Drucker Prager

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This command is used to construct an multi dimensional material object that has a Drucker-Prager yield criterium.

nDmaterial DruckerPrager $matTag $k $G $sigmaY $rho $rhoBar $Kinf $Ko $delta1 $delta2 $H $theta

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This Code has been Developed by: Peter Mackenzie, U Washington and the great Pedro Arduino, U Washington

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$matTag	integer tag identifying material
$k	bulk modulus
$G	shear modulus
$sigmaY	yield stress
$rho	frictional strength parameter
$rhoBar	controls evolution of plastic volume change, 0 ≤ $rhoBar ≤ $rho
$Kinf	nonlinear isotropic strain hardening parameter, $Kinf ≥ 0
$Ko	nonlinear isotropic strain hardening parameter, $Ko ≥ 0
$delta1	nonlinear isotropic strain hardening parameter, $delta1 ≥ 0
$delta2	tension softening parameter, $delta2 ≥ 0
$H	linear strain hardening parameter, $H ≥ 0
$theta	controls relative proportions of isotropic and kinematic hardening, 0 ≤ $theta ≤ 1

The material formulations for the Drucker-Prager object are "ThreeDimensional," "PlaneStrain," "Plane Stress," "AxiSymmetric".

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THEORY:

The yield condition for the Drucker-Prager model can be expressed as

<math> f\left(\mathbf{\sigma}, q^{iso}, \mathbf{q}^{kin}\right) = \left\| \mathbf{s} + \mathbf{q}^{kin} \right\| + \rho I_1 + \sqrt{\frac{2}{3}} q^{iso} - \sqrt{\frac{2}{3}} \sigma_Y^{} \leq 0

</math>

in which

<math> \mathbf{s} = \mathrm{dev} (\mathbf{\sigma}) = \mathbf{\sigma} - \frac{1}{3} I_1 \mathbf{1}

</math>

is the deviatoric stress tensor,

<math> I_1 = \mathrm{tr}(\mathbf{\sigma})

</math>

is the first invariant of the stress tensor, and the parameters <math>\rho_{}^{}</math> and <math>\sigma_Y^{}</math> are positive material constants.

The isotropic hardening stress is defined as

<math> q^{iso} = \theta H \alpha^{iso} + (K_{\infty} - K_o) \exp(-\delta_1 \alpha^{iso})

</math>

The kinematic hardening stress (or back-stress) is defined as

<math> \mathbf{q}^{kin} = -(1 - \theta) \frac{2}{3} H \mathbb{I}^{dev} : \mathbf{\alpha}^{kin}

</math>

The yield condition for the tension cutoff yield surface is defined as

<math> f_2(\mathbf{\sigma}, q^{ten}) = I_1 + q^{ten} \leq 0

</math>

where

<math> q^{ten} = T_o \exp(-\delta_2^{} \alpha^{ten})

</math>

and

<math> T_o = \sqrt{\frac{2}{3}} \frac{\sigma_Y}{\rho}

</math>

Further, general, information on theory for the Drucker-Prager yield criterion can be found at wikipedia here

Note: The Drucker-Prager strength parameters <math> \rho </math> and <math> \sigma_Y </math> can be related to the Mohr-Coulomb friction angle, <math> \phi </math>, and cohesive intercept, <math> c </math>, by evaluating the yield surfaces in a deviatoric plane as described by Chen and Saleeb (1994). By relating the two yield surfaces in triaxial compression, the following expressions are determined

<math> \rho = \frac{2 \sqrt{2} \sin \phi}{\sqrt{3} (3 - \sin \phi)}

</math>

<math> \sigma_Y = \frac{6 c \cos \phi}{\sqrt{2} (3 - \sin \phi)}

</math>

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EXAMPLE: Conventional Triaxial Compression Test

This example provides the input file and corresponding results for a confined triaxial compression (CTC) test using a single 8-node brick element and the Drucker-Prager Constitutive model.

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REFERENCES;

Drucker, D. C. and Prager, W., "Soil mechanics and plastic analysis for limit design. Quarterly of Applied Mathematics, vol. 10, no. 2, pp. 157–165, 1952.

Chen, W. F. and Saleeb, A. F., Constitutive Equations for Engineering Materials Volume I: Elasticity and Modeling. Elsevier Science B.V., Amsterdam, 1994.