Hill yield criterion

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 has the form
Here "F, G, H, L, M, N" are constants that have to be determined experimentally and formula_2 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
where formula_4 are the normal yield stresses with respect to the axes of anisotropy. Therefore we have
Similarly, if formula_8 are the yield stresses in shear (with respect to the axes of anisotropy), we have
Quadratic Hill yield criterion for plane stress.
The quadratic Hill yield criterion for thin rolled plates (plane stress conditions) can be expressed as
where the principal stresses formula_11 are assumed to be aligned with the axes of anisotropy with formula_12 in the rolling direction and formula_13 perpendicular to the rolling direction, formula_14, formula_15 is the R-value in the rolling direction, and formula_16 is the R-value perpendicular to the rolling direction.
For the special case of transverse isotropy we have formula_17 and we get
Generalized Hill yield criterion.
The generalized Hill yield criterion has the form
where formula_20 are the principal stresses (which are aligned with the directions of anisotropy), formula_21 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 formula_22 being the plane of symmetry, the generalized Hill yield criterion reduces to (with formula_23 and formula_24)
The R-value or Lankford coefficient can be determined by considering the situation where formula_26. The R-value is then given by
Under plane stress conditions and with some assumptions, the generalized Hill criterion can take several forms. 
Hill 1993 yield criterion.
In 1993, Hill proposed another yield criterion for plane stress problems with planar anisotropy. The Hill93 criterion has the form
where formula_41 is the uniaxial tensile yield stress in the rolling direction, formula_42 is the uniaxial tensile yield stress in the direction normal to the rolling direction, formula_43 is the yield stress under uniform biaxial tension, and formula_44 are parameters defined as
and formula_15 is the R-value for uniaxial tension in the rolling direction, and formula_16 is the R-value for uniaxial tension in the in-plane direction perpendicular to the rolling direction.
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.
The Caddell-Raghava-Atkins yield criterion.
An extension that allows for pressure dependence is Caddell-Raghava-Atkins (CRA) model which has the form
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 for honeycomb structures (used in sandwich composite construction). This yield criterion has the form