Hosford yield criterion

Last updated April 07, 2019

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

The von Mises yield criterion suggests that yielding of a ductile material begins when the second deviatoric stress invariant $reaches a critical value. It is part of plasticity theory that applies best to ductile materials, such as some metals. Prior to yield, material response can be assumed to be of a nonlinear elastic, viscoelastic, or linear elastic behavior.$

{\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

In mathematics, the L^p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue, although according to the Bourbaki group they were first introduced by Frigyes Riesz. L^p spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, finance, engineering, and other disciplines.

\ \|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 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]

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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.