Introduction
============

The values of the expansion coefficients are determined by dilatometry tests that take place starting from room temperature (:math:`0°C` or more generally :math:`20°C`). As a result, values of the expansion coefficient defined with respect to :math:`20°C` are generally available (temperature at which zero thermal deformation is assumed).

Some studies require taking a reference temperature different from the ambient temperature (zero thermal deformation for a temperature other than the ambient temperature). It is then necessary to change the frame of reference in the calculation of the thermal deformation (equation and figure below).


.. image:: images/100034F0000069BB0000374BB075F41294B1EA27.svg
    :width: 447
    :height: 234


.. _RefImage_100034F0000069BB0000374BB075F41294B1EA27.svg:


.. math::
 :label: eq-1

    {\ mathrm {\ epsilon}}} ^ {\ mathit {th}}}\ left (T\ right) = {\ mathrm {\ epsilon}} _ {m} ^ {\ mathit {th}}\ left (T\ right)}\ left (T\ right)} - {\ mathrm {\ epsilon}}\ left (T\ right) - {\ mathrm {\ epsilon}}} _ {m} ^ {\ mathit {th}}}\ left (T\ right)}\ left (T\ right) - {\ mathrm {\ epsilon}} - {\ mathrm {\ epsilon}}} _ {m} ^ {\ mathit {th}}}\ left (T\ right)}\ left (T\ right)\ mathit {ref}}\ right)

Where :math:`{\mathrm{\epsilon }}_{m}^{\mathit{th}}` is the measured thermal deformation (defined in relation to ambient temperature) and :math:`{\mathrm{\epsilon }}^{\mathit{th}}` is the calculated thermal deformation (defined in relation to a reference temperature).

In c*ode_aster*, thermal deformation is calculated by the following expression:

.. math::
 :label: eq-2

    {\ mathrm {\ epsilon}}} ^ {\ mathit {th}}\ left (T\ right) =\ widehat {\ mathrm {\ alpha}}\ left (T\ right)\ left (T\ right)\ left (T- {T}} _ {\ mathit {ref}}\ right)

where :math:`\widehat{\mathrm{\alpha }}\left(T\right)` is the average expansion coefficient (in the sense RCC_M) at the temperature :math:`T` determined with respect to the temperature :math:`{T}_{\mathit{ref}}` (:math:`{T}_{\mathit{ref}}` being the temperature at which it is considered that :math:`{\mathrm{\epsilon }}^{\mathit{th}}\left({T}_{\mathit{ref}}\right)=0`).

In THM, the thermal deformation calculation is different. Thermal deformation is evaluated by the following formula:

.. math::
 :label: eq-3

    {\ mathrm {\ epsilon}}} ^ {\ mathit {th}}\ left (T\ right) =\ mathrm {\ alpha}\ left (T- {T} _ {\ mathit {th}} _ {\ mathit {th}}\ right)

The thermal expansion coefficient :math:`\mathrm{\alpha }` is necessarily a constant. You can't shift the curve. It is therefore necessary that the thermal expansion coefficient has been evaluated at ambient temperature. Also, the reference temperature is given by THM_INIT/TEMP in DEFI_MATERIAU.