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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2410.22163 |
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| _version_ | 1866917866785210368 |
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| author | Federico, Salvatore Holthausen, Sebastian Husemann, Nina J. Neff, Patrizio |
| author_facet | Federico, Salvatore Holthausen, Sebastian Husemann, Nina J. Neff, Patrizio |
| contents | We recall in this note that the induced tangent stiffness tensor $\mathbb{H}^{\text{ZJ}}_τ(τ)$ appearing in a hypoelastic formulation based on the Zaremba-Jaumann corotational derivative and the rate constitutive equation for the Kirchhoff stress tensor $τ$ is minor and major symmetric if the Kirchhoff stress $τ$ is derived from an elastic potential $\mathrm{W}(F)$. This result is vaguely known in the literature. Here, we expose two different notational approaches which highlight the full symmetry of the tangent stiffness tensor $\mathbb{H}^{\text{ZJ}}_τ(τ)$. The first approach is based on the direct use of the definition of each symmetry (minor and major), i.e., via contractions of the tensor with the deformation rate tensor $D$. The second approach aims at finding an absolute expression of the tensor $\mathbb{H}^{\text{ZJ}}_τ(τ)$, by means of special tensor products and their symmetrisations. In some past works, the major symmetry of $\mathbb{H}^{\text{ZJ}}_τ(τ)$ has been missed because not all necessary symmetrisations were applied. The approach is exemplified for the isotropic Hencky energy. Corresponding stability checks of software packages are shortly discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22163 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Major symmetry of the induced tangent stiffness tensor for the Zaremba-Jaumann rate and Kirchhoff stress in hyperelasticity: two different approaches Federico, Salvatore Holthausen, Sebastian Husemann, Nina J. Neff, Patrizio Analysis of PDEs 74B20 We recall in this note that the induced tangent stiffness tensor $\mathbb{H}^{\text{ZJ}}_τ(τ)$ appearing in a hypoelastic formulation based on the Zaremba-Jaumann corotational derivative and the rate constitutive equation for the Kirchhoff stress tensor $τ$ is minor and major symmetric if the Kirchhoff stress $τ$ is derived from an elastic potential $\mathrm{W}(F)$. This result is vaguely known in the literature. Here, we expose two different notational approaches which highlight the full symmetry of the tangent stiffness tensor $\mathbb{H}^{\text{ZJ}}_τ(τ)$. The first approach is based on the direct use of the definition of each symmetry (minor and major), i.e., via contractions of the tensor with the deformation rate tensor $D$. The second approach aims at finding an absolute expression of the tensor $\mathbb{H}^{\text{ZJ}}_τ(τ)$, by means of special tensor products and their symmetrisations. In some past works, the major symmetry of $\mathbb{H}^{\text{ZJ}}_τ(τ)$ has been missed because not all necessary symmetrisations were applied. The approach is exemplified for the isotropic Hencky energy. Corresponding stability checks of software packages are shortly discussed. |
| title | Major symmetry of the induced tangent stiffness tensor for the Zaremba-Jaumann rate and Kirchhoff stress in hyperelasticity: two different approaches |
| topic | Analysis of PDEs 74B20 |
| url | https://arxiv.org/abs/2410.22163 |