Ductile Damage Model
In Brittle Damage: JohnsonCook Plasticity Model (LAW27), a damage model for brittle materials is presented. It is used in Radioss LAW27 valid for shell meshes. The damage is generated when the shell works in traction only. A generalized damage model for ductile materials is incorporated in Radioss LAW22 and LAW23. The damage is not only generated in traction but also in compression and shear. It is valid for solids and shells. The elasticplastic behavior is formulated by JohnsonCook model. The damage is introduced by the use of damage parameter, $\delta $ . The damage appears in the material when the strain is larger than a maximum value, ${\epsilon}_{dam}$ :
 If $\epsilon <{\epsilon}_{dam}\Rightarrow \delta =0$ LAW 22 is identical to LAW2.
 If $\epsilon \ge {\epsilon}_{dam}\Rightarrow {\epsilon}_{dam}=\left(1\delta \right)E$ and ${v}_{dam}=\frac{1}{2}\delta +\left(1\delta \right)v$
This implies an isotropic damage with the same effects in tension and compression. The inputs of the model are the starting damage strain ${\epsilon}_{dam}$ and the slope of the softening curve ${E}_{t}$ as shown in Figure 1.
(a) Strain rate effect on ${\sigma}_{\mathrm{max}}$  (b) No strain rate effect on ${\sigma}_{\mathrm{max}}$ 



$\sigma ={\sigma}_{y}\left(1+c\mathrm{ln}\left(\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{o}}\right)\right)$
${\sigma}_{\mathrm{max}}={\sigma}_{\mathrm{max}}^{0}\left(1+c.\mathrm{ln}\left(\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{0}}\right)\right)$ 
$\sigma ={\sigma}_{y}\left(1+c\mathrm{ln}\left(\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{o}}\right)\right)$
${\sigma}_{\mathrm{max}}={\sigma}_{\mathrm{max}}^{0}$ 