Int. J. Microstructure and Materials Properties, Vol. 1, No. 2, 2006 197

Copyright © 2006 Inderscience Enterprises Ltd.

Study and modelling of microstructural evolutions and thermomechanical behaviour during the tempering of steel

Yunning Wang*, Benoit Appolaire, Sabine Denis, Pierre Archambault and Bernard Dussoubs LSG2M, UMR 7584 CNR-INPL-UHP, Ecole des Mines de Nancy, Parc de Saurupt, 54042 Nancy, France E-mail: Yunning.Wang@mines.inpl-nancy.fr E-mail: benoit.appolaire@mines.inpl-nancy.fr E-mail: Sabine.Denis@mines.inpl-nancy.fr E-mail: pierre.archambault@mines.inpl-nancy.fr E-mail: bernard.dussoubs@mines.inpl-nancy.fr *Corresponding author

Abstract: A model for the evolution of microstructures and flow stress during the tempering of low alloyed steels has been developed. The competitive precipitations of ε carbide and cementite are simulated. The results are used to calculate the flow stress of the tempered martensite by a thermo-elasto-viscoplastic law that takes into account solid solution, dislocation and precipitation hardenings. The model is applied to a 80MnCr5 steel and the calculated results are compared with the experimental ones.

Keywords: tempering; ε carbide; cementite; precipitation; thermomechanical behaviour.

Reference to this paper should be made as follows: Wang, Y., Appolaire, B., Denis, S., Archambault, P. and Dussoubs, B. (2006) ‘Study and modelling of microstructural evolutions and thermomechanical behaviour during the tempering of steel’, Int. J. Microstructure and Materials Properties, Vol. 1, No. 2, pp.197–207.

Biographical notes: Yunning Wang received his BS and MS Degrees from Taiyuan University of Technology in Taiyuan, Shanxi, China, and will receive his PhD Degree from Institut National Polytechnique de Lorraine, France. He has coauthored approximately ten publications on various aspects of heat treating and phase transformations in alloys. Currently, he is a Researcher in LSG2M, Ecole des Mines de Nancy, France.

Benoît Appolaire is currently Associate Professor at INPL, and working in the field of Numerical Modelling of phase transformations in metallic alloys. He has received his PhD at Ecole des Mines de Nancy in 1999, and has written about 20 papers in the field of solidification and solid-solid phase transformation.

Sabine Denis received her PhD in Material Science and Engineering in 1980, she is a Professor at University Henri Poincaré in Nancy, works with LSG2M (Laboratoire de Science et Génie des Matériaux et de Métallurgie) in the field

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of couplings between phase transformations – temperature and stresses in metallic alloys.

Pierre Archambault is a Director of LSG2M (Ecole des Mines de Nancy, France). He is Doctor of Science of INPL in 1985 and Director of Research at CNRS. He has co-signed about 60 papers on heat treatment and solid phase transformations.

Bernard Dussoubs received his MS and PhD Degrees from University of Limoges, France in 1994 and 1998, respectively. At the time, his research domain was the numerical simulation of plasma spraying at atmospheric pressure. Since 1999, he works as a Research Engineer in LSG2M, Ecole des Mines de Nancy, France, and he has specialised in numerical simulation. He has coauthored approximately 30 papers on modelling and simulation in various domains, including surface treatment, chemical engineering, solidification, phase transformation both in alloys and ceramics.

The paper was presented at 1st International Conference on Distortion Engineering – IDE 2005, Bremen, Germany, September 14–16, 2005.

1 Introduction

The tempering process may have two main objectives: on one hand to get the microstructures that lead to service mechanical properties and, on the other hand, to promote the relaxation of the residual stresses generated during quenching. A better control of this process can be achieved through numerical simulation that requires two main steps: firstly, the study and the modelling of the microstructural evolutions during tempering and their consequences on the thermomechanical behaviour of the material at the scale of a volume element without gradients. Secondly, these models must be included in a finite element numerical simulation at the scale of a massive specimen (with thermal gradients) in order to predict the residual stresses after tempering.

In this paper, we present the results at the scale of the volume element. The models for the precipitation of carbides and the thermomechanical behaviour during tempering of martensite are described first. A complete experimental study of the metallurgical and thermomechanical behaviour of a 80MnCr5 steel during tempering has been associated with the modelling work. We present here some of the experimental results, focusing on the comparison with the calculated results.

