The biharmonic optimal support problem
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arXiv
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| Format: | Preprint |
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2024
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| author | Lemenant, Antoine Pakzad, Mohammad Reza |
| author_facet | Lemenant, Antoine Pakzad, Mohammad Reza |
| contents | We establish a $Γ$-convergence result for $h\to 0$ of a thin nonlinearly elastic 3D-plate of thickness $h>0$ which is assumed to be glued to a support region in the 2D-plane $x_3=0$ over the $h$-2D-neighborhood of a given closed set $K$. In the regime of very small vertical forces we identify the $Γ$-limit as being the bi-harmonic energy, with Dirichlet condition on the gluing region $K$, following a general strategy by Friesecke, James, and Müller that we have to adapt in presence of the glued region. Then we introduce a shape optimization problem that we call "optimal support problem" and which aims to find the best glued plate. In this problem the bi-harmonic energy is optimized among all possible glued regions $K$ that we assume to be connected and for which we penalize the length. By relating the dual problem with Griffith almost-minimizers, we are able to prove that any minimizer is $C^{1,α}$ regular outside a set of Hausdorff dimension strictly less then one. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_00689 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The biharmonic optimal support problem Lemenant, Antoine Pakzad, Mohammad Reza Analysis of PDEs Functional Analysis 49Q10 (primary), 74B20 (secondary), 74K20 (secondary), 35Q74 (secondary) We establish a $Γ$-convergence result for $h\to 0$ of a thin nonlinearly elastic 3D-plate of thickness $h>0$ which is assumed to be glued to a support region in the 2D-plane $x_3=0$ over the $h$-2D-neighborhood of a given closed set $K$. In the regime of very small vertical forces we identify the $Γ$-limit as being the bi-harmonic energy, with Dirichlet condition on the gluing region $K$, following a general strategy by Friesecke, James, and Müller that we have to adapt in presence of the glued region. Then we introduce a shape optimization problem that we call "optimal support problem" and which aims to find the best glued plate. In this problem the bi-harmonic energy is optimized among all possible glued regions $K$ that we assume to be connected and for which we penalize the length. By relating the dual problem with Griffith almost-minimizers, we are able to prove that any minimizer is $C^{1,α}$ regular outside a set of Hausdorff dimension strictly less then one. |
| title | The biharmonic optimal support problem |
| topic | Analysis of PDEs Functional Analysis 49Q10 (primary), 74B20 (secondary), 74K20 (secondary), 35Q74 (secondary) |
| url | https://arxiv.org/abs/2404.00689 |