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.
The quadratic Hill yield criterion[1] has the form :
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.
If the axes of material anisotropy are assumed to be orthogonal, we can write where
are the normal yield stresses with respect to the axes of anisotropy.
are the yield stresses in shear (with respect to the axes of anisotropy), we have The quadratic Hill yield criterion for thin rolled plates (plane stress conditions) can be expressed as where the principal stresses
are assumed to be aligned with the axes of anisotropy with
is the R-value perpendicular to the rolling direction.
For the special case of transverse isotropy we have
If we assume an associated flow rule we have This implies that For plane stress
is defined as the ratio of the in-plane and out-of-plane plastic strains under uniaxial stress
is the plastic strain ratio under uniaxial stress
, the yield condition can be written as which in turn may be expressed as This is of the same form as the required expression.
leads to which implies that Therefore, the plane stress form of the quadratic Hill yield criterion can be expressed as The generalized Hill yield criterion[2] has the form where
are the principal stresses (which are aligned with the directions of anisotropy),
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.
being the plane of symmetry, the generalized Hill yield criterion reduces to (with
) The R-value or Lankford coefficient can be determined by considering the situation where
The R-value is then given by Under plane stress conditions and with some assumptions, the generalized Hill criterion can take several forms.
[3] In 1993, Hill proposed another yield criterion[5] for plane stress problems with planar anisotropy.
is the uniaxial tensile yield stress in the rolling direction,
is the uniaxial tensile yield stress in the direction normal to the rolling direction,
is the yield stress under uniform biaxial tension, and
is the R-value for uniaxial tension in the rolling direction, and
is the R-value for uniaxial tension in the in-plane direction perpendicular to the rolling direction.
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.
An extension that allows for pressure dependence is Caddell–Raghava–Atkins (CRA) model[6] which has the form 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).