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:

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

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:

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

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

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:

(3)
$$a=2-2\sqrt{\left(\frac{{r}_{00}}{1+{r}_{00}}\right)\left(\frac{{r}_{90}}{1+{r}_{90}}\right)}$$

$c=2-a$

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

The width to thickness ratio for any angle

$\phi $
can be calculated:

^{1}(4)
$${R}_{\phi}=\frac{2m{\left({\sigma}_{e}\right)}^{m}}{\left(\frac{\partial F}{\partial {\sigma}_{xx}}+\frac{\partial F}{\partial {\sigma}_{yy}}\right){\sigma}_{\phi}}-1$$

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:

(5)
$$\frac{2m{\left({\sigma}_{e}\right)}^{m}}{\left(\frac{\partial F}{\partial {\sigma}_{xx}}+\frac{\partial F}{\partial {\sigma}_{yy}}\right){\sigma}_{45}}-1-{r}_{45}=0$$

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.