View source for Physics:Bresler Pister yield criterion

{{Continuum mechanics|cTopic=[[Physics:Solid mechanics|Solid mechanics]]}}
The '''Bresler-Pister yield criterion'''<ref>Bresler, B. and Pister, K.S., (1985), ''Strength of concrete under combined stresses'', ACI Journal, vol. 551, no. 9, pp. 321-345.</ref> is a function that was originally devised to predict the strength of [[Chemistry:Concrete|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
:<math>
  \sqrt{J_2} = A + B~I_1 + C~I_1^2
 </math>
where <math>I_1</math> is the first invariant of the Cauchy stress, <math>J_2</math> is the second invariant of the deviatoric part of the Cauchy stress, and <math>A, B, C</math> are material constants.

Yield criteria of this form have also been used for [[Chemistry:Polypropylene|polypropylene]] <ref>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.</ref> and [[Chemistry:Foam|polymeric foams]].<ref>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.</ref>

The parameters <math>A,B,C</math> have to be chosen with care for reasonably shaped [[Yield surface|yield surface]]s.  If <math>\sigma_c</math> is the yield stress in uniaxial compression, <math>\sigma_t</math> is the yield stress in uniaxial tension, and <math>\sigma_b</math> is the yield stress in biaxial compression, the parameters can be expressed as
:<math>
  \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}} + B\sigma_c -C\sigma_c^2
  \end{align}
 </math>

:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!Derivation of expressions for parameters A, B, C
|-
|The Bresler-Pister yield criterion in terms of the principal stresses <math>\sigma_1,\sigma_2,\sigma_3</math> is
:<math>
   \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~.
 </math>
If <math>\sigma_t = \sigma_1</math> is the yield stress in uniaxial tension, then
:<math>
   \cfrac{1}{\sqrt{3}}~\sigma_t - A - B\sigma_t - C\sigma_t^2 = 0 ~.
 </math>
If <math>-\sigma_c = \sigma_1</math> is the yield stress in uniaxial compression, then
:<math>
   \cfrac{1}{\sqrt{3}}~\sigma_c - A + B\sigma_c - C\sigma_c^2 = 0 ~.
 </math>
If <math>-\sigma_b = \sigma_1 = \sigma_2</math> is the yield stress in equibiaxial compression, then
:<math>
   \cfrac{1}{\sqrt{3}}~\sigma_b - A + 2B\sigma_b - 4C\sigma_b^2 = 0 ~.
 </math>
Solving these three equations for <math>A,B,C</math> (using Maple) gives us
:<math>
  \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}
 </math>
|}
{| border="0"
|-
| valign="bottom"|
[[Image:Bresler Pister Yield Surface 3Da.png|240px|none|thumb|Figure 1: View of the three-parameter Bresler-Pister yield surface in 3D space of principal stresses for <math>\sigma_c=1, \sigma_t=0.3, \sigma_b=1.7</math>]]
| valign="bottom"|
[[Image:Bresler Pister Yield Surface 3Db.png|260px|none|thumb|Figure 2: The three-parameter Bresler-Pister yield surface in the <math>\pi</math>-plane for <math>\sigma_c=1, \sigma_t=0.3, \sigma_b=1.7</math>]]
| valign="bottom"|
[[Image:Bresler Pister Yield Surface sig1sig2.png|240px|none|thumb|Figure 3: Trace of the three-parameter Bresler-Pister yield surface in the <math>\sigma_1-\sigma_2</math>-plane for <math>\sigma_c=1, \sigma_t=0.3, \sigma_b=1.7</math>]]
|-
|}

== Alternative forms of the Bresler-Pister yield criterion ==
In terms of the equivalent stress (<math>\sigma_e</math>) and the mean stress (<math>\sigma_m</math>), the Bresler-Pister yield criterion can be written as
:<math>
  \sigma_e = a + b~\sigma_m + c~\sigma_m^2 ~;~~ \sigma_e = \sqrt{3J_2} ~,~~ \sigma_m = I_1/3 ~.
 </math>

The Etse-Willam<ref>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.</ref> form of the Bresler-Pister yield criterion for concrete can be expressed as
:<math>
   \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 
 </math>
where <math>\sigma_c</math> is the yield stress in uniaxial compression and <math>\sigma_t</math> is the yield stress in uniaxial tension.

The GAZT yield criterion<ref>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.</ref> for plastic collapse of foams also has a form similar to the Bresler-Pister yield criterion and can be expressed as
:<math>
  \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}
 </math>
where <math>\rho</math> is the density of the foam and <math>\rho_m</math> is the density of the matrix material.

== References ==
<references/>

== See also ==
*[[Yield surface]]
*[[Engineering:Yield (engineering)|Yield (engineering)]]
*[[Physics:Plasticity|Plasticity]]

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