3. Identification and implementation of the cooling model#
3.1. Principle#
The identification of the model and the use of the experimental data that constitute diagrams TRC to determine the value taken by the function \(f\) in a state thermo-metallurgical \((T,\dot{T},z;d)\) given are based on the finding and the following hypotheses:
the thermo-metallurgical stories on a TRC diagram are all solutions particulars of the differential equation
eq-equaDiffPhasesFroides. They therefore make it possible to calculate in each thermodynamic state encountered experimentally and present in a diagram TRC the value taken by the function \(f\).the \(f\) function is regular; that is, if two dots \({E}_{k}\) and \({E}_{j}\) are neighbors with \({E}_{k}=\left\{T({t}_{K}),\dot{T}({t}_{K}),z({t}_{K});d({t}_{K})\right\}\), their rates of evolution in \(z\) are also similar, namely:
The rates of structural transformations of any state are then determined by interpolation among all the \(({E}_{k},f({E}_{k}))\) « couples » defined by the TRC diagrams.
3.2. Integration of experimental data into the model#
3.2.1. Principle#
In general, a TRC diagram defines in a \(\left[\ln(t)-T\right]\) coordinate system the structural transformations associated with a series of thermal stories traced on this diagram [r4.04.01-ExempleTRC]. The integration of experimental data then consists in identifying for each story these Diagram the successive values of \(\left ( T,\dot{T},z \right )\) so that for Any temperature \(T\) the model knows the values taken by the function \(f\) in \((T,\dot{T}(T),z(T))\).
In order to be able, from a small number of digital data, to reconstruct continuously thermometallurgical evolutions, we formulate some hypotheses on thermal evolutions and on the metallurgical behavior of steels.
3.2.2. Chart interpretation rules TRC#
Thermal evolutions
To define the thermo-metallurgical histories present in a TRC diagram you need characterize their thermal evolutions. It can be noted that, in a coordinate system \(\left[\ln(t)-T\right]\) and for temperatures below \(\mathrm{820}°C\), the thermal stories in TRC diagrams can, with a fairly good approximation, deduce one from the other by a horizontal translation [:ref:r4.04.01-ExempleTRC`].
It is therefore possible to define a thermal history \({T}^{i}(t)\) from the data of a pilot curve \({T}_{p}(t)\) and the moment (in seconds) for which this story crosses the 820°C isotherm by
: label: eq - Translation story
{t} ^ {i} (T) =expleft{left{lnleft{lnleft [{t}} (T)right] +lnleft [{t} ^ {i} (text {820}) {i} (text {820}) {i} (text {820}} {i} (text {820}} {i} (text {820}) {i} (text {820}) (text {820}) {i} (text {820}) (text {820}) {i} (text {820}) (text {820}) (text {820}) (text {820}) (text {820
where \({t}^{i}(T)\) and \({t}_{p}(t)\) refer to reciprocal functions of \({T}^{i}(t)\) and \({T}_{p}(t)\).
In fact, it is easier to obtain information on the cooling rates of
thermo-metallurgical stories of diagrams TRC that at moments of crossing
isotherm \(820°C\).
This is particularly the case with welding steels, whose TRC diagrams are plotted in a
« cooling rate at \(700°C\) -temperature » mark.
Given eq-translationHistoire, we can then express the moment of crossing of
the isotherm \(820°C\) according to \({T}_{p}(t)\) and
\({\dot{T}}^{i}(\text{700})\) and we get \({T}^{i}(t)\) as a characterization
with \(F(T)=\ln\left \{ {t}_{p}(T)\right \}\) and, in particular \({\dot{T}}^{i}(t(T))=\frac{1}{{F}^{\text{'}}(T){t}^{i}(T)}\)
Concretely, we interpolate function \(F(T)\) by a polynomial of degree 5.
An experimental thermal evolution is therefore completely defined by the coefficient data. of the polynomial characterizing its pilot curve and by its cooling rate at \(700°C\). The validation of this method of parametrizing thermal stories « read » on diagrams TRC is shown in [bib2] _.
