Lines Matching refs:S

16 …are stress and strain, respectively, in the small strain regime, while $\bm S$ and $\bm E$ are the…
20   {\overbrace{\bm S(\bm E)}^{\text{Finite Strain Hyperelastic}}}
22   {\overbrace{\bm S = \mathsf C \bm E}^{\text{St. Venant-Kirchoff}}} \\
23 …@V{\text{geometric}}V{\begin{smallmatrix}\bm E \to \bm \epsilon \\ \bm S \to \bm \sigma \end{small…
24 …@V{\begin{smallmatrix}\bm E \to \bm \epsilon \\ \bm S \to \bm \sigma \end{smallmatrix}}V{\text{geo…
245 \bm{P} = \bm{F} \, \bm{S},
248 where $\bm S$ is the *second Piola-Kirchhoff stress* tensor, a symmetric tensor defined entirely in…
249 Different constitutive models can define $\bm S$.
266 The constitutive models considered, appropriate for large deformations, express $\bm S$ as a functi…
294 \bm S = \lambda \log J \bm C^{-1} + \mu (\bm I_3 - \bm C^{-1}).
301 \bm S = \lambda \log J \bm C^{-1} + 2 \mu \bm C^{-1} \bm E,
332 \bm S &=  \lambda \log J \bm{C}^{-1} - (\mu_1 + 2\mu_2) \bm{C}^{-1} + \mu_1\bm I_3 + \mu_2(\mathbb{…
394 \bm S = \lambda (\trace \bm E) \bm I_3 + 2 \mu \bm E,
417 …utive nonlinearities in defining $\bm S(\bm E)$, as well as geometric nonlinearities through $\bm …
426 …diff \bm F = \diff \bm F\, \bm S + \bm F \underbrace{\frac{\partial \bm S}{\partial \bm E} \!:\! \…
436 The quantity ${\partial \bm S} / {\partial \bm E}$ is known as the incremental elasticity tensor, a…
437 We now evaluate $\diff \bm S$ for the Neo-Hookean model {eq}`neo-hookean-stress`,
440 \diff\bm S = \frac{\partial \bm S}{\partial \bm E} \!:\! \diff \bm E
460 \diff\bm S &= \lambda (\bm C^{-1} \tcolon \diff\bm E) \bm C^{-1} \\
475 \diff \bm P &= \diff \bm F\, \bm S
477 &= \diff \bm F\, \bm S
479 &= \diff \bm F\, \bm S
493 :::{dropdown} $\diff\bm S$ in index notation
509 …line of {eq}`eq-neo-hookean-incremental-stress-index` to compute $\diff \bm S$ while avoiding comp…
510 3. conclude by {eq}`eq-diff-P`, where $\bm S$ is either stored or recomputed from its definition ex…
526 Along with 6 entries for $\bm S$, this totals 27 entries of overhead compared to computing everythi…
527 This compares with 13 entries of overhead for direct storage of $\{ \bm S, \bm C^{-1}, \log J \}$, …