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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 ==
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[[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]]

Latest revision as of 21:08, 1 June 2014

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Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. The nominee in each category with the most votes was crowned the 2010 Parents Picks Awards WINNER and has been established as the best product, tip or place in that category. After age 35, 18% of pregnancies will end in miscarriage. Our skilled expertise, skillfulness and excellence have been well known all across the world. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration.

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