Overall, and considering the relative inaccuracies in the plot of diagrams TRC, of the reading \({T}^{i}(t)\) and the determination of \({\dot{T}}^{i}(\text{700})\), the agreement between the thermal stories read and recalculated seems very sufficient.
If we have the records of thermal changes in diagrams TRC, we can define each experimental thermal evolution considering that it is its own pilot curve. Moreover, in the case where the dilatometric tests defining diagram TRC used for the identification of the model is carried out with constant cooling rates, Characterize these cooling kinetics only by their cooling rates to \(700°C\) and an identically null function \(F\).
Start and end of processing temperatures
A TRC diagram provides, for a series of known thermal histories, the proportions of various metallurgical components that were formed during cooling as well as temperatures for which we observe on a dilatometric curve a significant variation of overall coefficient of expansion of the specimen [r4.04.01-Dilato]. These temperatures are then considered to be the start and end temperatures of transformations. More specifically:
the temperatures at the start of transformation shown in diagrams TRC correspond to 1% of a constituent already formed;
the temperatures at the end of transformation correspond to the final proportion of the constituent in training courses minus 1%
Kinetics of ferritic, pearlitic, and bainitic transformations
The observation of a dilatometric curve shows that, except in the vicinity of the initial temperatures and at the end of transformation, the change in deformation as a function of temperature is almost linear.
Taking into account equation eq-defoTherMulti1 the evolution of the quantity of phase transformed
as a function of the temperature is then not very far from an affine function and we assume
so that:
for ferritic, pearlitic and bainitic transformations, the transformation rate is, between the experimental temperatures at the beginning and the end of transformation, a function temperature linear;
- the speeds of these transformations are**twice as slow* at the beginning (from 0 to 1% of the constituent
transformed) and at the end of the transformation (from \({Z}^{f}-1 \%\) to \({Z}^{f}\)) that between the initial experimental temperatures and end of transformation.
Martensitic transformations
It is assumed that martensitic transformations are described by the Koïstinen-Marbürger law
eq-KoistinenMarburger and the phenomenological equation eq-TransfoMartensite expressing
\({M}_{s}\). We then use each TRC diagram to determine the coefficients
\(\beta\), \(A\) and \({Z}^{s}_{\gamma}\) as well as the temperature \({M}_{s,0}\).
Finally, to prevent the model from systematically converting the remaining austenite into martensite
when temperature \({M}_{s}\) is reached, an additional parameter is introduced,
called TPLM, characterizing (by its cooling rate to \(700°C\)) the slowest
cooling kinetics that cause a martensitic transformation.
More specifically [Transformation martensitique]:
\({M}_{s,0}\) is considered to be the transformation start temperature martensitic when it is total;
\(\beta\) is assumed to be constant and calculated in order to verify, in the case of a total martensitic transformation \({Z}_{4}(M^f_s)=\mathrm{0,99}\) where \(M^f_s\) is the experimental temperature at the end of transformation;
\(A\) and \({Z}^{s}_{\gamma}\) are determined by linear regression from experimental thermo-metallurgical stories leading to a transformation partial martensitic.
Effect of austenitic grain size on the kinetics of transformations during cooling. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~
Phase transformations proceed by germination and growth. The germination stage is mainly done on grain joints. The grain size of austenite therefore plays a role. important on cooling transformations. For this reason diagrams TRC are established for given austenitization conditions and should not be used only for similar austenitization conditions. Experimental results tend to show that austenitic grain size changes more transformation kinetics than the temperatures at which transformations begin and end, which is translated relatively well by translating diagram TRC along the time axis. Each point \(M\) in a TRC diagram corresponds to the n-tuple \((T,\dot{T},z)\). Translate The TRC along the time axis is the same as multiplying \(\dot{T}\) by a coefficient different from the unit (the time axis is given on a logarithmic scale) [bib15] _. We therefore define an « effective » cooling rate \({\dot{T}}_{\text{eff}}\):
with \({d}_{\text{ref}}\) the reference austenitic grain size from diagram TRC (homogeneous at a length) and \(a\) a material coefficient (homogeneous at the opposite of a length).
