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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2408.13110 |
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| _version_ | 1866916367675949056 |
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| author | Indergand, Roman Kochmann, Dennis Rüland, Angkana Tribuzio, Antonio Zillinger, Christian |
| author_facet | Indergand, Roman Kochmann, Dennis Rüland, Angkana Tribuzio, Antonio Zillinger, Christian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_13110 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a $T_3$-Structure in Geometrically Linearized Elasticity: Qualitative and Quantitative Analysis and Numerical Simulations Indergand, Roman Kochmann, Dennis Rüland, Angkana Tribuzio, Antonio Zillinger, Christian Analysis of PDEs 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. |
| title | On a $T_3$-Structure in Geometrically Linearized Elasticity: Qualitative and Quantitative Analysis and Numerical Simulations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.13110 |