Lines Matching full:diff

164 \diff \bm{\sigma} = \dfrac{\partial \bm{\sigma}}{\partial \bm{\epsilon}} \tcolon \diff \bm{\epsilon}
170 \diff \bm{\epsilon} = \dfrac{1}{2}\left( \nabla \diff \bm{u} + \nabla \diff \bm{u}^T \right)
176 \diff \nabla \bm{u} = \nabla \diff \bm{u} .
182 \diff \bm{\sigma}  = \bar{\lambda} \cdot \operatorname{trace} \diff \bm{\epsilon} \cdot \bm{I}_3 + …
197   \diff \sigma_{11} \\
198   \diff \sigma_{22} \\
199   \diff \sigma_{33} \\
200   \diff \sigma_{23} \\
201   \diff \sigma_{13} \\
202   \diff \sigma_{12}
213   \diff \epsilon_{11} \\
214   \diff \epsilon_{22} \\
215   \diff \epsilon_{33} \\
216   2 \diff \epsilon_{23} \\
217   2 \diff \epsilon_{13} \\
218   2 \diff \epsilon_{12}
426 …diff \bm P = \frac{\partial \bm P}{\partial \bm F} \!:\! \diff \bm F = \diff \bm F\, \bm S + \bm F…
427 $$ (eq-diff-P)
432 \diff \bm E = \frac{\partial \bm E}{\partial \bm F} \!:\! \diff \bm F = \frac 1 2 \Big( \diff \bm F…
435 and $\diff\bm F = \nabla_X\diff\bm u$.
437 We now evaluate $\diff \bm S$ for the Neo-Hookean model {eq}`neo-hookean-stress`,
440 \diff\bm S = \frac{\partial \bm S}{\partial \bm E} \!:\! \diff \bm E
441 = \lambda (\bm C^{-1} \!:\! \diff\bm E) \bm C^{-1}
442   + 2 (\mu - \lambda \log J) \bm C^{-1} \diff\bm E \, \bm C^{-1},
448 \diff \bm C^{-1} = \frac{\partial \bm C^{-1}}{\partial \bm E} \!:\! \diff\bm E = -2 \bm C^{-1} \dif…
460 \diff\bm S &= \lambda (\bm C^{-1} \tcolon \diff\bm E) \bm C^{-1} \\
461 &\quad + 2(\mu_1 + 2\mu_2 - \lambda \log J) \bm C^{-1} \diff\bm E \bm C^{-1} \\
462 &\quad + 2 \mu_2 \Big[ \trace (\diff\bm E) \bm I_3 - \diff\bm E\Big] .
471 …expense of symmetry) if we substitute {eq}`eq-neo-hookean-incremental-stress` into {eq}`eq-diff-P`,
475 \diff \bm P &= \diff \bm F\, \bm S
476 …+ \lambda (\bm C^{-1} : \diff \bm E) \bm F^{-T} + 2(\mu - \lambda \log J) \bm F^{-T} \diff\bm E \,…
477 &= \diff \bm F\, \bm S
478 …+ \lambda (\bm F^{-T} : \diff \bm F) \bm F^{-T} + (\mu - \lambda \log J) \bm F^{-T} (\bm F^T \diff…
479 &= \diff \bm F\, \bm S
480 …+ \lambda (\bm F^{-T} : \diff \bm F) \bm F^{-T} + (\mu - \lambda \log J) \Big( \diff \bm F\, \bm C…
482 $$ (eq-diff-P-dF)
487 …diff \bm E = \bm C_{IJ}^{-1} \diff \bm E_{IJ} &= \frac 1 2 \bm F_{Ik}^{-1} \bm F_{Jk}^{-1} (\bm F_…
490 We prefer to compute with {eq}`eq-neo-hookean-incremental-stress` because {eq}`eq-diff-P-dF` is mor…
493 :::{dropdown} $\diff\bm S$ in index notation
498 \diff\bm S_{IJ} &= \frac{\partial \bm S_{IJ}}{\partial \bm E_{KL}} \diff \bm E_{KL} \\
499 …&= \lambda (\bm C^{-1}_{KL} \diff\bm E_{KL}) \bm C^{-1}_{IJ} + 2 (\mu - \lambda \log J) \bm C^{-1}…
500 …mu - \lambda \log J) \bm C^{-1}_{IK} \bm C^{-1}_{JL} \Big)}_{\mathsf C_{IJKL}} \diff \bm E_{KL} \,,
506 … and solution increment $\diff \bm F = \nabla_X (\diff \bm u)$ (which we are solving for in the Ne…
509 …second line of {eq}`eq-neo-hookean-incremental-stress-index` to compute $\diff \bm S$ while avoidi…
510 3. conclude by {eq}`eq-diff-P`, where $\bm S$ is either stored or recomputed from its definition ex…
513 …-form-initial` may be written as a weak form for linear operators: find $\diff\bm u \in \mathcal V…
516 \int_{\Omega_0} \nabla_X \bm v \!:\! \diff\bm P dV = \text{rhs}, \quad \forall \bm v \in \mathcal V…
519 where $\diff \bm P$ is defined by {eq}`eq-diff-P` and {eq}`eq-neo-hookean-incremental-stress`, and …