Hypo-elasticity, Cauchy-elasticity, corotational stability and monotonicity in the logarithmic strain
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910625089716224 |
|---|---|
| author | Neff, Patrizio Holthausen, Sebastian d'Agostino, Marco Valerio Bernardini, Davide Sky, Adam Ghiba, Ionel-Dumitrel Martin, Robert J. |
| author_facet | Neff, Patrizio Holthausen, Sebastian d'Agostino, Marco Valerio Bernardini, Davide Sky, Adam Ghiba, Ionel-Dumitrel Martin, Robert J. |
| contents | We combine the rate-formulation for the objective, corotational Zaremba-Jaumann rate
\begin{align}
\frac{{\rm D}^{\rm ZJ}}{{\rm D} t} [σ] = \mathbb{H}^{\rm ZJ}(σ).D, \qquad D = {\rm sym} {\rm D} v\,,
\end{align} operating on the Cauchy stress $σ$, the Eulerian strain rate $D$ and the spatial velocity $v$ with the novel \enquote{corotational stability postulate} (CSP)\begin{equation}
\Bigl\langle \frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[σ], D \Bigr\rangle > 0 \qquad \forall \, D\in{\rm Sym}(3)\setminus\{0\} \end{equation} to show that for a given isotropic Cauchy-elastic constitutive law $B \mapsto σ(B)$ in terms of the left Cauchy-Green tensor $B = F F^T$, the induced fourth-order tangent stiffness tensor $\mathbb{H}^{\rm ZJ}(σ)$ is positive definite if and only if for $\widehatσ(\log B):=σ(B)$, the strong monotonicity condition (TSTS-M$^{++}$) in the logarithmic strain is satisfied. Thus (CSP) implies (TSTS-M^{++}) and vice-versa, and both imply the invertibility of the hypo-elastic material law between the stress and strain rates given by the tensor $\mathbb{H}^{\rm ZJ}(σ)$. The same characterization remains true for the corotational Green-Naghdi rate as well as the corotational logarithmic rate, conferring the corotational stability postulate (CSP) together with the monotonicity in the logarithmic strain tensor (TSTS-M^{++}) a far reaching generality. It is conjectured that this characterization of (CSP) holds for a large class of reasonable corotational rates. The result for the logarithmic rate is based on a novel chain rule for corotational derivatives of isotropic tensor functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_20051 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hypo-elasticity, Cauchy-elasticity, corotational stability and monotonicity in the logarithmic strain Neff, Patrizio Holthausen, Sebastian d'Agostino, Marco Valerio Bernardini, Davide Sky, Adam Ghiba, Ionel-Dumitrel Martin, Robert J. Analysis of PDEs Mathematical Physics 15A24, 73G05, 73G99, 74B20 We combine the rate-formulation for the objective, corotational Zaremba-Jaumann rate \begin{align} \frac{{\rm D}^{\rm ZJ}}{{\rm D} t} [σ] = \mathbb{H}^{\rm ZJ}(σ).D, \qquad D = {\rm sym} {\rm D} v\,, \end{align} operating on the Cauchy stress $σ$, the Eulerian strain rate $D$ and the spatial velocity $v$ with the novel \enquote{corotational stability postulate} (CSP)\begin{equation} \Bigl\langle \frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[σ], D \Bigr\rangle > 0 \qquad \forall \, D\in{\rm Sym}(3)\setminus\{0\} \end{equation} to show that for a given isotropic Cauchy-elastic constitutive law $B \mapsto σ(B)$ in terms of the left Cauchy-Green tensor $B = F F^T$, the induced fourth-order tangent stiffness tensor $\mathbb{H}^{\rm ZJ}(σ)$ is positive definite if and only if for $\widehatσ(\log B):=σ(B)$, the strong monotonicity condition (TSTS-M$^{++}$) in the logarithmic strain is satisfied. Thus (CSP) implies (TSTS-M^{++}) and vice-versa, and both imply the invertibility of the hypo-elastic material law between the stress and strain rates given by the tensor $\mathbb{H}^{\rm ZJ}(σ)$. The same characterization remains true for the corotational Green-Naghdi rate as well as the corotational logarithmic rate, conferring the corotational stability postulate (CSP) together with the monotonicity in the logarithmic strain tensor (TSTS-M^{++}) a far reaching generality. It is conjectured that this characterization of (CSP) holds for a large class of reasonable corotational rates. The result for the logarithmic rate is based on a novel chain rule for corotational derivatives of isotropic tensor functions. |
| title | Hypo-elasticity, Cauchy-elasticity, corotational stability and monotonicity in the logarithmic strain |
| topic | Analysis of PDEs Mathematical Physics 15A24, 73G05, 73G99, 74B20 |
| url | https://arxiv.org/abs/2409.20051 |