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


  \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 [2] and polymeric foams.[3]

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{align}
    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{align}
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[4] 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[5] 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.

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.

See also

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