1. Introduction

The values of the expansion coefficients are determined by dilatometry tests that take place starting from room temperature (\(0°C\) or more generally \(20°C\)). As a result, values of the expansion coefficient defined with respect to \(20°C\) 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).

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

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

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

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

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

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

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

The thermal expansion coefficient \(\mathrm{\alpha }\) 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.