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

$\begin{align} B = & \left(\cfrac{\sigma_t-\sigma_c}{\sqrt{3}(\sigma_t+\sigma_c)}\right) \left(\cfrac{4\sigma_b^2 - \sigma_b(\sigma_c+\sigma_t) + \sigma_c\sigma_t}{4\sigma_b^2 + 2\sigma_b(\sigma_t-\sigma_c) - \sigma_c\sigma_t} \right) \\ C = & \left(\cfrac{1}{\sqrt{3}(\sigma_t+\sigma_c)}\right) \left(\cfrac{\sigma_b(3\sigma_t-\sigma_c) -2\sigma_c\sigma_t}{4\sigma_b^2 + 2\sigma_b(\sigma_t-\sigma_c) - \sigma_c\sigma_t} \right) \\ A = & \cfrac{\sigma_c}{\sqrt{3}} + c_1\sigma_c -c_2\sigma_c^2 \end{align}$

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 $\begin{align} A := & \cfrac{1}{\sqrt{3}}~\cfrac{\sigma_c\sigma_t\sigma_b(\sigma_t+8\sigma_b-3\sigma_c)} {(\sigma_c+\sigma_t)(2\sigma_b-\sigma_c)(2\sigma_b+\sigma_t)} \\ B := & \cfrac{1}{\sqrt{3}}~\cfrac{(\sigma_c-\sigma_t)(\sigma_b\sigma_c+\sigma_b\sigma_t-\sigma_c\sigma_t-4\sigma_b^2)}{(\sigma_c+\sigma_t)(2\sigma_b-\sigma_c)(2\sigma_b+\sigma_t)} \\ C := & \cfrac{1}{\sqrt{3}}~\cfrac{3\sigma_b\sigma_t-\sigma_b\sigma_c-2\sigma_c\sigma_t}{(\sigma_c+\sigma_t)(2\sigma_b-\sigma_c)(2\sigma_b+\sigma_t)} \end{align}$

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

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