Lines Matching refs:E

16 …nd strain, respectively, in the small strain regime, while $\bm S$ and $\bm E$ are their finite-st…
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…
114 \begin{aligned} \lambda &= \frac{E \nu}{(1 + \nu)(1 - 2 \nu)} \\ \mu &= \frac{E}{2(1 + \nu)} \end{a…
262 \bm E = \frac 1 2 (\bm C - \bm I_3) = \frac 1 2 \Big( \nabla_X \bm u + (\nabla_X \bm u)^T + (\nabla…
266 …, appropriate for large deformations, express $\bm S$ as a function of $\bm E$, similar to the lin…
269 We will assume without loss of generality that $\bm E$ is diagonal and take its set of eigenvalues …
270 …alternate choices, such as $\operatorname{trace}(\bm E), \operatorname{trace}(\bm E^2), \lvert \bm…
271 …n in the literature for invariants to be taken from $\bm C = \bm I_3 + 2 \bm E$ instead of $\bm E$.
277 \Phi(\bm E) &= \frac{\lambda}{2}(\log J)^2 - \mu \log J + \frac \mu 2 (\operatorname{trace} \bm C -…
278   &= \frac{\lambda}{2}(\log J)^2 - \mu \log J + \mu \operatorname{trace} \bm E,
287 \frac{\partial J}{\partial \bm E} = \frac{\partial \sqrt{\lvert \bm C \rvert}}{\partial \bm E} = \l…
290 where the factor of $\frac 1 2$ has been absorbed due to $\bm C = \bm I_3 + 2 \bm E.$
301 \bm S = \lambda \log J \bm C^{-1} + 2 \mu \bm C^{-1} \bm E,
304 which is more numerically stable for small $\bm E$, and thus preferred for computation.
305 Note that the product $\bm C^{-1} \bm E$ is also symmetric, and that $\bm E$ should be computed usi…
321 …n just two scalar invariants, $\mathbb I_1 = \trace \bm C = 3 + 2\trace \bm E$ and $J$, Mooney-Riv…
341 …al \bm E} &= 2 \bm I_3, & \frac{\partial \mathbb{I_2}}{\partial \bm E} &= 2 \mathbb I_1 \bm I_3 - …
362 nh["parameters"] = "E=2.8, nu=0.4"
391 …ookean-stress` around $\bm E = 0$, for which $\bm C = \bm I_3 + 2 \bm E \to \bm I_3$ and $J \to 1 …
394 \bm S = \lambda (\trace \bm E) \bm I_3 + 2 \mu \bm E,
401 Alternatively, one can drop geometric nonlinearities, $\bm E \to \bm \epsilon$ and $\bm C \to \bm I…
417 …nlinearities in defining $\bm S(\bm E)$, as well as geometric nonlinearities through $\bm P = \bm …
426 … F\, \bm S + \bm F \underbrace{\frac{\partial \bm S}{\partial \bm E} \!:\! \diff \bm E}_{\diff \bm…
432 \diff \bm E = \frac{\partial \bm E}{\partial \bm F} \!:\! \diff \bm F = \frac 1 2 \Big( \diff \bm F…
436 The quantity ${\partial \bm S} / {\partial \bm E}$ is known as the incremental elasticity tensor, a…
440 \diff\bm S = \frac{\partial \bm S}{\partial \bm E} \!:\! \diff \bm E
441 = \lambda (\bm C^{-1} \!:\! \diff\bm E) \bm C^{-1}
442   + 2 (\mu - \lambda \log J) \bm C^{-1} \diff\bm E \, \bm C^{-1},
448 …\bm C^{-1} = \frac{\partial \bm C^{-1}}{\partial \bm E} \!:\! \diff\bm E = -2 \bm C^{-1} \diff \bm…
460 \diff\bm S &= \lambda (\bm C^{-1} \tcolon \diff\bm E) \bm C^{-1} \\
461 &\quad + 2(\mu_1 + 2\mu_2 - \lambda \log J) \bm C^{-1} \diff\bm E \bm C^{-1} \\
462 &\quad + 2 \mu_2 \Big[ \trace (\diff\bm E) \bm I_3 - \diff\bm E\Big] .
476 …+ \lambda (\bm C^{-1} : \diff \bm E) \bm F^{-T} + 2(\mu - \lambda \log J) \bm F^{-T} \diff\bm E \,…
487 \begin{aligned} \bm C^{-1} \!:\! \diff \bm E = \bm C_{IJ}^{-1} \diff \bm E_{IJ} &= \frac 1 2 \bm F_…