View source for Drucker Prager

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

{| 
| style="background:yellow; color:black; width:800px" | '''nDmaterial DruckerPrager $matTag $k $G $sigmaY $rho $rhoBar $Kinf $Ko $delta1 $delta2 $H $theta '''
|}


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This Code has been Developed by: <span style="color:blue">[http://www.ce.washington.edu/people/faculty/bios/mackenzie_p.html Peter Mackenzie, U Washington]</span> and the great <span style="color:blue">[http://www.ce.washington.edu/people/faculty/bios/arduino_p.html  Pedro Arduino, U Washington]</span>


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{|
|  style="width:150px" | '''$matTag ''' || integer tag identifying material
|-
|  '''$k ''' || bulk modulus
|-
|  '''$G ''' || shear modulus
|-
|  '''$sigmaY ''' || yield stress
|-
|  '''$rho ''' || frictional strength parameter
|-
|  '''$rhoBar ''' || controls evolution of plastic volume change,  0 &le; $rhoBar &le; $rho
|-
|  '''$Kinf ''' || nonlinear isotropic strain hardening parameter,  $Kinf &ge; 0
|-
|  '''$Ko ''' || nonlinear isotropic strain hardening parameter,  $Ko &ge; 0
|-
|  '''$delta1 ''' || nonlinear isotropic strain hardening parameter,  $delta1 &ge; 0
|-
|  '''$delta2 ''' || tension softening parameter,  $delta2 &ge; 0
|-
|  '''$H ''' || linear strain hardening parameter,  $H &ge; 0
|-
|  '''$theta ''' || controls relative proportions of isotropic and kinematic hardening,  0 &le; $theta &le; 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 [http://en.wikipedia.org/wiki/Drucker_Prager_yield_criterion here]

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EXAMPLE

An example like [[ZeroLengthContactNTS2D]] would be nice

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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.