Hill yield criterion

If the axes of material anisotropy are assumed to be orthogonal, we can write

${\displaystyle (G+H)~(\sigma _{1}^{y})^{2}=1~;~~(F+H)~(\sigma _{2}^{y})^{2}=1~;~~(F+G)~(\sigma _{3}^{y})^{2}=1}$

where ${\displaystyle \sigma _{1}^{y},\sigma _{2}^{y},\sigma _{3}^{y}}$ are the normal yield stresses with respect to the axes of anisotropy. Therefore we have

${\displaystyle F={\cfrac {1}{2}}\left[{\cfrac {1}{(\sigma _{2}^{y})^{2}}}+{\cfrac {1}{(\sigma _{3}^{y})^{2}}}-{\cfrac {1}{(\sigma _{1}^{y})^{2}}}\right]}$

${\displaystyle G={\cfrac {1}{2}}\left[{\cfrac {1}{(\sigma _{3}^{y})^{2}}}+{\cfrac {1}{(\sigma _{1}^{y})^{2}}}-{\cfrac {1}{(\sigma _{2}^{y})^{2}}}\right]}$

${\displaystyle H={\cfrac {1}{2}}\left[{\cfrac {1}{(\sigma _{1}^{y})^{2}}}+{\cfrac {1}{(\sigma _{2}^{y})^{2}}}-{\cfrac {1}{(\sigma _{3}^{y})^{2}}}\right]}$

Similarly, if ${\displaystyle \tau _{12}^{y},\tau _{23}^{y},\tau _{31}^{y}}$ are the yield stresses in shear (with respect to the axes of anisotropy), we have

${\displaystyle L={\cfrac {1}{2~(\tau _{23}^{y})^{2}}}~;~~M={\cfrac {1}{2~(\tau _{31}^{y})^{2}}}~;~~N={\cfrac {1}{2~(\tau _{12}^{y})^{2}}}}$

The quadratic Hill yield criterion for thin rolled plates (plane stress conditions) can be expressed as

${\displaystyle \sigma _{1}^{2}+{\cfrac {R_{0}~(1+R_{90})}{R_{90}~(1+R_{0})}}~\sigma _{2}^{2}-{\cfrac {2~R_{0}}{1+R_{0}}}~\sigma _{1}\sigma _{2}=(\sigma _{1}^{y})^{2}}$

where the principal stresses ${\displaystyle \sigma _{1},\sigma _{2}}$ are assumed to be aligned with the axes of anisotropy with ${\displaystyle \sigma _{1}}$ in the rolling direction and ${\displaystyle \sigma _{2}}$ perpendicular to the rolling direction, ${\displaystyle \sigma _{3}=0}$, ${\displaystyle R_{0}}$ is the R-value in the rolling direction, and ${\displaystyle R_{90}}$ is the R-value perpendicular to the rolling direction.

For the special case of transverse isotropy we have ${\displaystyle R=R_{0}=R_{90}}$ and we get

${\displaystyle \sigma _{1}^{2}+\sigma _{2}^{2}-{\cfrac {2~R}{1+R}}~\sigma _{1}\sigma _{2}=(\sigma _{1}^{y})^{2}}$

