On Exponents of Thickness in Geometry Rigidity Inequality for Shells
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910890172874752 |
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| author | Chen, Liang-Biao Yao, Peng-Fei |
| author_facet | Chen, Liang-Biao Yao, Peng-Fei |
| contents | We study exponents of thickness in Frieseck-James-Müller's inequalities for shells. We derive the following results: (a) the exponent of thickness $μ(S)\leq15/8$ if the middle surface $S$ is parabolic; (b) the exponent of thickness $μ(S)\leq11/6$ if the middle surface $S$ is a minimal surface with negative curvature; (c) the exponent of thickness $μ(S)\leq11/6$ if the middle surface $S$ is a ruled surface with negative curvature. The exponents of thickness in Frieseck-James-Müller's inequalities for thin shells represent the relationship between rigidity and thickness $h$ of a shell when the large deformations take place, i. e., the rigidity of the shell related to the thickness $h$ is $$Ch^{μ(S)}.$$ Thus the above results of $μ(S)<2$ show that those shells are strictly more rigid than plates since $μ(S)=2$ for plates. Moreover, we present another result which shows that when $μ(S)<2,$ any $W^{2,2}$ isometry of the middle surface is rigid. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_18411 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Exponents of Thickness in Geometry Rigidity Inequality for Shells Chen, Liang-Biao Yao, Peng-Fei Analysis of PDEs 74K20(primary), 74B20(secondary) We study exponents of thickness in Frieseck-James-Müller's inequalities for shells. We derive the following results: (a) the exponent of thickness $μ(S)\leq15/8$ if the middle surface $S$ is parabolic; (b) the exponent of thickness $μ(S)\leq11/6$ if the middle surface $S$ is a minimal surface with negative curvature; (c) the exponent of thickness $μ(S)\leq11/6$ if the middle surface $S$ is a ruled surface with negative curvature. The exponents of thickness in Frieseck-James-Müller's inequalities for thin shells represent the relationship between rigidity and thickness $h$ of a shell when the large deformations take place, i. e., the rigidity of the shell related to the thickness $h$ is $$Ch^{μ(S)}.$$ Thus the above results of $μ(S)<2$ show that those shells are strictly more rigid than plates since $μ(S)=2$ for plates. Moreover, we present another result which shows that when $μ(S)<2,$ any $W^{2,2}$ isometry of the middle surface is rigid. |
| title | On Exponents of Thickness in Geometry Rigidity Inequality for Shells |
| topic | Analysis of PDEs 74K20(primary), 74B20(secondary) |
| url | https://arxiv.org/abs/2503.18411 |