2 Modelling of microstructural evolutions

A model describing the precipitation during the tempering of martensite including nucleation, growth and coarsening processes of the carbides (transition carbides as well as cementite) has been already developed for Fe-C alloys (Wang et al., 2004). This model has been further improved by taking into account the effects of the alloying elements (Wang, 2006). Our model is based on the models by Kampmann and Wagner (1984) and Myhr and Grong (2000). The nucleation rates are described by the classical nucleation theory. For ε carbide (metastable carbide with composition Fe2.4C), homogeneous

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nucleation is assumed. Furthermore we have taken into account the contribution of the coherent elastic strain energy to the nucleation driving force. For cementite, which composition is obtained with the ThermoCalc software, heterogeneous nucleation is considered.

The growth rates are calculated using the Laplace approximation for the diffusion fields of the alloying elements, and considering the Gibbs-Thomson effect. In addition for ε carbide, the effect of elastic energy on the carbide composition is taken into account. For ε carbide, only carbon diffusion is considered. For cementite, we assume that local equilibrium holds at the precipitate/matrix interface. These equilibrium conditions are described by a solubility product (calculated with ThermoCalc). The main results of this model are, at each time step (either in isothermal or continuous heating conditions), the volume fractions of precipitate, the size distributions, the mean sizes of the precipitates and the chemical composition of the solid solution.

3 Modelling of the thermomechanical behaviour

3.1 Behaviour law of the material

In the literature, models can be found which predict the yield stress of tempered martensite by taking into account the hardening effects due to different elements of the microstructure (precipitates, dislocations …) (Young and Bhadeshia, 1994). But these models apply only to the yield stress at room temperature. For our purpose, we need a model able to describe the behaviour of the steel in the whole range of temperatures covered during tempering. By the way, it must take into account not only the effect of microstructure, but also the effects of temperature and strain rate on the mechanical behaviours.

So, we have used an additive thermo-elasto-viscoplastic behaviour law with isotropic hardening (Lemaitre and Chaboche, 1988) to describe the flow stress of tempered martensite:

1/0

mnP Pσ σ Hε Kε= + + (1)

where σ0 is the threshold stress, H the strain hardening constant, εP the plastic strain, K the viscous hardening constant, Pε the plastic strain rate, and n, m are constants.

Generally, σ0, H, K, n and m are temperature and microstructure dependent. In our approach, we have assumed that microstructural evolutions affect only the threshold stress but not the strain hardening and the viscous stress. Thus, the threshold stress varies with both the microstructure and the deformation temperature; and the other parameters are only dependent on the deformation temperature.

The terms in equation (1) are determined from the experimental stress-strain curves by using an optimization procedure in the software ZeBuLoN (Besson et al., 1998), and σ0 is calculated as described in the following section.

3.2 Coupling with the precipitation

The threshold stress of tempered martensite can be expressed as the sum of different hardening contributions:

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0 DisFe C SS Pσ σ σ σ σ σ= + + + + (2)

where σFe is the friction stress in pure Fe, σC the solid solution hardening due to carbon atoms, σSS the hardening due to the substitutional solid solution atoms, σP the hardening due to the precipitates, and σDis the hardening due to dislocations or martensite laths and grain boundaries.

The value of σFe found in the literature at room temperature shows a large dispersion (Smith, 1977; Naylor, 1979; Young and Bhadeshia, 1994), and varies from 13 MPa to 218 MPa probably due to the grain size effect. We have used the experimental data from Liebaut (1988) for ferrite at high temperature to obtain the following expression:

e 78 0.023 MPaFσ T= − × (3)

where T is the absolute temperature. σC is calculated as (Dilip, 2001):

MPaC Cσ Kµ X= (4)

where µ is the shear modulus; K has been calculated from the data of (Speich, 1968) as K = 0.0167; XC is the molar fraction of carbon in the matrix that is calculated by the microstructural model.

Concerning the other elements in solid solution, their contributions are generally assumed proportional to their concentrations (Béranger, 1994). For the steel considered in this study (80MnCr5), the contributions of Cr and Mn to the yield stress are nearly inverse according to Constant (1986). So, we have assumed σSS = 0 MPa.

The hardening contribution of precipitates depends on the interaction mechanism between dislocations and precipitates, either shear or by-passing. The strengthening by shear is expressed as (Deschamps and Brechet, 1999):

1 1, 0.02 / MPaP V mσ K f R K Mµ b= = (5)

where fV is the volume fraction of precipitates, Rm the precipitate mean radius that are predicted by the microstructural model. M is the Taylor factor and b the Burgers vector.

The strengthening by by-passing is written as:

2 2/ , 0.6 MPa.P V mσ K f R K Mµb= = (6)

The transition between shear and by-passing occurs for a radius estimated as 10b (Brown, 1971).

The contribution due to the dislocations can be calculated by (Friedel, 1964):

Dis MPaσ αµb ρ= (7)

where ρ is the dislocation density in the matrix, and α a coefficient that may depend on temperature and strain rate (Kocks, 1985). It is usually about 1/3 at room temperature (Dieter, 1988). In our calculation, α is adjusted as:

40.4 4.42 10α T−= − × × (8)

where T is the absolute temperature.