The law of evolution adopted is therefore written as:
This writing has the advantage of limiting the interpolation to a single reference diagram TRC.
- note
It is hypothesized that the germination and growth of martensite are considered to be instantaneous and that the density of the nucleation sites has little influence on this transformation. The effect of grain size therefore does not concern the evolution of the martensitic phase and is here only on the evolution of the ferritic, pearlitic and bainitic phases.
3.2.3. Entering diagrams TRC#
Taking into account the previous hypotheses, entering the experimental data contained in diagrams TRC therefore includes:
For diagram TRC:
the value of the austenitic grain size \({d}_{\text{ref}}\) from the diagram and which will be The reference grain size
the translation coefficient \(a\) to take into account the effect of grain size austenitic
the temperature at the start of total martensitic transformation \({M}_{s,0}\)
the value of the coefficient \(\beta\) of the Koïstinen-Marbürger law
eq-KoistinenMarburgerthe value of the coefficients \(A\) and \({Z}^{s}_{\gamma}\) used in equation
eq-TransfoMartensitethe values of the six coefficients of the polynomial of degree five interpolating the function \(\ln\left[{t}_{p}(T)\right]\) (if we know the thermal stories explicitly, each of them is considered to be its own pilot curve and the definition of its six coefficients must be renewed for each story)
For each thermal story in a TRC diagram:
the cooling rate to \(700°C\)
the final proportions of ferrite, pearlite, and bainite \(Z^{f}_{1}\), \(Z^{f}_{2}\), and \(Z^{f}_{3}\)
the start temperatures of each \(T^{d}_{1}\), \(T^{d}_{2}\), and \(T^{d}_{3}\) transformation
the end temperatures of each \(T^{f}_{1}\), \(T^{f}_{2}\) and \(T^{f}_{3}\) transformation
Diagrams TRC are introduced by the DEFI_TRC command.
The complete definition of metallurgical behavior models (parameter values) \({Ar}_{3}\), « quasi-static » ferritic transformation temperature, of \({M}_{s,0}\), and the complete definition of the heating and austenitic grain growth model) is carried out within the DEFI_MATERIAU command under the keyword factor META_ACIER.
3.3. Evaluation of the evolutionary function based on experimental data#
3.3.1. Evaluating the evolutionary function for experimental stories#
Taking into account the hypotheses concerning the evolution of structural transformations associated with thermo-metallurgical stories \({H}_{i}\) of a TRC diagram, so we have a set of particular solutions parameterized by \({d}_{\text{ref}}\) of the differential equation (for \(T \ge {M}_{s}\)):
which allow for any thermo-metallurgical condition \({E}_{k}=\left\{T,\dot{T},z;{d}_{\text{ref}}\right\}\) of an experimental story \({H}_{i}\) to calculate:
In fact:
However, taking into account the linearity assumptions on the evolution of \({Z}_{i}(t)\) between two consecutive states \({E}_{k}^{i}\) and \({E}_{k}^{i+1}\) of the same discretized story:
where \(\dot{T}({E}_{k})\) can be estimated by deriving the selected analytic expression to represent \({T}_{i}(t)\).
Thus, for any temperature \(T\), we can know the values taken by the function \(f\) in thermo-metallurgical states \({E}_{i}=\left\{T,{\dot{T}}_{i}(T),{Z}_{i}(T);{d}_{\text{ref}}\right\}\) where the clue \(i\) refers to stories that are known experimentally.
