title: Hosford yield criterion
text: # 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.
  
  