Derivation of Hill's criterion for plane stress

For the situation where the principal stresses are aligned with the directions of anisotropy we have ${\displaystyle f:=F(\sigma _{2}-\sigma _{3})^{2}+G(\sigma _{3}-\sigma _{1})^{2}+H(\sigma _{1}-\sigma _{2})^{2}-1=0\,}$ where ${\displaystyle \sigma _{1},\sigma _{2},\sigma _{3}}$ are the principal stresses. If we assume an associated flow rule we have ${\displaystyle {\dot {\varepsilon }}_{i}^{p}={\dot {\lambda }}~{\cfrac {\partial f}{\partial \sigma _{i}}}\qquad \implies \qquad {\cfrac {d\varepsilon _{i}^{p}}{d\lambda }}={\cfrac {\partial f}{\partial \sigma _{i}}}~.}$ This implies that ${\displaystyle {\begin{aligned}{\cfrac {d\varepsilon _{1}^{p}}{d\lambda }}&=2(G+H)\sigma _{1}-2H\sigma _{2}-2G\sigma _{3}\\{\cfrac {d\varepsilon _{2}^{p}}{d\lambda }}&=2(F+H)\sigma _{2}-2H\sigma _{1}-2F\sigma _{3}\\{\cfrac {d\varepsilon _{3}^{p}}{d\lambda }}&=2(F+G)\sigma _{3}-2G\sigma _{1}-2F\sigma _{2}~.\end{aligned}}}$ For plane stress ${\displaystyle \sigma _{3}=0}$, which gives ${\displaystyle {\begin{aligned}{\cfrac {d\varepsilon _{1}^{p}}{d\lambda }}&=2(G+H)\sigma _{1}-2H\sigma _{2}\\{\cfrac {d\varepsilon _{2}^{p}}{d\lambda }}&=2(F+H)\sigma _{2}-2H\sigma _{1}\\{\cfrac {d\varepsilon _{3}^{p}}{d\lambda }}&=-2G\sigma _{1}-2F\sigma _{2}~.\end{aligned}}}$ The R-value ${\displaystyle R_{0}}$ is defined as the ratio of the in-plane and out-of-plane plastic strains under uniaxial stress ${\displaystyle \sigma _{1}}$. The quantity ${\displaystyle R_{90}}$ is the plastic strain ratio under uniaxial stress ${\displaystyle \sigma _{2}}$. Therefore, we have ${\displaystyle R_{0}={\cfrac {d\varepsilon _{2}^{p}}{d\varepsilon _{3}^{p}}}={\cfrac {H}{G}}~;~~R_{90}={\cfrac {d\varepsilon _{1}^{p}}{d\varepsilon _{3}^{p}}}={\cfrac {H}{F}}~.}$ Then, using ${\displaystyle H=R_{0}G}$ and ${\displaystyle \sigma _{3}=0}$, the yield condition can be written as ${\displaystyle f:=F\sigma _{2}^{2}+G\sigma _{1}^{2}+R_{0}G(\sigma _{1}-\sigma _{2})^{2}-1=0\,}$ which in turn may be expressed as ${\displaystyle \sigma _{1}^{2}+{\cfrac {F+R_{0}G}{G(1+R_{0})}}~\sigma _{2}^{2}-{\cfrac {2R_{0}}{1+R_{0}}}~\sigma _{1}\sigma _{2}={\cfrac {1}{(1+R_{0})G}}~.}$ This is of the same form as the required expression. All we have to do is to express ${\displaystyle F,G}$ in terms of ${\displaystyle \sigma _{1}^{y}}$. Recall that, ${\displaystyle {\begin{aligned}F&={\cfrac {1}{2}}\left[{\cfrac {1}{(\sigma _{2}^{y})^{2}}}+{\cfrac {1}{(\sigma _{3}^{y})^{2}}}-{\cfrac {1}{(\sigma _{1}^{y})^{2}}}\right]\\G&={\cfrac {1}{2}}\left[{\cfrac {1}{(\sigma _{3}^{y})^{2}}}+{\cfrac {1}{(\sigma _{1}^{y})^{2}}}-{\cfrac {1}{(\sigma _{2}^{y})^{2}}}\right]\\H&={\cfrac {1}{2}}\left[{\cfrac {1}{(\sigma _{1}^{y})^{2}}}+{\cfrac {1}{(\sigma _{2}^{y})^{2}}}-{\cfrac {1}{(\sigma _{3}^{y})^{2}}}\right]\end{aligned}}}$ We can use these to obtain ${\displaystyle {\begin{aligned}R_{0}={\cfrac {H}{G}}&\implies (1+R_{0}){\cfrac {1}{(\sigma _{3}^{y})^{2}}}-(1+R_{0}){\cfrac {1}{(\sigma _{2}^{y})^{2}}}=(1-R_{0}){\cfrac {1}{(\sigma _{1}^{y})^{2}}}\\R_{90}={\cfrac {H}{F}}&\implies (1+R_{90}){\cfrac {1}{(\sigma _{3}^{y})^{2}}}-(1-R_{90}){\cfrac {1}{(\sigma _{2}^{y})^{2}}}=(1+R_{90}){\cfrac {1}{(\sigma _{1}^{y})^{2}}}\end{aligned}}}$ Solving for ${\displaystyle {\cfrac {1}{(\sigma _{3}^{y})^{2}}},{\cfrac {1}{(\sigma _{2}^{y})^{2}}}}$ gives us ${\displaystyle {\cfrac {1}{(\sigma _{3}^{y})^{2}}}={\cfrac {R_{0}+R_{90}}{(1+R_{0})~R_{90}}}~{\cfrac {1}{(\sigma _{1}^{y})^{2}}}~;~~{\cfrac {1}{(\sigma _{2}^{y})^{2}}}={\cfrac {R_{0}(1+R_{90})}{(1+R_{0})~R_{90}}}~{\cfrac {1}{(\sigma _{1}^{y})^{2}}}}$ Plugging back into the expressions for ${\displaystyle F,G}$ leads to ${\displaystyle F={\cfrac {R_{0}}{(1+R_{0})~R_{90}}}~{\cfrac {1}{(\sigma _{1}^{y})^{2}}}~;~~G={\cfrac {1}{1+R_{0}}}~{\cfrac {1}{(\sigma _{1}^{y})^{2}}}}$ which implies that ${\displaystyle {\cfrac {1}{G(1+R_{0})}}=(\sigma _{1}^{y})^{2}~;~~{\cfrac {F+R_{0}G}{G(1+R_{0})}}={\cfrac {R_{0}(1+R_{90})}{R_{90}(1+R_{0})}}~.}$ Therefore the plane stress form of the quadratic Hill yield criterion can be expressed as ${\displaystyle \sigma _{1}^{2}+{\cfrac {R_{0}~(1+R_{90})}{R_{90}~(1+R_{0})}}~\sigma _{2}^{2}-{\cfrac {2~R_{0}}{1+R_{0}}}~\sigma _{1}\sigma _{2}=(\sigma _{1}^{y})^{2}}$

