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arXiv · 2408.13110

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

Roman Indergand·Dennis Kochmann·Angkana Rüland·Antonio Tribuzio·Christian Zillinger
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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.

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Roman Indergand, Dennis Kochmann, Angkana Rüland, Antonio Tribuzio, Christian Zillinger. 2024-08-23. On a $T_3$-Structure in Geometrically Linearized Elasticity: Qualitative and Quantitative Analysis and Numerical Simulations. https://arxiv.org/abs/2408.13110

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