Accepted Paper

Inserted: 2 sep 2024
Last Updated: 29 jan 2026

Journal: Math. Models Methods Appl. Sci.
Year: 2026
Doi: https://doi.org/10.1142/S021820252650017X

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.