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 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 {\displaystyle {\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 σ i {\displaystyle \sigma _{i}} , i=1,2,3 are the principal stresses, n {\displaystyle n} is a material-dependent exponent and σ y {\displaystyle \sigma _{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 . {\displaystyle \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 Lp norm which is defined as
‖ x ‖ p = ( | x 1 | p + | x 2 | p + ⋯ + | x n | p ) 1 / p . {\displaystyle \ \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots +|x_{n}|^{p}\right)^{1/p}\,.}
When p = ∞ {\displaystyle p=\infty } , the we get the L∞ norm,
‖ x ‖ ∞ = max { | x 1 | , | x 2 | , … , | x n | } {\displaystyle \ \|x\|_{\infty }=\max \left\{|x_{1}|,|x_{2}|,\ldots ,|x_{n}|\right\}} . 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 | ) . {\displaystyle (\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
1 2 ( | σ 1 | n + | σ 2 | n ) + 1 2 | σ 1 − σ 2 | n = σ y n {\displaystyle {\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 ≥ 1 {\displaystyle n\geq 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 {\displaystyle 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, σ i {\displaystyle \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. Accepted values of n {\displaystyle 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 {\displaystyle {\cfrac {1}{1+R}}(|\sigma _{1}|^{n}+|\sigma _{2}|^{n})+{\cfrac {R}{1+R}}|\sigma _{1}-\sigma _{2}|^{n}=\sigma _{y}^{n}}
where R {\displaystyle R} is the R-value and σ y {\displaystyle \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 {\displaystyle n} that are less than 2, the yield locus exhibits corners and such values are not recommended.
References
See also Yield surface Yield (engineering) Plasticity (physics) Stress (physics)