Non-interpenetration of rods derived by $Γ$-limits
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866912063473844224 |
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| author | Benešová, Barbora Campbell, Daniel Hencl, Stanislav Kružík, Martin |
| author_facet | Benešová, Barbora Campbell, Daniel Hencl, Stanislav Kružík, Martin |
| contents | Ensuring non-interpenetration of matter is a fundamental prerequisite when modeling the deformation response of solid materials. In this contribution, we thoroughly examine how this requirement, equivalent to the injectivity of deformations within bulk structures, manifests itself in dimensional-reduction problems. Specifically, we focus on the case of rods embedded in a two-dimensional plane. Our results focus on $Γ$-limits of energy functionals that enforce an admissible deformation to be a homeomorphism. These $Γ$-limits are evaluated along a passage from the bulk configuration to the rod arrangement. The proofs rely on the equivalence between the weak and strong closures of the set of homeomorphisms from $\mathbb{R}$ to $\mathbb{R}^2$, a result that is of independent interest and that we establish in this paper, too. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05601 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-interpenetration of rods derived by $Γ$-limits Benešová, Barbora Campbell, Daniel Hencl, Stanislav Kružík, Martin Analysis of PDEs 49J45, 35Q74, 74B20 Ensuring non-interpenetration of matter is a fundamental prerequisite when modeling the deformation response of solid materials. In this contribution, we thoroughly examine how this requirement, equivalent to the injectivity of deformations within bulk structures, manifests itself in dimensional-reduction problems. Specifically, we focus on the case of rods embedded in a two-dimensional plane. Our results focus on $Γ$-limits of energy functionals that enforce an admissible deformation to be a homeomorphism. These $Γ$-limits are evaluated along a passage from the bulk configuration to the rod arrangement. The proofs rely on the equivalence between the weak and strong closures of the set of homeomorphisms from $\mathbb{R}$ to $\mathbb{R}^2$, a result that is of independent interest and that we establish in this paper, too. |
| title | Non-interpenetration of rods derived by $Γ$-limits |
| topic | Analysis of PDEs 49J45, 35Q74, 74B20 |
| url | https://arxiv.org/abs/2402.05601 |