Hosford yield criterion explained

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^[1] is a generalization of the von Mises yield criterion. It has the form

\tfrac{ 1 }{ 2 }|\sigma _{2-\sigma} ^{n} ^{3 |} + \tfrac{ 1 }{ 2 }|\sigma _{3-\sigma} ^{n} ^{1 |} + \tfrac{ 1 }{ 2 }|\sigma _{1-\sigma} ^{n} ^{2 |} = \sigma ^{n} _{y} 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} ^{n} ^{3 |} + \tfrac{ 1 }{ 2 }|\sigma _{3-\sigma} ^{n} ^{1 |} + \tfrac{ 1 }{ 2 }|\sigma _{1-\sigma} ^{n\right)} ^{2 |} ^{1/n} . This expression has the form of an L^p norm which is defined as\|x\| _{p=\left(|x} ^{p\right)} ^{n|} ^{1/p} . When p = inf ty , the we get the L^∞ norm, \|x\| _{infty=max} \left\{|x _{1|,} |x _{2|,} \ldots, |x _{n|\right\}} . Comparing this with the Hosford criterionindicates that if n = ∞, we have(\sigma _{y)} _{n → 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 ^{n} _{1 |} + |\sigma ^{n\right)} _{2 |} + \cfrac{ 1 }{ 2 }|\sigma _{1-\sigma} ^{n} ^{2 |} = \sigma ^{n} _{y} A plot of the yield locus in plane stress for various values of the exponent n \ge 1 is shown in the adjacent figure.

Logan-Hosford yield criterion for anisotropic plasticity

The Logan-Hosford yield criterion for anisotropic plasticity^{[2] [3]} is similar to Hill's generalized yield criterion and has the form

F|\sigma _{2-\sigma} ^{n} ^{3 |} + G|\sigma _{3-\sigma} ^{n} ^{1 |} + H|\sigma _{1-\sigma} ^{n} ^{2 |} = 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.^[4] 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 ^{n} _{1 |} + |\sigma ^{n)} _{2 |} + \cfrac{R}{ 1+ R} |\sigma _{1-\sigma} ^{n} ^{2 |} = \sigma ^{n} _{y} 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.^[4]

References

1. Hosford, W. F. (1972). A generalized isotropic yield criterion, Journal of Applied Mechanics, v. 39, n. 2, pp. 607-609.
2. Hosford, W. F., (1979), On yield loci of anisotropic cubic metals, Proc. 7th North American Metalworking Conf., SME, Dearborn, MI.
3. 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.
4. Hosford, W. F., (2005), Mechanical Behavior of Materials, p. 92, Cambridge University Press.

See also

• Yield surface
• Yield (engineering)
• Plasticity (physics)
• Stress (physics)