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	non-associative parameter
$Kinf	nonlinear isotropic strain hardening parameter
$Ko	nonlinear isotropic strain hardening parameter
$delta1	nonlinear isotropic strain hardening parameter
$delta2	tension softening parameter
$H	linear kinematic strain hardening parameter
$theta	linear kinematic strain hardening parameter

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

---

EXAMPLE

An example like ZeroLengthContactNTS2D would be nice

---

THEORY:

The theory for the Drucker-Prager yield criterion can be found at wikipedia here

The the isotropic hardening modulus is defined using the indicated input paramaters as:

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

and the kinematic hardening modulus is defined as:

<math> H^\prime(\alpha) = (1 - \theta) H </math>

Tension softening is defined as:

<math> T(\alpha_2) = T_o \exp(-\delta_2 \alpha_2) </math>

in which

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

---

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.