Algorithmic stiffness

$\displaystyle d$ $\displaystyle \mbox{\boldmath$\sigma$}$ $\displaystyle _{n+1}$	$\displaystyle =$	$\displaystyle \mbox{\boldmath$D$}$ $\displaystyle \left(d\mbox{\boldmath$\varepsilon$}_{n+1}-d\mbox{\boldmath$\varepsilon$}^p_{n+1}\right)$	(4.24)
$\displaystyle d$ $\displaystyle \mbox{\boldmath$\varepsilon $}$ $\displaystyle ^p_{n+1}$	$\displaystyle =$	$\displaystyle \sum\left(\lambda^i\partial_{\sigma\sigma}gd\mbox{\boldmath$\sigm... ...a}\mbox{\boldmath$\kappa$}d\lambda^i\right)+d\lambda^i\partial_{\sigma}g\right)$	(4.25)

$\displaystyle d$ $\displaystyle \mbox{\boldmath$\sigma$}$ $\displaystyle =$ $\displaystyle \mbox{\boldmath$\Xi$}$ $\displaystyle _{n+1} \left\{d\mbox{\boldmath$\varepsilon$}_{n+1}-\sum\lambda^i\... ...}\mbox{\boldmath$\kappa$}d\lambda^i - \sum d\lambda^i\partial_{\sigma}g\right\}$

(4.26)

$\displaystyle \mbox{\boldmath$\Xi$}$ $\displaystyle _{n+1}=\left[\mbox{\boldmath$D$}^{-1}+\sum\lambda^i\partial_{\sig... ...\lambda\partial_{\sigma\kappa}g\partial_{\sigma}\mbox{\boldmath$\kappa$}\right]$

(4.27)

$\displaystyle \partial_\sigma f^i d$ $\displaystyle \mbox{\boldmath$\sigma$}$ $\displaystyle + \partial_\kappa f^i (\partial_\sigma$ $\displaystyle \mbox{\boldmath$\kappa$}$ $\displaystyle d$ $\displaystyle \mbox{\boldmath$\sigma$}$ $\displaystyle + \partial_\lambda$ $\displaystyle \mbox{\boldmath$\kappa$}$ $\displaystyle d$ $\displaystyle \mbox{\boldmath$\lambda$}$ $\displaystyle ) = 0$

(4.28)

$\displaystyle d$ $\displaystyle \mbox{\boldmath$\lambda$}$ $\displaystyle =$ $\displaystyle \mbox{\boldmath$G$}$ $\displaystyle \left\{\mbox{\boldmath$V$}\mbox{\boldmath$\Xi$}d\mbox{\boldmath$\varepsilon$}\right\}$

(4.29)

$\displaystyle \mbox{\boldmath$G$}$ $\displaystyle =\left[\mbox{\boldmath$V$}^T\mbox{\boldmath$\Xi$}\mbox{\boldmath$... ...kappa}\mbox{\boldmath$f$}\partial_{\lambda}\mbox{\boldmath$\kappa$}\right]^{-1}$

(4.30)

$\displaystyle \mbox{$\displaystyle\frac{\rm {d}\mbox{\boldmath$\sigma$}}{\rm {d}\mbox{\boldmath$\varepsilon$}}$}$ $\displaystyle \vert_{n+1}=$ $\displaystyle \mbox{\boldmath$\Xi$}$ $\displaystyle -$ $\displaystyle \mbox{\boldmath$\Xi$}$ $\displaystyle \mbox{\boldmath$U$}$ $\displaystyle \left(\mbox{\boldmath$V$}\mbox{\boldmath$\Xi$}\mbox{\boldmath$U$}... ...ambda}\mbox{\boldmath$\kappa$}]\right) \mbox{\boldmath$V$}\mbox{\boldmath$\Xi$}$

(4.31)