Lines Matching refs:stress

11 The stress-strain relationship (constitutive law) for each of the material models is provided.
16 …sketched in the diagram below, where $\bm \sigma$ and $\bm \epsilon$ are stress and strain, respec…
50 where $\bm{\sigma}$ and $\bm{g}$ are stress and forcing functions, respectively.
105 The constitutive law (stress-strain relationship) is therefore given by its gradient,
117 The constitutive law (stress-strain relationship) can also be written as
121 $$ (linear-stress-strain)
238 $\bm{P}$ and $\bm{g}$ are the *first Piola-Kirchhoff stress* tensor and the prescribed forcing func…
248 where $\bm S$ is the *second Piola-Kirchhoff stress* tensor, a symmetric tensor defined entirely in…
266 …n of $\bm E$, similar to the linear case, shown in equation  {eq}`linear-stress-strain`, which  ex…
295 $$ (neo-hookean-stress)
298 An equivalent form of {eq}`neo-hookean-stress` is
302 $$ (neo-hookean-stress-stable)
328 We differentiate $\Phi$ as in the Neo-Hookean case {eq}`neo-hookean-stress` to yield the second Pio…
391 …stress` around $\bm E = 0$, for which $\bm C = \bm I_3 + 2 \bm E \to \bm I_3$ and $J \to 1 + \oper…
437 We now evaluate $\diff \bm S$ for the Neo-Hookean model {eq}`neo-hookean-stress`,
443 $$ (eq-neo-hookean-incremental-stress)
452 …_3$ and $\log J \to 0$, thereby reducing {eq}`eq-neo-hookean-incremental-stress` to the St. Venant…
456 Similar to {eq}`eq-neo-hookean-incremental-stress`, we differentiate {eq}`mooney-rivlin-stress_coup…
466 Note that this agrees with {eq}`eq-neo-hookean-incremental-stress` if $\mu_1 = \mu, \mu_2 = 0$.
471 …he expense of symmetry) if we substitute {eq}`eq-neo-hookean-incremental-stress` into {eq}`eq-diff…
490 We prefer to compute with {eq}`eq-neo-hookean-incremental-stress` because {eq}`eq-diff-P-dF` is mor…
494 It is sometimes useful to express {eq}`eq-neo-hookean-incremental-stress` in index notation,
502 $$ (eq-neo-hookean-incremental-stress-index)
509 …cts as in {eq}`eq-neo-hookean-incremental-stress` or the second line of {eq}`eq-neo-hookean-increm…
519 where $\diff \bm P$ is defined by {eq}`eq-diff-P` and {eq}`eq-neo-hookean-incremental-stress`, and …
525 … C_{IJKL} = \mathsf C_{KLIJ}$ evident in {eq}`eq-neo-hookean-incremental-stress-index`, thus $\mat…