3.3.2. Calculating the progress of transformations for any state#
Knowing \(T,\dot{T},z,{M}_{s}\) and \(d\) at a given time \(t\), it’s about determine the values of the metallurgical variables at the next instant \((t+\Delta t)\). More specifically:
If \(T(t) \ge {Ar}_{3}\) or if \(\dot{T}>0\), then the transformation model metallurgical when cooled is inactive
If \({Ar}_{3}>T(t)\ge {M}_{s}(t)\)
so \(\dot{z}(t)=f(T,\dot{T},z;d)=f(T,{\dot{T}}_{\text{eff}},z;{d}_{\text{ref}})\) and \(z(t+\Delta t)=z(t)+\dot{z}(t)\Delta t\)
then the martensitic transformation temperature is updated:
If \(T(t+\Delta t)\ge {M}_{s}(t+\Delta t)\) then \({Z}_{4}(t+\Delta t)={Z}_{4}(t)\)
Otherwise
If \(T(t)<{M}_{s}(t)\), then
so \(z(t+\Delta t)=z(t)\) and \({M}_{s}(t+\Delta t)={M}_{s}(t)\)
and
In the case where \({Ar}_{3}>T(t) \ge {M}_{s}(t)\), we determine (thanks to the regularity hypothesis) of \(f\)) the value taken by \(f\) in \((T,\dot{T},z;d)\) from knowledge, for any temperature \(T\), values taken by \(f\) in the states thermo-metallurgical \({E}_{i}\left\{T,{\dot{T}}_{i}(T),{Z}_{i}(T);{d}_{\text{ref}}(T)\right\}\) of stories known experimentally, where \({{\dot{T}}_{i}}(T)\) is the cooling rate for story \({H}_{i}\) at temperature \(T\) (obtained by interpolation).
More precisely, we will determine a linear approximation of \(f\) in the vicinity of \((T,\dot{T},z;d)\). \(f\) is a function of \({\mathbb{R}}^{5}\) (because the dependency by with respect to parameter \(d\) is included in the possible change in the speed of current cooling) in \(\mathbb{R}\), determine a linear approximation of \(f\) in the vicinity of \((T,\dot{T},z;d)\) is the same as determining the equation of a hyperplane in \({\mathbb{R}}^{6}\) and therefore to have the value taken by \(f\) in six points \(\left\{{E}_{i}{f}({E}_{i})\right\}\) « close » to \((T,\dot{T},z;d)\).
Concretely, the steps of this interpolation of the values of \(f\) into \((T,\dot{T},z;d)\) are as follows:
- we calculate an « effective » temperature \({\dot{T}}_{\text{eff}}\) to hold
account for the effect of austenitic grain size if it is different from that of diagram, and we then look for the value taken by \(f\) by \((T,{\dot{T}}_{\text{eff}},z;{d}_{\text{ref}})\)
- we calculate for all known \({H}_{i}\) experimental stories
the values taken by function \(f\) in the following thermometallurgical states (in order to know a set of values of \(f\) in a neighborhood of \((T,{\dot{T}}_{\text{eff}},z;{d}_{\text{ref}})\) quite dense in temperature):
- we determine the six closest neighbors of
\(E(t)=\left\{T(t),{\dot{T}}_{\text{eff}}(t),z(t);{d}_{\text{ref}}(t)\right\}\) among all The \({E}_{i}^{j}(t)(j=\mathrm{1,3})\) defining the metallurgical behavior of material in the vicinity of temperature \(T(t)\) by minimizing the distance from \(E(t)\) to each of the \({E}_{i}^{j}(t)\)
- we calculate the barycentric coordinates of \(E(t)\) in relation to its closest
neighbors \({E}^{v}(t)(v=\mathrm{1,6})\). To do this, we solve the linear system associated with this calculation in the sense of least squares and by choosing the minimum norm solution in the case where its determinant is zero (this is the case when the nearest neighbors belong to an affine manifold with dimensions less than six, to solve such a system, see [Résolution de systèmes non réguliers par une méthode de décomposition en valeurs singulières])
- we only remember neighbors \({E}^{w}(t)(w \le 6)\) such as all coordinates
barycentrics \({\lambda }_{w}\) of \(E(t)\) are positive (so that \(E(t)\) be located inside the convex polyhedron based on these points);
we then calculate:
finally, we calculate \(z\) at the following time step \(z(t+\Delta t)\) according to the following explicit diagram
- note
The definition of a distance used in the proximity criterion is not obvious, count given the non-dimensionless nature of the \(\left\{T,\dot{T},z,d\right\}\) space. Currently, the search for the nearest neighbors is carried out by simply dimensionalizing each of the variables but it could be envisaged to introduce weighting coefficients in each « direction » (\(T\), \(\dot{T}\), or \(z\)) in order to report on a preponderant role played by this or that variable.