On a T$_3$-structure in geometrically linearized elasticity: Qualitative and quantitative analysis and numerical simulations

We study the rigidity properties of the T$_3$-structure for the symmetrized gradient from the work of Bhattacharya, Firoozye, James and Kohn [Restrictions on microstructure, Proc. Roy. Soc. Edinburgh Sect. A124 (1994) 843–878; BFJK] qualitatively, quantitatively and numerically. More precisely, we complement the flexibility result for approximate solutions of the associated differential inclusion which was deduced in BFJK by a rigidity result on the level of exact solutions and by a quantitative rigidity estimate and scaling result. The T3-structure for the symmetrized gradient can hence be regarded as a symmetrized gradient analogue of the Tartar square for the gradient. As such a structure cannot exist in R$^{2×2}_{sym}$ the example from BFJK is in this sense minimal. We complement our theoretical findings with numerical simulations of the resulting microstructure.