# Bresler Pister yield criterion

The Bresler-Pister yield criterion[1] 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

${\displaystyle {\sqrt {J_{2}}}=A+B~I_{1}+C~I_{1}^{2}}$

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

Yield criteria of this form have also been used for polypropylene [2] and polymeric foams.[3]

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

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

${\displaystyle \sigma _{e}=a+b~\sigma _{m}+c~\sigma _{m}^{2}~;~~\sigma _{e}={\sqrt {3J_{2}}}~,~~\sigma _{m}=I_{1}/3~.}$

The Etse-Willam[4] form of the Bresler-Pister yield criterion for concrete can be expressed as

${\displaystyle {\sqrt {J_{2}}}={\cfrac {1}{\sqrt {3}}}~I_{1}-{\cfrac {1}{2{\sqrt {3}}}}~\left({\cfrac {\sigma _{t}}{\sigma _{c}^{2}-\sigma _{t}^{2}}}\right)~I_{1}^{2}}$

where ${\displaystyle \sigma _{c}}$ is the yield stress in uniaxial compression and ${\displaystyle \sigma _{t}}$ is the yield stress in uniaxial tension.

The GAZT yield criterion[5] for plastic collapse of foams also has a form similar to the Bresler-Pister yield criterion and can be expressed as

${\displaystyle {\sqrt {J_{2}}}={\begin{cases}{\cfrac {1}{\sqrt {3}}}~\sigma _{t}-0.03{\sqrt {3}}{\cfrac {\rho }{\rho _{m}~\sigma _{t}}}~I_{1}^{2}\\-{\cfrac {1}{\sqrt {3}}}~\sigma _{c}+0.03{\sqrt {3}}{\cfrac {\rho }{\rho _{m}~\sigma _{c}}}~I_{1}^{2}\end{cases}}}$

where ${\displaystyle \rho }$ is the density of the foam and ${\displaystyle \rho _{m}}$ is the density of the matrix material.

## References

1. Bresler, B. and Pister, K.S., (1985), Strength of concrete under combined stresses, ACI Journal, vol. 551, no. 9, pp. 321-345.
2. Pae, K. D., (1977), The macroscopic yield behavior of polymers in multiaxial stress fields, Journal of Materials Science, vol. 12, no. 6, pp. 1209-1214.
3. Kim, Y. and Kang, S., (2003), Development of experimental method to characterize pressure-dependent yield criteria for polymeric foams. Polymer Testing, vol. 22, no. 2, pp. 197-202.
4. Etse, G. and Willam, K., (1994), Fracture energy formulation for inelastic behavior of plain concrete, Journal of Engineering Mechanics, vol. 120, no. 9, pp. 1983-2011.
5. Gibson, L. J., Ashby, M. F., Zhang, J., and Triantafillou, T. C. (1989). Failure surfaces for cellular materials under multiaxial loads. I. Modelling. International Journal of Mechanical Sciences, vol. 31, no. 9, pp. 635–663.