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:(1)
$F=a{|{K}_{1}+{K}_{2}|}^{m}+a{|{K}_{1}-{K}_{2}|}^{m}+c{|2{K}_{2}|}^{m}-2{\left({\sigma }_{e}\right)}^{m}$
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:(2)
$\begin{array}{l}{K}_{1}=\frac{{\sigma }_{xx}-h{\sigma }_{yy}}{2}\\ {K}_{2}=\sqrt{{\left(\frac{{\sigma }_{xx}-h{\sigma }_{yy}}{2}\right)}^{2}+{p}^{2}{\left({\sigma }_{xy}\right)}^{2}}\end{array}$
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)
$\begin{array}{l}a=2-2\sqrt{\left(\frac{{R}_{00}}{1+{R}_{00}}\right)\left(\frac{{R}_{90}}{1+{R}_{90}}\right)}\\ c=2-a\\ h=\sqrt{\left(\frac{{R}_{00}}{1+{R}_{00}}\right)\left(\frac{1+{R}_{90}}{{R}_{90}}\right)}\end{array}$
The width to thickness ratio for any angle $\varphi$ can be calculated: 1(4)
${R}_{\varphi }=\frac{2m{\left({\sigma }_{e}\right)}^{m}}{\left(\frac{\partial F}{\partial {\sigma }_{xx}}+\frac{\partial F}{\partial {\sigma }_{yy}}\right){\sigma }_{\varphi }}-1$
Where, ${\sigma }_{\varphi }$ is the uniaxial tension in the $\varphi$ direction. Let $\varphi$ = 45°, 式 4 gives an equation from which the anisotropy parameter $p$ can be computed implicitly by using an iterative procedure:(5)
$\frac{2m{\left({\sigma }_{e}\right)}^{m}}{\left(\frac{\partial F}{\partial {\sigma }_{xx}}+\frac{\partial F}{\partial {\sigma }_{yy}}\right){\sigma }_{45}}-1-{R}_{45}=0$

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.