Английская Википедия: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[1] 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 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 Lp 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∞ 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 = ∞, 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
The Logan-Hosford yield criterion for anisotropic plasticity[2][3] is similar to 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.[4] Accepted values of <math>n</math> 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
- <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 and <math>\sigma_y</math> 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 <math> n </math> that are less than 2, the yield locus exhibits corners and such values are not recommended.[4]
References
- ↑ Hosford, W. F. (1972). A generalized isotropic yield criterion, Journal of Applied Mechanics, v. 39, n. 2, pp. 607-609.
- ↑ Hosford, W. F., (1979), On yield loci of anisotropic cubic metals, Proc. 7th North American Metalworking Conf., SME, Dearborn, MI.
- ↑ 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.
- ↑ 4,0 4,1 Hosford, W. F., (2005), Mechanical Behavior of Materials, p. 92, Cambridge University Press.
See also
