# 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
    ${\tfrac {1}{2}}|\sigma _{2}-\sigma _{3}|^{n}+{\tfrac {1}{2}}|\sigma _{3}-\sigma _{1}|^{n}+{\tfrac {1}{2}}|\sigma _{1}-\sigma _{2}|^{n}=\sigma _{y}^{n}\,$
where $\sigma _{i}$, i=1,2,3 are the principal stresses, $n$ is a material-dependent exponent and $\sigma _{y}$ is the yield stress in uniaxial tension/compression.
Alternatively, the yield criterion may be written as
    $\sigma _{y}=\left({\tfrac {1}{2}}|\sigma _{2}-\sigma _{3}|^{n}+{\tfrac {1}{2}}|\sigma _{3}-\sigma _{1}|^{n}+{\tfrac {1}{2}}|\sigma _{1}-\sigma _{2}|^{n}\right)^{1/n}\,.$
This expression has the form of an L$^p$ norm which is defined as
    $\ \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots +|x_{n}|^{p}\right)^{1/p}\,.$
When $p=\infty$, the we get the L$^∞$ norm,
    $\ \|x\|_{\infty }=\max \left\{|x_{1}|,|x_{2}|,\ldots ,|x_{n}|\right\}$. Comparing this with the Hosford criterion
indicates that if n = ∞, we have
    $(\sigma _{y})_{n\rightarrow \infty }=\max \left(|\sigma _{2}-\sigma _{3}|,|\sigma _{3}-\sigma _{1}|,|\sigma _{1}-\sigma _{2}|\right)\,.$
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
    ${\cfrac {1}{2}}\left(|\sigma _{1}|^{n}+|\sigma _{2}|^{n}\right)+{\cfrac {1}{2}}|\sigma _{1}-\sigma _{2}|^{n}=\sigma _{y}^{n}\,$
A plot of the yield locus in plane stress for various values of the exponent $n\geq 1$ 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
    $F|\sigma _{2}-\sigma _{3}|^{n}+G|\sigma _{3}-\sigma _{1}|^{n}+H|\sigma _{1}-\sigma _{2}|^{n}=1\,$
where F,G,H are constants, $\sigma _{i}$ 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 $n$ 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
    ${\cfrac {1}{1+R}}(|\sigma _{1}|^{n}+|\sigma _{2}|^{n})+{\cfrac {R}{1+R}}|\sigma _{1}-\sigma _{2}|^{n}=\sigma _{y}^{n}$
where $R$ is the R-value and $\sigma _{y}$ 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 $n$ that are less than 2, the yield locus exhibits corners and such values are not recommended.