This is constitutive model for ductile metals with optional thermal softening and strain rate hardening. The effective plastic flow stress is defined as:

$\displaystyle{ \sigma_y = f(\varepsilon_{eff}^p) \cdot g(D) \cdot \left( 1 - \left( \frac{T-T_0}{T_m - T_0}\right)^m \right) \cdot \left( 1 + \frac{\dot{\varepsilon}_{eff}^p}{\dot{\varepsilon}_0} \right)^c}$

where $f(\varepsilon_{eff}^p)$ is a user defined CURVE or FUNCTION, $g(D)$ is an optional damage softening and $T$ is the current temperature. $g(D)$, where $D$ is the damage level, is defined as:

$g(D) = \left\{ \begin{array}{cc} 1 & D \leq s_0 \\ \displaystyle{ 1 + \frac{D-s_0}{1-s_0} \cdot (s_1-1)} & D \gt s_0 \end{array} \right. $

That is, $g(D)$ drops linearly from 1 at $D=s_0$ to $s_1$ at $D=1$ (full damage). There is a special SPH (PARTICLE_SPH) version of this material model, with an optional strain and temperature independent strain rate hardening. It is activated by specifying $\mu \gt 0$:

$\displaystyle{ \sigma_y = f(\varepsilon_{eff}^p) \cdot g(D) \cdot \left( 1 - \left( \frac{T-T_0}{T_m - T_0}\right)^m \right) + \mu \left( \frac{\dot{\varepsilon}_{eff}^p}{\dot{\varepsilon}_0} \right)^c}$

This variant has been developed specifically for shaped charge jets that can exhibit strong strain rate effects at high temperatures.

The hydrostatic pressure $p$ is defined as:

$p = -K \varepsilon_v + 3K \alpha_T (T-T_{ref})$

where $K$ is the bulk modulus, $\varepsilon_v$ is the volumetric strain. $\alpha_T$ is the thermal expansion coefficient and $T_{ref}$ is the reference temperature (see PROP_THERMAL).