7. Income model#

7.1. Introduction#

A heat treatment called « tempering » is a heat treatment carried out at temperatures. lower than the allotropic transformation temperatures of metals. In the case of steels, it This is the austenization temperature.

In the context of welding, this is an operation carried out voluntarily to relax the constraints at the end of the process. In a micro-structural way, the cooling of steel and its transition from a \(\gamma\) phase (austenite) to \(\alpha\) phases (ferrite), bainite, pearlite, martensite) is metastable.

We then pass from a « crude tempering » phase to a so-called « tempering » phase. In practice, only bainite and martensite are affected by these metallurgical changes and the properties The mechanics of these two phases are different (tempering properties similar to ferrite).

7.2. Principles of the model#

The proposed model is based on the work of Y. Vincent [bib16] _. We consider the following temperature kinetics representative of a thermal kinetics of welding observed at one point on the figure [Cinétique thermique de revenu]

_images/revenuCinetiqueThermique.png
: name: R4.04.01 - Thermal kinetic income
width:

50%

Thermal tempering kinetics

The tempering phase is a heating phase that starts at a temperature \(T^r\) ( \(T^r \approx 550°C\)), which reaches a temperature of around 700°C (lower at austenitization temperature \(Ac_3\)) and which decreases back to \(T^r\).

The mathematical model used to model this pseudotransformation is identical to that used for the kinetics of diffusional transformations developed by Leblond [bib11] _.

We note:

  • \({Z}^{r}(t)\) the proportion of phase that returned at each moment of the previous thermal cycle

  • \({Z}^{b}(t)\) the proportion of raw tempering phase at each moment of the thermal cycle previous

  • \({\tilde{Z}}^{r} = \frac{{Z}^{r}}{{Z}^{b}}\) is the proportion of phase returned to In the raw phase

We consider a Johnson-Mehl-Avrami law to describe the evolution of the percentage From phase returned \({\tilde{Z}}^{r}\) as a function of the holding time \(t^{*}\) to a Temperature \(T^{m}\)

(7.1)#\[ {\ tilde {Z}} ^ {r} (t^ {*}) =\ left (1 -\ exp\ left\ {-b ({t^ {*}}}) ^n\ right {*}}) ^n\ right\}\ right)\]

with \(b=b(T^{m})\) and \(n(T^{m})\) of the characteristics of the material to be identified who depend on holding temperature \(T^{m}\). Assume that the temperature is constant between two moments.

Let’s be the list of moments \(t_i\) with \(i \in [0, N]\) allowing a description discretized by temperature kinetics \(T\). We note \(\Delta t = t_{i+1} - t_{i}\) Under the hypothesis \(T_{i+1} = T(t_{i+1})\) between \(t_i\) and \(t_{i+1}\), the value of Percentage of phase returned to instant \(t_{i+1}\) is given by the following expression

(7.2)#\[ {\ tilde {Z}} ^ {r} (t_ {i+1}) =\ left (1 -\ exp\ left\ {-b (T_ {i+1}) ({\ tau_ {i+1}}} +\ Delta t}} +\ Delta t) ^ {n (T_ {i+1})}\ right\}\ right)\]

with

(7.3)#\[ \ tau_ {i+1} =\ left (\ frac {- -\ log\ left (1 - {\ tilde {Z}}} ^ {r} (t_ {i})\ right)} {b (T_ {i+1})}}\ right)}\ right)}\ right) ^ {\ frac {1} {1} {n (T_ {i+1})}}\]

7.3. Identifying model parameters#

A limited number \(N_c\) of pairs of values is assumed to be known \(\left( {\tilde{Z}}^{r}_k, t^{*}_k \right )_{k=1,N_c}\) that combine a percentage of phase returned to a thermal maintenance time applied at the same given temperature \(T_d\).

The application of eq-JMA allows you to have an estimate of the percentage of phase returned to Moment \(t=t_1\):

(7.4)#\[ {\ tilde {Z}} ^ {r} (t=t_1) = {\ tilde {Z}} ^ {r} (t^ {*} =\ Delta t) =\ left (1 -\ exp\ left\ left\ {-left\ {-b ({t^ {*}}}) ^n\ right\}\ right)\]

So we find a relationship between \({\tilde{Z}}^{r}\) and \(t^{*}\)

\[\]

: label: EQ-time equivalent

logleft (t^ {*}right) =frac {1} {n}logleft (-logleft (1- {tilde {Z}}} ^ {r}right)right)right)right)right)right) -frac {1} {n}logleft (bright)

We change the variable by noting \(x = \log\left( -\log\left( 1- {\tilde{Z}}^{r} \right) \right )\) and \(y=\log\left( t^{*} \right )\), we can determine by linear regression the line \(y=\frac{1}{n} x + \frac{1}{n} \log (b)\).

The available data come from the thesis of Yannick Vincent [bib16] _. This data characterize the evolution of the hardness of bainite and that of martensite during maintenance isotherm lasting one hour, for various maintenance temperatures between 546°C and 700°C.

