# Ductile Damage Model

In Brittle Damage: Johnson-Cook 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 elastic-plastic behavior is formulated by Johnson-Cook 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, ${\epsilon }_{dam}$ :

$0\le \delta \le 1$
• If $\epsilon <{\epsilon }_{dam}⇒\delta =0$ LAW 22 is identical to LAW2.
• If $\epsilon \ge {\epsilon }_{dam}⇒{\epsilon }_{dam}=\left(1-\delta \right)E$ 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.

For brick elements the damage law can be only applied to the deviatoric part of stress tensor ${s}_{ij}$ and ${G}_{dam}=\frac{{E}_{dam}}{2\left(1+{v}_{dam}\right)}$ . This is the case of LAW22 in Radioss. However, if the application of damage law to stress tensor ${\sigma }_{ij}$ is expected, Radioss LAW23 may be used.

The strain rate definition and filtering for these laws are explained in Johnson-Cook Plasticity Model (LAW2). The strain rate $\stackrel{˙}{\epsilon }$ may or may not affect the maximum stress value ${\sigma }_{\mathrm{max}}$ according to the user's choice, as shown in Table 1.

Table 1. Strain Rate Dependency
(a) Strain rate effect on ${\sigma }_{\mathrm{max}}$ (b) No strain rate effect on ${\sigma }_{\mathrm{max}}$

$\sigma ={\sigma }_{y}\left(1+c\mathrm{ln}\left(\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{o}}\right)\right)$

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

$\sigma ={\sigma }_{y}\left(1+c\mathrm{ln}\left(\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{o}}\right)\right)$

${\sigma }_{\mathrm{max}}={\sigma }_{\mathrm{max}}^{0}$