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Hosford yield criterion

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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

The Hosford yield criterion for isotropic materials is a generalization of the von Mises yield criterion. It has the form

1 2 | σ 2 − σ 3 | n + 1 2 | σ 3 − σ 1 | n + 1 2 | σ 1 − σ 2 | n = σ y n

where σ i , i=1,2,3 are the principal stresses, n is a material-dependent exponent and σ y is the yield stress in uniaxial tension/compression.

Alternatively, the yield criterion may be written as

σ y = ( 1 2 | σ 2 − σ 3 | n + 1 2 | σ 3 − σ 1 | n + 1 2 | σ 1 − σ 2 | n ) 1 / n .

This expression has the form of an Lp norm which is defined as

  ∥ x ∥ p = ( | x 1 | p + | x 2 | p + ⋯ + | x n | p ) 1 / p .

When p = ∞ , the we get the L∞ norm,

  ∥ x ∥ ∞ = max { | x 1 | , | x 2 | , … , | x n | } . Comparing this with the Hosford criterion

indicates that if n = ∞, we have

( σ y ) n → ∞ = max ( | σ 2 − σ 3 | , | σ 3 − σ 1 | , | σ 1 − σ 2 | ) .

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

1 2 ( | σ 1 | n + | σ 2 | n ) + 1 2 | σ 1 − σ 2 | n = σ y n

A plot of the yield locus in plane stress for various values of the exponent n ≥ 1 is shown in the adjacent figure.

Logan-Hosford yield criterion for anisotropic plasticity

The Logan-Hosford yield criterion for anisotropic plasticity is similar to Hill's generalized yield criterion and has the form

F | σ 2 − σ 3 | n + G | σ 3 − σ 1 | n + H | σ 1 − σ 2 | n = 1

where F,G,H are constants, σ 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. 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

1 1 + R ( | σ 1 | n + | σ 2 | n ) + R 1 + R | σ 1 − σ 2 | n = σ y n

where R is the R-value and σ 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.

References

Hosford yield criterion Wikipedia
(Text) CC BY-SA


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