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[[Image:Drucker Prager Yield Surface 3Da.png|300px|right|thumb|Figure 1: View of Drucker–Prager yield surface in 3D space of principal stresses for <math>c=2, \phi=-20^\circ</math>]]

The '''Drucker–Prager yield criterion'''<ref>Drucker, D. C. and Prager, W. (1952). ''Soil mechanics and plastic analysis for limit design''. Quarterly of Applied Mathematics, vol. 10, no. 2, pp. 157–165.</ref> is a pressure-dependent model for determining whether a material has failed or undergone plastic yielding.  The criterion was introduced to deal with the plastic deformation of soils.  It and its many variants have been applied to rock, concrete, polymers, foams, and other pressure-dependent materials.

The [[Daniel C. Drucker|Drucker]]–[[William Prager|Prager]] yield criterion has the form
:<math>
   \sqrt{J_2} = A + B~I_1
 </math>
where <math>I_1</math> is the [[Stress_(physics)#Principal_stresses_and_stress_invariants|first invariant]] of the [[Stress (physics)|Cauchy stress]] and <math>J_2</math> is the [[Stress_(physics)#Invariants_of_the_stress_deviator_tensor|second invariant]] of the [[Stress_(physics)#Stress_deviator_tensor|deviatoric]] part of the [[Stress (physics)|Cauchy stress]].  The constants <math>A, B </math> are determined from experiments.  

In terms of the [[von Mises stress|equivalent stress]] (or [[von Mises stress]]) and the [[hydrostatic stress|hydrostatic (or mean) stress]], the Drucker–Prager criterion can be expressed as
:<math>
   \sigma_e = a + b~\sigma_m
 </math>
where <math>\sigma_e</math> is the equivalent stress, <math>\sigma_m</math> is the hydrostatic stress, and
<math>a,b</math> are material constants.  The Drucker–Prager yield criterion expressed in [[Yield_surface#Invariants_used_to_describe_yield_surfaces|Haigh–Westergaard coordinates]] is
:<math>
   \tfrac{1}{\sqrt{2}}\rho - \sqrt{3}~B\xi = A 
 </math>

The [[Yield surface#Drucker.E2.80.93Prager_yield_surface|Drucker–Prager yield surface]] is a smooth version of the [[Yield surface#Mohr.E2.80.93Coulomb_yield_surface|Mohr–Coulomb yield surface]].

== Expressions for A and B ==
The Drucker–Prager model can be written in terms of the [[stress (physics)#Principal_stresses_and_stress_invariants|principal stresses]] as
:<math>
  \sqrt{\cfrac{1}{6}\left[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2\right]} = A + B~(\sigma_1+\sigma_2+\sigma_3) ~.
 </math>
If <math>\sigma_t</math> is the yield stress in uniaxial tension, the Drucker–Prager criterion implies
:<math>
   \cfrac{1}{\sqrt{3}}~\sigma_t = A + B~\sigma_t ~.
 </math>
If <math>\sigma_c</math> is the yield stress in uniaxial compression, the Drucker–Prager criterion implies
:<math>
   \cfrac{1}{\sqrt{3}}~\sigma_c = A - B~\sigma_c ~.
 </math>
Solving these two equations gives
:<math>
   A = \cfrac{2}{\sqrt{3}}~\left(\cfrac{\sigma_c~\sigma_t}{\sigma_c+\sigma_t}\right) ~;~~ B = \cfrac{1}{\sqrt{3}}~\left(\cfrac{\sigma_t-\sigma_c}{\sigma_c+\sigma_t}\right) ~.
 </math>

=== Uniaxial asymmetry ratio ===
Different uniaxial yield stresses in tension and in compression are predicted by the Drucker–Prager model.  The uniaxial asymmetry ratio for the Drucker–Prager model is 
:<math>
   \beta = \cfrac{\sigma_\mathrm{c}}{\sigma_\mathrm{t}} = \cfrac{1 - \sqrt{3}~B}{1 + \sqrt{3}~B} ~.
 </math>

