Hosford yield criterion

The Hosford yield criterion is a function that is used to determine whether a material has undergone plastic yielding under the action of stress.
Hosford yield criterion for isotropic plasticity.
The Hosford yield criterion for isotropic materials is a generalization of the von Mises yield criterion. It has the form
where formula_2, i=1,2,3 are the principal stresses, formula_3 is a material-dependent exponent and formula_4 is the yield stress in uniaxial tension/compression.
Alternatively, the yield criterion may be written as
This expression has the form of an "L""p" norm which is defined as
When formula_7, the we get the "L"∞ norm, 
indicates that if "n" = ∞, we have
This is identical to the Tresca yield criterion.
Therefore, when "n = 1" or "n" goes to infinity the Hosford criterion reduces to the Tresca yield criterion. When "n = 2" the Hosford criterion reduces to the von Mises yield criterion.
Note that the exponent "n" does not need to be an integer.
Hosford yield criterion for plane stress.
For the practically important situation of plane stress, the Hosford yield criterion takes the form
A plot of the yield locus in plane stress for various values of the exponent formula_11 is shown in the adjacent figure.
Logan-Hosford yield criterion for anisotropic plasticity.
The Logan-Hosford yield criterion for anisotropic plasticity is similar to Hill's generalized yield criterion and has the form
where "F,G,H" are constants, formula_2 are the principal stresses, and the exponent "n" depends on the type of crystal (bcc, fcc, hcp, etc.) and has a value much greater than 2. Accepted values of formula_3 are 6 for bcc materials and 8 for fcc materials.
Though the form is similar to Hill's generalized yield criterion, the exponent "n" is independent of the R-value unlike the Hill's criterion.
Logan-Hosford criterion in plane stress.
Under plane stress conditions, the Logan-Hosford criterion can be expressed as
where formula_16 is the R-value and formula_4 is the yield stress in uniaxial tension/compression. For a derivation of this relation see Hill's yield criteria for plane stress. A plot of the yield locus for the anisotropic Hosford criterion is shown in the adjacent figure. For values of formula_18 that are less than 2, the yield locus exhibits corners and such values are not recommended.