For transversely isotropic materials with ${\displaystyle 1-2}$ being the plane of symmetry, the generalized Hill yield criterion reduces to (with ${\displaystyle F=G}$ and ${\displaystyle L=M}$)

${\displaystyle {\begin{aligned}f:=&F|\sigma _{2}-\sigma _{3}|^{m}+G|\sigma _{3}-\sigma _{1}|^{m}+H|\sigma _{1}-\sigma _{2}|^{m}+L|2\sigma _{1}-\sigma _{2}-\sigma _{3}|^{m}\\&+L|2\sigma _{2}-\sigma _{3}-\sigma _{1}|^{m}+N|2\sigma _{3}-\sigma _{1}-\sigma _{2}|^{m}-\sigma _{y}^{m}\leq 0\end{aligned}}}$

The R-value or Lankford coefficient can be determined by considering the situation where ${\displaystyle \sigma _{1}>(\sigma _{2}=\sigma _{3}=0)}$. The R-value is then given by

${\displaystyle R={\cfrac {(2^{m-1}+2)L-N+H}{(2^{m-1}-1)L+2N+F}}~.}$

Under plane stress conditions and with some assumptions, the generalized Hill criterion can take several forms.[3]

• Case 1: ${\displaystyle L=0,H=0.}$

${\displaystyle f:={\cfrac {1+2R}{1+R}}(|\sigma _{1}|^{m}+|\sigma _{2}|^{m})-{\cfrac {R}{1+R}}|\sigma _{1}+\sigma _{2}|^{m}-\sigma _{y}^{m}\leq 0}$

• Case 2: ${\displaystyle N=0,F=0.}$

${\displaystyle f:={\cfrac {2^{m-1}(1-R)+(R+2)}{(1-2^{m-1})(1+R)}}|\sigma _{1}-\sigma _{2}|^{m}-{\cfrac {1}{(1-2^{m-1})(1+R)}}(|2\sigma _{1}-\sigma _{2}|^{m}+|2\sigma _{2}-\sigma _{1}|^{m})-\sigma _{y}^{m}\leq 0}$

• Case 3: ${\displaystyle N=0,H=0.}$

${\displaystyle f:={\cfrac {2^{m-1}(1-R)+(R+2)}{(2+2^{m-1})(1+R)}}(|\sigma _{1}|^{m}-|\sigma _{2}|^{m})+{\cfrac {R}{(2+2^{m-1})(1+R)}}(|2\sigma _{1}-\sigma _{2}|^{m}+|2\sigma _{2}-\sigma _{1}|^{m})-\sigma _{y}^{m}\leq 0}$

• Case 4: ${\displaystyle L=0,F=0.}$

${\displaystyle f:={\cfrac {1+2R}{2(1+R)}}|\sigma _{1}-\sigma _{2}|^{m}+{\cfrac {1}{2(1+R)}}|\sigma _{1}+\sigma _{2}|^{m}-\sigma _{y}^{m}\leq 0}$

• Case 5: ${\displaystyle L=0,N=0.}$. This is the Hosford yield criterion.

${\displaystyle f:={\cfrac {1}{1+R}}(|\sigma _{1}|^{m}+|\sigma _{2}|^{m})+{\cfrac {R}{1+R}}|\sigma _{1}-\sigma _{2}|^{m}-\sigma _{y}^{m}\leq 0}$

Care must be exercised in using these forms of the generalized Hill yield criterion because the yield surfaces become concave (sometimes even unbounded) for certain combinations of ${\displaystyle R}$ and ${\displaystyle m}$.[4]

An extension that allows for pressure dependence is Caddell–Raghava–Atkins (CRA) model [6] which has the form

${\displaystyle F(\sigma _{22}-\sigma _{33})^{2}+G(\sigma _{33}-\sigma _{11})^{2}+H(\sigma _{11}-\sigma _{22})^{2}+2L\sigma _{23}^{2}+2M\sigma _{31}^{2}+2N\sigma _{12}^{2}+I\sigma _{11}+J\sigma _{22}+K\sigma _{33}=1~.}$

Another pressure-dependent extension of Hill's quadratic yield criterion which has a form similar to the Bresler Pister yield criterion is the Deshpande, Fleck and Ashby (DFA) yield criterion [7] for honeycomb structures (used in sandwich composite construction). This yield criterion has the form

${\displaystyle F(\sigma _{22}-\sigma _{33})^{2}+G(\sigma _{33}-\sigma _{11})^{2}+H(\sigma _{11}-\sigma _{22})^{2}+2L\sigma _{23}^{2}+2M\sigma _{31}^{2}+2N\sigma _{12}^{2}+K(\sigma _{11}+\sigma _{22}+\sigma _{33})^{2}=1~.}$