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

TL;DR

The paper analyzes the quasiconvex relaxation of planar Biot-type energies, focusing on the determinant constraint by contrasting relaxations on $\mathrm{GL}^+(2)$ with those on the full matrix space $\mathbb{R}^{2\times2}$. It derives explicit envelopes: for $W_{\text{Biot}}$ on $\mathbb{R}^{2\times2}$, $QW_{\text{Biot}}(F)=\sum_{k=1}^2 [\lambda_k-1]_+^2$, and for $W_{\text{dist}}(F)=\dist^2(F,\mathrm{SO}(2))$ on $\mathbb{R}^{2\times2}$, a Brighi-based piecewise formula in terms of $q(F)=\|F\|^2+2\det F$; on $\mathrm{GL}^+(2)$ the constrained envelope equals the unconstrained envelope of the distance form. The study shows that the two relaxations do not coincide in general and validates these analytic results with numerical relaxations via finite elements, physics-informed neural networks, and rank-one lamination methods, highlighting the crucial role of the determinant constraint in obtaining correct energy landscapes and microstructures. The findings provide explicit, computable envelopes for planar Biot-type energies and demonstrate how determinant constraints shape stability and microstructure outcomes in nonlinear elasticity. Overall, the work clarifies the relationship between Biot-type measures, distance-based energies, and their relaxations, offering practical guidance for selecting relaxation approaches in simulations and for understanding material stability in the plane.

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.

Figures (18)

• Figure 1: Comparison between the energies $W_{\subtext{dist}}(x)=\dist^2(x,\SO(1))=\dist^2(x,{1})=\abs{x-1}^2$ (black) and $W_{\subtext{Biot}}(x)=\bigabs{\abs{x}-1}^2$ (red) in the one-dimensional case.
• Figure 2: Left: Energy behaviour of $W_{\subtext{dist}}$ and its relaxation for volumetric deformations, with the mapping $\alpha\mapsto W_{\subtext{dist}}(\alpha\cdot \id_2) = \dist^2(\alpha\cdot\id_2,\SO(2))$ shown in red vs. the relaxed energy mapping $\alpha\mapsto QW_{\subtext{dist}}(\alpha\cdot \id_2)$ in blue. Right: For simple shear, the mapping $\gamma\mapsto W_{\subtext{dist}}(\matr{1\gamma\\01})$ is equal to the relaxed mapping $\gamma\mapsto QW_{\subtext{dist}}(\matr{1\gamma\\01})$.
• Figure 3: $\beta=0$
• Figure 4: $\beta=1$
• Figure 5: $\beta=\frac{1}{4}$

Theorems & Definitions (22)

• definition 1
• theorem 1: Brighi97
• proposition 1
• proof
• proposition 2
• proof
• corollary 1
• proof
• proposition 3
• proof