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


Geometry
---------

We consider an infinite hollow cylinder.


.. image:: images/Object_13.png
    :width: 3.598in
    :height: 2.6307in


.. _RefSchema_Object_13.png:

Figure 1: Hollow cylinder geometry

*Modeling A:*

The geometric properties are:


* Inner radius :math:`{R}_{i}\mathrm{=}10\mathit{mm}`


* External radius :math:`{R}_{e}\mathrm{=}20\mathit{mm}`

*B modeling:*

The geometric properties are:


* Inner radius :math:`{R}_{i}\mathrm{=}100\mathit{mm}`


* External radius :math:`{R}_{e}\mathrm{=}200\mathit{mm}`


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

The materials used in the calculations use the VMIS_JOHN_COOK law of behavior. Its work hardening function is written as follows [:ref:`1 <1>`]:

:math:`R(p,\dot{p},T)\mathrm{=}(A+B{p}^{n})(1+C\mathrm{ln}(\frac{\dot{p}}{\dot{{p}_{0}}}))(1\mathrm{-}(\frac{T\mathrm{-}{T}_{\mathit{room}}}{{T}_{\mathit{melt}}\mathrm{-}{T}_{\mathit{room}}}))`

We will not use terms related to temperature dependence and the rate of plastic deformation. We will then work with:

:math:`R(p)\mathrm{=}A+B{p}^{n}`

*Modeling A:*

An elastic law of behavior is modelled. The following mechanical and thermal parameters are used:


* Young's module :math:`E\mathrm{=}206900\mathit{MPa}`


* Poisson's ratio :math:`\nu \mathrm{=}0.29`


* Elastic limit :math:`A\mathrm{=}5{.10}^{100}\mathit{Pa}`


* Post-elastic slope :math:`B\mathrm{=}2{.10}^{10}\mathit{Pa}`


* Post-elastic power :math:`n\mathrm{=}1`


* Density :math:`\rho \mathrm{=}7800\mathit{kg}\mathrm{/}{m}^{3}`


* Expansion coefficient :math:`\alpha \mathrm{=}\mathrm{1,5}{.10}^{\mathrm{-}5}{K}^{\mathrm{-}1}`


* Conductivity :math:`\lambda \mathrm{=}45W\mathrm{/}(\mathit{m.K})`


* Heat capacity :math:`{C}_{p}\mathrm{=}460J\mathrm{/}(\mathit{kg.K})`


* Reference temperature :math:`{T}_{\mathit{ref}}\mathrm{=}293K`

*B modeling:*

An elastoplastic behavior law equivalent to a VMIS_ISOT_LINE law is modelled. The following mechanical and thermal parameters are used:


* Young's module :math:`E\mathrm{=}70000\mathit{MPa}`


* Poisson's ratio :math:`\nu \mathrm{=}0.3`


* Yield strength as a function of temperature :math:`A(T)\mathrm{=}A({T}_{\mathit{ref}})(1\mathrm{-}w(T\mathrm{-}{T}_{\mathit{ref}}))`


* Yield strength at reference temperature :math:`A({T}_{\mathit{ref}})\mathrm{=}70\mathit{MPa}`


* Linear softening coefficient :math:`w\mathrm{=}3{.10}^{\mathrm{-}4}{K}^{\mathrm{-}1}`


* Post-elastic slope :math:`B\mathrm{=}210\mathit{MPa}`


* Post-elastic power :math:`n\mathrm{=}1`


* Density :math:`\rho \mathrm{=}2700\mathit{kg}\mathrm{/}{m}^{3}`


* Expansion coefficient :math:`\alpha \mathrm{=}\mathrm{2,38}{.10}^{\mathrm{-}5}{K}^{\mathrm{-}1}`


* Conductivity :math:`\lambda \mathrm{=}150W\mathrm{/}(\mathit{m.K})`


* Heat capacity :math:`{C}_{p}\mathrm{=}900J\mathrm{/}(\mathit{kg.K})`


* Reference temperature :math:`{T}_{\mathit{ref}}\mathrm{=}293K`


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

*Modeling A:*

A radial displacement of :math:`20\mathit{mm}` is imposed on the inner face of the cylinder at the speed of :math:`\mathrm{1mm}\mathrm{/}s`. The temperature of the external face is set to :math:`{T}_{\mathit{ref}}`. The other faces of the cylinder are considered to be thermally insulated. The behavior is thermoelastic and the expansion of a cylinder therefore causes it to cool.

*B modeling:*

The surface of the cylinder is considered to be thermally insulated. The displacement of the inner face of the cylinder is :math:`130\mathit{mm}` applied in :math:`\mathrm{1,3}s`. The behavior is thermoelastoplastic and the plastic dissipation effect compensates for the previous thermo-elastic cooling effect to cause the overall heating of the cylinder.


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

The cylinder temperature is equal to :math:`{T}_{\mathit{ref}}`.