Drucker–Prager yield criterion

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Figure 1: View of Drucker–Prager yield surface in 3D space of principal stresses for c=2, \phi=-20^\circ

The Drucker–Prager yield criterion[1] 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 DruckerPrager yield criterion has the form


   \sqrt{J_2} = A + B~I_1

where I_1 is the first invariant of the Cauchy stress and J_2 is the second invariant of the deviatoric part of the Cauchy stress. The constants A, B are determined from experiments.

In terms of the equivalent stress (or von Mises stress) and the hydrostatic (or mean) stress, the Drucker–Prager criterion can be expressed as


   \sigma_e = a + b~\sigma_m

where \sigma_e is the equivalent stress, \sigma_m is the hydrostatic stress, and a,b are material constants. The Drucker–Prager yield criterion expressed in Haigh–Westergaard coordinates is


   \tfrac{1}{\sqrt{2}}\rho - \sqrt{3}~B\xi = A

The Drucker–Prager yield surface is a smooth version of the Mohr–Coulomb yield surface.

Expressions for A and B[edit]

The Drucker–Prager model can be written in terms of the principal stresses as


  \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) ~.

If \sigma_t is the yield stress in uniaxial tension, the Drucker–Prager criterion implies


   \cfrac{1}{\sqrt{3}}~\sigma_t = A + B~\sigma_t ~.

If \sigma_c is the yield stress in uniaxial compression, the Drucker–Prager criterion implies


   \cfrac{1}{\sqrt{3}}~\sigma_c = A - B~\sigma_c ~.

Solving these two equations gives


   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) ~.

Uniaxial asymmetry ratio[edit]

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


   \beta = \cfrac{\sigma_\mathrm{c}}{\sigma_\mathrm{t}} = \cfrac{1 - \sqrt{3}~B}{1 + \sqrt{3}~B} ~.

Expressions in terms of cohesion and friction angle[edit]

Since the Drucker–Prager yield surface is a smooth version of the Mohr–Coulomb yield surface, it is often expressed in terms of the cohesion (c) and the angle of internal friction (\phi) that are used to describe the Mohr–Coulomb yield surface. If we assume that the Drucker–Prager yield surface circumscribes the Mohr–Coulomb yield surface then the expressions for A and B are


   A = \cfrac{6~c~\cos\phi}{\sqrt{3}(3+\sin\phi)} ~;~~
   B = \cfrac{2~\sin\phi}{\sqrt{3}(3+\sin\phi)}

If the Drucker–Prager yield surface inscribes the Mohr–Coulomb yield surface then


   A = \cfrac{6~c~\cos\phi}{\sqrt{3}(3-\sin\phi)} ~;~~
   B = \cfrac{2~\sin\phi}{\sqrt{3}(3-\sin\phi)}
Figure 2: Drucker–Prager yield surface in the \pi-plane for c = 2, \phi = 20^\circ
Figure 3: Trace of the Drucker–Prager and Mohr–Coulomb yield surfaces in the \sigma_1-\sigma_2-plane for c = 2, \phi = 20^\circ. Yellow = Mohr–Coulomb, Cyan = Drucker–Prager.

Drucker–Prager model for polymers[edit]

The Drucker–Prager model has been used to model polymers such as polyoxymethylene and polypropylene[citation needed].[2] 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[edit]

For foams, the GAZT model [3] uses


   A = \pm \cfrac{\sigma_y}{\sqrt{3}} ~;~~ B = \mp \cfrac{1}{\sqrt{3}}~\left(\cfrac{\rho}{5~\rho_s}\right)

where \sigma_{y} is a critical stress for failure in tension or compression, \rho is the density of the foam, and \rho_s is the density of the base material.

Extensions of the isotropic Drucker–Prager model[edit]

The Drucker–Prager criterion can also be expressed in the alternative form


  J_2 = (A + B~I_1)^2 = a + b~I_1 + c~I_1^2 ~.

Deshpande–Fleck yield criterion or isotropic foam yield criterion[edit]

The Deshpande–Fleck yield criterion[4] for foams has the form given in above equation. The parameters a, b, c for the Deshpande–Fleck criterion are


  a = (1 + \beta^2)~\sigma_y^2 ~,~~
  b = 0 ~,~~
  c = -\cfrac{\beta^2}{3}

where \beta is a parameter[5] that determines the shape of the yield surface, and \sigma_y is the yield stress in tension or compression.

Anisotropic Drucker–Prager yield criterion[edit]

An anisotropic form of the Drucker–Prager yield criterion is the Liu–Huang–Stout yield criterion.[6] This yield criterion is an extension of the generalized Hill yield criterion and has the form


  \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}

The coefficients F,G,H,L,M,N,I,J,K are


  \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}

where


   \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}}

and \sigma_{ic}, i=1,2,3 are the uniaxial yield stresses in compression in the three principal directions of anisotropy, \sigma_{it}, i=1,2,3 are the uniaxial yield stresses in tension, and \sigma_{23}^y, \sigma_{31}^y, \sigma_{12}^y are the yield stresses in pure shear. It has been assumed in the above that the quantities \sigma_{1c},\sigma_{2c},\sigma_{3c} are positive and \sigma_{1t},\sigma_{2t},\sigma_{3t} are negative.

The Drucker yield criterion[edit]

The Drucker–Prager criterion should not be confused with the earlier Drucker criterion [7] which is independent of the pressure (I_1). The Drucker yield criterion has the form


   f := J_2^3 - \alpha~J_3^2 - k^2 \le 0

where J_2 is the second invariant of the deviatoric stress, J_3 is the third invariant of the deviatoric stress, \alpha is a constant that lies between -27/8 and 9/4 (for the yield surface to be convex), k is a constant that varies with the value of \alpha. For \alpha=0, k^2 = \cfrac{\sigma_y^6}{27} where \sigma_y is the yield stress in uniaxial tension.

Anisotropic Drucker Criterion[edit]

An anisotropic version of the Drucker yield criterion is the Cazacu–Barlat (CZ) yield criterion [8] which has the form


   f := (J_2^0)^3 - \alpha~(J_3^0)^2 - k^2 \le 0

where J_2^0, J_3^0 are generalized forms of the deviatoric stress and are defined as


   \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}

Cazacu–Barlat yield criterion for plane stress[edit]

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


   \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}

For thin sheets of metals and alloys, the parameters of the Cazacu–Barlat yield criterion are

Table 1. Cazacu–Barlat yield criterion parameters for sheet metals and alloys
Material a_1 a_2 a_3 a_6 b_1 b_2 b_3 b_4 b_5 b_{10} \alpha
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[edit]

References[edit]

  1. ^ 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.
  2. ^ Abrate, S. (2008). Criteria for yielding or failure of cellular materials. Journal of Sandwich Structures and Materials, vol. 10. pp. 5–51.
  3. ^ Gibson, L.J., 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.
  4. ^ V. S. Deshpande, and Fleck, N. A. (2001). Multi-axial yield behaviour of polymer foams. Acta Materialia, vol. 49, no. 10, pp. 1859–1866.
  5. ^ \beta= \alpha/3 where \alpha is the quantity used by Deshpande–Fleck
  6. ^ 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
  7. ^ Drucker, D. C. (1949) Relations of experiments to mathematical theories of plasticity, Journal of Applied Mechanics, vol. 16, pp. 349–357.
  8. ^ 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.