=== Expressions in terms of cohesion and friction angle ===
Since the Drucker–Prager [[yield surface]] is a smooth version of the [[Mohr-Coulomb theory|Mohr–Coulomb yield surface]], it is often expressed in terms of the cohesion (<math>c</math>) and the angle of internal friction (<math>\phi</math>) that are used to describe the [[Mohr-Coulomb theory|Mohr–Coulomb yield surface]].  If we assume that the Drucker–Prager yield surface '''circumscribes''' the Mohr–Coulomb yield surface then the expressions for <math>A</math> and <math>B</math> are
:<math>
   A = \cfrac{6~c~\cos\phi}{\sqrt{3}(3+\sin\phi)} ~;~~
   B = \cfrac{2~\sin\phi}{\sqrt{3}(3+\sin\phi)}
 </math>
If the Drucker–Prager yield surface '''inscribes''' the Mohr–Coulomb yield surface then
:<math>
   A = \cfrac{6~c~\cos\phi}{\sqrt{3}(3-\sin\phi)} ~;~~
   B = \cfrac{2~\sin\phi}{\sqrt{3}(3-\sin\phi)}
 </math>
:{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
!Derivation of expressions for <math>A,B</math> in terms of <math>c,\phi</math>
|-
|The expression for the [[Mohr-Coulomb theory|Mohr–Coulomb yield criterionddadsa]] in [[Yield_surface#Invariants_used_to_describe_yield_surfaces|Haigh–Westergaard space]] is
:<math>
   \left[\sqrt{3}~\sin\left(\theta+\tfrac{\pi}{3}\right) - \sin\phi\cos\left(\theta+\tfrac{\pi}{3}\right)\right]\rho - \sqrt{2}\sin(\phi)\xi = \sqrt{6} c \cos\phi
 </math>
If we assume that the Drucker–Prager yield surface '''circumscribes''' the Mohr–Coulomb yield surface such that the two surfaces coincide at <math>\theta=\tfrac{\pi}{3}</math>, then at those points the Mohr–Coulomb yield surface can be expressed as
:<math>
   \left[\sqrt{3}~\sin\tfrac{2\pi}{3} - \sin\phi\cos\tfrac{2\pi}{3}\right]\rho - \sqrt{2}\sin(\phi)\xi = \sqrt{6} c \cos\phi
 </math>
or,
:<math>
   \tfrac{1}{\sqrt{2}}\rho - \cfrac{2\sin\phi}{3+\sin\phi}\xi = \cfrac{\sqrt{12} c \cos\phi}{3+\sin\phi}  \qquad \qquad (1.1)
 </math>

The Drucker–Prager yield criterion expressed in [[Yield_surface#Invariants_used_to_describe_yield_surfaces|Haigh–Westergaard coordinates]] is
:<math>
   \tfrac{1}{\sqrt{2}}\rho - \sqrt{3}~B\xi = A \qquad \qquad (1.2)
 </math>
Comparing equations (1.1) and (1.2), we have
:<math>
   A = \cfrac{\sqrt{12} c \cos\phi}{3+\sin\phi} = \cfrac{6 c \cos\phi}{\sqrt{3}(3+\sin\phi)} ~;~~ B = \cfrac{2\sin\phi}{\sqrt{3}(3+\sin\phi)}
 </math>
These are the expressions for <math>A,B</math> in terms of <math>c,\phi</math>.  

On the other hand if the Drucker–Prager surface inscribes the Mohr–Coulomb surface, then matching the two surfaces at <math>\theta=0</math> gives
:<math>
   A = \cfrac{6 c \cos\phi}{\sqrt{3}(3-\sin\phi)} ~;~~ B = \cfrac{2\sin\phi}{\sqrt{3}(3-\sin\phi)}
 </math>
[[Image:MC DP Yield Surface 3Da.png|300px|left|thumb|Comparison of Drucker–Prager and Mohr–Coulomb (inscribed) yield surfaces in the <math>\pi</math>-plane for <math>c = 2, \phi = 20^\circ</math>]]
[[Image:MC DP Yield Surface 3Db.png|300px|none|thumb|Comparison of Drucker–Prager and Mohr–Coulomb (circumscribed) yield surfaces in the <math>\pi</math>-plane for <math>c = 2, \phi = 20^\circ</math>]]
|}
{| border="0"
|-
| valign="bottom"|
[[Image:Drucker Prager Yield Surface 3Db.png|300px|none|thumb|Figure 2: Drucker–Prager yield surface in the <math>\pi</math>-plane for <math>c = 2, \phi = 20^\circ</math>]]
|
|
|valign="bottom"|
[[Image:MC DP Yield Surface sig1sig2.png|300px|none|thumb|Figure 3: Trace of the Drucker–Prager and Mohr–Coulomb yield surfaces in the <math>\sigma_1-\sigma_2</math>-plane for <math>c = 2, \phi = 20^\circ</math>. Yellow = Mohr–Coulomb, Cyan = Drucker–Prager.]]
|-
|}

