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,

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

the stress strain law

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

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

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

the incremental definition of cumulative plastic strain

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

the linear hardening law

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

the evolution law for the plastic strain

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

the loading-unloading conditions

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

and the damage law

$$\omega(\kappa) = \omega_c(1- ^{-a\kappa}),$$ (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.