Hill yield criteria

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Rodney Hill has developed several yield criteria for anisotropic plastic deformations. The earliest version was a straightforward extension of the von Mises yield criterion and had a quadratic form. This model was later generalized by allowing for an exponent m. Variations of these criteria are in wide use for metals, polymers, and certain composites.

1 Quadratic Hill yield criterion
- 1.1 Expressions for F, G, H, L, M, N
- 1.2 Quadratic Hill yield criterion for plane stress
2 Generalized Hill yield criterion
- 2.1 Generalized Hill yield criterion for plane stress
3 Hill 1993 yield criterion
4 Extensions of Hill's yield criteria
- 4.1 The Caddell-Raghava-Atkins yield criterion
- 4.2 The Deshpande-Fleck-Ashby yield criterion
5 References
6 External links

[edit] Quadratic Hill yield criterion

The quadratic Hill yield criterion ^[1]. has the form

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

Here F, G, H, L, M, N are constants that have to be determined experimentally and $σ i j$ are the stresses. The quadratic Hill yield criterion depends only on the deviatoric/shear stresses and is pressure independent. It predicts the same yield stress in tension and in compression.

[edit] Expressions for F, G, H, L, M, N

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

$(G + H)~(\sigma_1^y)^2 = 1 ~;~~ (F + H)~(\sigma_2^y)^2 = 1 ~;~~ (F + G)~(\sigma_3^y)^2 = 1$

where $\sigma_1^y, \sigma_2^y, \sigma_3^y$ are the normal yield stresses with respect to the axes of anisotropy. Therefore we have

$F = \cfrac{1}{2}\left[\cfrac{1}{(\sigma_2^y)^2} + \cfrac{1}{(\sigma_3^y)^2} - \cfrac{1}{(\sigma_1^y)^2}\right]$

$G = \cfrac{1}{2}\left[\cfrac{1}{(\sigma_3^y)^2} + \cfrac{1}{(\sigma_1^y)^2} - \cfrac{1}{(\sigma_2^y)^2}\right]$

$H = \cfrac{1}{2}\left[\cfrac{1}{(\sigma_1^y)^2} + \cfrac{1}{(\sigma_2^y)^2} - \cfrac{1}{(\sigma_3^y)^2}\right]$

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

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

[edit] Quadratic Hill yield criterion for plane stress

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

$\sigma_1^2 + \cfrac{R_0~(1+R_{90})}{R_{90}~(1+R_0)}~\sigma_2^2 - \cfrac{2~R_0}{1+R_0}~\sigma_1\sigma_2 = (\sigma_1^y)^2$

where the principal stresses $σ 1,σ 2$ are assumed to be aligned with the axes of anisotropy with $σ 1$ in the rolling direction and $σ 2$ perpendicular to the rolling direction, $σ 3 = 0$ , $R 0$ is the R-value in the rolling direction, and $R 90$ is the R-value perpendicular to the rolling direction.

For the special case of transverse isotropy we have $R = R 0 = R 90$ and we get

$\sigma_1^2 + \sigma_2^2 - \cfrac{2~R}{1+R}~\sigma_1\sigma_2 = (\sigma_1^y)^2$

Derivation of Hill's criterion for plane stress

For the situation where the principal stresses are aligned with the directions of anisotropy we have

$f := F(\sigma_2-\sigma_3)^2 + G(\sigma_3-\sigma_1)^2 + H(\sigma_1-\sigma_2)^2 - 1 = 0 \,$

where $σ 1,σ 2,σ 3$ are the principal stresses. If we assume an associated flow rule we have

$\epsilon^p_i = \lambda~\cfrac{\partial f}{\partial \sigma_i} \qquad \implies \qquad \cfrac{d\epsilon^p_i}{d\lambda} = \cfrac{\partial f}{\partial \sigma_i} ~.$

