Lines Matching refs:C
16 …in tensor, respectively) defined in the initial configuration, and $\mathsf C$ is a linearized con…
22 {\overbrace{\bm S = \mathsf C \bm E}^{\text{St. Venant-Kirchoff}}} \\
27 {\underbrace{\bm \sigma = \mathsf C \bm \epsilon}_\text{Linear Elastic}}
120 \bm{\sigma} = \mathsf{C} \!:\! \bm{\epsilon}.
124 Hence, the fourth order elasticity tensor $\mathsf C$ (also known as elastic moduli tensor or mater…
127 \mathsf C = \begin{pmatrix}
137 …sible limit $\nu \to \frac 1 2$ causes $\lambda \to \infty$, and thus $\mathsf C$ becomes singular.
256 \bm C = \bm F^T \bm F
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…
271 It is common in the literature for invariants to be taken from $\bm C = \bm I_3 + 2 \bm E$ instead …
277 \Phi(\bm E) &= \frac{\lambda}{2}(\log J)^2 - \mu \log J + \frac \mu 2 (\operatorname{trace} \bm C -…
282 where $J = \lvert \bm F \rvert = \sqrt{\lvert \bm C \rvert}$ is the determinant of deformation (i.e…
287 …c{\partial \sqrt{\lvert \bm C \rvert}}{\partial \bm E} = \lvert \bm C \rvert^{-1/2} \lvert \bm C \…
290 where the factor of $\frac 1 2$ has been absorbed due to $\bm C = \bm I_3 + 2 \bm E.$
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,
305 Note that the product $\bm C^{-1} \bm E$ is also symmetric, and that $\bm E$ should be computed usi…
316 For example, if $u_{i,j} \sim 10^{-8}$, then naive computation of $\bm I_3 - \bm C^{-1}$ and $\log …
321 … C = 3 + 2\trace \bm E$ and $J$, Mooney-Rivlin models depend on the additional invariant, $\mathbb…
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{…
333 &= (\lambda \log J - \mu_1 - 2\mu_2) \bm C^{-1} + (\mu_1 + \mu_2 \mathbb I_1) \bm I_3 - \mu_2 \bm C,
341 …{\partial \bm E} &= 2 \mathbb I_1 \bm I_3 - 2 \bm C, & \frac{\partial \log J}{\partial \bm E} &= \…
391 One can linearize {eq}`neo-hookean-stress` around $\bm E = 0$, for which $\bm C = \bm I_3 + 2 \bm E…
401 Alternatively, one can drop geometric nonlinearities, $\bm E \to \bm \epsilon$ and $\bm C \to \bm I…
436 …lasticity tensor, and is analogous to the linear elasticity tensor $\mathsf C$ of {eq}`linear-elas…
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…
452 In the small-strain limit, $\bm C \to \bm I_3$ and $\log J \to 0$, thereby reducing {eq}`eq-neo-hoo…
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} \\
476 …+ \lambda (\bm C^{-1} : \diff \bm E) \bm F^{-T} + 2(\mu - \lambda \log J) \bm F^{-T} \diff\bm E \,…
478 …^{-T} + (\mu - \lambda \log J) \bm F^{-T} (\bm F^T \diff \bm F + \diff \bm F^T \bm F) \bm C^{-1} \\
480 … : \diff \bm F) \bm F^{-T} + (\mu - \lambda \log J) \Big( \diff \bm F\, \bm C^{-1} + \bm F^{-T} \d…
484 where we have exploited $\bm F \bm C^{-1} = \bm F^{-T}$ and
487 \begin{aligned} \bm C^{-1} \!:\! \diff \bm E = \bm C_{IJ}^{-1} \diff \bm E_{IJ} &= \frac 1 2 \bm F_…
490 …ring access to (non-symmetric) $\bm F^{-1}$ in addition to (symmetric) $\bm C^{-1} = \bm F^{-1} \b…
499 …&= \lambda (\bm C^{-1}_{KL} \diff\bm E_{KL}) \bm C^{-1}_{IJ} + 2 (\mu - \lambda \log J) \bm C^{-1}…
500 …&= \underbrace{\Big( \lambda \bm C^{-1}_{IJ} \bm C^{-1}_{KL} + 2 (\mu - \lambda \log J) \bm C^{-1}…
504 where we have identified the effective elasticity tensor $\mathsf C = \mathsf C_{IJKL}$.
505 It is generally not desirable to store $\mathsf C$, but rather to use the earlier expressions so th…
508 1. recover $\bm C^{-1}$ and $\log J$ (either stored at quadrature points or recomputed),
525 …J}$ evident in {eq}`eq-neo-hookean-incremental-stress-index`, thus $\mathsf C$ can be stored as a …
527 This compares with 13 entries of overhead for direct storage of $\{ \bm S, \bm C^{-1}, \log J \}$, …