For bainite, we have

Tableau 7.1 Variation de la dureté de la bainite en fonction de la durée et de la température de maintien isotherme#

Temperature

Hold time

Vickers hardness
widths:

10

30

30

546°C

1 h

\(270\)

550°C

1h

\(268\)

600°C

1 h

\(243\)

625°C

1 h

\(232\)

650°C

1 h

\(221\)

675°C

1 h

\(211\)

700°C

1 h

\(201\)

700°C

100 hours

\(168\)

We have \(H_{V,b}^{min}=168\) and \(H_{V,b}^{max}=271\). For martensite:

Tableau 7.2 Variation de la dureté de la martensite en fonction de la durée et de la température isothermal holder :header: « Temperature », « Hold time », « Vickers hardness » :widths: 10, 30, 30#

546 °C

1 h

330

550 °C

1 h

327

575 °C

1 h

306

600 °C

1 h

286

625 °C

1 h

268

650 °C

1 h

250

675 °C

1 h

234

700 °C

1 h

218

700 °C

10 h

191

with \(H_{V,m}^{max}=405\) and \(H_{V,m}^{min}=165\). To find the proportion of phase revenue \({\tilde{Z}}^{r}_{{\alpha_{\alpha}}}\), we apply the Next formula starting with hardness \(H_{V,{\alpha_{\alpha}}}\) (with \({{\alpha_{\alpha}}}=b \text{ ou } {{\alpha_{\alpha}}}=m\) for bainite and for martensite) for a time of Maintain \(t^{*}\) and temperature \(T^m\)

\[\]

: label: EQ-PhasesFromHV

{tilde {Z}} ^ {r} _ {{alpha_ {alpha_ {alpha}}} (T^m, t^ {*}) =frac {H_ {V, {alpha_ {alpha}}}}}} ^ {alpha}}}} {H_ {V, {alpha_ {alpha}}} (T^m, t^ {*}})} {H_ {V, {alpha_ {alpha}}}}} ^ {max} -H_ {V} ^ {min, {alpha_ {alpha}}}}

But this data is insufficient to determine the parameters of the income model given that for the same maintenance temperature, few maintenance times or even a single duration are associated. These data are then completed by considering the following relationship, called equivalence. temperature-time.

\[\]

: label: EQ - equivalencetimeTemperature

P_ {a} (T, t^ {*}) =left (frac {1} {T} -frac {n R} {Delta H}logleft (frac {t^ {*}}) =left (frac {t^ {*}}}} {t_ {0}} -frac {n R} {Delta H}logleft (frac {t^ {*}}}} {t_ {0}}right)right) ^ {-1}}

with \(t_0\) the unit of time (to size the \(\frac{t^{*}}{t_{0}}\) ratio), \(n=\log(10)\approx 2,3\) the Nepalese logarithm of 2, \(R\) the ideal gas constant and \(\Delta H\) the heat of activation of the phenomenon under study. Note \(c_0 = \frac{n R}{\Delta H}\) the activation coefficient (without units).

This equation makes it possible to express the holding time \(t^{*}_{i}\) at a temperature \(T_i\) to observe the same phenomenon as those obtained (experimentally for example) by a time of maintaining \(t^{*}_{j}\) at a temperature \(T_j\).

\[\]

: label: eq - EquivalenceTimeTemperature example

t^ {*} _i =exp {left (frac {1} {c_0}left (frac {1} {T_i} -frac {1} {P_ {a} {P_ {a} (T_j, t^ {*} _j)}right)right)}

By setting a maintenance temperature that is different (for example 610° C.) from that of the previous tests, We have enough points to determine \(b_{{\alpha_{\alpha}}}(610)\) and \(n_{{\alpha_{\alpha}}}(610)\).

7.4. Resolution algorithm#

Income therefore introduces two additional phases: returned bainite and returned martensite. By convention, « raw » hardened bainite (or martensite) will be referred to as bainite (or martensite) of the no income model.

The algorithm for this post-processing is based on the following elements: which describe the conditions for activating the income phenomenon:

  • The temperature is assumed to be constant over a time increment \(\Delta t = t_{i+1} - t_{i}\) and it equals \(T = T(t_{i+1})\)

  • The temperature range concerned is \(T \in \left[T^r, {Ac}_3 \right]\)

  • The income is only activated after a complete thermal welding cycle that would have generated a martensitic or bainitic phase called crude tempering. To do this, we introduce a indicator, noted \({I}^{th}\), which will take the following values:

    The cycle is :math:`{I}^{th} = 2` if we have seen a complete thermal cycle,*i.e. \(T^{max} > {Ac}_3\) and

    the proportion of crude tempering phase (bainite or martensite) is strictly greater than Zero \(Z^{b}_{\alpha_{\alpha}} > 0\)

    • The cycle is \({I}^{th} = 1\) if the first thermal cycle is in progress and the raw tempering phase is in progress is different from 0 \(Z^{b}_{\alpha_{\alpha}} > 0\)

    • The cycle is \({I}^{th} = 0\) for other cases.

