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)

$$F=a{\left|K_1+K_2\right|}^m+a{\left|K_1-K_2\right|}^m+c{\left|2K_2\right|}^m-2{\left({\sigma }_e\right)}^m$$

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)

$$\begin{array}{l}{K}_1=\frac{{\sigma }_{xx}-h{\sigma }_{yy}}{2}\\ K_2=\sqrt{{\left(\frac{{\sigma }_{xx}-h{\sigma }_{yy}}{2}\right)}^2+p^2{\left({\sigma }_{xy}\right)}^2}\end{array}$$

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°, Equation 4 gives an equation from which the anisotropy parameter $p$ can be computed implicitly by using an iterative procedure:(5)

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