Small-strain formulation:

The small-strain version of hardening Mises plasticity can be combined with isotropic damage. The basic equations include the additive decomposition of strain into elastic and plastic parts,

$\displaystyle \mbox{\boldmath$\varepsilon$}$$\displaystyle =$   $\displaystyle \mbox{\boldmath$\varepsilon$}$$\displaystyle _{\rm {e}} +$   $\displaystyle \mbox{\boldmath$\varepsilon$}$$\displaystyle _{\rm {p}},$ (38)

the stress strain law

$\displaystyle \mbox{\boldmath$\sigma$}$$\displaystyle = (1-\omega)\bar{\mbox{\boldmath$\sigma$}}=(1-\omega)\mbox{\boldmath$D$} :(\mbox{\boldmath$\varepsilon$}-\mbox{\boldmath$\varepsilon$}_{\rm {p}}),$ (39)

the definition of the yield function in terms of the effective stress,

$\displaystyle f(\bar{\mbox{\boldmath$s$}},\kappa) = \sqrt{\frac{3}{2}\bar{\mbox...
...igma_Y(\kappa) = \sqrt{3 J_2(\bar{\mbox{\boldmath$\sigma$}})}-\sigma_Y(\kappa),$ (40)

the incremental definition of cumulative plastic strain

$\displaystyle \dot{\kappa} = \Vert \dot{\mbox{\boldmath$\varepsilon$}}_{\mathrm{p}}\Vert,$ (41)

the linear hardening law

$\displaystyle \sigma_Y(\kappa) = \sigma_0 + H\kappa,$ (42)

the evolution law for the plastic strain

$\displaystyle \dot{\mbox{\boldmath$\varepsilon$}}_{\mathrm{p}}= \dot{\lambda}\frac{\partial f}{\partial \bar{\mbox{\boldmath$s$}}},$ (43)

the loading-unloading conditions

$\displaystyle \dot{\lambda} > 0 \qquad f(\bar{\mbox{\boldmath$s$}},\kappa)\leq 0 \qquad \dot{\lambda} f(\bar{\mbox{\boldmath$s$}},\kappa) = 0.$ (44)

and the damage law

$\displaystyle \omega(\kappa) = \omega_c(1-$e$\displaystyle ^{-a\kappa}),$ (45)

In the equations above, $ \varepsilon$ is the strain tensor, $ \varepsilon$$ _{\rm {e}}$ is the elastic strain tensor, $ \varepsilon$$ _{p}$ is the plastic strain tensor, $ D$ is the elastic stiffness tensor, $ \sigma$ is the nominal stress tensor, $ \bar{\mbox{\boldmath $\sigma$}}$ is the effective stress tensor, $ \bar{\mbox{\boldmath $s$}}$ is the effective deviatoric stress tensor, $ \sigma_Y$ is the magnitude of stress at yielding under uniaxial tension (or compression), $ \kappa$ is the cumulated plastic strain, $ H$ is the hardening modulus, $ \lambda$ is the plastic multiplier, $ \omega$ is the damage variable, $ \omega_c$ is critical damage and $ a$ is a positive dimensionless parameter.

Borek Patzak