.. _AdvancedExampleModifiedCamClay:


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Modified CamClay Model: Triaxial Driver versus Semi-Analytical Solution 
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Problem description
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This example uses the Triaxial Driver to simulate an elasto-plastic oedometric compression test of a 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 :math:`\Delta p` and :math:`\Delta q` can be derived from the imposed vertical strain variation by solving the following equation system:

.. math::
   \Delta \varepsilon_V = \Delta p(\frac{1}{K} + \frac{1}{h}\frac{\partial F}{\partial p}\frac{\partial G}{\partial p}) + \Delta q \frac{1}{h}\frac{\partial F}{\partial q}\frac{\partial G}{\partial p}

.. math::
   \Delta \varepsilon_V = \Delta p \frac{3}{2h}\frac{\partial F}{\partial p}\frac{\partial G}{\partial q} + \Delta q(\frac{1}{2\mu} + \frac{1}{h}\frac{\partial F}{\partial q}\frac{\partial G}{\partial q})

where :math:`K` and :math:`\mu` are elastic bulk and shear moduli, :math:`F` and :math:`G` are the plastic yield surface and the plastic potential, and :math:`h` the is hardening rate defined by:

.. math::
   h = -\frac{\partial F}{\partial \varepsilon^{vp}_{vol}}\frac{\partial G}{\partial p}

in which :math:`\varepsilon^{vp}_{vol}` is the volumetric visco-plastic strain. 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:

.. code-block:: console

  inputFiles/triaxialDriver/triaxialDriver_base.xml

and

.. code-block:: console

  inputFiles/triaxialDriver/triaxialDriver_ModifiedCamClay.xml

It also uses a set of table files located at:

.. code-block:: console

  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:

.. code-block:: console

  src/docs/sphinx/advancedExamples/validationStudies/viscoplasticity/ModifiedCamClay/TriaxialDriver_vs_SemiAnalytic_ModifiedCamClay.py

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

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

.. literalinclude:: ../../../../../../../inputFiles/triaxialDriver/triaxialDriver_ModifiedCamClay.xml
    :language: xml
    :start-after: <!-- SPHINX_TASK -->
    :end-before: <!-- SPHINX_TASK_END -->

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Constitutive laws
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The elasto-plastic parameters are defined as


.. literalinclude:: ../../../../../../../inputFiles/triaxialDriver/triaxialDriver_base.xml
    :language: xml
    :start-after: <!-- SPHINX_MATERIAL_MODIFIED_CAMCLAY -->
    :end-before: <!-- SPHINX_MATERIAL_MODIFIED_CAMCLAY_END -->


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

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A comparison between GEOS results and semi-analytical results
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The simulation results are saved in a text file, named ``ModifiedCamClayResults.txt``. A perfect comparison between the results given by the TriaxialDriver solver in GEOS and the semi-analytical results presented above is shown below: 


.. plot:: docs/sphinx/advancedExamples/validationStudies/viscoplasticity/ModifiedCamClay/TriaxialDriver_vs_SemiAnalytic_ModifiedCamClay.py


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To go further
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**Feedback on this example**

For any feedback on this example, please submit a `GitHub issue on the project's GitHub page <https://github.com/GEOS-DEV/GEOS/issues>`_.