2. Calculation of the coefficient of thermal expansion

2.1. From the reference temperature

The values of the thermal expansion coefficient were determined by dilatometry tests carried out starting at temperature \({T}_{\mathit{ref}}\). In this case, the TEMP_DEF_ALPHA keyword should not be specified in the DEFI_MATERIAU [U4.23.01] command. The equation is simplified, since \({\mathrm{\epsilon }}_{m}^{\mathit{th}}\left({T}_{\mathit{ref}}\right)=0\). From where:

\[\]

: label: eq-4

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

Where expansion coefficient values \(\widehat{\mathrm{\alpha }}\left(T\right)\) are entered under the keyword ALPHA (or ALPHA_ *) in DEFI_MATERIAU.

2.2. From a temperature different from the reference temperature

The values of the thermal expansion coefficient were determined by dilatometry tests that took place using a temperature \({T}_{\text{def}}\) different from the reference temperature \({T}_{\mathit{ref}}\).

In fact, in general, values of the expansion coefficient defined with respect to ambient temperature, \(0°C\) or more generally \(20°C\), are available, and some studies require taking a reference temperature different from the ambient temperature.

It is then necessary to change the frame of reference (through the equation). In this case, the user enters the value of the temperature \({T}_{\text{def}}\) under the keyword TEMP_DEF_ALPHA of the command DEFI_MATERIAU, and under the keyword ALPHA (or ALPHA_ *) the values of the expansion coefficient \(\mathrm{\alpha }\left(T\right)\) (defined in relation to the temperature \({T}_{\text{def}}\)). In the AFFE_MATERIAU command under the TEMP_REF keyword, it indicates the value of the reference temperature \({T}_{\mathit{ref}}\). The calculation for \({\mathrm{\epsilon }}^{\mathit{th}}\left(T\right)\) is done using the formula:

\[\]

: label: eq-5

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

The calculation of \({\mathrm{\epsilon }}^{\mathit{th}}\left(T\right)\) requires the prior calculation of the values of the \(\widehat{\mathrm{\alpha }}\left(T\right)\) function.

The function \(\widehat{\mathrm{\alpha }}\left(T\right)\) remains defined (or filled in) for the same values of \(T\) as \(\mathrm{\alpha }\left(T\right)\) and keeps the same attributes (same type of interpolation (“LIN”, “LOG”) and same type of extension (“CONSTANT”, “LINEAIRE”, “EXCLUS”).

2.2.1. Calculation at temperatures different from the reference (with precision)

We get the expression for \(\widehat{\mathrm{\alpha }}({T}_{i})\) using the equation.

:math:`widehat{mathrm{alpha }}left({T}_{i}right)=frac{mathrm{alpha }left({T}_{i}right)left({T}_{i}-{T}_{text{def}}right)-mathrm{alpha }left({T}_{mathit{ref}}right)left({T}_{mathit{ref}}-{T}_{text{def}}right)}{left({T}_{i}-{T}_{mathit{ref}}right)}phantom{rule{6em}{0ex}}forall iphantom{rule{2em}{0ex}}mathrm{tel}mathrm{que}phantom{rule{2em}{0ex}}

{T}_{i}-{T}_{mathit{ref}}

phantom{rule{2em}{0ex}}ge phantom{rule{2em}{0ex}}text{Prec}`

The precision value is:

• be specified by the user under the keyword PRECISION of the keyword factor ELAS_FO (command DEFI_MATERIAU [U4.23.01]),

• be equal to \(1.\): value by default.

2.2.2. Calculation at temperatures close to the reference (with precision)

You can’t use the equation directly. We derive the equation with respect to temperature:

(2.1)\[ \ frac {\ partial {\ mathrm {\ epsilon}}} ^ {\ mathit {th}}} {\ partial T} =\ mathrm {\ alpha}\ text {'} (T) (T) (T) (T- {T} _ _ {\ text {def}} _ {\ text {def}}) +\ mathrm {\ def}}) +\ mathrm {\ alpha} (T) =\ widehat {\ mathrm {\ alpha}} (T) (T) (T) (T) (T) (T) (T- {T} _ _ {\ text {def}}) +\ mathrm {\ def}}) +\ mathrm {\ alpha} (T) =\ widehat {\ mathrm {'} (T) (T- {T} _ {\ mathit {ref}}) +\ widehat {\ mathrm {\ alpha}} (T)\]

We take the value of the derivative at temperature \({T}_{\mathit{ref}}\), we get:

