Visco Modified CamClay model: Triaxial Driver versus Semi-Analytical Solution

Problem description

This example uses the Triaxial Driver to simulate a visco-elasto-plastic oedometric compression test of a Visco Modified CamClay solid. Oedometric condition with zero lateral strain together with loading/unloading axial strain periods are imposed. Semi-analytical results for the mean and shear stress variations \(\Delta p\) and \(\Delta q\) can be established, considering the Perzyna approach, for the imposed oedometric boundary conditions as (Runesson et al. 1999) :

\[\Delta p = K(\Delta\varepsilon_{V} - \Delta\lambda \frac{\partial G}{\partial p})\]

\[\Delta q = 2\mu(\Delta\varepsilon_{V} - \Delta\lambda \frac{3}{2}\frac{\partial G}{\partial q})\]

where \(K\) and \(\mu\) are elastic bulk and shear moduli, \(G\) is the plastic potential and \(\Delta\lambda\) is the visco-plastic multiplier that can be approximated by:

\[\Delta\lambda = \frac{\Delta t}{t_*} \frac{F}{3\mu\frac{\partial F}{\partial q}\frac{\partial G}{\partial q} + K\frac{\partial F}{\partial p}\frac{\partial G}{\partial p} + h}\]

in which \(\Delta t\) is the time increment, \(t_*\) is the relaxation time, \(F\) is the stress function defining the visco-plastic yield surface and \(h\) is the hardening rate defined by:

\[h = -\frac{\partial F}{\partial \lambda}\]

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_ViscoModifiedCamClay.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/ViscoModifiedCamClay/TriaxialDriver_vs_SemiAnalytic_ViscoModifiedCamClay.py

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

Task

The imposed axial strain loading/unloading periods, the lateral zero strain, and the initial stress are defined in the Task block as:

  <Tasks>
   <TriaxialDriver
      name="triaxialDriver"
      material="ViscoModifiedCamClay"
      mode="strainControl" 
      axialControl="strainFunction"
      radialControl="zeroStrain"
      initialStress="-1e5"
      steps="200" 
      output="ViscoModifiedCamClayResults.txt" />
  </Tasks>

Constitutive laws

The elasto-visco-plastic parameters are defined as:

    <ViscoModifiedCamClay 
      name="ViscoModifiedCamClay"
      defaultDensity="2700"
      defaultRefPressure="-1e5"
      defaultRefStrainVol="0.0"
      defaultShearModulus="5e7"
      defaultPreConsolidationPressure="-1.5e5"
      defaultCslSlope="1.2"
      defaultRecompressionIndex="0.002"
      defaultVirginCompressionIndex="0.003" 
      relaxationTime="0.1"
    />

All constitutive parameters such as density, viscosity, and the 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 ViscoModifiedCamClayResults.txt. A comparison between the results given by the TriaxialDriver solver in GEOS and the semi-analytical results presented above is shown below. The discrepancy between these results may due to the difference between the Duvaut-Lions approach and the Perzyna approach for time dependant behavior when applying for the Modified CamClay model as discussed by Runesson et al. (1999).

(Source code)

To go further

Feedback on this example

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