Drucker Prager

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

---

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

---

$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".

---

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

---

EXAMPLE

An example like ZeroLengthContactNTS2D would be nice

---

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.