Derivations of two one-dimensional models for transversely curved shallow shells: one leads to relaxation
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| Format: | Preprint |
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2025
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| _version_ | 1866913968345317376 |
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| author | Roberto, Paroni Marco, Picchi Scardaoni |
| author_facet | Roberto, Paroni Marco, Picchi Scardaoni |
| contents | We study the $Γ$-limit of sequences of variational problems for straight, transversely curved shallow shells, as the width of the planform $\varepsilon$ goes to zero.
The energy is of von Kármán type for shallow shells under suitable boundary conditions. What distinguishes the various regimes is the scaling of the stretching energy $\sim \varepsilon^{2β}$, with $β$ a positive number. We derive two one-dimensional models as $β$ ranges in $(0, 2]$. Remarkably, boundary conditions are essential to get compactness.
We show that for $β\in (0, 2)$ the $Γ$-limit leads to relaxation: the limit membrane energy vanishes on compression. For $β=2$ there is no relaxation, and the limit model is a nonlinear energy coupling four kinematical descriptors in a nontrivial way.
As special cases of the latter limit model, a nonlinear Vlasov torsion theory and a nonlinear Euler-Bernoulli beam theory can be deduced. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_23545 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Derivations of two one-dimensional models for transversely curved shallow shells: one leads to relaxation Roberto, Paroni Marco, Picchi Scardaoni Mathematical Physics Analysis of PDEs We study the $Γ$-limit of sequences of variational problems for straight, transversely curved shallow shells, as the width of the planform $\varepsilon$ goes to zero. The energy is of von Kármán type for shallow shells under suitable boundary conditions. What distinguishes the various regimes is the scaling of the stretching energy $\sim \varepsilon^{2β}$, with $β$ a positive number. We derive two one-dimensional models as $β$ ranges in $(0, 2]$. Remarkably, boundary conditions are essential to get compactness. We show that for $β\in (0, 2)$ the $Γ$-limit leads to relaxation: the limit membrane energy vanishes on compression. For $β=2$ there is no relaxation, and the limit model is a nonlinear energy coupling four kinematical descriptors in a nontrivial way. As special cases of the latter limit model, a nonlinear Vlasov torsion theory and a nonlinear Euler-Bernoulli beam theory can be deduced. |
| title | Derivations of two one-dimensional models for transversely curved shallow shells: one leads to relaxation |
| topic | Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2507.23545 |