Lines Matching refs:F

245 \bm{P} = \bm{F} \, \bm{S},
248 … a symmetric tensor defined entirely in the initial configuration, and $\bm{F} = \bm I_3 + \nabla_…
256 \bm C = \bm F^T \bm F
282 where $J = \lvert \bm F \rvert = \sqrt{\lvert \bm C \rvert}$ is the determinant of deformation (i.e…
308 To sketch this idea, suppose we have the $2\times 2$ non-symmetric matrix $\bm{F} = \left( \begin{s…
417 …m E)$, as well as geometric nonlinearities through $\bm P = \bm F\, \bm S$, $\bm E(\bm F)$, and th…
426 \diff \bm P = \frac{\partial \bm P}{\partial \bm F} \!:\! \diff \bm F = \diff \bm F\, \bm S + \bm F…
432 … \frac{\partial \bm E}{\partial \bm F} \!:\! \diff \bm F = \frac 1 2 \Big( \diff \bm F^T \bm F + \…
435 and $\diff\bm F = \nabla_X\diff\bm u$.
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…
484 where we have exploited $\bm F \bm C^{-1} = \bm F^{-T}$ and
487 …m F_{\ell I} \Big) \\ &= \bm F_{Ik}^{-1} \diff \bm F_{kI} = \bm F^{-T} \!:\! \diff \bm F. \end{ali…
490 …e, requiring access to (non-symmetric) $\bm F^{-1}$ in addition to (symmetric) $\bm C^{-1} = \bm F…
506 That is, given the linearization point $\bm F$ and solution increment $\diff \bm F = \nabla_X (\dif…
522 The decision of whether to recompute or store functions of the current state $\bm F$ depends on a r…
526 …ries for $\bm S$, this totals 27 entries of overhead compared to computing everything from $\bm F$.