The quadratic Hill yield criterion[1] has the form :
Here F, G, H, L, M, N are constants that have to be determined experimentally and
are the stresses. The quadratic Hill yield criterion depends only on the deviatoric stresses and is pressure independent. It predicts the same yield stress in tension and in compression.
Expressions for F, G, H, L, M, N
If the axes of material anisotropy are assumed to be orthogonal, we can write

where
are the normal yield stresses with respect to the axes of anisotropy. Therefore, we have
![{\displaystyle F={\cfrac {1}{2}}\left[{\cfrac {1}{(\sigma _{2}^{y})^{2}}}+{\cfrac {1}{(\sigma _{3}^{y})^{2}}}-{\cfrac {1}{(\sigma _{1}^{y})^{2}}}\right]}](//wikimedia.org/api/rest_v1/media/math/render/svg/cfaed307481211b4a91f7d9e7c9f949afde88cb4)
![{\displaystyle G={\cfrac {1}{2}}\left[{\cfrac {1}{(\sigma _{3}^{y})^{2}}}+{\cfrac {1}{(\sigma _{1}^{y})^{2}}}-{\cfrac {1}{(\sigma _{2}^{y})^{2}}}\right]}](//wikimedia.org/api/rest_v1/media/math/render/svg/0fb6bb43d23a5654cba1c82b82adbd0fba9d9250)
![{\displaystyle H={\cfrac {1}{2}}\left[{\cfrac {1}{(\sigma _{1}^{y})^{2}}}+{\cfrac {1}{(\sigma _{2}^{y})^{2}}}-{\cfrac {1}{(\sigma _{3}^{y})^{2}}}\right]}](//wikimedia.org/api/rest_v1/media/math/render/svg/cac2750191c5937e64bc62b7634ffd86ef3dc934)
Similarly, if
are the yield stresses in shear (with respect to the axes of anisotropy), we have

Quadratic Hill yield criterion for plane stress
The quadratic Hill yield criterion for thin rolled plates (plane stress conditions) can be expressed as

where the principal stresses
are assumed to be aligned with the axes of anisotropy with
in the rolling direction and
perpendicular to the rolling direction,
,
is the R-value in the rolling direction, and
is the R-value perpendicular to the rolling direction.
For the special case of transverse isotropy we have
and we get

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, where ...
Derivation of Hill's criterion for plane stress |
For the situation where the principal stresses are aligned with the directions of anisotropy we have

where are the principal stresses. If we assume an associated flow rule we have

This implies that

For plane stress , which gives

The R-value is defined as the ratio of the in-plane and out-of-plane plastic strains under uniaxial stress . The quantity is the plastic strain ratio under uniaxial stress . Therefore, we have

Then, using and , the yield condition can be written as

which in turn may be expressed as

This is of the same form as the required expression. All we have to do is to express in terms of . Recall that,
![{\displaystyle {\begin{aligned}F&={\cfrac {1}{2}}\left[{\cfrac {1}{(\sigma _{2}^{y})^{2}}}+{\cfrac {1}{(\sigma _{3}^{y})^{2}}}-{\cfrac {1}{(\sigma _{1}^{y})^{2}}}\right]\\G&={\cfrac {1}{2}}\left[{\cfrac {1}{(\sigma _{3}^{y})^{2}}}+{\cfrac {1}{(\sigma _{1}^{y})^{2}}}-{\cfrac {1}{(\sigma _{2}^{y})^{2}}}\right]\\H&={\cfrac {1}{2}}\left[{\cfrac {1}{(\sigma _{1}^{y})^{2}}}+{\cfrac {1}{(\sigma _{2}^{y})^{2}}}-{\cfrac {1}{(\sigma _{3}^{y})^{2}}}\right]\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/470b0bc61070673f267b0632ad9a61e60db5167a)
We can use these to obtain

Solving for gives us

Plugging back into the expressions for leads to

which implies that

Therefore, the plane stress form of the quadratic Hill yield criterion can be expressed as

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