Settings
==========

Classification
--------------

Eleven parameters are required to define model CSSM. These are classified in the :numref:`r7.01.44-parametres`.

.. _r7.01.44-parametres:

.. list-table:: Paramètres.

 *-**Intervention**
   - **Appellation**
   - **Definition**
   - **Unit**
   - **Interval**
   - **Symbol**
 * - Elasticity
   - | *BulkModulus*
     | *ShearModulus*
     | *ShearModulusRatio*
   - | Compressibility module
     | Shear module
     | Report of the shear modulus of component 1 to the shear modulus
   - | [Not]
     | [Not]
     | [-]
   - |:math: `\ mathbb {R} ^ {+*}`
     |:math: `\ mathbb {R} ^ {+*}`
     |:math: `[0; 1]`
   - |:math: `K`
     |:math: `\ mu`
     |:math: `\ rho`
 * - 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 
   - | [-]
     | [Not]
     | [-]
     | [-]
     | [-]
   - |:math: `\ mathbb {R} ^ {+*}`
     |:math: `\ mathbb {R} ^ {+*}`
     |:math: `\ mathbb {R} ^ {+}`
     |:math: `[0; 1 [`
     |:math: `\ mathbb {R} ^ {+}`
   - |:math: `M`
     |:math: `p_ {c0} `
     |:math: `\ beta`
     |:math: `\ eta`
     |:math: `\ omega`
 * - Component 2
   - | *HypDistortion*
     | *HyperExponent*
     | *miNcritPress*
   - | Reference distortion of the "modified hyperbolic" relationship:eq: `calibration_shear_3`
     | Curvature parameter for the "modified hyperbolic" relationship:eq: `calibration_shear_3`
     | Minimum pressure at which the critical state:eq: `systeme_critical_status` is achievable
   - | [-]
     | [-]
     | [Not]
   - |:math: `\ mathbb {R} ^ {+*}`
     |:math: `[0; 1 [`
     |:math: `\ mathbb {R} ^ {+*}`
   - |:math: `\ gamma_ {\ mathrm {hyp}}`
     |:math: `n_ {\ mathrm {hyp}} `
     |:math: `C`

Special cases
----------------

By design, model CSSM is a hybrid model that combines two modified Cam-Clay (component 1) and Iwan (component 2) models.
This combination is achieved by weighting the shear behavior using the parameter :math:`\rho\in [0;1]`. So the cases 
individuals can be deducted.

* If :math:`\rho=1`: the initial elasticity domain of the CSSM model is characterized by the intersection :math:`\left\{f_{0}\leq 0 \cap F_{1,0}\leq 0\right\}` where according to :eq:`critere_plasticite_1_initial` 
  and :eq:`critere_plasticite_2_initial`:

  .. math:
    f_ {0} =\ sqrt {\ left (\ frac {\ sigma_ {eq}} {M}\ right) ^2 +\ left (\ sigma_m+p_ {c0} (1-\ eta)\ left (1-\ eta)\ right) ^2} -p_ {c0} (1-\ eta),
    \ quad 
    F_ {1.0} =\ left|\ frac {R_1} {C} {C}\ left\ langle\ sigma_M+C\ right\ rangle\ right|-R_1
    :label: crite_plasticite_initial_rho_1

  It is easy to see that :math:`\left\{f_{0}\leq 0 \cap F_{1,0}\leq 0\right\}=\left\{f_{0}\leq 0\right\}` and that only component 1 of the CSSM model is acting.

* If :math:`\rho=0`: the initial elasticity domain of the CSSM model is, on the other hand, characterized by :math:`\left\{f_{0}\leq 0 \cap F_{1,0}\leq 0\right\}` with:

  .. math:
    f_ {0} =\ left|\ sigma_m+p_ {c0} (1-\ eta)\ right|-p_ {c0} (1-\ eta),
    \ quad
    F_ {1.0} =\ sqrt {\ left (\ sigma_ {eq}\ right) ^2 +\ left (\ frac {R_1} {C}\ left\ langle\ sigma_M+C\ sigma_M+C\ right\ rangle\ right) ^2} -R_1
    :label: crite_plasticite_initial_rho_0

  In this case, as long as we are only interested in shear behavior, only component 2 of model CSSM acts.

We illustrate in :numref:`r7.01.44-cas_tests` that selecting :math:`\rho\in\{0,1\}` does not make it possible to predict satisfactorily both the behavior of a ground material under monotonic and cyclic loads, 
ranging from very low to average levels of deformation. It is therefore the combination of the two components that improves the predictions, and in fact justifies the interest of the model.
CSSM.