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

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

Derivation of expressions for parameters A, B, C

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

Figure 1: View of the three-parameter Bresler–Pister yield surface in 3D space of principal stresses for ${\displaystyle \sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7}$	Figure 2: The three-parameter Bresler–Pister yield surface in the ${\displaystyle \pi }$-plane for ${\displaystyle \sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7}$	Figure 3: Trace of the three-parameter Bresler–Pister yield surface in the ${\displaystyle \sigma _{1}-\sigma _{2}}$-plane for ${\displaystyle \sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7}$