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During the tempering, recovery occurs in the martensite. The evolution of the dislocation density can be estimated by (Hou and Lu, 1989):

d exp( / )d gCρ Q R Ttρ = − − (9)

where C is a constant, ρ the dislocation density, Q the activation energy for the moving of defaults (134kJ/mol) related to the pipe diffusion of iron atomes along dislocations (Gjostein, 1973), T the absolute temperature, Rg the gas constant, and t the tempering time.

By integrating equation (9), we obtain:

1 2exp( exp( / ) )).gρ C Q R T t C= − − + (10)

The constants C1, C2 have been estimated from the literature data (Hou and Lu, 1989; Hoffmann et al., 1997): C1 = –2.1 × 10–4 and C2 = 36.84.

4 Application to 80MnCr5 steel

4.1 Experimental procedure

The material studied is a 80MnCr5 steel (0.8C-1.25Mn-1.25Cr). All the samples have been austenitised at 880°C for 1 hour and quenched by gas (helium) in order to obtain a martensitic transformation, and then cooled in liquid N2 to transform the retained austenite. The final amount of the retained austenite is about 5%.

The tempering kinetics has been measured in isothermal conditions and during continuous heating by dilatometry. The tests have been carried out on an in-house dilatometer on cylindrical samples (outer diameter 4 mm, inner diameter 3 mm, length 30 mm) under vacuum. The isothermal tests have been performed at temperatures ranging from 100°C to 600°C and for holding times of 30 seconds to 4 hours (heating rate was 50°C/s). After the isothermal tempering, the samples have been analysed by TEM to get the precipitate sizes (about 200 particles for each measurement) and the matrix chemical composition (by EDAX). For the continuous heating tests, the heating rates ranged from 10°C/min to 50°C/s up to 600°C.

For getting the mechanical behaviour during tempering, tensile tests have been performed in the in-house thermomechanical simulator DITHEM on samples with 3 mm in diameter and 20 mm gauge length. The tensile tests have been done on one hand, at different tempering temperatures (heating rate was 10°C/s, ranging from 250°C to 600°C) after different holding times (from 5 min to 4 hours). Two deformation rates (10–3s–1, 10–4s–1) have been used in order to quantify the viscous effects. On the other hand, tensile tests have been performed at room temperature after tempering (above mentioned tempering conditions). All the results will be discussed thoroughly in Wang (2006). Hereafter, we have chosen to present only the results for tempering at 500°C.

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4.2 Comparison between experimental and simulation results

The results are discussed first for the microstructural evolution (Figure 1) and then for the mechanical behaviour (Figures 3 and 4) for a tempering treatment at 500°C characterised with a heating rate of 10°C/s and different holding times at 500°C.

Figure 1 Evolution of microstructure during the heating and holding at 500°C (a) dilatometric curve during heating; (b) kinetics of precipitations for ε carbide and cementite; (c) volume fractions of ε carbide and cementite; (d) matrix composition; (e) critical radius and mean radius of ε carbide and (f) critical radius and mean radius of cementite

(a) (b)

(c) (d)

(e) (f)

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Microstructure evolutions

From the dilatometric test (Figure 1(a)), the experimental precipitation kinetics of ε carbide and cementite have been obtained (Aubry, 1997). The calculated results are reported too in Figure 1(b). (Note that they have been normalised to the maximum volume fractions formed, 9% for ε carbides and 12% for cementite, see Figure 1(C)). They show, as the experimental ones, that the precipitation of ε carbides is the first to proceed and is approximately finished when the cementite precipitation occurs. It should be underlined that nucleation and growth of ε carbide and cementite start simultaneously because the driving force is available for both processes, but rates for cementite are much lower than for ε carbide. The calculates results also show that ε carbide dissolution in the matrix is concomitant with the precipitation of cementite (Figure 1(c)).

It can be seen that the calculation overestimates the precipitation beginning temperatures and the transformation rates of cementite and to a lower extent those of ε carbides (Figure 1(b)). This can be attributed to the model assumptions and parameters: on one hand, the interface and elastic strain energies that affect strongly the nucleation rates; on the other hand the overestimation of the diffusion coefficients since their dependencies on the concentrations in alloying elements are not taken into account.

The calculation gives also the evolution of the matrix chemical composition (Figure 1(d)). The carbon content decreases as soon as ε carbides precipitate. As the mean composition of the matrix becomes lower than their equilibrium composition, ε carbides dissolve. During the precipitation of the alloyed cementite (Fe, Cr, Mn)3C, Cr and Mn contents decrease and remain approximately constant at their equilibrium values once the precipitation is finished.

