Hill yield criterion explained

The Hill yield criterion developed by Rodney Hill, is one of several yield criteria for describing 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.

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} + 2 L\sigma ^{2} _{23} + 2 M\sigma ^{2} _{31} + 2 N\sigma ^{2} _{12} = 1 ~. Here F, G, H, L, M, N are constants that have to be determined experimentally and \sigma _{ij} are the stresses. The quadratic Hill yield criterion depends only on the deviatoric stresses and is pressure independent. It predicts the same yield stress in tension and in compression.

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 ^{y)} _{1} ^{2} = 1 ~;~~ (F + H)~(\sigma ^{y)} _{2} ^{2} = 1 ~;~~ (F + G)~(\sigma ^{y)} _{3} ^{2} = 1 where \sigma ^{y,} _{1} \sigma ^{y,} _{2} \sigma ^{y} _{3} are the normal yield stresses with respect to the axes of anisotropy. Therefore we haveF = \cfrac{ 1 }{ 2 }\left[\cfrac{ 1 }{(\sigma ^{y)} _{2} ^{2}} + \cfrac{ 1 }{(\sigma ^{y)} _{3} ^{2}} - \cfrac{ 1 }{(\sigma ^{y)} _{1} ^{2}\right]} G = \cfrac{ 1 }{ 2 }\left[\cfrac{ 1 }{(\sigma ^{y)} _{3} ^{2}} + \cfrac{ 1 }{(\sigma ^{y)} _{1} ^{2}} - \cfrac{ 1 }{(\sigma ^{y)} _{2} ^{2}\right]} H = \cfrac{ 1 }{ 2 }\left[\cfrac{ 1 }{(\sigma ^{y)} _{1} ^{2}} + \cfrac{ 1 }{(\sigma ^{y)} _{2} ^{2}} - \cfrac{ 1 }{(\sigma ^{y)} _{3} ^{2}\right]} Similarly, if \tau ^{y,} _{12} \tau ^{y,} _{23} \tau ^{y} _{31} are the yield stresses in shear (with respect to the axes of anisotropy), we haveL = \cfrac{ 1 }{ 2 ~(\tau ^{y)} _{23} ^{2}} ~;~~ M = \cfrac{ 1 }{ 2 ~(\tau ^{y)} _{31} ^{2}} ~;~~ N = \cfrac{ 1 }{ 2 ~(\tau ^{y)} _{12} ^{2}}

Quadratic Hill yield criterion for plane stress

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

\sigma ^{2} _{1} + \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 ^{y)} _{1} ^{2} where the principal stresses \sigma _{1,} \sigma _{2} are assumed to be aligned with the axes of anisotropy with \sigma _{1} in the rolling direction and \sigma _{2} perpendicular to the rolling direction, \sigma _{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 ^{2} _{1} + \sigma ^{2} _{2} - \cfrac{ 2 ~R}{ 1+ R}~\sigma _{1\sigma} _{2} = (\sigma ^{y)} _{1} ^{2}

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 ^{m} _{3 |} \\ & + M| 2 \sigma _{2} - \sigma _{3} - \sigma ^{m} _{1 |} + N| 2 \sigma _{3} - \sigma _{1} - \sigma ^{m} _{2 |} = \sigma ^{m} _{y} ~. \end{align} where \sigma _{i} are the principal stresses (which are aligned with the directions of anisotropy), \sigma _{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.

