# Elastic-plastic Anisotropic Shells (Barlat's Law)

Barlat's 3- parameter plasticity model is developed in F. Barlat, J. Lian ^{1} for modeling of sheet under plane stress assumption with
an anisotropic plasticity model. The anisotropic yield stress criterion for plane stress is
defined as:

Where, ${\sigma}_{e}$ is the yield stress, $a$ and $c$ are anisotropic material constants, $m$ exponent and ${K}_{1}$ and ${K}_{2}$ are defined by:

${K}_{2}=\sqrt{{\left(\frac{{\sigma}_{xx}-h{\sigma}_{yy}}{2}\right)}^{2}+{p}^{2}{\left({\sigma}_{xy}\right)}^{2}}$

Where, $h$ and $p$ are additional anisotropic material constants. All anisotropic material constants, except for $p$ which is obtained implicitly, are determined from Barlat width to thickness strain ratio $R$ from:

$c=2-a$

$h=\sqrt{\left(\frac{{r}_{00}}{1+{r}_{00}}\right)\left(\frac{1+{r}_{90}}{{r}_{90}}\right)}$

The width to thickness ratio for any angle
$\phi $
can be calculated: ^{1}

^{1}Barlat F. and Lian J.,

Plastic behavior and stretchability of sheet metals, Part I: A yield function for orthoropic sheets under plane stress conditions, International Journal of Plasticity, Vol. 5, pp. 51-66, 1989.