This implies that

$\begin{align} \cfrac{d\epsilon^p_1}{d\lambda} &= 2(G+H)\sigma_1 - 2H\sigma_2 - 2G\sigma_3 \\ \cfrac{d\epsilon^p_2}{d\lambda} &= 2(F+H)\sigma_2 - 2H\sigma_1 - 2F\sigma_3 \\ \cfrac{d\epsilon^p_1}{d\lambda} &= 2(F+G)\sigma_3 - 2G\sigma_1 - 2F\sigma_2 ~. \end{align}$

For plane stress $σ 3 = 0$ , which gives

$\begin{align} \cfrac{d\epsilon^p_1}{d\lambda} &= 2(G+H)\sigma_1 - 2H\sigma_2\\ \cfrac{d\epsilon^p_2}{d\lambda} &= 2(F+H)\sigma_2 - 2H\sigma_1\\ \cfrac{d\epsilon^p_1}{d\lambda} &= - 2G\sigma_1 - 2F\sigma_2 ~. \end{align}$

The R-value $R 0$ is defined as the ratio of the in-plane and out-of-plane plastic strains under uniaxial stress $σ 1$ . The quantity $R 90$ is the plastic strain ratio under uniaxial stress $σ 2$ . Therefore, we have

$R_0 = \cfrac{d\epsilon^p_2}{d\epsilon^p_3} = \cfrac{H}{G} ~;~~ R_{90} = \cfrac{d\epsilon^p_1}{d\epsilon^p_3} = \cfrac{H}{F} ~.$

Then, using $H = R 0 G$ and $s i g m a 3 = 0$ , the yield condition can be written as

$f := F \sigma_2^2 + G \sigma_1^2 + R_0 G(\sigma_1-\sigma_2)^2 - 1 = 0 \,$

which in turn may be expressed as

$\sigma_1^2 + \cfrac{F+R_0 G}{G(1+R_0)}~\sigma_2^2 - \cfrac{2R_0}{1+R_0}~\sigma_1\sigma_2 = \cfrac{1}{(1+R_0)G}~.$

This is of the same form as the required expression. All we have to do is to express $F, G$ in terms of $\sigma_1^y$ . Recall that,

$\begin{align} F & = \cfrac{1}{2}\left[\cfrac{1}{(\sigma_2^y)^2} + \cfrac{1}{(\sigma_3^y)^2} - \cfrac{1}{(\sigma_1^y)^2} \right] \\ G & = \cfrac{1}{2}\left[\cfrac{1}{(\sigma_3^y)^2} + \cfrac{1}{(\sigma_1^y)^2} - \cfrac{1}{(\sigma_2^y)^2} \right] \\ H & = \cfrac{1}{2}\left[\cfrac{1}{(\sigma_1^y)^2} + \cfrac{1}{(\sigma_2^y)^2} - \cfrac{1}{(\sigma_3^y)^2} \right] \end{align}$

We can use these to obtain

$\begin{align} R_0 = \cfrac{H}{G} & \implies (1+R_0)\cfrac{1}{(\sigma_3^y)^2} - (1+R_0)\cfrac{1}{(\sigma_2^y)^2} = (1-R_0)\cfrac{1}{(\sigma_1^y)^2} \\ R_{90} = \cfrac{H}{F} & \implies (1+R_{90})\cfrac{1}{(\sigma_3^y)^2} - (1-R_{90})\cfrac{1}{(\sigma_2^y)^2} = (1+R_{90})\cfrac{1}{(\sigma_1^y)^2} \end{align}$

Solving for $\cfrac{1}{(\sigma_3^y)^2}, \cfrac{1}{(\sigma_2^y)^2}$ gives us

$\cfrac{1}{(\sigma_3^y)^2} = \cfrac{R_0+R_{90}}{(1+R_0)~R_{90}}~\cfrac{1}{(\sigma_1^y)^2} ~;~~ \cfrac{1}{(\sigma_2^y)^2} = \cfrac{R_0(1+R_{90)}}{(1+R_0)~R_{90}}~\cfrac{1}{(\sigma_1^y)^2}$