Generalized Hill yield criterion for anisotropic material

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} ^{m} ^{3 |} + G|\sigma _{3-\sigma} ^{m} ^{1 |} + H|\sigma _{1-\sigma} ^{m} ^{2 |} + L| 2 \sigma _{1} - \sigma _{2} - \sigma ^{m} _{3 |} \\ & + L| 2 \sigma _{2-\sigma} _{3-\sigma} ^{m} ^{1 |} + N| 2 \sigma _{3-\sigma} _{1-\sigma} ^{m} ^{2 |} - \sigma ^{m} _{y} \le 0 \end{align} The R-value or Lankford coefficient can be determined by considering the situation where \sigma _{1} > (\sigma _{2} = \sigma _{3} = 0 ) . The R-value is then given byR = \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+2 R}{ 1+ R}(|\sigma ^{m} _{1 |} + |\sigma ^{m)} _{2 |} - \cfrac{R}{ 1+ R} |\sigma _{1} + \sigma ^{m} _{2 |} - \sigma ^{m} _{y} \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 ^{m} _{2 |} - \cfrac{ 1 }{( 1-2 ^{m-1} )( 1+ R)} (| 2 \sigma _{1} - \sigma ^{m} _{2 |} + | 2 \sigma _{2-\sigma} ^{m) -} ^{1 |} \sigma ^{m} _{y} \le 0

• Case 3:

N = 0 , H = 0 . f: = \cfrac{ 2 ^{m-1} ( 1- R) + (R +2 )}{( 2+2 ^{m-1} )( 1+ R)}(|\sigma ^{m} _{1 |} - |\sigma ^{m)} _{2 |} + \cfrac{R}{( 2+2 ^{m-1} )( 1+ R)} (| 2 \sigma _{1} - \sigma ^{m} _{2 |} + | 2 \sigma _{2-\sigma} ^{m) -} ^{1 |} \sigma ^{m} _{y} \le 0

• Case 4:

L = 0 , F = 0 . f: = \cfrac{ 1+2 R}{ 2 ( 1+ R)}|\sigma _{1} - \sigma ^{m} _{2 |} + \cfrac{ 1 }{ 2 ( 1+ R)} |\sigma _{1} + \sigma ^{m} _{2 |} - \sigma ^{m} _{y} \le 0

• Case 5:

L = 0 , N = 0 . . This is the Hosford yield criterion.f : = \cfrac{ 1 }{ 1+ R}(|\sigma ^{m} _{1 |} + |\sigma ^{m)} _{2 |} + \cfrac{R}{ 1+ R}|\sigma _{1-\sigma} ^{m} ^{2 |} - \sigma ^{m} _{y} \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]

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} ^{2} ^{0 }\right)} + \left(\cfrac{\sigma _{2}{\sigma} _{90} }\right)^2 + \left[(p + q - c) - \cfrac{p\sigma_1+q\sigma_2}{\sigma_b}\right]\left(\cfrac\right) = 1 where \sigma _{0} is the uniaxial tensile yield stress in the rolling direction, \sigma _{90} is the uniaxial tensile yield stress in the direction normal to the rolling direction, \sigma _{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 - \cfrac \\ \left(\cfrac+\cfrac-\cfrac\right)~p & = \cfrac - \cfrac + \cfrac \\ \left(\cfrac+\cfrac-\cfrac\right)~q & = \cfrac - \cfrac + \cfrac \end 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.

Extensions of Hill's yield criterion

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

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 ^{2} _{23} + 2 M \sigma ^{2} _{31} + 2 N\sigma ^{2} _{12} + I \sigma _{11} + J \sigma _{22} + K \sigma _{33} = 1 ~.

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 ^{2} _{23} + 2 M \sigma ^{2} _{31} + 2 N\sigma ^{2} _{12} + K (\sigma _{11} + \sigma _{22} + \sigma _{33} ) ^{2} = 1 ~.

See also

• Logan-Hosford yield criterion for anisotropic plasticity

References

1. R. Hill. (1948). A theory of the yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. London, 193:281–297
2. R. Hill. (1979). Theoretical plasticity of textured aggregates. Math. Proc. Camb. Phil. Soc., 85(1):179–191.
3. Chu, E. (1995). Generalization of Hill's 1979 anisotropic yield criteria. Journal of Materials Processing Technology, vol. 50, pp. 207–215.
4. 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, p. 733.
5. Hill. R. (1993). User-friendly theory of orthotropic plasticity in sheet metals. International Journal of Mechanical Sciences, vol. 35, no. 1, pp. 19–25.
6. 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.
7. 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.

External links

• Yield criteria for aluminum