TheBresler–Pister yield criterion^[1] is a function that was originally devised to predict the strength ofconcrete under multiaxial stress states. This yield criterion is an extension of theDrucker–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 forpolypropylene^[2] andpolymeric foams.^[3]

The parameters${\displaystyle A,B,C}$ have to be chosen with care for reasonably shapedyield 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

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 ${\displaystyle \frac{1}{\sqrt{6}}{\left[({\sigma }_1-{\sigma }_2{)}^2+({\sigma }_2-{\sigma }_3{)}^2+({\sigma }_3-{\sigma }_1{)}^2\right]}^{1/2}-A-B({\sigma }_1+{\sigma }_2+{\sigma }_3)-C({\sigma }_1+{\sigma }_2+{\sigma }_3{)}^2=0.}$ If${\displaystyle {\sigma }_t={\sigma }_1}$ is the yield stress in uniaxial tension, then ${\displaystyle \frac{1}{\sqrt{3}}{\sigma }_t-A-B{\sigma }_t-C{\sigma }_t^2=0.}$ If${\displaystyle -{\sigma }_c={\sigma }_1}$ is the yield stress in uniaxial compression, then ${\displaystyle \frac{1}{\sqrt{3}}{\sigma }_c-A+B{\sigma }_c-C{\sigma }_c^2=0.}$ If${\displaystyle -{\sigma }_b={\sigma }_1={\sigma }_2}$ is the yield stress in equibiaxial compression, then ${\displaystyle \frac{1}{\sqrt{3}}{\sigma }_b-A+2B{\sigma }_b-4C{\sigma }_b^2=0.}$ Solving these three equations for${\displaystyle A,B,C}$ (using Maple) gives us

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

${\displaystyle \sqrt{J_2}=\frac{1}{\sqrt{3}}{I}_1-\frac{1}{2\sqrt{3}}\left(\frac{{\sigma }_t}{{\sigma }_c^2-{\sigma }_t^2}\right)I_1^2}$

where${\displaystyle {\sigma }_c}$ is the yield stress in uniaxial compression and${\displaystyle {\sigma }_t}$ is the yield stress in uniaxial tension.

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

${\displaystyle \sqrt{J_2}=\{\begin{array}{l}\frac{1}{\sqrt{3}}{\sigma }_t-0.03\sqrt{3}\frac{\rho }{{\rho }_m{\sigma }_t}{I}_1^2\\ -\frac{1}{\sqrt{3}}{\sigma }_c+0.03\sqrt{3}\frac{\rho }{{\rho }_m{\sigma }_c}{I}_1^2\end{array}}$

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

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

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• Yield criteria

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