5. Identification of the heating model

5.1. Determining function \({Z}_{\mathrm{eq}}(T)\)

The \({Z}_{\mathrm{eq}}(T)\) function can be considered to be the quasistatic solution of the differential equation eq-modeleChauffage and we choose to define it (as in [bib11] _) by the eq-modeleChauffageZeq relationship.

In this expression, temperatures \({Ac}_{1}\) and \({Ac}_{3}\) are temperatures « theoretical » quasistatics at the beginning and end of austenitic transformation that correspond to a level of austenite formed that is still equal to zero or already equal to one.

In fact, these temperatures are difficult to determine experimentally and we consider Generally that the quasistatic temperatures at the beginning and end of austenitic transformation determined experimentally corresponding, respectively, to 5 and 95% of austenite formed.

In other words, if we write down these temperatures \(\widehat{Ac}_{1}\) and \(\widehat{Ac}_{3}\), They check

\[\]

: label: EQ-TempTransfo

{Z} _ {mathrm {eq}} (widehat {Ac} _ {1}) =mathrm {0.05}text {and} {Z} _ {mathrm {eq}} (widehat {Ac} _ {1}) =mathrm {0.95}} _ {3}) =mathrm {0.95}

To determine \(\widehat{Ac}_{1}\) and \(\widehat{Ac}_{3}\), you can use tests of dilatometry at low heating rates or apply formulas from the literature relating the quasistatic temperatures at the beginning and end of austenitic transformation at the composition steels. In general, these temperatures are also shown on diagrams TRC used for identification of the cooling transformation model or can be estimated using formulas that know the composition of steel [bib4] _.

Finally, knowing \(\widehat{Ac}_{1}\) and \(\widehat{Ac}_{3}\), we can then determine temperatures \({Ac}_{1}\) and \({Ac}_{3}\) defining the function \({Z}_{\mathrm{eq}}(T)\) based on the two eq-tempTransfo equations above.

A complete example of identifying the austenitic transformation model is shown in [bib4] _.

5.2. Determining function \(\tau(T)\)

In general, it is not easy to find a simple and quick way of identifying the function \(\tau (T)\). This is the reason why it is proposed to adopt for this function the simplified form below.

If \({Ac}_{1}\le T \le {Ac}_{3}\):

\[\ tau (T) = {\ tau} _ {1} +\ frac {T- {Ac} _ {1}} {{Ac} _ {3} - {Ac} _ {1}} ({\ tau} _ {1}}} ({\ tau} _ {1}}) ({\ tau} _ {1}})\]

where \({\tau }_{1}\) and \({\tau }_{3}\) are positive constants.

For the identification phase, we first consider the particular case where \(\tau\) is constant between \({Ac}_{1}\) and \({Ac}_{3}\). Two types of identification are then proposed allowing to determine either a value \({\tau }_{1}\) of \(\tau\) consistent with experimental temperatures at the start of transformation, i.e. a value \({\tau }_{3}\) of \(\tau\) consistent with the experimental temperatures at the end of transformation.

We present in [bib4] _ the results obtained by these two identifications and we show (with no other form of theoretical justification) that the function \(\tau (T)\) affine defined with the values \({\tau }_{1}\) and \({\tau }_{3}\) previously determined make it possible to obtain an agreement with experience quite comparable to that obtained with the Leblond model.

5.2.1. Identifying \({\tau }_{3}\) from \(\widehat{Ac}_{3}\)

For \(\dot{T}\) and \(\tau\) constants and the initial condition \({Z}_{\gamma }({Ac}_{1})=0\), the solution to the eq-modeleChauffage evolution equation is (as long as \({Z}_{\mathrm{eq}}^{\text{'}}(T)\) is constant, i.e. as long as \(T\le {Ac}_{3}\)):

\[{Z} _ {\ gamma} (T) = {Z} _ {\ mathrm {eq}} (T) -\ tau {Z} _ {\ mathrm {eq}}} ^ {\ text {'}} (T)\ dot {T}} (T)\ dot {T}} (T)\ dot {T}\ dot {T}\ left (1-\ exp\ left) [\ frac {{Ac} _ {1} -T} {\ text {'}}} (T)\ dot {T} (T)\ dot {T}}\ right]\ right)\]

In particular, for \(T=\widehat{Ac}_{3}\), we therefore have:

\[\ text {0.95} = {Z} _ {\ mathrm {eq}}} (`\ widehat {Ac} _ {3}`) -\ tau {Z} _ {\ mathrm {eq}}} ^ {\ text {'}}} ^ {\ text {'}}}} ({Ac}}}} ({Ac}}} ({Ac}}} ({Ac} _ {3})\ dot {T}\ left (1-\ exp\ left [\ frac {{Ac}}}} ^ {\ text {'}}} ^ {\ text {'}}} ^ {\ text {'}}} ({Ac}}} ({Ac}}} ({Ac} _ {3})\ dot {T}\ 1} -\ widehat {Ac} _ {3}} {\ tau\ dot {T}}\ right]\ right)\]

A dilatometry test at a constant heating rate (and not very low) then makes it possible to determine the \({\tau }_{3}\) value of \(\tau\) to achieve the agreement between experimental and calculated values of \(\widehat{Ac}_{3}\). In [bib4] _ we present comparisons between experiment and calculation obtained by identifying the function in this way \(\tau\) considered to be constant.

5.2.2. Identifying \({\tau }_{1}\) from \(\widehat{Ac}_{1}\)

In the same way as before, we can also write, for \(T=\widehat{Ac}_{1}\):

\[\begin{split}\ text {0.05} = {Z} _ {\ mathrm {eq}}} (\ widehat {Ac} _ {1}) -\ tau {Z} _ {\ mathrm {eq}}} ^ {\ text {'}}} (T)\\ text {'}}} (T)\ dot {T}}} (T)\ dot {T}}\ left (1-\ exp\ left [\ frac {{Ac}} _ {1}} ^ {\ text {'}}} ^ {\ text {'}}} (T)\ dot {T}} (T)\ dot {T}\ left (1-\ exp\ left [\ frac {{Ac}} _ {1}} ^ {\ text {'}}} ^ {\ text {Ac} _ {1}} {\ tau\ dot {T}}\ right]\ right)\end{split}\]

Again, using a test at a constant heating rate, the previous equation makes it possible to determine a \({\tau }_{1}\) value of \(\tau\) to obtain a good agreement on the calculated and experimental temperatures \(\widehat{Ac}_{1}\).