Stress potentials for anisotropic metal plasticity

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The Mises stress potential is

$f (σ) = q,$

where

$q = \sqrt{\frac{3}{2} S : S},$

in which $S$ is the deviatoric stress:

$S = σ - \frac{1}{3} trace (σ) I = σ - \frac{1}{3} I I : (σ) .$

The potential is a circle in the plane normal to the hydrostatic axis in principal stress space. For this function,

$\frac{\partial f}{\partial σ} = \frac{1}{q} \frac{3}{2} S,$

and

$\frac{\partial^{2} f}{\partial σ \partial σ} = \frac{1}{q} (\frac{3}{2} ℑ - \frac{1}{2} I I - \frac{\partial f}{\partial σ} \frac{\partial f}{\partial σ})$

in which $ℑ$ is the fourth-order unit tensor.

Hill's stress function is a simple extension of the Mises function to allow anisotropic behavior. The function is

$f (σ) = \sqrt{F {(σ_{y} - σ_{z})}^{2} + G {(σ_{z} - σ_{x})}^{2} + H {(σ_{x} - σ_{y})}^{2} + 2 L τ_{y z}^{2} + 2 M τ_{z x}^{2} + 2 N τ_{x y}^{2}},$

in terms of rectangular Cartesian stress components, where $F, G, H, L, M, N$ are constants obtained by tests of the material in different orientations. They are defined as

$F = \frac{σ_{0}^{2}}{2} (\frac{1}{{\bar{σ}}_{22}^{2}} + \frac{1}{{\bar{σ}}_{33}^{2}} - \frac{1}{{\bar{σ}}_{11}^{2}}),$

$G = \frac{σ_{0}^{2}}{2} (\frac{1}{{\bar{σ}}_{33}^{2}} + \frac{1}{{\bar{σ}}_{11}^{2}} - \frac{1}{{\bar{σ}}_{22}^{2}}),$

$H = \frac{σ_{0}^{2}}{2} (\frac{1}{{\bar{σ}}_{11}^{2}} + \frac{1}{{\bar{σ}}_{22}^{2}} - \frac{1}{{\bar{σ}}_{33}^{2}}),$

$L = \frac{3}{2} {(\frac{τ_{0}}{{\bar{τ}}_{23}})}^{2},$

$M = \frac{3}{2} {(\frac{τ_{0}}{{\bar{τ}}_{13}})}^{2},$

$N = \frac{3}{2} {(\frac{τ_{0}}{{\bar{τ}}_{12}})}^{2},$

where $σ_{0}, {\bar{σ}}_{11}, {\bar{σ}}_{22}, {\bar{σ}}_{33}, {\bar{τ}}_{12}, {\bar{τ}}_{23}, {\bar{τ}}_{13}$ are specified by the user and $τ_{0} = σ_{0} / \sqrt{3}$ . $\bar{σ}$ and $\bar{τ}$ are the values of stress that make the potential equal to $σ_{0}$ if only one stress component is nonzero.

For this function

$\frac{\partial f}{\partial σ} = \frac{1}{f} b,$

where

$b = [\begin{matrix} - G (σ_{z} - σ_{x}) + H (σ_{x} - σ_{y}) \\ F (σ_{y} - σ_{z}) - H (σ_{x} - σ_{y}) \\ - F (σ_{y} - σ_{z}) + G (σ_{z} - σ_{x}) \\ 2 N τ_{x y} \\ 2 M τ_{z x} \\ 2 L τ_{y z} \end{matrix}] .$

In addition,

$\frac{\partial^{2} f}{\partial σ \partial σ} = \frac{1}{f} (\frac{\partial b}{\partial σ} - \frac{1}{f^{2}} b b),$

where

$\frac{\partial b}{\partial σ} = [\begin{matrix} G + H & - H & - G & 0 & 0 & 0 \\ - H & F + H & - F & 0 & 0 & 0 \\ - G & - F & F + G & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 N & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 M & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 L \end{matrix}] .$