4. Model of metallurgical behavior under heating#
4.1. Assumptions#
During heating, the only transformation likely to occur is the transformation into austenite, whose rate is assumed to be independent of the heating rate. Moreover, it is also assumed that all of the ferritic, pearlitic, bainitic and martensitic are transformed into austenite in an identical manner. These hypotheses are generally common to all austenitization models [bib9] _, [bib10] _ and [bib11] _. Consequently, the model selected is of the form:
It is recalled that the metallurgical transformation model proposed by Leblond and Devaux and implanted in the Sysweld code [bib11] _ is of the form (for transformations to heating and cooling):
where, for the austenitic transformation, the parameter \(\lambda\) is taken to be constant.
The elements of comparison to the experiment presented in [bib11] _, [bi612] _ and [bib13] _ show that, provided that the functions \({Z}_{\mathrm{eq}}(T)\) and \(\tau (T)\) have been identified Based on tests at different heating rates, this model allows a complete description satisfactory austenitic transformation of steels. However, it seems that identifying function \(\tau (T)\) remains difficult [bib4] _.
The austenitic transformation model is of the form:
but with a simple form for function \(\tau (T)\), in order to keep a set of metallurgical models for easy and quick identification.
4.2. Shape of the selected model#
In its continuous form, the model selected is such that:
: label: EQ - Heating model
{dot {z}} _ {gamma} (T, {Z}} (T, {Z}} _ {gamma}) =frac {{Z}} _ {mathrm {eq}} (T) - {Z}} _ {gamma}} {tau (T)} (T)} (T) - {Z} _ {Z}} (T) - {Z} _ {Z}} (T) - {Z} _ {Z} _ {gamma}}
where \({Z}_{\gamma }\) refers to the proportion of austenite. The \({Z}_{\mathrm{eq}}(T)\) function is such that
: label: EQ - ZEQ heating model
{Z} _ {mathrm {eq}} (T) =left{begin {array} {cl} 0&text {si} Tle {Ac} _ {1}\ frac {T- {Ac} {T- {Ac} _ {Ac} _ {1}} (T) =left{mathrm {eq}}} (T) =left{begin {array} {cl} 0&text {si} {Ac} _ {1}}frac {T- {Ac} _ {Ac} _ {1}} (T) =left{mathrm {eq}}} (T) =left{begin {array} {cl} 0&text {si} {Ac} Tle {Ac} _ {3}\ 1&1&text {si} Tge {Ac} _ {3}end {array}right.
with \({Ac}_{1}\) and \({Ac}_{3}\) positive constants. The \(\tau (T)\) function is such that
: label: EQ - TAUT heating model
tau (T) =left{begin {array} {cl} {cl} {tau} _ {1} &text {si} Tle {Ac} _ {1} _ {1} +frac {T- {1}} +frac {T- {1}} {Ac} _ {1}} +frac {T- {1}} +frac {T- {1}} +frac {T- {1}} +frac {T- {Ac} _ {1}} +frac {T- {1}} +frac {T- {Ac} _ {1}} +frac {T- {1}} +frac {T- {1}} +frac {T- {Ac}}} _ {1}) &text {si} {Ac} _ {1}le Tle {Ac} _ {3}\ {tau} _ {3} &text {si} Tge {Ac} _ {1}le Tge {Ac} _ {1}le Tle {Ac} _ {1} _ {3}text {si} Tge {Ac} _ {Ac} _ {3}end {array}right.
with \({\tau }_{1}\) and \({\tau }_{3}\) positive constants.
The definition of function \({Z}_{\mathrm{eq}}(T)\) is the same as that given by
Leblond and Devaux in [bib11] _ and [bi612] _. It corresponds to the evolution of the austenite level
transformed for very low heating rates.
Indeed, at a temperature \(T\) fixed, \({Z}_{\mathrm{eq}}(T)\) is the solution
asymptotic towards which the solution of the differential equation eq-modeleChauffage tends
with the time constant \(\tau (T)\).
For low heating rates, the asymptotic solution can be considered as
reached at all times and \({Z}_{\mathrm{eq}}(T)\) therefore corresponds to the evolution of the rate
of austenite transformed during « quasi-static » evolutions. The function
\({Z}_{\mathrm{eq}}(T)\) is therefore entirely defined by the data in \({Ac}_{1}\) and
\({Ac}_{3}\) which is done under the simple keywords AC1 and AC3 under the keyword
factor META_ACIER of the DEFI_MATERIAU command.
In the model proposed by Leblond and Devaux, the form of function \(\tau (T)\) is not
not specified and this function is identified in order to obtain satisfactory agreement between
the experimental and calculated start and end temperatures of transformation.
