# Tabulated Piecewise Linear and Quadratic Elasto-plastic Laws (LAW36 and LAW60)

The elastic-plastic behavior of isotropic material is modeled with user-defined functions for work hardening curve.

The elastic portion of the material stress-strain curve is modeled using the elastic modulus, E, and Poisson's ratio, $\upsilon$ . The hardening behavior of the material is defined in function of plastic strain for a given strain rate (図 1). An arbitrary number of material plasticity curves can be defined for different strain rates. For a given strain rate, a linear interpolation of stress for plastic strain change, can be used. This is the case of LAW36 in Radioss. However, in LAW60 a quadratic interpolation of the functions allows to better simulate the strain rate effects on the behavior of material as it is developed in LAW60. For a given plastic strain, a linear interpolation of stress for strain rate change is used. Compared to Johnson-Cook model (LAW2), there is no maximum value for the stress. The curves are extrapolated if the plastic deformation is larger than the maximum plastic strain. The hardening model may be isotropic, kinematic or a combination of the two models as described in Cowper-Symonds Plasticity Model (LAW44). The material failure model is the same as in Zhao law.

For some kinds of steels the yield stress dependence to pressure has to be incorporated especially for massive structures. The yield stress variation is then given by:(1)
${\sigma }_{\gamma }={\sigma }_{\gamma }^{0}\text{\hspace{0.17em}}\left({\epsilon }_{p}\right)×f\left(p\right)$
Where, $p$ is the pressure defined by Stresses in Solids, 式 2. Drücker-Prager model described in Drücker-Prager (LAW10 and LAW21) gives a nonlinear function for $f\left(p\right)$ . However, for steel type materials where the dependence to pressure is low, a simple linear function may be considered:(2)
${\sigma }_{y}={\sigma }_{y}^{0}\text{\hspace{0.17em}}\left({\epsilon }_{p}\right)×C×p\left({\epsilon }_{p}\right)$
Where,
$C$
User-defined constant
$p$
Computed pressure for a given deformed configuration
Chard in /MAT/LAW36 is same like in /MAT/LAW44. For more detail on Chard, see Cowper-Symonds Plasticity Model (LAW44).
The principal strain rate is used for the strain rate definition:(3)
$\frac{d\epsilon }{dt}=\frac{1}{2}\left(\frac{d{\epsilon }_{x}}{dt}+\frac{d{\epsilon }_{y}}{dt}+\sqrt{{\left(\frac{d{\epsilon }_{x}}{dt}-\frac{d{\epsilon }_{y}}{dt}\right)}^{2}+{\left(\frac{d{\gamma }_{xy}}{dt}\right)}^{2}}\right)$

For strain rate filtering, refer to Strain Rate Filtering.