== Drucker–Prager model for polymers ==
The Drucker–Prager model has been used to model polymers such as [[polyoxymethylene]] and [[polypropylene]]{{Citation needed|date=September 2011}}.<ref>Abrate, S. (2008). ''Criteria for yielding or failure of cellular materials''. Journal of Sandwich Structures and Materials, vol. 10. pp. 5–51.</ref>   For [[polyoxymethylene]] the yield stress is a linear function of the pressure.  However,  [[polypropylene]] shows a quadratic pressure-dependence of the yield stress.

== Drucker–Prager model for foams ==
For foams, the GAZT model <ref>Gibson, L.J., [[M. F. Ashby|Ashby, M.F.]], Zhang, J. and Triantafilliou, T.C. (1989). ''Failure surfaces for cellular materials under multi-axial loads. I. Modeling''. International Journal of
Mechanical Sciences, vol. 31, no. 9, pp. 635–665.</ref> uses
:<math>
   A = \pm \cfrac{\sigma_y}{\sqrt{3}} ~;~~ B = \mp \cfrac{1}{\sqrt{3}}~\left(\cfrac{\rho}{5~\rho_s}\right)
 </math>
where <math>\sigma_{y}</math> is a critical stress for failure in tension or compression, <math>\rho</math> is the density of the foam, and <math>\rho_s</math> is the density of the base material.

== Extensions of the isotropic Drucker–Prager model ==
The Drucker–Prager criterion can also be expressed in the alternative form
:<math>
  J_2 = (A + B~I_1)^2 = a + b~I_1 + c~I_1^2 ~.
</math>

=== Deshpande–Fleck yield criterion ===
The Deshpande–Fleck yield criterion<ref>V. S. Deshpande, and Fleck, N. A. (2001). ''Multi-axial yield behaviour of polymer foams.'' Acta Materialia, vol. 49, no. 10, pp. 1859–1866.</ref> for foams has the form given in above equation.  The parameters <math>a, b, c</math>  for the Deshpande–Fleck criterion are
:<math>
  a = (1 + \beta^2)~\sigma_y^2 ~,~~
  b = 0 ~,~~
  c = -\cfrac{\beta^2}{3}
</math>
where <math>\beta</math> is a parameter<ref><math>\beta= \alpha/3</math> where <math>\alpha</math> is the 
quantity used by Deshpande–Fleck</ref> that determines the shape of the yield surface, and <math>\sigma_y</math> is the yield stress in tension or compression.

== Anisotropic Drucker–Prager yield criterion ==
An anisotropic form of the Drucker–Prager yield criterion is the Liu–Huang–Stout yield criterion.<ref>Liu, C., Huang, Y., and Stout, M. G. (1997). ''On the asymmetric yield surface of plastically orthotropic materials: A phenomenological study.'' Acta Materialia, vol. 45, no. 6, pp. 2397–2406</ref>  This yield criterion is an extension of the [[Hill yield criteria|generalized Hill yield criterion]] and has the form
:<math>
  \begin{align}
    f := & \sqrt{F(\sigma_{22}-\sigma_{33})^2+G(\sigma_{33}-\sigma_{11})^2+H(\sigma_{11}-\sigma_{22})^2  
         + 2L\sigma_{23}^2+2M\sigma_{31}^2+2N\sigma_{12}^2}\\
         &  + I\sigma_{11}+J\sigma_{22}+K\sigma_{33} - 1 \le 0
  \end{align}
 </math>

