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

Geometry
---------

This is a cubic sample in :math:`1/8` representation using a :math:`\text{HEXA20}` element.


.. image:: images/Object_1.svg
    :width: 484
    :height: 248


.. _RefImage_Object_1.svg:


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

The elastic properties are:


* isotropic compressibility module: :math:`K=516200\mathrm{kPa}`;


* shear modulus: :math:`\mu =238200\mathrm{kPa}`;


* density [1] _
: :math:`{\rho }_{s}=2500\mathrm{kg}/{m}^{3}`.

The anelastic properties of the cyclic Hujeux law come from the document referenced at the following internet address:
< http://www.mssmat.ecp.fr/IMG /pdf/resp_loph40.pdf >` http://www.mssmat.ecp.fr/IMG /pdf/resp_loph40.pdf`_
< http://www.mssmat.ecp.fr/IMG /pdf/resp_loph40.pdf >``_. These are the parameters relating to Hostun sand:


* power of the nonlinear elastic law: :math:`{n}_{e}=0.4`;


* :math:`\beta =24`;


* :math:`d=2.5`;


* :math:`b=0.2`;


* friction angle: :math:`\varphi =33°`;


* characteristic angle: :math:`\psi =33°`;


* critical pressure: :math:`{P}_{\mathrm{c0}}=-1\mathrm{MPa}`;


* reference pressure: :math:`{P}_{\mathrm{ref}}=-1\mathrm{MPa}`;


* elastic radius of isotropic mechanisms: :math:`{r}_{\mathrm{ela}}^{s}=0.001`;


* elastic radius of deviatory mechanisms: :math:`{r}_{\mathrm{ela}}^{d}=0.005`;


* :math:`{a}_{\mathrm{mon}}=0.008`;


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


* :math:`{c}_{\mathrm{mon}}=0.2`;


* :math:`{c}_{\mathrm{cyc}}=0.1`;


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


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


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


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

The hydraulic properties are:


* Biot coefficient: :math:`B=1`;


* the density of water: :math:`{\rho }_{e}=1000\mathrm{kg}/{m}^{3}`;


* viscosity: :math:`\nu =0.001`;


* intrinsic permeability: :math:`{K}^{\text{int}}={1.E}^{-8}{m}^{3}/\mathrm{kg}/s`;


* the compressibility module of water: :math:`{K}_{e}={1.E}^{+12}\mathrm{Pa}`.


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


Boundary conditions
~~~~~~~~~~~~~~~~~~~~~~~~

These are the symmetry conditions on the element, which represents :math:`1/8` of the sample. The movements are blocked on the front panels (

.. image:: images/Object_2.svg
    :width: 484
    :height: 248


.. _RefImage_Object_2.svg:

), left lateral (

.. image:: images/Object_3.svg
    :width: 484
    :height: 248


.. _RefImage_Object_3.svg:

) and lower (

.. image:: images/Object_4.svg
    :width: 484
    :height: 248


.. _RefImage_Object_4.svg:

).


Loading
~~~~~~~~~~

**Phase 1: consolidation of the sample up to the confinement pressure p0**

The sample is brought to a homogeneous state of isostatic *actual* stresses :math:`{\sigma }_{\mathrm{xx}}^{0}={\sigma }_{\mathrm{yy}}^{0}={\sigma }_{\mathrm{zz}}^{0}={\sigma }^{0}=-30\mathrm{kPa}`, by imposing pressure :math:`{\sigma }_{0}` on the rear, right lateral, and upper faces of the element, while maintaining zero water pressures :math:`\text{PRE1}` in the sample.

**Phase 2: undrained triaxial load**

To obtain the non-drained conditions, zero hydraulic flows are imposed on all faces.

By maintaining a pressure equal to :math:`{\sigma }_{0}` on the rear and right lateral faces, an alternating pressure loading with an amplitude :math:`\Delta \sigma` equal to :math:`15\mathit{kPa}` is applied to the upper face, so as to obtain a variation in the vertical stress :math:`{\sigma }_{\mathrm{zz}}` in the sample included in the interval :math:`\mathrm{-}45\mathit{kPa}\mathrm{\le }{\sigma }_{\mathit{zz}}\mathrm{\le }\mathrm{-}15\mathit{kPa}` (with the minus sign convention for compression).


Results
---------

The solutions are post-treated at point :math:`C`, in terms of effective isotropic pressure :math:`P` (:math:`\mathrm{=}\text{tr}(\sigma \text{'})\mathrm{/}3`), plastic volume deformation :math:`{\varepsilon }_{v}^{p}` and isotropic work hardening coefficients :math:`({r}_{\mathrm{iso}}^{m}+{r}_{\mathrm{ela}}^{\mathrm{iso}})` and :math:`({r}_{\mathit{iso}}^{c}+{r}_{\mathit{ela}}^{\mathit{iso}})` and deviations :math:`({r}_{d}^{m}+{r}_{\mathrm{ela}}^{d})` and :math:`({r}_{d}^{c}+{r}_{\mathrm{ela}}^{d})`.

Validation is carried out by comparison with solutions GEFDYNfournies by Ecole Centrale Paris.

.. [1] In the absence of gravity, the densities of soil and water do not intervene in the problem.