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 {(σ_{e})}^{m}

Where,

σ_{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{σ_{x x} + h σ_{y y}}{2}

$K_{2} = \sqrt{{(\frac{σ_{x x} - h σ_{y y}}{2})}^{2} + p^{2} {(σ_{x y})}^{2}}$

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{(\frac{r_{00}}{1 + r_{00}}) (\frac{r_{90}}{1 + r_{90}})}

$c = 2 - a$

$h = \sqrt{(\frac{r_{00}}{1 + r_{00}}) (\frac{1 + r_{90}}{r_{90}})}$

The width to thickness ratio for any angle

φ

can be calculated: ¹

(4)

R_{φ} = \frac{2 m {(σ_{e})}^{m}}{(\frac{\partial F}{\partial σ_{x x}} + \frac{\partial F}{\partial σ_{y y}}) σ_{φ}} - 1

Where,

σ_{φ}

is the uniaxial tension in the

φ

direction. Let

φ

= 45°, Equation 4 gives an equation from which the anisotropy parameter

p

can be computed implicitly by using an iterative procedure:(5)

\frac{2 m {(σ_{e})}^{m}}{(\frac{\partial F}{\partial σ_{x x}} + \frac{\partial F}{\partial σ_{y y}}) σ_{45}} - 1 - r_{45} = 0

Note: Barlat's law reduces to Hill's law when using

m

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.