The quadratic Hill yield criterion has the form
F ( σ 22 − σ 33 ) 2 + G ( σ 33 − σ 11 ) 2 + H ( σ 11 − σ 22 ) 2 + 2 L σ 23 2 + 2 M σ 31 2 + 2 N σ 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 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
( G + H ) ( σ 1 y ) 2 = 1 ; ( F + H ) ( σ 2 y ) 2 = 1 ; ( F + G ) ( σ 3 y ) 2 = 1 where σ 1 y , σ 2 y , σ 3 y are the normal yield stresses with respect to the axes of anisotropy. Therefore we have
F = 1 2 [ 1 ( σ 2 y ) 2 + 1 ( σ 3 y ) 2 − 1 ( σ 1 y ) 2 ] G = 1 2 [ 1 ( σ 3 y ) 2 + 1 ( σ 1 y ) 2 − 1 ( σ 2 y ) 2 ] H = 1 2 [ 1 ( σ 1 y ) 2 + 1 ( σ 2 y ) 2 − 1 ( σ 3 y ) 2 ] Similarly, if τ 12 y , τ 23 y , τ 31 y are the yield stresses in shear (with respect to the axes of anisotropy), we have
L = 1 2 ( τ 23 y ) 2 ; M = 1 2 ( τ 31 y ) 2 ; N = 1 2 ( τ 12 y ) 2 The quadratic Hill yield criterion for thin rolled plates (plane stress conditions) can be expressed as
σ 1 2 + R 0 ( 1 + R 90 ) R 90 ( 1 + R 0 ) σ 2 2 − 2 R 0 1 + R 0 σ 1 σ 2 = ( σ 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
σ 1 2 + σ 2 2 − 2 R 1 + R σ 1 σ 2 = ( σ 1 y ) 2 The generalized Hill yield criterion has the form
F | σ 2 − σ 3 | m + G | σ 3 − σ 1 | m + H | σ 1 − σ 2 | m + L | 2 σ 1 − σ 2 − σ 3 | m + M | 2 σ 2 − σ 3 − σ 1 | m + N | 2 σ 3 − σ 1 − σ 2 | m = σ y m . 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.
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 )
f := F | σ 2 − σ 3 | m + F | σ 3 − σ 1 | m + H | σ 1 − σ 2 | m + L | 2 σ 1 − σ 2 − σ 3 | m + L | 2 σ 2 − σ 3 − σ 1 | m + N | 2 σ 3 − σ 1 − σ 2 | m − σ y m ≤ 0 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 = ( 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.
Case 1: L = 0 , H = 0. f := 1 + 2 R 1 + R ( | σ 1 | m + | σ 2 | m ) − R 1 + R | σ 1 + σ 2 | m − σ y m ≤ 0 Case 2: N = 0 , F = 0. f := 2 m − 1 ( 1 − R ) + ( R + 2 ) ( 1 − 2 m − 1 ) ( 1 + R ) | σ 1 − σ 2 | m − 1 ( 1 − 2 m − 1 ) ( 1 + R ) ( | 2 σ 1 − σ 2 | m + | 2 σ 2 − σ 1 | m ) − σ y m ≤ 0 Case 3: N = 0 , H = 0. f := 2 m − 1 ( 1 − R ) + ( R + 2 ) ( 2 + 2 m − 1 ) ( 1 + R ) ( | σ 1 | m − | σ 2 | m ) + R ( 2 + 2 m − 1 ) ( 1 + R ) ( | 2 σ 1 − σ 2 | m + | 2 σ 2 − σ 1 | m ) − σ y m ≤ 0 Case 4: L = 0 , F = 0. f := 1 + 2 R 2 ( 1 + R ) | σ 1 − σ 2 | m + 1 2 ( 1 + R ) | σ 1 + σ 2 | m − σ y m ≤ 0 Case 5: L = 0 , N = 0. . This is the Hosford yield criterion. f := 1 1 + R ( | σ 1 | m + | σ 2 | m ) + R 1 + R | σ 1 − σ 2 | m − σ y m ≤ 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 .
In 1993, Hill proposed another yield criterion for plane stress problems with planar anisotropy. The Hill93 criterion has the form
( σ 1 σ 0 ) 2 + ( σ 2 σ 90 ) 2 + [ ( p + q − c ) − p σ 1 + q σ 2 σ b ] ( σ 1 σ 2 σ 0 σ 90 ) = 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
c = σ 0 σ 90 + σ 90 σ 0 − σ 0 σ 90 σ b 2 ( 1 σ 0 + 1 σ 90 − 1 σ b ) p = 2 R 0 ( σ b − σ 90 ) ( 1 + R 0 ) σ 0 2 − 2 R 90 σ b ( 1 + R 90 ) σ 90 2 + c σ 0 ( 1 σ 0 + 1 σ 90 − 1 σ b ) q = 2 R 90 ( σ b − σ 0 ) ( 1 + R 90 ) σ 90 2 − 2 R 0 σ b ( 1 + R 0 ) σ 0 2 + c σ 90 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.
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.
An extension that allows for pressure dependence is Caddell-Raghava-Atkins (CRA) model which has the form
F ( σ 22 − σ 33 ) 2 + G ( σ 33 − σ 11 ) 2 + H ( σ 11 − σ 22 ) 2 + 2 L σ 23 2 + 2 M σ 31 2 + 2 N σ 12 2 + I σ 11 + J σ 22 + K σ 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 for honeycomb structures (used in sandwich composite construction). This yield criterion has the form
F ( σ 22 − σ 33 ) 2 + G ( σ 33 − σ 11 ) 2 + H ( σ 11 − σ 22 ) 2 + 2 L σ 23 2 + 2 M σ 31 2 + 2 N σ 12 2 + K ( σ 11 + σ 22 + σ 33 ) 2 = 1 . 