Plugging back into the expressions for $F, G$ leads to

$F = \cfrac{R_0}{(1+R_0)~R_{90}}~\cfrac{1}{(\sigma_1^y)^2} ~;~~ G = \cfrac{1}{1+R_0}~\cfrac{1}{(\sigma_1^y)^2}$

which implies that

$\cfrac{1}{G(1+R_0)} = (\sigma_1^y)^2 ~;~~ \cfrac{F+R_0 G}{G(1+R_0)} = \cfrac{R_0(1+R_{90})}{R_{90}(1+R_0)} ~.$

Therefore the plane stress form of the quadratic Hill yield criterion can be expressed as

$\sigma_1^2 + \cfrac{R_0~(1+R_{90})}{R_{90}~(1+R_0)}~\sigma_2^2 - \cfrac{2~R_0}{1+R_0}~\sigma_1\sigma_2 = (\sigma_1^y)^2$

[edit] Generalized Hill yield criterion

The generalized Hill yield criterion^[2] has the form

$\begin{align} F|\sigma_{2}-\sigma_{3}|^m & + G|\sigma_{3}-\sigma_{1}|^m + H|\sigma_{1}-\sigma_{2}|^m + L|2\sigma_1 - \sigma_2 - \sigma_3|^m \\ & + M|2\sigma_2 - \sigma_3 - \sigma_1|^m + N|2\sigma_3 - \sigma_1 - \sigma_2|^m = \sigma_y^m ~. \end{align}$

where $σ i$ are the principal stresses (which are aligned with the directions of anisotropy), $σ y$ is the yield stress, and F, G, H, L, M, N are constants. The value of m is determined by the degree of anisotropy of the material and must be greater than 1 to ensure convexity of the yield surface.

[edit] Generalized Hill yield criterion for plane stress

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

$\begin{align} f := & F|\sigma_2-\sigma_3|^m + F|\sigma_3-\sigma_1|^m + H|\sigma_1-\sigma_2|^m + L|2\sigma_1 - \sigma_2 - \sigma_3|^m \\ & + L|2\sigma_2-\sigma_3-\sigma_1|^m + N|2\sigma_3-\sigma_1-\sigma_2|^m - \sigma_y^m \le 0 \end{align}$

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

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

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

Case 1: $L = 0, H = 0.$

$f:= \cfrac{1+2R}{1+R}(|\sigma_1|^m + |\sigma_2|^m) - \cfrac{R}{1+R} |\sigma_1 + \sigma_2|^m - \sigma_y^m \le 0$

Case 2: $N = 0, F = 0.$

$f:= \cfrac{2^{m-1}(1-R)+(R+2)}{(1-2^{m-1})(1+R)}|\sigma_1 -\sigma_2|^m - \cfrac{1}{(1-2^{m-1})(1+R)} (|2\sigma_1 - \sigma_2|^m + |2\sigma_2-\sigma_1|^m)- \sigma_y^m \le 0$

Case 3: $N = 0, H = 0.$

$f:= \cfrac{2^{m-1}(1-R)+(R+2)}{(2+2^{m-1})(1+R)}(|\sigma_1|^m -|\sigma_2|^m) + \cfrac{R}{(2+2^{m-1})(1+R)} (|2\sigma_1 - \sigma_2|^m + |2\sigma_2-\sigma_1|^m)- \sigma_y^m \le 0$

Case 4: $L = 0, F = 0.$

$f:= \cfrac{1+2R}{2(1+R)}|\sigma_1 - \sigma_2|^m + \cfrac{1}{2(1+R)} |\sigma_1 + \sigma_2|^m - \sigma_y^m \le 0$

Case 5: $L = 0, N = 0.$ . This is the Hosford yield criterion.

