Modeling A
==============


Characteristics of modeling
-----------------------------------

AXIS, :math:`\mathit{Q8}` mesh


.. csv-table::

    "Breakdown:", "10 elements depending on the length"
    "", "1 element in the thickness"


Boundary conditions:


.. csv-table::

    "in :math:`A`, :math:`B` "," DDL_IMPO = (GROUP_NO = 'A', DY= 0.)"
    "", "DDL_IMPO = (GROUP_NO = 'B', DY= 0.)"


Pressure + shape effect: field :math:`U`


.. csv-table::

    "PRES_REP:", "(GROUP_MA = cont_pr, PRES = 2.E8)"
    "FORCE_CONTOUR:", "(GROUP_MA = effond, FY= 1.95E9)"


Thermal expansion: field :math:`{U}_{1}`


.. csv-table::

    "char_no:", ""
    "CREA_CHAMP ", "(AFFE = (TOUT = 'OUI', NOM_CMP = 'TEMP', VALE = 100.))"



.. csv-table::

    "char_th:", ""
    "AFFE_MATERIAU ", "(AFFE_VARC = F (TOUT = 'OUI', CHAM_GD = '', = CHAR_NO, VALE_REF = 0., NOM_VARC = 'TEMP',)"


Pre-deformations: field :math:`{U}_{2}`


.. csv-table::

    "PRE_EPSI:", "(TOUT = 'OUI', EPXX = 1.2E-3, EPYY = 1.2E-3,"
    "", "EPZZ = 1.2E-3, EPXY = 0.)"


Node names:


.. csv-table::

    ":math:`A\mathrm{=}\mathit{N1}` "," :math:`B\mathrm{=}\mathit{N2}` "," :math:`C\mathrm{=}\mathit{N3}` "," :math:`D\mathrm{=}\mathit{N4}`"



Characteristics of the mesh
----------------------------

Number of knots: 53

Number of meshes and types: 10 QUAD8, 22 SEG3


Tested sizes and results
------------------------------

Results for fields :math:`{U}_{1}`, :math:`{U}_{2}`, :math:`U`


.. csv-table::

    "**Field**", "**Location**", "**Variables**", "**Reference**"
    "Thermal field :math:`{U}_{1}` "," :math:`A` "," "," :math:`\mathit{Ur}(\mathit{DX})` ", "5.7 x 10-5"
    "", ":math:`B` "," :math:`\mathit{Ur}(\mathit{DX})` ", "6 x 10-5"
    "", ":math:`C` "," :math:`\mathit{Ur}(\mathit{DX})` ", "6 x 10-5"
    "", "", ":math:`\mathit{DY}` ", "1.2 x 10-3"
    "", ":math:`D` "," :math:`\mathit{Ur}(\mathit{DX})` ", "5.7 x 10-5"
    "", "", ":math:`U(\mathit{DY})` ", "1.2 x 10-3"
    "", ":math:`A`, mesh :math:`\mathit{M1}` "," :math:`{\varepsilon }_{\mathit{rr}}` "," 1.2 x 10-3"
    "", "", ":math:`{\varepsilon }_{\theta \theta }` ", "1.2 x 10-3"
    "", "", ":math:`{\varepsilon }_{\mathit{zz}}` ", "1.2 x 10-3"
    "", ":math:`B`, mesh :math:`\mathit{M1}` "," :math:`{\varepsilon }_{\mathit{rr}}` "," 1.2 x 10-3"
    "", "", ":math:`{\varepsilon }_{\theta \theta }` ", "1.2 x 10-3"
    "", "", ":math:`{\varepsilon }_{\mathit{zz}}` ", "1.2 x 10-3"
    "", ":math:`C`, mesh :math:`\mathit{M10}` "," :math:`{\varepsilon }_{\mathit{rr}}` "," 1.2 x 10-3"
    "", "", ":math:`{\varepsilon }_{\theta \theta }` ", "1.2 x 10-3"
    "", "", ":math:`{\varepsilon }_{\mathit{zz}}` ", "1.2 x 10-3"
    "", ":math:`D`, mesh :math:`\mathit{M10}` "," :math:`{\varepsilon }_{\mathit{rr}}` "," 1.2 x 10-3"
    "", "", ":math:`{\varepsilon }_{\theta \theta }` ", "1.2 x 10-3"
    "", "", ":math:`{\varepsilon }_{\mathit{zz}}` ", "1.2 x 10-3"
    "Pressure field :math:`U` "," :math:`C` "," "," :math:`{U}_{\theta }(\mathit{DY})` ", "3.714 x 10-3"
    "", ":math:`D` "," :math:`{U}_{\theta }(\mathit{DY})` ", "3.714 x 10-3"
    "", ":math:`C`, mesh :math:`\mathit{M10}` "," :math:`{\varepsilon }_{\theta \theta }` ", "3.714 x 10-3"
    "", ":math:`D`, mesh :math:`\mathit{M10}` "," :math:`{\varepsilon }_{\theta \theta }` ", "3.714 x 10-3"
    "Field :math:`{U}_{2}` "," :math:`C` "," :math:`{U}_{\theta \theta }` ", "4.914 x 10-3"
    "", ":math:`D` "," :math:`{U}_{\theta \theta }` ", "4.914 x 10-3"
    "", ":math:`C`, mesh", ":math:`{\varepsilon }_{\theta \theta }` ", "4.914 x 10-3"
    "", ":math:`D`, mesh", ":math:`{\varepsilon }_{\theta \theta }` ", "4.914 x 10-3"



notes
---------


* The aim of the test is not to obtain high precision in terms of the results, but simply to verify the relationship: :math:`{U}_{2}\mathrm{=}U+{U}_{1}`; therefore, the calculation was only carried out with a rough mesh.


* It can be seen that the relationship sought is well verified at the free end of the cylinder.


* It is also verified that the deformation field resulting from thermal expansion is uniformly equal to :math:`1.2\mathrm{\times }{10}^{\mathrm{-}3}`.