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

${\begin{aligned}B=&\left({\cfrac {\sigma _{t}-\sigma _{c}}{{\sqrt {3}}(\sigma _{t}+\sigma _{c})}}\right)\left({\cfrac {4\sigma _{b}^{2}-\sigma _{b}(\sigma _{c}+\sigma _{t})+\sigma _{c}\sigma _{t}}{4\sigma _{b}^{2}+2\sigma _{b}(\sigma _{t}-\sigma _{c})-\sigma _{c}\sigma _{t}}}\right)\\C=&\left({\cfrac {1}{{\sqrt {3}}(\sigma _{t}+\sigma _{c})}}\right)\left({\cfrac {\sigma _{b}(3\sigma _{t}-\sigma _{c})-2\sigma _{c}\sigma _{t}}{4\sigma _{b}^{2}+2\sigma _{b}(\sigma _{t}-\sigma _{c})-\sigma _{c}\sigma _{t}}}\right)\\A=&{\cfrac {\sigma _{c}}{\sqrt {3}}}+c_{1}\sigma _{c}-c_{2}\sigma _{c}^{2}\end{aligned}}$

Derivation of expressions for parameters A, B, C

The Bresler-Pister yield criterion in terms of the principal stresses $\sigma _{1},\sigma _{2},\sigma _{3}$ is ${\cfrac {1}{\sqrt {6}}}\left[(\sigma _{1}-\sigma _{2})^{2}+(\sigma _{2}-\sigma _{3})^{2}+(\sigma _{3}-\sigma _{1})^{2}\right]^{1/2}-A-B~(\sigma _{1}+\sigma _{2}+\sigma _{3})-C~(\sigma _{1}+\sigma _{2}+\sigma _{3})^{2}=0~.$ If $\sigma _{t}=\sigma _{1}$ is the yield stress in uniaxial tension, then ${\cfrac {1}{\sqrt {3}}}~\sigma _{t}-A-B\sigma _{t}-C\sigma _{t}^{2}=0~.$ If $-\sigma _{c}=\sigma _{1}$ is the yield stress in uniaxial compression, then ${\cfrac {1}{\sqrt {3}}}~\sigma _{c}-A+B\sigma _{c}-C\sigma _{c}^{2}=0~.$ If $-\sigma _{b}=\sigma _{1}=\sigma _{2}$ is the yield stress in equibiaxial compression, then ${\cfrac {1}{\sqrt {3}}}~\sigma _{b}-A+2B\sigma _{b}-4C\sigma _{b}^{2}=0~.$ Solving these three equations for $A,B,C$ (using Maple) gives us ${\begin{aligned}A:=&{\cfrac {1}{\sqrt {3}}}~{\cfrac {\sigma _{c}\sigma _{t}\sigma _{b}(\sigma _{t}+8\sigma _{b}-3\sigma _{c})}{(\sigma _{c}+\sigma _{t})(2\sigma _{b}-\sigma _{c})(2\sigma _{b}+\sigma _{t})}}\\B:=&{\cfrac {1}{\sqrt {3}}}~{\cfrac {(\sigma _{c}-\sigma _{t})(\sigma _{b}\sigma _{c}+\sigma _{b}\sigma _{t}-\sigma _{c}\sigma _{t}-4\sigma _{b}^{2})}{(\sigma _{c}+\sigma _{t})(2\sigma _{b}-\sigma _{c})(2\sigma _{b}+\sigma _{t})}}\\C:=&{\cfrac {1}{\sqrt {3}}}~{\cfrac {3\sigma _{b}\sigma _{t}-\sigma _{b}\sigma _{c}-2\sigma _{c}\sigma _{t}}{(\sigma _{c}+\sigma _{t})(2\sigma _{b}-\sigma _{c})(2\sigma _{b}+\sigma _{t})}}\end{aligned}}$

Figure 1: View of the three-parameter Bresler-Pister yield surface in 3D space of principal stresses for $\sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7$

Figure 2: The three-parameter Bresler-Pister yield surface in the $\pi$-plane for $\sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7$

Figure 3: Trace of the three-parameter Bresler-Pister yield surface in the $\sigma _{1}-\sigma _{2}$-plane for $\sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7$

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.

See also

• Yield surface
• Yield (engineering)
• Plasticity (physics)