(2.2)\[ \ widehat {\ mathrm {\ alpha}} ({T}} _ {\ mathit {ref}}) =\ mathrm {\ alpha}\ text {'} ({T} _ {\ mathit {ref}}) ({\ mathit {ref}}}) ({T} _ {\ mathit {ref}}}) +\ mathrm {\ alpha}}) +\ mathrm {\ alpha}} ({T}) +\ mathrm {\ alpha}} ({T}}) +\ mathrm {\ alpha}} ({T})} _ {\ mathit {ref}})\]

It is considered that:

:math:`widehat{mathrm{alpha }}({T}_{i})=widehat{mathrm{alpha }}({T}_{mathit{ref}})phantom{rule{6em}{0ex}}forall iphantom{rule{2em}{0ex}}mathrm{tel}mathrm{que}phantom{rule{2em}{0ex}}

{T}_{i}-{T}_{mathit{ref}}

phantom{rule{2em}{0ex}}ge phantom{rule{2em}{0ex}}text{Prec}`

The precision value is:

• be specified by the user under the keyword PRECISION of the keyword factor ELAS_FO (command DEFI_MATERIAU [U4.23.01]),

• be equal to \(1.\): value by default.

Also, to calculate \(\widehat{\mathrm{\alpha }}({T}_{i})\) you must first calculate \(\mathrm{\alpha }\text{'}({T}_{\mathit{ref}})\).

2.2.3. Calculation of the derivative of the coefficient of thermal expansion

The calculation of the derivative of the thermal expansion coefficient is done by an algorithm that treats three cases.

First case:

+—————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————++ |:math: `\ forall i\ phantom {\ rule {2em} {2em} {0em}} {0ex}}\ text {such as}\ phantom {\ rule {2em} {0ex}} | {T} - {T} - {T} _ {\ rule {2em}} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _} _ {T} _ {T} _ {T} _ {T} _ {Prec}\ text {with} i\ne 1, i\ne n`:Math:`\ mathrm {\ alpha}\ text {'} ({T} _ {\ mathit {ref}}) =\ frac {1} {2}\) =\ frac {1} {2}}\ left (\ frac {\ mathrm {\ alpha}\ left (\ frac {\ mathrm {\ alpha}} ({T} _ {i+1}}) =\ frac {1} {ref}}) =\ frac {1} {2}\ left (\ frac {\ mathrm {\ alpha}}) -\ mathrm {\ alpha} ({T} _ {\ mathit {ref}})})} {{T} _ {i+1} - {T} _ {\ mathit {ref}}}} +\ frac {\ mathrm {\ alpha} ({T}} ({T} _ {\ mathit {ref}})} {\ mathit {ref}})} {{T}} ({T}} _ {\ mathit {ref}})} {{T} _ {\ mathit {ref}})} {{T} _ {\ mathit {ref}})} {{T} _ {\ mathit {ref}})} {{T}} _ {\ mathit {ref}}}} - {T} _ {i-1}}}\ right) `|| +—————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————++

Second case:

+—————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————-++ |:math: `\ forall i\ phantom {\ rule {2em} {2em} {0em}} {0ex}}\ text {such as}\ phantom {\ rule {2em} {0ex}} | {T} - {T} - {T} _ {\ rule {2em}} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _} _ {T} _ {T} _ {T} _ {T} _ {Prec}\ text {with} i=N`:math:`\ mathrm {\ alpha}\ text {'} ({T} _ {\ mathit {ref}}) =\ frac {\ mathrm {\ alpha} ({\ alpha}} ({T}}) -\ mathrm {\ alpha}} ({T} _ {i-1})} {{T})} {{T})} {{T} _ {\ mathit {ref}} - {T} _ {i-1}} `|| +—————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————-++

Third case:

+———————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————–++ |:math: `\ forall i\ phantom {\ rule {2em} {2em} {0em}} {0ex}}\ text {such as}\ phantom {\ rule {2em} {0ex}} | {T} - {T} - {T} _ {\ rule {2em}} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _} _ {T} _ {T} _ {T} _ {T} _ {Prec}\ text {with} i=1`:math:`\ mathrm {\ alpha}\ text {'} ({T} _ {\ mathit {ref}}) =\ frac {\ mathrm {\ alpha} ({\ alpha}) ({T} _ {i+1}) -\ mathrm {\ alpha}\) -\ mathrm {\ alpha} ({T} _ {ref}}) =\ frac {\ mathrm {\ ref}}) {\ alpha} ({T}}) {{T})} {{T}) _ {i+1} - {T} _ {\ mathit {ref}}} `|| +———————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————–++