Quasiconvex relaxation of planar Biot-type energies and the role of determinant constraints
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| Format: | Preprint |
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2025
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| author | Martin, Robert J. Ghiba, Ionel-Dumitrel Köhler, Maximilian Balzani, Daniel Sander, Oliver Neff, Patrizio |
| author_facet | Martin, Robert J. Ghiba, Ionel-Dumitrel Köhler, Maximilian Balzani, Daniel Sander, Oliver Neff, Patrizio |
| contents | We derive the quasiconvex relaxation of the Biot-type energy density $\lVert\sqrt{\operatorname{D}φ^T \operatorname{D}φ}-I_2\rVert^2$ for planar mappings $φ\colon\mathbb{R}^2\to \mathbb{R}^2$ in two different scenarios. First, we consider the case $\operatorname{D}φ\in\textrm{GL}^+(2)$, in which the energy can be expressed as the squared Euclidean distance $\operatorname{dist}^2(\operatorname{D}φ,\textrm{SO}(2))$ to the special orthogonal group $\textrm{SO}(2)$. We then allow for planar mappings with arbitrary $\operatorname{D}φ\in\mathbb{R}^{2\times 2}$; in the context of solid mechanics, this lack of determinant constraints on the deformation gradient would allow for self-interpenetration of matter. We demonstrate that the two resulting relaxations do not coincide and compare the analytical findings to numerical results for different relaxation approaches, including a rank-one sequential lamination algorithm, trust-region FEM calculations of representative microstructures and physics-informed neural networks. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_10853 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasiconvex relaxation of planar Biot-type energies and the role of determinant constraints Martin, Robert J. Ghiba, Ionel-Dumitrel Köhler, Maximilian Balzani, Daniel Sander, Oliver Neff, Patrizio Analysis of PDEs Mathematical Physics 74A05, 74A60, 74B20, 74G65 We derive the quasiconvex relaxation of the Biot-type energy density $\lVert\sqrt{\operatorname{D}φ^T \operatorname{D}φ}-I_2\rVert^2$ for planar mappings $φ\colon\mathbb{R}^2\to \mathbb{R}^2$ in two different scenarios. First, we consider the case $\operatorname{D}φ\in\textrm{GL}^+(2)$, in which the energy can be expressed as the squared Euclidean distance $\operatorname{dist}^2(\operatorname{D}φ,\textrm{SO}(2))$ to the special orthogonal group $\textrm{SO}(2)$. We then allow for planar mappings with arbitrary $\operatorname{D}φ\in\mathbb{R}^{2\times 2}$; in the context of solid mechanics, this lack of determinant constraints on the deformation gradient would allow for self-interpenetration of matter. We demonstrate that the two resulting relaxations do not coincide and compare the analytical findings to numerical results for different relaxation approaches, including a rank-one sequential lamination algorithm, trust-region FEM calculations of representative microstructures and physics-informed neural networks. |
| title | Quasiconvex relaxation of planar Biot-type energies and the role of determinant constraints |
| topic | Analysis of PDEs Mathematical Physics 74A05, 74A60, 74B20, 74G65 |
| url | https://arxiv.org/abs/2501.10853 |