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


Geometry
---------

Axisymmetric cylinder (models :math:`A` and :math:`E`) or rectangular plate (modeling :math:`B`) or straight pipe (models :math:`C` and :math:`D`), or beam (modeling :math:`F`), or parallelepiped :math:`\mathrm{3D}`


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(models :math:`G`, :math:`H` and :math:`J`).

Cylinder geometry :math:`(\mathrm{mm})`:


* :math:`a=1`


* :math:`b=2`


* :math:`H=4`


Property of the materials
-----------------------

For all models:

Young's module: :math:`E=200000\mathrm{MPa}`

Tangent module: :math:`{E}_{t}=50000\mathrm{MPa}`

Poisson's ratio: :math:`\nu =0.3`

:math:`{\sigma }_{0}=400\mathrm{MPa}`

:math:`s=1.0{E}^{-2}°{C}^{-1}`

Coefficient of thermal expansion: :math:`\alpha =1.0{E}^{-5}°{C}^{-1}`

Volume heat: :math:`{C}^{p}=0{\mathrm{J.mm}}^{-3}\mathrm{.}°{C}^{-1}`

Thermal conductivity: :math:`\lambda =1.0{E}^{-3}{\mathrm{W.mm}}^{-1}\mathrm{.}°{C}^{-1}`

For the isotropic material declared orthotropic, it comes:

E_L= E_T= E_N= :math:`E`

nu_LT = nu_LN= nu_TN= :math:`\nu`

G_LT= G_LN= G_TN= :math:`75000\mathrm{MPa}`

ALPHA_L = ALPHA_T = ALPHA_N = :math:`\alpha`

For models from :math:`A` to :math:`G`:

:math:`{\sigma }_{y}(T)={\sigma }_{0}(1-\mathrm{s.}(T-{T}_{0}))`

For models :math:`H` and :math:`I`:

:math:`{\sigma }_{y}(T)\mathrm{=}{\sigma }_{0}` :math:`(s=0)`


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Figure: Material tensile curve


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


* Models :math:`A`, :math:`E` and :math:`K`: :math:`\mathrm{uz}=0` on the sides :math:`\mathrm{AB}` and :math:`\mathrm{CD}` (Fixed :math:`\mathrm{Oz}` axis)


* Models :math:`B` and :math:`I`: :math:`\mathrm{uy}=0` on the sides :math:`\mathrm{AB}` and :math:`\mathrm{CD}`, :math:`\mathrm{ux}=0` in :math:`A`


* Models :math:`C,D` and :math:`F`: embedding in :math:`A`, :math:`\mathrm{Uy}=0` in :math:`C`


* Modelings :math:`G`, :math:`H`, :math:`L`, and :math:`M`: :math:`\mathrm{uy}=0` on the sides :math:`\mathrm{AB}` and :math:`\mathrm{CD}`, :math:`\mathrm{ux}=\mathrm{uz}=0` (node :math:`\mathrm{N3}`), :math:`\mathrm{uz}=0` (node): on the sides and :math:`\mathrm{N4}`


* :math:`T(t)=\gamma t+\mathrm{T0}` with: :math:`\gamma =1°C/s` and :math:`\mathrm{T0}=0°C`.


* Models :math:`H` and :math:`I`: Initial deformation fields: :math:`\varepsilon =\alpha (T-{T}_{0})\mathrm{Id}`


* Modeling :math:`J`: Field of imposed deformations: :math:`{\varepsilon }_{\mathrm{yy}}=0`