Algorithmic stiffness

Differentiation of the elastic stress-strain relations (64) and the discrete flow rule (65) yields

$$d\mbox{\boldmath$\sigma$}_{n+1} = \mbox{\boldmath$D$}\left(d\mbox{\boldmath$\varepsilon$}_{n+1}-d\mbox{\boldmath$\varepsilon$}^p_{n+1}\right)$$ (80)

$$d\mbox{\boldmath$\varepsilon$}^p_{n+1} = \sum\left(\lambda^i\partial_{\sigma\sigma}gd\mbox{\boldmath$\sigm... ...a}\mbox{\boldmath$\kappa$}d\lambda^i\right)+d\lambda^i\partial_{\sigma}g\right)$$ (81)

Combining this two equations, one obtains following relation

$$d\mbox{\boldmath$\sigma$}= \mbox{\boldmath$\Xi$}_{n+1} \lef... ...\kappa$}d\lambda^i - \sum d\lambda^i\partial_{\sigma}g\right\}$$ (82)

where $\mbox{\boldmath$\Xi$}_{n+1}$ is the algorithmic moduli defined as

$$\mbox{\boldmath$\Xi$}_{n+1}=\left[\mbox{\boldmath$D$}^{-1}+... ...\sigma\kappa}g\partial_{\sigma}\mbox{\boldmath$\kappa$}\right]$$ (83)

Differentiation of discrete consistency condition yields

$$\partial_\sigma f^i d\mbox{\boldmath$\sigma$}+ \partial_\... ...lambda \mbox{\boldmath$\kappa$}d\mbox{\boldmath$\lambda$}) = 0$$ (84)

By substitution of (82) into (84) the following relation is obtained

$$d\mbox{\boldmath$\lambda$} = \mbox{\boldmath$G$} \left\{\mb... ...V$}\mbox{\boldmath$\Xi$}d\mbox{\boldmath$\varepsilon$}\right\}$$ (85)

where matrix $\mbox{\boldmath$G$}$ is defined as

$$\mbox{\boldmath$G$}=\left[\mbox{\boldmath$V$}^T\mbox{\bol... ...math$f$}\partial_{\lambda}\mbox{\boldmath$\kappa$}\right]^{-1}$$ (86)

Finally, by substitution of (86) into (82) one obtains the algorithmic elastoplastic tangent moduli

$$\mbox{$\displaystyle\frac{\rm {d}\mbox{\boldmath$\sigma$}}{... ...math$\kappa$}]\right) \mbox{\boldmath$V$}\mbox{\boldmath$\Xi$}$$ (87)