Visco Extended Drucker-Prager Model: Triaxial Driver versus Semi-Analytical Solution

Problem description

This example uses the Triaxial Driver to simulate a triaxial compression test of a Visco Extended Drucker-Prager solid. Constant lateral confining stress together with loading/unloading axial strain periods are imposed. Imposed axial strain range are high enough for allowing plastic yield in both loading and unloading period. This complicated scenario is used for verifying the numerical convergence and accuracy of the Visco Extended Drucker-Prager constitutive model implemented in GEOS.

Semi analytical result for axial stress variation $\Delta\sigma_{V}$ and lateral strain variation $\Delta\varepsilon_{V}$ can be established for the imposed triaxial boundary conditions following the theoretical basis of the Perzyna time dependent approach presented by (Runesson et al. 1999) as:

$\Delta\sigma_{V} = (\Delta\varepsilon_{V}-\Delta\lambda\frac{\theta b-3}{3}) E$

$\Delta\varepsilon_{H} = \Delta\varepsilon_{V} - \frac{\Delta\sigma_{V}}{2\mu} + \frac{3}{2}\Delta\lambda$

where $E$ and $\mu$ are the elastic Young and shear moduli. The visco-plastic multiplier $\Delta\lambda$ can be approximated by:

$\Delta\lambda = \frac{\Delta t}{t_*} \frac{F}{3\mu + K\theta b^2 + h}$

in which $\Delta t$ is the time increment, $t_*$ is the relaxation time, $F$ is the stress function defining the visco-plastic yield surface, $K$ is the elastic bulk modulus, $b$ is the frictional parameter defining the visco-plastic yield surface, $\theta$ is the dilation ratio defining the plastic potential and $h$ is the hardening rate. The hardening rate $h$ is defined by

$h = \frac{\partial F}{\partial b} \frac{\partial b}{\partial \lambda}$

These solutions were established for a positive shear stress $q = -(\sigma_{V} - \sigma_{H})$ (negative sign convention for compression stress). For the case when the plastic yield occurs at a negative shear stress, we have

$\Delta\sigma_{V} = (\Delta\varepsilon_{V}-\Delta\lambda\frac{\theta b+3}{3}) E$

$\Delta\varepsilon_{H} = \Delta\varepsilon_{V} - \frac{\Delta\sigma_{V}}{2\mu} - \frac{3}{2}\Delta\lambda$

These solutions are implemented in a Python script associated to this example for verifying GEOS results.

Input files

This validation example uses two GEOS xml files that are located at:

inputFiles/triaxialDriver/triaxialDriver_base.xml

and

inputFiles/triaxialDriver/triaxialDriver_ViscoExtendedDruckerPrager.xml

It also uses a set of table files located at:

inputFiles/triaxialDriver/tables/

A Python script for the semi-analytical solutions presented above as well as for post-processing the GEOS results is provided at:

src/docs/sphinx/advancedExamples/validationStudies/viscoplasticity/ViscoExtendedDruckerPrager/TriaxialDriver_vs_SemiAnalytic_ViscoExtendedDruckerPrager.py

For this example, we focus on the Task and the Constitutive tags.

Task

The imposed axial strain loading/unloading periods, the constant lateral confining stress as well as the initial stress are defined in the Task block as

  <Tasks>
    <TriaxialDriver
      name="triaxialDriver"
      material="ViscoExtendedDruckerPrager"
      mode="mixedControl" 
      axialControl="strainFunction"
      radialControl="stressFunction"
      initialStress="-10.e6"
      steps="200" 
      output="ViscoExtendedDruckerPragerResults.txt" />
  </Tasks>

Constitutive laws

The elasto-visco-plastic parameters are defined as

    <ViscoExtendedDruckerPrager 
      name="ViscoExtendedDruckerPrager"
      defaultDensity="2700"
      defaultBulkModulus="10.0e9"
      defaultShearModulus="6.0e9"
      defaultCohesion="0.1e6"
      defaultInitialFrictionAngle="6.0"
      defaultResidualFrictionAngle="10.0"
      defaultDilationRatio="0.5"
      defaultHardening="0.0001"
      relaxationTime="0.1"
    />  

All constitutive parameters such as density, viscosity, and bulk and shear moduli are specified in the International System of Units.

A comparison between GEOS results and semi-analytical results

The simulation results are saved in a text file, named ViscoExtendedDruckerPragerResults.txt. A comparison between the results given by the TriaxialDriver solver in GEOS and the approximated semi-analytical results presented above is shown below. Interestingly we observed that the Duvaut-Lions approach implemented in GEOS can fit perfectly with the Perzyna approach that was considered for deriving the analytical results. This consistency between these time dependence approaches is because of the linear hardening law of the considered constitutive model as already discussed by (Runesson et al. 1999) .

(Source code)

../../../../../../_images/TriaxialDriver_vs_SemiAnalytic_ViscoExtendedDruckerPrager.png

To go further

Feedback on this example

For any feedback on this example, please submit a GitHub issue on the project’s GitHub page.