On a $T_3$-Structure in Geometrically Linearized Elasticity: Qualitative and Quantitative Analysis and Numerical Simulations

TL;DR

This paper investigates the rigidity and flexibility of the $T_3$-structure in geometrically linearized elasticity, providing new theoretical results and numerical simulations to understand its properties and limitations.

Contribution

It offers a rigidity result for exact solutions, a quantitative estimate, and numerical analysis, extending prior flexibility results for the $T_3$-structure in elasticity.

Findings

Rigidity result for exact solutions

Quantitative rigidity estimate and scaling

Numerical simulations of microstructure

Abstract

We study the rigidity properties of the $T_3$-structure for the symmetrized gradient from \cite{BFJK94} qualitatively, quantitatively and numerically. More precisely, we complement the flexibility result for approximate solutions of the associated differential inclusion which was deduced in \cite{BFJK94} by a rigidity result on the level of exact solutions and by a quantitative rigidity estimate and scaling result. The $T_3$-structure for the symmetrized gradient from \cite{BFJK94} can hence be regarded as a symmetrized gradient analogue of the Tartar square for the gradient. As such a structure cannot exist in $\mathbb{R}^{2\times 2}_{sym}$ the example from \cite{BFJK94} is in this sense minimal. We complement our theoretical findings with numerical simulations of the resulting microstructure.