Drucker Prager yield criterion

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

The DruckerPrager 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 I1 is the first invariant of the Cauchy stress and J2 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 DruckerPrager criterion can be expressed as


   \sigma_e = a + b~\sigma_m

where σe is the equivalent stress, σm is the hydrostatic stress, and a,b are material constants. The DruckerPrager yield criterion expressed in HaighWestergaard coordinates is


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

The DruckerPrager yield surface is a smooth version of the MohrCoulomb yield surface.

Contents

Expressions for A and B

The DruckerPrager 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 σt is the yield stress in uniaxial tension, the DruckerPrager criterion implies


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

If σc is the yield stress in uniaxial compression, the DruckerPrager 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

Different uniaxial yield stresses in tension and in compression are predicted by the DruckerPrager model. The uniaxial asymmetry ratio for the DruckerPrager 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

Since the DruckerPrager yield surface is a smooth version of the MohrCoulomb yield surface, it is often expressed in terms of the cohesion (c) and the angle of internal friction (ϕ) that are used to describe the MohrCoulomb yield surface. If we assume that the DruckerPrager yield surface circumscribes the MohrCoulomb 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 DruckerPrager yield surface inscribes the MohrCoulomb 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: DruckerPrager yield surface in the π-plane for c = 2, \phi = -20^\circ
Figure 3: Trace of the DruckerPrager and MohrCoulomb yield surfaces in the σ1 σ2-plane for c = 2, \phi = -20^\circ. Yellow = MohrCoulomb, Cyan = DruckerPrager.

DruckerPrager model for polymers

The DruckerPrager 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.

DruckerPrager model for foams

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 σy is a critical stress for failure in tension or compression, ρ is the density of the foam, and ρs is the density of the base material.

Extensions of the isotropic DruckerPrager model

The DruckerPrager criterion can also be expressed in the alternative form


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

DeshpandeFleck yield criterion

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


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

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

Anisotropic DruckerPrager yield criterion

An anisotropic form of the DruckerPrager yield criterion is the LiuHuangStout 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_{11}-\sigma_{22})^2+G(\sigma_{22}-\sigma_{33})^2+H(\sigma_{33}-\sigma_{11})^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 σic,i = 1,2,3 are the uniaxial yield stresses in compression in the three principal directions of anisotropy, σit,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.

The Drucker yield criterion

The DruckerPrager criterion should not be confused with the earlier Drucker criterion [7] which is independent of the pressure (I1). The Drucker yield criterion has the form


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

where J2 is the second invariant of the deviatoric stress, J3 is the third invariant of the deviatoric stress, α 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 α. For α = 0, k^2 = \cfrac{\sigma_y^6}{27} where σy is the yield stress in uniaxial tension.

Anisotropic Drucker Criterion

An anisotropic version of the Drucker yield criterion is the CazacuBarlat (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}

CazacuBarlat yield criterion for plane stress

For thin sheet metals, the state of stress can be approximated as plane stress. In that case the CazacuBarlat 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 CazacuBarlat yield criterion are

Table 1. CazacuBarlat yield criterion parameters for sheet metals and alloys
Material a1 a2 a3 a6 b1 b2 b3 b4 b5 b10 α
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

References

  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. 157165.
  2. ^ Abrate, S. (2008). Criteria for yielding or failure of cellular materials. Journal of Sandwich Structures and Materials, vol. 10. pp. 551.
  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. 635665.
  4. ^ V. S. Deshpande, and Fleck, N. A. (2001). Multi-axial yield behaviour of polymer foams. Acta Materialia, vol. 49, no. 10, pp. 18591866.
  5. ^ β = α / 3 where α is the quantity used by DeshpandeFleck
  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. 23972406
  7. ^ Drucker, D. C. (1949) Relations of experiments to mathematical theories of plasticity, Journal of Applied Mechanics, vol. 16, pp. 349357.
  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. 613630.

See also


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