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


Geometry
---------

It is a material point, representative of a state of homogeneous stresses and deformations.


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


Umat data
~~~~~~~~~~~~

The coefficients of Umat behavior are (cf. [:ref:`U4.43.01 <U4.43.01>`]):

:math:`\mathrm{C1}=\lambda =\frac{E\nu }{(1+\nu )(1-2\nu )}`

:math:`\mathrm{C2}=\mu =\frac{E}{2(1+\nu )}`

:math:`\mathrm{C3}=\tilde{\lambda }=\frac{\lambda }{20}`

:math:`\mathrm{C4}=\tilde{\mu }=\frac{\mu }{20}`

:math:`\mathrm{C5}=\tilde{\nu }=0`

We will use: DEFI_MATERIAU/UMAT =_F (LISTE_COEF =( C1, C2, C3, C4, C5)),


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

The load is the same as in tests COMP001, see [:external:ref:`V6.07.101 <V6.07.101>`].


Characteristics of loading paths
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The proposed loading causes each component of the deformation tensor to vary in a decoupled manner by successive step. A cyclic load-discharge path is proposed by covering the states of traction and compression as well as an inversion of the signs of shear in order to test a wide range of values.

Schematically, it follows a course on 8 segments :math:`[O-A-B-C-O-C’-B’-A’-O]` where the second part of the path :math:`[O-C’-B’-A’-O]` is symmetric with respect to the origin of the first :math:`[O-A-B-C-O]`.


Application of requests
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

We come back to the study of a material point (using the macro-command SIMU_POINT_MAT [:ref:`U4.51.12 <U4.51.12>`]) by stressing an element in a homogeneous manner by imposing in :math:`\mathrm{3D}`, the 6 components of the deformation tensor:

:math:`\stackrel{ˉ}{\varepsilon }=\left[\begin{array}{ccc}{\varepsilon }_{\mathrm{xx}}& {\varepsilon }_{\mathrm{xy}}& {\varepsilon }_{\mathrm{xz}}\\ {\varepsilon }_{\mathrm{xy}}& {\varepsilon }_{\mathrm{yy}}& {\varepsilon }_{\mathrm{yz}}\\ {\varepsilon }_{\mathrm{xz}}& {\varepsilon }_{\mathrm{yz}}& {\varepsilon }_{\mathrm{zz}}\end{array}\right]`

For a more general description, the imposed deformation tensor will be decomposed into a hydrostatic and deviatoric part on shear bases:

:math:`\stackrel{ˉ}{\varepsilon }=\left[\begin{array}{ccc}{\varepsilon }_{\mathrm{xx}}& {\varepsilon }_{\mathrm{xy}}& {\varepsilon }_{\mathrm{xz}}\\ {\varepsilon }_{\mathrm{xy}}& {\varepsilon }_{\mathrm{yy}}& {\varepsilon }_{\mathrm{yz}}\\ {\varepsilon }_{\mathrm{xz}}& {\varepsilon }_{\mathrm{yz}}& {\varepsilon }_{\mathrm{zz}}\end{array}\right]=p\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]+{d}_{1}\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 0\end{array}\right]+{d}_{2}\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]+\left[\begin{array}{ccc}0& {\varepsilon }_{\mathrm{xy}}& {\varepsilon }_{\mathrm{xz}}\\ {\varepsilon }_{\mathrm{xy}}& 0& {\varepsilon }_{\mathrm{yz}}\\ {\varepsilon }_{\mathrm{xz}}& {\varepsilon }_{\mathrm{yz}}& 0\end{array}\right]` in 3D.


Description of the imposed deformation path in 3D
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The path applied is described in the table below, the deformation values applied are calibrated with respect to the elastic modulus:

.. csv-table::

    "Segment number", "1", "2", "2", "3", "3", "3", "4", "5", "6", "7", "8"
    "Segment", ":math:`0-A` "," :math:`A-B` "," "," :math:`B-C` "," "," :math:`O` "," :math:`C’` "," :math:`B’` "," :math:`A’` "," :math:`O`"
    ":math:`{\varepsilon }_{\mathrm{xx}}\ast E` ", "787.5", "1050", "1050", "350", "350", "0", "-350", "-1050", "-787.5", "0"
    ":math:`{\varepsilon }_{\mathrm{yy}}\ast E` ", "525.0", "-175", "-175", "-350", "-350", "175", "525", "0"
    ":math:`{\varepsilon }_{\mathrm{zz}}\ast E` ", "262.5", "700", "700", "-525", "-525", "525", "-700", "-262.5", "0"
    ":math:`{\varepsilon }_{\mathrm{xy}}\ast E/(1+\nu )` ", "700", "350", "350", "1050", "1050", "-1050", "-350", "-700", "0"
    ":math:`{\varepsilon }_{\mathrm{xz}}\ast E/(1+\nu )` ", "-350", "350", "350", "700", "700", "0", "-700", "700", "0"
    ":math:`{\varepsilon }_{\mathrm{yz}}\ast E/(1+\nu )` ", "0", "700", "-350", "-350", "0", "350", "-700", "0", "0"
    ":math:`P` ", "525", "525", "525", "-175", "-175", "-525", "-525", "0"
    ":math:`\mathrm{d1}` ", "262.5", "525", "525", "525", "0", "-525", "-525", "-262.5", "0"
    ":math:`\mathrm{d2}` ", "262.5", "-175", "-175", "350", "350", "0", "-350", "175", "-262.5", "0"


This path is illustrated by the following graph:


.. image:: images/1000000000000318000002630F3FC811B946DA5C.png
    :width: 5.6091in
    :height: 4.3909in


.. _RefImage_1000000000000318000002630F3FC811B946DA5C.png:


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

Zero stresses and deformations.