# Bresler Pister yield criterion

The Bresler-Pister yield criterion is a function that was originally devised to predict the strength of concrete under multiaxial stress states. This yield criterion is an extension of the Drucker-Prager yield criterion and can be expressed on terms of the stress invariants as

${\sqrt {J_{2}}}=A+B~I_{1}+C~I_{1}^{2}$ where $I_{1}$ is the first invariant of the Cauchy stress, $J_{2}$ is the second invariant of the deviatoric part of the Cauchy stress, and $A,B,C$ are material constants.

Yield criteria of this form have also been used for polypropylene  and polymeric foams.

The parameters $A,B,C$ have to be chosen with care for reasonably shaped yield surfaces. If $\sigma _{c}$ is the yield stress in uniaxial compression, $\sigma _{t}$ is the yield stress in uniaxial tension, and $\sigma _{b}$ is the yield stress in biaxial compression, the parameters can be expressed as

{\begin{aligned}B=&\left({\cfrac {\sigma _{t}-\sigma _{c}}{{\sqrt {3}}(\sigma _{t}+\sigma _{c})}}\right)\left({\cfrac {4\sigma _{b}^{2}-\sigma _{b}(\sigma _{c}+\sigma _{t})+\sigma _{c}\sigma _{t}}{4\sigma _{b}^{2}+2\sigma _{b}(\sigma _{t}-\sigma _{c})-\sigma _{c}\sigma _{t}}}\right)\\C=&\left({\cfrac {1}{{\sqrt {3}}(\sigma _{t}+\sigma _{c})}}\right)\left({\cfrac {\sigma _{b}(3\sigma _{t}-\sigma _{c})-2\sigma _{c}\sigma _{t}}{4\sigma _{b}^{2}+2\sigma _{b}(\sigma _{t}-\sigma _{c})-\sigma _{c}\sigma _{t}}}\right)\\A=&{\cfrac {\sigma _{c}}{\sqrt {3}}}+B\sigma _{c}-C\sigma _{c}^{2}\end{aligned}} ## Alternative forms of the Bresler-Pister yield criterion

In terms of the equivalent stress ($\sigma _{e}$ ) and the mean stress ($\sigma _{m}$ ), the Bresler-Pister yield criterion can be written as

$\sigma _{e}=a+b~\sigma _{m}+c~\sigma _{m}^{2}~;~~\sigma _{e}={\sqrt {3J_{2}}}~,~~\sigma _{m}=I_{1}/3~.$ The Etse-Willam form of the Bresler-Pister yield criterion for concrete can be expressed as

${\sqrt {J_{2}}}={\cfrac {1}{\sqrt {3}}}~I_{1}-{\cfrac {1}{2{\sqrt {3}}}}~\left({\cfrac {\sigma _{t}}{\sigma _{c}^{2}-\sigma _{t}^{2}}}\right)~I_{1}^{2}$ where $\sigma _{c}$ is the yield stress in uniaxial compression and $\sigma _{t}$ is the yield stress in uniaxial tension.

The GAZT yield criterion for plastic collapse of foams also has a form similar to the Bresler-Pister yield criterion and can be expressed as

${\sqrt {J_{2}}}={\begin{cases}{\cfrac {1}{\sqrt {3}}}~\sigma _{t}-0.03{\sqrt {3}}{\cfrac {\rho }{\rho _{m}~\sigma _{t}}}~I_{1}^{2}\\-{\cfrac {1}{\sqrt {3}}}~\sigma _{c}+0.03{\sqrt {3}}{\cfrac {\rho }{\rho _{m}~\sigma _{c}}}~I_{1}^{2}\end{cases}}$ where $\rho$ is the density of the foam and $\rho _{m}$ is the density of the matrix material.