A natural requirement for objective corotational rates -- on structure preserving corotational rates
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909329196580864 |
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| author | Neff, Patrizio Holthause, Sebastian Korobeynikov, Sergey N. Ghiba, Ionel-Dumitrel Martin, Robert J. |
| author_facet | Neff, Patrizio Holthause, Sebastian Korobeynikov, Sergey N. Ghiba, Ionel-Dumitrel Martin, Robert J. |
| contents | We investigate objective corotational rates satisfying an additional, physically plausible assumption. More precisely, we require for \begin{equation*} \frac{{\rm D}^{\circ}}{{\rm D} t}[B] = \mathbb{A}^{\circ}(B).D \end{equation*} that $\mathbb{A}^{\circ}(B)$ is positive definite. Here, $B = F \, F^T$ is the left Cauchy-Green tensor, $\frac{{\rm D}^{\circ}}{{\rm D}t}$ is a specific objective corotational rate, $D = {\rm sym} \, {\rm D} v$ is the Eulerian stretching and $\mathbb{A}^{\circ}(B)$ is the corresponding induced fourth order tangent stiffness tensor. Well known corotational rates like the Zaremba-Jaumann rate, the Green-Naghdi rate and the logarithmic rate belong to this family of ``positive'' corotational rates. For general objective corotational rates $\frac{{\rm D}^{\circ}}{{\rm D} t}$ we determine several conditions characterizing positivity. Among them an explicit condition on the material spin-functions of Xiao, Bruhns and Meyers (2004). We also give a geometrical motivation for invertibility and positivity and highlight the structure preserving properties of corotational rates that distinguish them from more general objective stress rates. Applications of this novel concept are indicated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19707 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A natural requirement for objective corotational rates -- on structure preserving corotational rates Neff, Patrizio Holthause, Sebastian Korobeynikov, Sergey N. Ghiba, Ionel-Dumitrel Martin, Robert J. Analysis of PDEs Mathematical Physics 15A24, 73G05, 73G99, 74B20 We investigate objective corotational rates satisfying an additional, physically plausible assumption. More precisely, we require for \begin{equation*} \frac{{\rm D}^{\circ}}{{\rm D} t}[B] = \mathbb{A}^{\circ}(B).D \end{equation*} that $\mathbb{A}^{\circ}(B)$ is positive definite. Here, $B = F \, F^T$ is the left Cauchy-Green tensor, $\frac{{\rm D}^{\circ}}{{\rm D}t}$ is a specific objective corotational rate, $D = {\rm sym} \, {\rm D} v$ is the Eulerian stretching and $\mathbb{A}^{\circ}(B)$ is the corresponding induced fourth order tangent stiffness tensor. Well known corotational rates like the Zaremba-Jaumann rate, the Green-Naghdi rate and the logarithmic rate belong to this family of ``positive'' corotational rates. For general objective corotational rates $\frac{{\rm D}^{\circ}}{{\rm D} t}$ we determine several conditions characterizing positivity. Among them an explicit condition on the material spin-functions of Xiao, Bruhns and Meyers (2004). We also give a geometrical motivation for invertibility and positivity and highlight the structure preserving properties of corotational rates that distinguish them from more general objective stress rates. Applications of this novel concept are indicated. |
| title | A natural requirement for objective corotational rates -- on structure preserving corotational rates |
| topic | Analysis of PDEs Mathematical Physics 15A24, 73G05, 73G99, 74B20 |
| url | https://arxiv.org/abs/2409.19707 |