In order to obtain a simple and fast identification model, we have chosen a simple form.
for function \(\tau (T)\). More precisely, to be able to integrate the equation
of evolution eq-modeleChauffage, we first considered the case where the function
\(\tau (T)\) is constant. In this case, we can then propose two possibilities
simple identification of this \(\tau\) constant function.
The first possibility is to identify a \({\tau }_{1}\) value from \(\tau\)
allowing to correctly describe the beginning of the transformations while the second consists of
identify a \({\tau }_{3}\) value of \(\tau\) to correctly describe the
end of the transformations. We then tested the model obtained with a function \(\tau (T)\)
affine defined based on the values \({\tau }_{1}\) and \({\tau }_{3}\) defined
above. The results obtained are completely satisfactory and comparable to those obtained.
with the model available in Sysweld, we chose to introduce a model where the function
\(\tau (T)\) is affine and is defined by \({\tau }_{1}\) and \({\tau }_{3}\) which
are filled in with AC1 and AC3.
4.3. Integration of the evolution equation#
We chose to integrate the evolution equation eq-modeleChauffage exactly by
\({Z}_{\gamma }\) and explicitly in \(\dot{T}\) and \(\tau\) on each time step
(i.e. considering \(\dot{T}\) and \(\tau\) to be constant on the step and equal to their
values at the start of the time step). We then obtain:
The resulting change in the proportions of all the other metallurgical components is then defined by:
In other words, each of the phases present is transformed into austenite up to its proportion at the beginning of the time step.
4.4. Evolution of austenitic grain size upon heating#
Once austenized, steel sees its grain size increase more or less rapidly depending on of the temperature, but this growth always takes place since austenite occurs with a zero grain size. Austenitic grain growth is a thermally activated process. The growth model chosen is that of Grey and Higgins, suitable for treating material during processing [bib15] _
Growth model:
Growth during processing, austenite appearing with zero grain size:
Where we have \(\lambda ={\lambda }_{0}\cdot \exp \left(\frac{{Q}_{\text{app}}}{R T} \right )\) and \({d}_{\text{lim}}={d}_{\text{10}}\exp \left (-\frac{{W}_{\text{app}}}{R T} \right )\). With the following settings:
\(z\) is the proportion of the austenitic phase
\(d\) is the austenitic grain diameter (homogeneous at one length)
\({d}_{\text{lim}}\) is the grain size limit, depending on \({d}_{\text{10}}\), Material parameter homogeneous at one length
\({Q}_{\text{app}}\) and \({W}_{\text{app}}\) are material parameters that are homogeneous to activation energies (\({{J.mol}}^{-1}\))
\(R\) is the ideal gas constant (\(\text{8,314 } {{J.K}}^{-1}.{{mol}}^{-1}\))
\({d}_{\text{10}}\) is a material parameter that is homogeneous to seconds per unit length
The material parameters are to be entered under the keyword META_ACIER of DEFI_MATERIAU.
4.4.1. Digital processing#
The grain size calculation is performed after the phase proportion calculation and the integration of the evolution equation is done according to an implicit pattern in \(d\)
with increment \(\Delta d\)
We solve a quadratic equation in \(d\).
4.5. Meaning of metallurgical evolution#
In a structural calculation, some areas may be heated while others are cool. In addition, under certain conditions, an austenitic transformation initiated during heating may continue at the beginning of cooling. So it does not exist, strictly speaking Talk, an austenitic transformation model and a cooling transformation model but a single model of metallurgical transformations which, depending on the temperature under consideration, and the Sign of the rate of thermal evolution is described either by the decomposition model of austenite, or by the austenite formation model.
The direction of metallurgical evolution (i.e. formation or decomposition of austenite) is determined as follows:
If the temperature is such as \(T(t+\Delta t) <{Ac}_{1}\), there are two cases:
If the speed is (strictly) positive \(\dot{T}(t) > 0\), we use the model of Formation of austenite
If the speed is negative or zero \(\dot{T}(t) \le 0\), we use the model of decomposition of austenite
If the temperature is such as \(T(t+\Delta t) \in \left[ {Ac}_{1}, {Ar}_{3} \right ]\), we have four cases:
If the speed is positive \(\dot{T}(t) > 0\), the austenite decomposition model is used
If the speed is zero \(\dot{T}(t) > 0\) and if \(z_{\gamma} \ge {Z}_{\mathrm{eq}}\), we Use the austenite decomposition model
If the speed is zero \(\dot{T}(t) > 0\) and if \(z_{\gamma} < {Z}_{\mathrm{eq}}\), we Use the austenite formation model
If the speed is negative \(\dot{T}(t) < 0\), the austenite formation model is used
If the temperature is such as \(T(t+\Delta t) > {Ar}_{3}\), we use the model of Formation of austenite
- note
\({Ar}_{3}\) is also a characteristic of metallurgical behavior at cooling already defined by the cooling transformation model.