$f := \cfrac{1}{1+R}(|\sigma_1|^m + |\sigma_2|^m) + \cfrac{R}{1+R}|\sigma_1-\sigma_2|^m - \sigma_y^m \le 0$

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

R

and

m

^[4].

[edit] Hill 1993 yield criterion

In 1993, Hill proposed another yield criterion ^[5] for plane stress problems with planar anisotropy. The Hill93 criterion has the form

$\left(\cfrac{\sigma_1}{\sigma_0}\right)^2 + \left(\cfrac{\sigma_2}{\sigma_{90}}\right)^2 + \left[ (p + q - c) - \cfrac{p\sigma_1+q\sigma_2}{\sigma_b}\right]\left(\cfrac{\sigma_1\sigma_2}{\sigma_0\sigma_{90}}\right) = 1$

where $σ 0$ is the uniaxial tensile yield stress in the rolling direction, $σ 90$ is the uniaxial tensile yield stress in the direction normal to the rolling direction, $σ b$ is the yield stress under uniform biaxial tension, and $c, p, q$ are parameters defined as

$\begin{align} c & = \cfrac{\sigma_0}{\sigma_{90}} + \cfrac{\sigma_{90}}{\sigma_0} - \cfrac{\sigma_0\sigma_{90}}{\sigma_b^2} \\ \left(\cfrac{1}{\sigma_0}+\cfrac{1}{\sigma_{90}}-\cfrac{1}{\sigma_b}\right)~p & = \cfrac{2 R_0 (\sigma_b-\sigma_{90})}{(1+R_0)\sigma_0^2} - \cfrac{2 R_{90} \sigma_b}{(1+R_{90})\sigma_{90}^2} + \cfrac{c}{\sigma_0} \\ \left(\cfrac{1}{\sigma_0}+\cfrac{1}{\sigma_{90}}-\cfrac{1}{\sigma_b}\right)~q & = \cfrac{2 R_{90} (\sigma_b-\sigma_{0})}{(1+R_{90})\sigma_{90}^2} - \cfrac{2 R_{0} \sigma_b}{(1+R_{0})\sigma_{0}^2} + \cfrac{c}{\sigma_{90}} \end{align}$

and $R 0$ is the R-value for uniaxial tension in the rolling direction, and $R 90$ is the R-value for uniaxial tension in the in-plane direction perpendicular to the rolling direction.

[edit] Extensions of Hill's yield criteria

The original versions of Hill's yield criteria were designed for material that did not have pressure-dependent yield surfaces which are needed to model polymers and foams.

[edit] The Caddell-Raghava-Atkins yield criterion

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

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

[edit] The Deshpande-Fleck-Ashby yield criterion

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

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

[edit] References

^ R. Hill. (1948). A theory of the yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. London, 193:281–297
^ R. Hill. (1979). Theoretical plasticity of textured aggregates. Math. Proc. Camb. Phil. Soc., 85(1):179–191.
^ Chu, E. (1995). Generalization of Hill's 1979 anisotropic yield criteria. Journal of Materials Processing Technology, vol. 50, pp. 207-215.
^ Zhu, Y., Dodd, B., Caddell, R. M. and Hosford, W. F. (1987). Limitations of Hill's 1979 anisotropic yield criterion. International Journal of Mechanical Sciences, vol. 29, pp. 733.
^ Hill. R. (1993). User-friendly theory of orthotropic plasticity in sheet metals. International Journal of Mechanical Sciences, vol. 35, no. 1, pp. 19–25.
^ Caddell, R. M., Raghava, R. S. and Atkins, A. G., (1973), Yield criterion for anisotropic and pressure dependent solids such as oriented polymers. Journal of Materials Science, vol. 8, no. 11, pp. 1641-1646.
^ Deshpande, V. S., Fleck, N. A. and Ashby, M. F. (2001). Effective properties of the octet-truss lattice material. Journal of the Mechanics and Physics of Solids, vol. 49, no. 8, pp. 1747-1769.