The value of this indicator is estimated at each node and for each moment. It is a variable internal to the model.

The algorithm is as follows, for a bainitic or martensitic phase noted in a general manner by \({\alpha_{\alpha}}\). We are at moment \(t_{i+1}\) and at temperature \(T_{i+1}\) (on each node), with \(\Delta t\) the time increment and \(\Delta T = T_{i+1}-T_{i}\) the time increment temperature. Note \(T^r\) the tempering temperature and \(T^m\) the holding temperature (different) of that of the experimental data). A first phase calculated the raw phases (by applying the Waeckel model) \({Z}^{b}_{{\alpha_{\alpha}}}(t+1)\)

  • If \(T_{i+1} > Ac_3\), then we are in the austenitic phase: \({I}_{i+1}^{th} = 0\) and \(\tilde{Z}_{{\alpha_{\alpha}}} = 0\)

  • If \(T_{i+1} > T^r\), then we are in the income phase

    • If \({I}^{th}_{i} = 2\), we saw a complete thermal cycle. We calculate \(\tau_{i+1}\) by eq-JMADiscretN, then the time/temperature equivalence parameter \(P_{a}\) of eq-equivalenceTempsTemperature, we deduce the equivalent time \({\Delta t}_{eq}\) by eq-tempsEquivalent. All that’s left to do is apply eq-JMADiscret to get \({\tilde{Z}}^{r}_{{\alpha_{\alpha}}}\)

    • Otherwise, income is not activated and \({\tilde{Z}}^{r}_{{\alpha_{\alpha}}} = 0\)

  • Otherwise, we are not in an income phase, we just maintain the phases and Detects cycles

    • If \({I}^{th}_{i} = 2\), then \({I}^{th}_{i+1} = 2\), \({\tilde{Z}}^{r}_{{\alpha_{\alpha}}} = \frac{{{Z}}^{r}_{{\alpha_{\alpha}}}(t)}{{{Z}}^{b}_{{\alpha_{\alpha}}}(t)}\)

    • If \({I}^{th}_{i} = 1\), then \({\tilde{Z}}^{r}_{{\alpha_{\alpha}}} = 0\). By Elsewhere, if \(\Delta T > 0\), it’s because we warm up and we have \({I}^{th}_{i+1} = 2\)

    • If \({I}^{th}_{i} = 0\), then \({\tilde{Z}}^{r}_{{\alpha_{\alpha}}} = 0\). By elsewhere, if \(\Delta T > 0\) and \({Z}^:{r}_{{\alpha_{\alpha}}} = 0\), then \({I}^{th}_{i+1} = 1\)

We can calculate the income phases:

\[{Z} ^ {r} _ {{\ alpha_ {\ alpha_ {\ alpha}}}} (t+1) = {Z} ^ {b} _ {{\ alpha_ {\ alpha}}} (t) {\ tilde {Z}}} (t) {\ tilde {Z}}}} {\ tilde {Z}}}\]

And the raw phases

\[{Z} ^ {b} _ {{\ alpha_ {\ alpha_ {\ alpha}}}} (t+1) = {Z} ^ {b} _ {{\ alpha_ {\ alpha}}} (t) - {Z}}} (t) - {Z}} ^ {r}} _ {{\ alpha_ {\ alpha}}} (t+1)\]

To activate the calculation of income in the CALC_META operator, we activate the REVENU keyword by specifying « RELATION = » ACIER_REVENU «  » and the model « LOI_META = » JMA « ``. The material parameters must be entered in DEFI_MATERIAU, in the keyword META_ACIER_REVENU.

The ACIER_REVENU relationship of the CALC_META operator produces twelve internal variables:

Tableau 7.3 Variables internes pour le calcul des phases de revenu de l’acier#

Variable

Quantity

Description
widths:

10

20

50

\(\mathrm{V1}\)

\({Z_1}\)

ferrite proportion

\(\mathrm{V2}\)

\({Z_2}\)

proportion of pearlite

\(\mathrm{V3}\)

\({Z^b_3}\)

proportion of bainite (raw tempering phase)

\(\mathrm{V4}\)

\({Z^r_3}\)

proportion of bainite (tempering phase)

\(\mathrm{V5}\)

\({Z^b_4}\)

proportion of martensite (raw tempering phase)

\(\mathrm{V6}\)

\({Z^r_4}\)

proportion of bainite (tempering phase)

\(\mathrm{V7}\)

\({Z_\gamma}\)

proportion of austenite (hot phase)

\(\mathrm{V8}\)

\({Z_c}\)

proportion of cold phases

\(\mathrm{V9}\)

\(d\)

austenitic grain size

\(\mathrm{V10}\)

\(T_g\)

temperature at Gauss points

\(\mathrm{V11}\)

\({M}_{s}\)

martensitic transformation temperature

\(\mathrm{V12}\)

\({I}^{th}\)

thermal cycle indicator