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The '''Hosford yield criterion''' is a function that is used to determine whether a material has undergone plastic yielding under the action of stress.
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== Hosford yield criterion for isotropic plasticity ==
[[Image:Hosford plane stress.png|280px|right|thumb|The plane stress, isotropic, Hosford yield surface for three values of ''n'']]
The Hosford yield criterion for isotropic materials <ref>Hosford, W. F. (1972). ''A generalized isotropic yield criterion'', Journal of Applied Mechanics, v. 39, n. 2, pp. 607-609.</ref> is a generalization of the [[von Mises yield criterion]]. It has the form
:<math>
  \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 \,
</math>
where <math>\sigma_i</math>, i=1,2,3 are the [[Stress_%28physics%29#Principal_stresses_and_stress_invariants|principal stresses]], <math>n</math> is a material-dependent exponent and <math>\sigma_y</math> is the [[yield stress]] in uniaxial tension/compression.
 
Alternatively, the yield criterion may be written as
:<math>
  \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} \,.
</math>
This expression has the form of an ''L''<sup>''p''</sup> [[Lp space|norm]] which is defined as
:<math>\ \|x\|_p=\left(|x_1|^p+|x_2|^p+\cdots+|x_n|^p\right)^{1/p} \,.</math>
When <math>p = \infty</math>, the we get the ''L''<sup>∞</sup> norm,  
:<math>\ \|x\|_\infty=\max \left\{|x_1|, |x_2|, \ldots, |x_n|\right\}</math>. Comparing this with the Hosford criterion
indicates that if ''n''&nbsp;= ∞, we have
:<math>
  (\sigma_y)_{n\rightarrow\infty} = \max \left(|\sigma_2-\sigma_3|, |\sigma_3-\sigma_1|,|\sigma_1-\sigma_2|\right) \,.
</math>
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
:<math>
  \cfrac{1}{2}\left(|\sigma_1|^n + |\sigma_2|^n\right) + \cfrac{1}{2}|\sigma_1-\sigma_2|^n = \sigma_y^n \,
</math>
A plot of the yield locus in plane stress for various values of the exponent <math>n \ge 1</math> is shown in the adjacent figure.
 
== Logan-Hosford yield criterion for anisotropic plasticity ==
[[Image:Hosford aniso plane stress.png|280px|right|thumb|The plane stress, anisotropic, Hosford yield surface for four values of ''n'' and R=2.0]]
The Logan-Hosford yield criterion for anisotropic plasticity <ref>Hosford, W. F., (1979), ''On yield loci of anisotropic cubic metals'', Proc. 7th North American Metalworking Conf., SME, Dearborn, MI.</ref><ref>Logan, R. W. and Hosford, W. F., (1980), '' Upper-Bound Anisotropic Yield Locus Calculations Assuming< 111>-Pencil Glide'', International Journal of Mechanical Sciences, v. 22, n. 7, pp. 419-430.</ref> is similar to [[Hill yield criteria|Hill's generalized yield criterion]] and has the form
:<math>
  F|\sigma_2-\sigma_3|^n + G|\sigma_3-\sigma_1|^n + H|\sigma_1-\sigma_2|^n = 1 \,
</math>
where ''F,G,H'' are constants, <math>\sigma_i</math> 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.<ref name=HosfordBook>Hosford, W. F., (2005), '''Mechanical Behavior of Materials''', p. 92, Cambridge University Press.</ref>  Accepted values of <math>n</math> are 6 for [[Body-centered cubic|bcc]] materials and 8 for [[Face-centred cubic|fcc]] materials.
 
Though the form is similar to [[Hill yield criteria|Hill's generalized yield criterion]], the exponent ''n'' is independent of the [[R-value (plasticity)|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
:<math>
  \cfrac{1}{1+R} (|\sigma_1|^n + |\sigma_2|^n) + \cfrac{R}{1+R} |\sigma_1-\sigma_2|^n = \sigma_y^n
</math>
where <math>R</math> is the [[R-value (plasticity)|R-value]] and <math>\sigma_y</math> is the yield stress in uniaxial tension/compression.  For a derivation of this relation see [[Hill_yield_criteria#Generalized_Hill_yield_criterion_for_plane_stress|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 <math> n </math> that are less than 2, the yield locus exhibits corners and such values are not recommended.<ref name=HosfordBook/>
 
== References ==
<references/>
 
== See also ==
*[[Yield surface]]
*[[Yield (engineering)]]
*[[Plasticity (physics)]]
*[[Stress (physics)]]
 
[[Category:Plasticity]]
[[Category:Solid mechanics]]
[[Category:Mechanics]]
[[Category:Yield criteria]]

Revision as of 19:56, 23 February 2014

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