Modeling I
==============

Characteristics of modeling
-----------------------------------

Simulation at the hardware point. The objective is to validate the compatibility of CALC_ESSAI_GEOMECA with the CSSM behavior model. 
To do this, several drained cyclic shear tests with imposed displacement ESSAI_CISA_DR_C_D are carried out. 
The number of cycles applied for ten distortion amplitudes :math:`\gamma=2\varepsilon_{xy}\in [10^{-6};10^{-2}]`
is three; the confinement pressure is equal to :math:`200` kPa; 
The convergence criterion is :math:`10^{-10}`.

The parameters for model CSSM are given in :numref:`v6.07.112-table_parametres_CSSM`.

.. _v6.07.112-table_parametres_CSSM:

.. list-table:: Paramètres du modèle MCC utilisés dans la modélisation I.
  
 *-**Intervention**
   - **Appellation**
   - **Definition**
   - **Symbol**
   - **Value**
 * - Elasticity
   - | *BulkModulus*
     | *ShearModulus*
     | *ShearModulusRatio*
   - | Total compressibility module
     | Total shear modulus
     | Ratio of the shear modulus of component 1 to the total shear modulus
   - |:math: `K`
     |:math: `\ mu`
     |:math: `\ rho`
   - | 516 MPa
     | 238 MPa
     | 0.1
 * - Component 1
   - | *critstateSlope*
     | *initCritPress*
     | *IncoplastIndex*
     | *IsoHardRatio*
     | *isoHardIndex*
   - | Critical state slope
     | Initial critical pressure
     | Plastic incompressibility index
     | Homothetic reduction ratio of the initial elasticity domain
     | Hardening index by homothetic enlargement of the initial elasticity domain 
   - |:math: `M`
     |:math: `p_ {c0} `
     |:math: `\ beta`
     |:math: `\ eta`
     |:math: `\ omega`
   - |:math: `1.38`
     |:math: `100` kPa
     |:math: `30`
     |:math: `0`
     |:math: `32`
 * - Component 2
   - | *HypDistortion*
     | *HyperExponent*
     | *miNcritPress*
   - | Reference distortion of the "modified hyperbolic" relationship
     | Curvature parameter for the "modified hyperbolic" relationship
     | Minimum pressure at which the critical state is attainable
   - |:math: `\ gamma_ {\ mathrm {hyp}}`
     |:math: `n_ {\ mathrm {hyp}} `
     |:math: `C`
   - |:math: `2.10^ {-4} `
     |:math: `0.78`
     |:math: `448` kPa

..
   How to get the gray box here-under
..

 **Note**

 With the parameters of :numref:`v6.07.112-table_parametres_CSSM`, the initial elastic limit in isotropic compression is :math:`2p_{c0}(1-\eta)=200` kPa
 (see [:ref:`r7.01.44 <R7.01.49>`]). Thus, the state of the stresses, before applying shear, is located at the limit of the initial elasticity domain. 
 of the model. 

Tested sizes and results
------------------------------

Two non-regression tests are carried out on component SIXY at distortion amplitudes :math:`\gamma` equal to
:math:`10^{-6}` and :math:`10^{-2}` after having imposed the three loading cycles.

.. list-table::
  
 *-**Distortion amplitude**
   - **Aster code**
 * - :math:`10^{-6}`
   - :math:`-237.9999955384302` Not
 * - :math:`10^{-2}`
   - :math:`-165087.2891707449` Not
   
:numref:`v6.07.112-modelisation_I_hysteresis` shows hysteresis cycles. 
Positive work hardening is observed, which stabilizes during the three cycles imposed at the same level of distortion.
The :numref:`v6.07.112-modelisation_I_mu_secant` and :numref:`v6.07.112-modelisation_I_amortissement` 
trace the changes in the standardized secant shear modulus and hysteretic damping (reduced damping).
These are comparable to the predictions of the Hujeux and Iwan models. 

.. _v6.07.112-modelisation_I_hysteresis:

.. figure:: images/modelisation_I_hysteresis.png
 :align: center
 :width: 600

 Shear stress responses (Pa).
         
.. _v6.07.112-modelisation_I_mu_secant:

.. figure:: images/modelisation_I_mu_secant.png
 :align: center
 :width: 600

 Evolution of the normalized secant shear modulus.

.. _v6.07.112-modelisation_I_amortissement:

.. figure:: images/modelisation_I_amortissement.png
 :align: center
 :width: 600

 Evolution of hysteretic damping.