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


Geometry
----------

For modeling A, the shear test is carried out on a single isoparametric finite element of square shape QUAD8, a cell group called :math:`\mathrm{BLOC}`. The length of each edge is :math:`\mathrm{1m}`. The different sides of this square are mesh groups named :math:`\mathrm{HAUT}`, :math:`\mathrm{BAS}`, :math:`\mathrm{DROIT}`, and :math:`\mathrm{GAUCHE}`. The group of elements :math:`\mathrm{COTE}` also contains the mesh groups :math:`\mathrm{DROIT}` and :math:`\mathrm{GAUCHE}`; the mesh group :math:`\mathrm{APPUI}` the mesh group :math:`\mathrm{BAS}`.


.. image:: images/Shape1.gif


.. _RefSchema_Shape1.gif:

For modeling B the test is carried out on a single isoparametric finite element of cubic shape HEXA8. The length of each edge is 1. The different facets of this cube are mesh groups named :math:`\mathrm{HAUT}`, :math:`\mathrm{BAS}`,, :math:`\mathrm{DEVANT}`, :math:`\mathrm{ARRIERE}`, :math:`\mathrm{DROIT}`, and :math:`\mathrm{GAUCHE}`.


.. image:: images/100000000000051400000456D838525F86D79D41.png
    :width: 3.25in
    :height: 2.7752in


.. _RefImage_100000000000051400000456D838525F86D79D41.png:


Material properties of Hostun sand
-----------------------------------------

The elastic properties are:


* isotropic compressibility module: :math:`K=148\mathit{MPa}`


* shear modulus: :math:`\mathrm{\mu }=68\mathit{MPa}`

The equivalent orthotropic elastic properties are:


* :math:`{E}_{L}={E}_{T}={E}_{N}=176.906250\mathit{MPa}`


* :math:`{G}_{L}={G}_{T}={G}_{N}=68\mathit{MPa}`


* :math:`{\mathrm{\nu }}_{L}={\mathrm{\nu }}_{T}={\mathrm{\nu }}_{N}=0.30078125`

The anelastic properties (Hujeux model) come from K.Hamadi's thesis thesis [:ref:` **2 <**2>`] ** and correspond to low-density Hostun sand:


* power of the nonlinear elastic law: :math:`{n}_{e}=0.` (linear elastic)


* :math:`\beta =30.`


* :math:`d=2.5`


* :math:`b=0.2`


* friction angle: :math:`\phi =33°`


* characteristic angle: :math:`\Psi =33°`


* critical pressure: :math:`{P}_{\mathit{CO}}=-400\mathit{kPa}`


* reference pressure: :math:`{P}_{\mathrm{ref}}=-1000\mathrm{kPa}`


* elastic radius of the isotropic mechanism: :math:`{r}_{\mathrm{ela}}^{s}={10}^{-4}`


* elastic radius of the deviatory mechanism: :math:`{r}_{\mathrm{ela}}^{d}=0.01`


* :math:`{a}_{\mathrm{mon}}=0.017`


* :math:`{a}_{\mathrm{cyc}}=0.0001`


* :math:`{c}_{\mathrm{mon}}=0.08`


* :math:`{c}_{\mathrm{cyc}}=0.04`


* :math:`{r}_{\mathrm{hys}}=0.05`


* :math:`{r}_{\mathrm{mob}}=0.9`


* :math:`{x}_{m}=1.`


* :math:`\mathrm{dila}=1.`


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

The shear test presented here is carried out using D_ PLAN modeling (A modeling) and 3D modeling (B modeling).

For modeling A, the movements normal to the study plan are zero. A vertical displacement is imposed on the specimen while maintaining the lateral pressure constant in the study design. In the model under consideration, the square element represents a quarter of the sample. The boundary conditions are therefore as follows:

Symmetry conditions:


* :math:`{u}_{y}=0.` on mesh group :math:`\mathrm{BAS}`


* :math:`{u}_{x}=0.` on mesh group :math:`\mathrm{GAUCHE}`

Lateral pressure conditions:


* :math:`{P}_{n}=100\mathit{kPa}` on mesh group :math:`\mathrm{COTE}`

Loading conditions:


* :math:`{P}_{n}=100\mathit{kPa}` on mesh group :math:`\mathrm{HAUT}`


* :math:`{u}_{x}=-1\mathit{mm}` on mesh group :math:`\mathrm{HAUT}`

Charging is carried out in two phases:


* An isotropic stress state, :math:`{P}_{o}=100\mathrm{kPa}`, is initially assigned to mesh :math:`\mathrm{BLOC}`;


* A horizontal displacement is imposed on the group of elements :math:`\mathrm{HAUT}` and varies between :math:`t=0.` and :math:`t=10.` from :math:`{u}_{y}=0.` and :math:`{u}_{x}=-1\mathit{mm}`.

For modeling B, an embedment is imposed on the specimen on the face :math:`\mathrm{BAS}` and a horizontal movement along X on the face :math:`\mathrm{HAUT}`, while maintaining the constant lateral pressure in the lateral faces. The boundary conditions are therefore as follows:

The embedding condition:


* :math:`{u}_{x}={u}_{y}={u}_{z}=0` on mesh group :math:`\mathit{BAS}`

Lateral pressure conditions:


* :math:`{P}_{n}=100\mathit{kPa}` on mesh groups :math:`\mathrm{DEVANT}`, :math:`\mathrm{ARRIERE}`,, :math:`\mathrm{DROIT}`, and :math:`\mathrm{GAUCHE}`.

Loading conditions:


* :math:`{P}_{n}=100\mathit{kPa}` on the :math:`\mathrm{HAUT}` mesh group applied from the first moment of calculation and maintained thereafter


* :math:`{u}_{x}=-\mathrm{0,2}m` on the :math:`\mathrm{HAUT}` mesh group applied progressively by a ramp function

Charging is therefore carried out in two phases:


* An isotropic stress state, :math:`{P}_{o}=100\mathrm{kPa}`, is initially assigned to cells :math:`\mathrm{HAUT}`, :math:`\mathrm{DEVANT}`, :math:`\mathrm{ARRIERE}`, :math:`\mathrm{DROIT}` and :math:`\mathrm{GAUCHE}`. ;


* A horizontal displacement according to X is imposed on the group of elements :math:`\mathrm{HAUT}` and varies between :math:`t=0.` and :math:`t=10.` from :math:`{u}_{x}=0.` and :math:`{u}_{x}=-0.2m`.


Results
---------

The solutions are post-treated at point :math:`C`, in terms of stress :math:`{\mathrm{\sigma }}_{\mathit{xy}}`, total volume deformation :math:`{\varepsilon }_{v}` and isotropic work-hardening coefficients :math:`\left({r}_{\text{ela}}^{\text{iso},m}+{r}_{\text{iso}}^{m}\right)` and deviation :math:`\left({r}_{\text{ela}}^{d,m}+{r}_{\text{dev}}^{m}\right)`.

Bibliographical references
---------------------------

[:ref:`1 <1>`] Foucault A. "*Modeling the cyclical behavior of earthen structures integrating regulation techniques* *" .* Doctoral thesis, École Centrale Paris, École Centrale Paris, Châtenay Malabry, France, 2010.

[:ref:`2 <2>`] **Hamadi K.** "*Modeling bifurcations and instabilities in geomaterials*". PhD thesis, École Centrale Paris, Châtenay Malabry, France, 2006