The coefficients <math>F,G,H,L,M,N,I,J,K</math> are 
:<math>
  \begin{align}
    F = & \cfrac{1}{2}\left[\Sigma_2^2 + \Sigma_3^2 - \Sigma_1^2\right] ~;~~
    G = \cfrac{1}{2}\left[\Sigma_3^2 + \Sigma_1^2 - \Sigma_2^2\right] ~;~~
    H = \cfrac{1}{2}\left[\Sigma_1^2 + \Sigma_2^2 - \Sigma_3^2\right] \\
    L = & \cfrac{1}{2(\sigma_{23}^y)^2} ~;~~
    M =  \cfrac{1}{2(\sigma_{31}^y)^2} ~;~~
    N =  \cfrac{1}{2(\sigma_{12}^y)^2} \\
    I = & \cfrac{\sigma_{1c}-\sigma_{1t}}{2\sigma_{1c}\sigma_{1t}} ~;~~
    J = \cfrac{\sigma_{2c}-\sigma_{2t}}{2\sigma_{2c}\sigma_{2t}} ~;~~
    K = \cfrac{\sigma_{3c}-\sigma_{3t}}{2\sigma_{3c}\sigma_{3t}} 
  \end{align}
 </math>
where
:<math>
   \Sigma_1 := \cfrac{\sigma_{1c}+\sigma_{1t}}{2\sigma_{1c}\sigma_{1t}} ~;~~
   \Sigma_2 := \cfrac{\sigma_{2c}+\sigma_{2t}}{2\sigma_{2c}\sigma_{2t}} ~;~~
   \Sigma_3 := \cfrac{\sigma_{3c}+\sigma_{3t}}{2\sigma_{3c}\sigma_{3t}}
 </math>
and <math>\sigma_{ic}, i=1,2,3</math> are the uniaxial yield stresses in '''compression''' in the three principal directions of anisotropy, <math>\sigma_{it}, i=1,2,3</math> are the uniaxial yield stresses in '''tension''', and  <math>\sigma_{23}^y, \sigma_{31}^y, \sigma_{12}^y</math> are the yield stresses in pure shear.  It has been assumed in the above that the quantities <math>\sigma_{1c},\sigma_{2c},\sigma_{3c}</math> are positive and <math>\sigma_{1t},\sigma_{2t},\sigma_{3t}</math> are negative.

== The Drucker yield criterion ==
The Drucker–Prager criterion should not be confused with the earlier Drucker criterion <ref>Drucker, D. C. (1949) '' Relations of experiments to mathematical theories of plasticity'', Journal of Applied Mechanics, vol. 16, pp. 349–357.</ref> which is independent of the pressure (<math>I_1</math>).  The Drucker yield criterion has the form
:<math>
   f := J_2^3 - \alpha~J_3^2 - k^2 \le 0
 </math>
where <math>J_2</math> is the second invariant of the deviatoric stress, <math>J_3</math> is the third invariant of the deviatoric stress, <math>\alpha</math> is a constant that lies between -27/8 and 9/4 (for the yield surface to be convex), <math>k</math> is a constant that varies with the value of <math>\alpha</math>.  For <math>\alpha=0</math>, <math>k^2 = \cfrac{\sigma_y^6}{27}</math> where <math>\sigma_y</math> is the yield stress in uniaxial tension.

