# 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:

$F=a{|{K}_{1}+{K}_{2}|}^{m}+a{|{K}_{1}-{K}_{2}|}^{m}+c{|2{K}_{2}|}^{m}-2{\left({\sigma }_{e}\right)}^{m}$

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:

${K}_{1}=\frac{{\sigma }_{xx}+h{\sigma }_{yy}}{2}$

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

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:

$a=2-2\sqrt{\left(\frac{{r}_{00}}{1+{r}_{00}}\right)\left(\frac{{r}_{90}}{1+{r}_{90}}\right)}$

$c=2-a$

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

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

${R}_{\phi }=\frac{2m{\left({\sigma }_{e}\right)}^{m}}{\left(\frac{\partial F}{\partial {\sigma }_{xx}}+\frac{\partial F}{\partial {\sigma }_{yy}}\right){\sigma }_{\phi }}-1$

Where, ${\sigma }_{\phi }$ is the uniaxial tension in the $\phi$ direction. Let $\phi$ = 45°, Equation 4 gives an equation from which the anisotropy parameter $p$ can be computed implicitly by using an iterative procedure:
$\frac{2m{\left({\sigma }_{e}\right)}^{m}}{\left(\frac{\partial F}{\partial {\sigma }_{xx}}+\frac{\partial F}{\partial {\sigma }_{yy}}\right){\sigma }_{45}}-1-{r}_{45}=0$
Note: Barlat's law reduces to Hill's law when using $m$ =2
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.