Damage

In the presence of materi_damage $d$, the materi_stress follows:

$$\sigma_{ij}^{\rm damaged} = (1-d) \sigma_{ij}^{\rm undamaged}$$

For the damage, the group_materi_damage_mazars model is available:

$$d = d_t ~ \alpha^{\beta} ~ + ~ d_c ~ (1 - \alpha )^{\beta}$$

where

$$d_t = 1. - (1-a_t) ~ \frac{\epsilon^0}{\epsilon^{\rm eq}} - a_t ~ e ^ { -b_t(\epsilon^{\rm eq} - {\epsilon^0}) }$$

and

$$d_c = 1. - (1-a_c) ~ \frac{\epsilon^0}{\epsilon^{\rm eq}} - a_c ~ e ^ { -b_t(\epsilon^{\rm eq} - {\epsilon^0}) }$$

Here $\epsilon^{\rm eq}$ contains the positive principal strains. The parameter $\alpha$ is given by the ratio $\frac{\epsilon^{\rm eq}}{\epsilon}$, where $\epsilon$ contains the total strains (both negative and positive). The parameter $\epsilon^0$ is the strain threshold for damage; other material parameters are $\beta ~,~ a_t ~,~ b_t ~,~ a_c ~,~ b_c$. Typically for concrete:

$$1.e-4 < \epsilon^0 < 3.e-4 ~~;~~ \beta = 1. ~~ ; ~~ 1 < a_t < ... ...~~ ; ~~ 500 < b_t < 2000 ~~ ; ~~ 0.7 < a_c < 1.2 ~~ ; ~~ e^4 < b_c < 5 e^4$$

You can combine damage freely with plasticity models or other material behavior.