== Anisotropic Drucker Criterion ==
An anisotropic version of the Drucker yield criterion is the Cazacu–Barlat (CZ) yield criterion <ref>Cazacu, O. and Barlat, F. (2001). ''Generalization of Drucker's yield criterion to orthotropy.'' Mathematics and Mechanics of Solids, vol. 6, no. 6, pp. 613–630.</ref> which has the form
:<math>
   f := (J_2^0)^3 - \alpha~(J_3^0)^2 - k^2 \le 0
 </math>
where <math>J_2^0, J_3^0</math> are generalized forms of the deviatoric stress and are defined as
:<math>
   \begin{align}
     J_2^0  := & \cfrac{1}{6}\left[a_1(\sigma_{22}-\sigma_{33})^2+a_2(\sigma_{33}-\sigma_{11})^2 +a_3(\sigma_{11}-\sigma_{22})^2\right] + a_4\sigma_{23}^2 + a_5\sigma_{31}^2 + a_6\sigma_{12}^2 \\
     J_3^0  := & \cfrac{1}{27}\left[(b_1+b_2)\sigma_{11}^3 +(b_3+b_4)\sigma_{22}^3 + \{2(b_1+b_4)-(b_2+b_3)\}\sigma_{33}^3\right] \\
      & -\cfrac{1}{9}\left[(b_1\sigma_{22}+b_2\sigma_{33})\sigma_{11}^2+(b_3\sigma_{33}+b_4\sigma_{11})\sigma_{22}^2
   + \{(b_1-b_2+b_4)\sigma_{11}+(b_1-b_3+b_4)\sigma_{22}\}\sigma_{33}^2\right] \\
     & + \cfrac{2}{9}(b_1+b_4)\sigma_{11}\sigma_{22}\sigma_{33} + 2 b_{11}\sigma_{12}\sigma_{23}\sigma_{31}\\
     & - \cfrac{1}{3}\left[\{2b_9\sigma_{22}-b_8\sigma_{33}-(2b_9-b_8)\sigma_{11}\}\sigma_{31}^2+
       \{2b_{10}\sigma_{33}-b_5\sigma_{22}-(2b_{10}-b_5)\sigma_{11}\}\sigma_{12}^2 \right.\\
      & \qquad \qquad\left. \{(b_6+b_7)\sigma_{11} - b_6\sigma_{22}-b_7\sigma_{33}\}\sigma_{23}^2
     \right]
   \end{align}
 </math>

=== Cazacu–Barlat yield criterion for plane stress ===
For thin sheet metals, the state of stress can be approximated as [[plane stress]].  In that case the Cazacu–Barlat yield criterion reduces to its two-dimensional version with
:<math>
   \begin{align}
     J_2^0  = & \cfrac{1}{6}\left[(a_2+a_3)\sigma_{11}^2+(a_1+a_3)\sigma_{22}^2-2a_3\sigma_1\sigma_2\right]+ a_6\sigma_{12}^2 \\
     J_3^0  = & \cfrac{1}{27}\left[(b_1+b_2)\sigma_{11}^3 +(b_3+b_4)\sigma_{22}^3 \right]
       -\cfrac{1}{9}\left[b_1\sigma_{11}+b_4\sigma_{22}\right]\sigma_{11}\sigma_{22} 
       + \cfrac{1}{3}\left[b_5\sigma_{22}+(2b_{10}-b_5)\sigma_{11}\right]\sigma_{12}^2 
   \end{align}
 </math>

For thin sheets of metals and alloys, the parameters of the Cazacu–Barlat yield criterion are
{| border="1"
|+ Table 1. '''Cazacu–Barlat yield criterion parameters for sheet metals and alloys'''
! Material !! <math>a_1</math> !! <math>a_2</math> !! <math>a_3</math> !! <math>a_6</math> !! <math>b_1</math> !! <math>b_2</math> !! <math>b_3</math> !! <math>b_4</math> !! <math>b_5</math> !! <math>b_{10}</math> !! <math>\alpha</math>
|-
! 6016-T4 Aluminum Alloy
| 0.815 || 0.815 || 0.334 || 0.42 || 0.04 || -1.205 || -0.958 || 0.306 || 0.153 || -0.02 || 1.4
|-
! 2090-T3 Aluminum Alloy
| 1.05 || 0.823 || 0.586 || 0.96 || 1.44 || 0.061 || -1.302 || -0.281 || -0.375 || 0.445 || 1.285
|}

== See also ==
{{Continuum mechanics|cTopic=[[Solid mechanics]]}}
*[[Yield surface]]
*[[Yield (engineering)]]
*[[Plasticity (physics)]]
*[[Material failure theory]]
*[[Daniel C. Drucker]]
*[[William Prager]]

== References ==
<references/>

{{DEFAULTSORT:Drucker-Prager yield criterion}}
[[Category:Plasticity]]
[[Category:Soil mechanics]]
[[Category:Solid mechanics]]
[[Category:Yield criteria]]