\text{GL}}^{+}(2)$, in which the energy can be expressed as the squared Euclidean distance ${\mathrm{dist}}^2(D\ddot{I}†,\text{SO}(2))$ to the special orthogonal group $\text{SO}(2)$. We then allow for planar mappings with arbitrary $D\ddot{I}†\in {\mathbb{R}}^{2\times 2}$; 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.