We can see that the measured Cr and Mn concentrations are higher than the calculated ones. This may be explained by some experimental errors, in particular because the measured zone in the matrix may contain some small precipitates.

Figure 1(e) and 1(f) give respectively the evolutions vs. time of the mean radii and critical radii of ε carbide and cementite. It can be seen that the coarsening of ε carbide is predicted to occur between 25 s and 40 s (after nucleation and growth between 0 s and 25 s and before dissolution at about 40 s). For cementite (Figure 1(f)), our model predicts that the coarsening leads to a mean particle size of 90 nm after holding at 500°C for 4 hours. The calculated are in good agreement with the TEM measurements.

Figure 2 shows clearly the morphology and size evolutions for tempering at 500°C respectively for 1 hour and 4 hours.

Figure 2 Morphology of the particles of cementite after a tempering at 500°C (a) 1 hour and (b) 4 hours

(a) (b)

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Thermomechanical behaviour

As already mentioned, our methodology consists first in identifying the parameters σ0, H, n, K, and m of the mechanical behaviour law as well as Young’s modulus E. For the tensile tests performed at tempering temperature after different holding times, the parameters H, n, K, m and E depend on temperature, and σ0 depends on tempering temperature and holding time. According to our assumption, for the tensile tests at room temperature, H and n are constants and σ0 depends too on tempering conditions (temperature and time). At room temperature, the viscous effects have been neglected.

The stress-strain curves at room temperature after tempering at 500°C for holding times ranging from 5 minutes to 4 hours (Figures 3(a)) show, as expected, a decrease of the flow stress when the holding time increases due to the microstructural evolutions. The calculated curves, with the parameters determined above, fit well the experimental curves (the maximum difference is about 6%). In order to highlight the viscous effects at high tempering temperatures, Figure 3(b) shows the measured stress-strain curves at 500°C after the holding time of one hour for two deformation rates 10–3s–1 and 10–4s–1. We can see again a good agreement between the calculated and experimental curves. It must be mentioned that considering all the results, maximum discrepancies of about 10% are generally observed for the tests at higher temperatures.

Figure 3 Comparisons between the stress-strain curves measured and simulated by ZeBuLoN (a) tensile tests at room temperature and (b) tensile tests at 500°C by different strain rates

(a) (b)

In a second step, the threshold stress σ0 has been calculated including the different hardening contributions and compared to the σ0 obtained from the identifications above. Figure 4 shows the results during heating and holding at 500°C (Figure 4(a)), and at room temperature after tempering (Figure 4(b)).

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Figure 4 Threshold stresses of tempered martensite identified by ZeBuLoN (Sim.) and calculated by the model (Cal.) with the different hardening contributions after heating or holding for different time at 500°C (a) tensile tests at the actual tempering temperature and (b) tensile tests at room temperature

(a) (b)

We can see that the contributions of the original martensite to σ0 are mainly the dislocations hardening and the carbon solid solution hardening. During the tempering process (Figure 4(a)), as expected, the dislocations hardening decreases with the increase of temperature. As the precipitation of coherent ε carbides proceeds, an increase of the precipitate hardening contribution is observed that leads to a small increase of σ0. Then a rapid decrease of the carbon solid solution hardening as well as the growth of the incoherent cementite particles lead to a large decrease of σ0. When the holding temperature of 500°C is reached the variations of σ0 are smaller due to the smaller variations of the different hardening contributions with time.

At room temperature, the variations of σ0 can be linked in the same way to the variations of the hardening contributions. It comes out from the model that the relatively large decrease of σ0 that occurs after tempering at 500°C for durations larger than 1 hour is mainly due to the decrease of the dislocation density. We can notice a good agreement between the identified σ0 and the calculated one at both high and room temperatures.

5 Conclusions

A model to predict the evolutions of the microstructure and the thermomechanical behaviour of low alloyed steels during the tempering process has been developed. It allows to predict the precipitation kinetics of ε carbide and cementite, the sizes of the precipitates, the matrix composition and the flow stress of the steel using a thermo-elasto-viscoplastic behaviour law that includes the solute, precipitate and dislocation hardening mechanisms.

From the comparison with the experimental results for a 80MnCr5 steel, it comes out that the model predicts well the mean sizes of the precipitates for the different tempering conditions, despite some discrepancies on the global kinetics of precipitation. The evolutions of the flow stress of the steel are also correctly predicted in the range where experimental results are available. Finally, the main interest of the model is its ability to quantify the hardening contributions of the microstructural evolutions.

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The model is presently implemented into the finite element software ZeBuLoN in order to predict the evolutions during tempering of the residual stress states generated by quenching of steel parts.

Acknowledgement

Support from the French program SIMULFORGE is gratefully acknowledged.

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