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 plane stress, isotropic, Hosford yield surface for three values of n

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 _{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 plane stress, anisotropic, Hosford yield surface for four values of n and R=2.0

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 _{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.^[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 _{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.^[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.
1 2 Hosford, W. F., (2005), Mechanical Behavior of Materials, p. 92, Cambridge University Press.