A constitutive condition for idealized isotropic Cauchy elasticity involving the logarithmic strain
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| Format: | Preprint |
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2024
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| _version_ | 1866910588431499264 |
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| author | d'Agostino, Marco Valerio Holthausen, Sebastian Bernardini, Davide Sky, Adam Ghiba, Ionel-Dumitrel Martin, Robert J. Neff, Patrizio |
| author_facet | d'Agostino, Marco Valerio Holthausen, Sebastian Bernardini, Davide Sky, Adam Ghiba, Ionel-Dumitrel Martin, Robert J. Neff, Patrizio |
| contents | Following Hill and Leblond, the aim of our work is to show, for isotropic nonlinear elasticity, a relation between the corotational Zaremba-Jaumann objective derivative of the Cauchy stress $σ$, i.e.
\begin{equation}
\frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[σ] = \frac{\rm d}{{\rm d}{t}}[σ] - W \, σ+ σ\, W, \qquad W = {\rm skew}(\dot F \, F^{-1})
\end{equation} and a constitutive requirement involving the logarithmic strain tensor. Given the deformation tensor $F ={\rm D} φ$, the left Cauchy-Green tensor $B = F \, F^T$, and the strain-rate tensor $D = {\rm sym}(\dot F \, F^{-1})$, we show that
\begin{equation}
\label{eqCPSdef}
\begin{alignedat}{2}
\forall \,D\in{\rm Sym}(3) \! \setminus \! \{0\}:
~
\langle{\frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[σ]},{D}\rangle > 0
\quad
&\iff
\quad
\log B
\longmapsto
\widehatσ(\log B)
\;\textrm{is strongly Hilbert-monotone}
&\iff \quad {\rm sym} {\rm D}_{\log B} \widehat σ(\log B) \in{\rm Sym}^{++}_4(6) \quad \text{(TSTS-M$^{++}$)},
\end{alignedat} \tag{1}
\end{equation} where ${\rm Sym}^{++}_4(6)$ denotes the set of positive definite, (minor and major) symmetric fourth order tensors. We call the first inequality ``corotational stability postulate'' (CSP), a novel concept, which implies the \textbf{T}rue-\textbf{S}tress \textbf{T}rue-\textbf{S}train strict Hilbert-\textbf{M}onotonicity (TSTS-M$^+$) for $B \mapsto σ(B) = \widehat σ(\log B)$, i.e. \begin{equation}
\langle \widehatσ(\log B_1)-\widehatσ(\log B_2),{\log B_1-\log B_2} \rangle> 0
\qquad
\forall \, B_1\neq B_2\in{\rm Sym}^{++}(3) \, . \end{equation} In this paper we expand on the ideas of Hill and Leblond, extending Leblonds calculus to the Cauchy elastic case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01811 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A constitutive condition for idealized isotropic Cauchy elasticity involving the logarithmic strain d'Agostino, Marco Valerio Holthausen, Sebastian Bernardini, Davide Sky, Adam Ghiba, Ionel-Dumitrel Martin, Robert J. Neff, Patrizio Analysis of PDEs Mathematical Physics 15A24, 73G05, 73G99, 74B20 Following Hill and Leblond, the aim of our work is to show, for isotropic nonlinear elasticity, a relation between the corotational Zaremba-Jaumann objective derivative of the Cauchy stress $σ$, i.e. \begin{equation} \frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[σ] = \frac{\rm d}{{\rm d}{t}}[σ] - W \, σ+ σ\, W, \qquad W = {\rm skew}(\dot F \, F^{-1}) \end{equation} and a constitutive requirement involving the logarithmic strain tensor. Given the deformation tensor $F ={\rm D} φ$, the left Cauchy-Green tensor $B = F \, F^T$, and the strain-rate tensor $D = {\rm sym}(\dot F \, F^{-1})$, we show that \begin{equation} \label{eqCPSdef} \begin{alignedat}{2} \forall \,D\in{\rm Sym}(3) \! \setminus \! \{0\}: ~ \langle{\frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[σ]},{D}\rangle > 0 \quad &\iff \quad \log B \longmapsto \widehatσ(\log B) \;\textrm{is strongly Hilbert-monotone} &\iff \quad {\rm sym} {\rm D}_{\log B} \widehat σ(\log B) \in{\rm Sym}^{++}_4(6) \quad \text{(TSTS-M$^{++}$)}, \end{alignedat} \tag{1} \end{equation} where ${\rm Sym}^{++}_4(6)$ denotes the set of positive definite, (minor and major) symmetric fourth order tensors. We call the first inequality ``corotational stability postulate'' (CSP), a novel concept, which implies the \textbf{T}rue-\textbf{S}tress \textbf{T}rue-\textbf{S}train strict Hilbert-\textbf{M}onotonicity (TSTS-M$^+$) for $B \mapsto σ(B) = \widehat σ(\log B)$, i.e. \begin{equation} \langle \widehatσ(\log B_1)-\widehatσ(\log B_2),{\log B_1-\log B_2} \rangle> 0 \qquad \forall \, B_1\neq B_2\in{\rm Sym}^{++}(3) \, . \end{equation} In this paper we expand on the ideas of Hill and Leblond, extending Leblonds calculus to the Cauchy elastic case. |
| title | A constitutive condition for idealized isotropic Cauchy elasticity involving the logarithmic strain |
| topic | Analysis of PDEs Mathematical Physics 15A24, 73G05, 73G99, 74B20 |
| url | https://arxiv.org/abs/2409.01811 |