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

TL;DR

This paper derives the quasiconvex relaxation of a Biot-type energy for planar mappings, comparing cases with and without determinant constraints, and validates findings through numerical methods including neural networks and microstructure simulations.

Contribution

It introduces the first explicit quasiconvex relaxation formulas for Biot-type energies in planar mappings, highlighting differences caused by determinant constraints.

Findings

Relaxations differ with and without determinant constraints.

Numerical methods validate analytical relaxation formulas.

Comparison of relaxation approaches reveals their effectiveness.

Abstract

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.