Reference problem
=====================

Geometry
---------

We consider an element as a material point.

Material properties
----------------------

Hujeux and Iwan models of behavior
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The isotropic elastic properties of the material are :math:`E=619.33` MPa and :math:`\nu =0.3`.


The anelastic parameters specific to the Hujeux model (modeling A) are:


* :math:`n=0.4`, :math:`\mathrm{\beta }=24`, :math:`b=0.2`, :math:`d=2.5`.
* :math:`\mathrm{\varphi }=33°`, :math:`\mathrm{\psi }=33°`, :math:`{P}_{c0}=-1` MPa, :math:`{P}_{\mathit{ref}}=-1` MPa
* :math:`{a}_{\mathit{cyc}}=1\times10^{-4}`, :math:`{a}_{\mathit{mon}}=8\times10^{-3}`
* :math:`{c}_{\mathit{cyc}}=1\times10^{-1}`, :math:`{c}_{\mathit{mon}}=2\times10^{-1}`
* :math:`{r}_{\mathit{ela}}^{d}=5\times10^{-3}`, :math:`{r}_{\mathit{ela}}^{i}=1\times10^{-3}`, :math:`{r}_{\mathit{ela}}^{d,c}=5\times10^{-3}`, :math:`{r}_{\mathit{ela}}^{i,c}=1\times10^{-3}`
* :math:`{r}_{\mathit{hys}}=5\times10^{-2}`, :math:`{r}_{\mathit{mob}}=9\times10^{-1}`
* :math:`{x}_{m}=1`, :math:`\alpha=1`

The anelastic parameters of the Iwan model (modeling B) are for their part :math:`\gamma_{\mathrm{ref}}=2\times10^{-4}` and :math:`n=0.78`.

 **Note:**

 Using the *CALC_ESSAI_GEOMECA tool,* we are able to compare the behavior curves for the two models in terms of reducing the normalized secant shear modulus 
 and reduced depreciation (see :numref:`v6.04.205-module_secant_normalise` and :numref:`v6.04.205-amortissement_reduit`).

.. _v6.04.205-module_secant_normalise:

.. figure:: images/100002010000032000000258B5CDD4F1675D5855.png
 :align: center
 :width: 400

 Predictions of the secant shear modulus normalized with distortion by Hujeux and Iwan models.

.. _v6.04.205-amortissement_reduit:

.. figure:: images/1000020100000320000002589A53DF01E1E9C3F4.png
 :align: center
 :width: 400

 Predictions of reduced damping with distortion by Hujeux and Iwan models.

Behaviour model CSSM
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The parameters specific to the model CSSM [:ref:`r7.01.44 <R7.01.49>`] are grouped together in the :numref:`v6.04.160-table_parametres_CSSM` concerning the C modeling.

.. _v6.04.160-table_parametres_CSSM:
 
.. list-table:: Paramètres du modèle CSSM utilisés dans la modélisation C.

 *-**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: `0.99`
     |: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

Boundary conditions and loads
-------------------------------------

As a reminder, the use of the SIMU_POINT_MAT command makes it possible to directly impose a field of deformations and/or constraints.

A zero evolution during loading is imposed for the following components of the stress and strain tensors:

* :math:`{\mathrm{d}\sigma}_{xx}={\mathrm{d}\sigma}_{yy}={\mathrm{d}\sigma}_{zz}=0`
* :math:`{\mathrm{d}\varepsilon}_{yz}={\mathrm{d}\varepsilon}_{zx}=0`

The evolution of :numref:`v6.04.205-image_chargement` is imposed for :math:`\varepsilon_{xy}` shear deformations:

.. _v6.04.205-image_chargement:

.. figure:: images/image_chargement.png
 :align: center
 :width: 400
 
 Imposed loading chronology for the :math:`\varepsilon_{xy}` (standardized) deformation component.


Several calculations are carried out by varying the amplitude of deformations according to the values :math:`\left\{2\times10^{-5},2\times10^{-4},2\times10^{-3}\right\}`.


Initial conditions
--------------------

The initial stress state is isotropic and corresponds to a pressure equal to :math:`50` kPa.