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Simulation of steel microstructure evolution during induction heating
Materials Science and Engineering A 527 (2010) 2978–2984
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Simulation of steel microstructure evolution during induction heating B.J. Yang ∗ , A. Hattiangadi, W.Z. Li, G.F. Zhou, T.E. McGreevy Virtual Product Development Technology, Technical Center, Caterpillar Inc., USA
a r t i c l e
i n f o
Article history: Received 29 October 2009 Received in revised form 11 January 2010 Accepted 12 January 2010
Keywords: Induction heating Microstructure evolution Cellular automaton
a b s t r a c t An effective model and approach have been developed and validated to simulate the microstructure evolution and composition variation during induction heating. The basic method is described, and the validation and application are addressed in the paper. In this model, a real initial microstructure is used as an input to simulate austenitization, and the intrinsic chemical difference is utilized to describe the ferrite and pearlite phases. A set of temperature curves is applied to the corresponding locations in an induction heating part and the dynamic variation in microstructure, grain size and carbon content is revealed through modeling. The shift of the start temperature of austenite formation (Ac1 ) is taken into account and the correlation between Ac1 and the heating rate is presented. This model is being used in industrial practice to optimize existing processes and support new process development. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Induction heating is one of rapid heating processes, which is widely used for surface hardening to high-volume continuous heat treatment operations. Modeling is an efficient way to aid engineers in developing optimal induction hardening operations and to develop insights about thermal processes and physical transformations. The first essential issue for induction hardening is rapid heating, which could shift the critical temperatures of phase transformation, such as Ac1 and Ac3 , and consequently impact the microstructure prior to quenching. The microstructure state prior to quenching significantly affects the final microstructure and mechanical properties of surface-hardened workpieces; development of an integrated tool to simulate microstructure evolution during the induction heating for austenitization is important to achieve the desired final microstructure and properties. Heating is different from cooling because during heating both solute diffusion and the driving force for austenite formation increase with increasing temperature. On cooling, diffusion decreases whereas the driving force for transformation to martensite or pearlite/bainite increases. The start temperature for austenite formation, Ac1 , is a function of the heating rate, initial microstructure and composition, and usually shifts to higher temperature with increasing heating rates [1]. When the temperature reaches Ac1 , austenite nucleation and grain growth will occur simultaneously. Due to temperature gradients in a workpiece dur-
∗ Corresponding author at: Virtual Product Development Technology, Technical Center, Caterpillar Inc., Old Galene, Peoria, IL 61656, USA. Tel.: +1 309 578 0047; fax: +1 309 578 3322. E-mail address: yang bing [email protected] (B.J. Yang). 0921-5093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2010.01.038
ing heating, austenitization occurs at different rates at different locations, so the microstructure and the grain size distribution will be non-uniform from the surface to the core. This means an accurate evaluation of Ac1 is very important for rapid heating processes. A typical equilibrium condition for hypoeutectoid carbon steel consists of two distinctive microstructures—pearlite and proeutectoid ferrite. The mechanism and the kinetics of austenitization have been widely discussed in the literature [2–4]. The transformation occurs in two ways—pearlite transforms to austenite and proeutectoid ferrite transforms into austenite. Pearlite transforms to austenite in a relatively short time due to the short distance for carbon diffusion in pearlite colonies. Jacot and Rappaz demonstrated this using a combined model [5]. If pearlite transformation is ignored, the description of microstructure evolution during heating is incomplete. The rate of austenitization depends on the heating rate, the start temperature of austenite formation (Ac1 ), composition and initial microstructure, austenite nucleation rate, and grain growth. Although the direct observation of austenitization is difficult, modeling and simulation of microstructural evolution is possible and realistically achievable for industrial use. Computer modeling using cellular automata (CA) is an applicable and powerful way to simulate microstructural evolution during both austenitization and cooling processes. Cellular automata can often express local conditions and structural detail more efficiently than typical continuum models. In recent years, CA methods have been successfully applied to simulate microstructural evolution of dendritic structures in solidification [6], recrystallization [7,8], austenite to ferrite transformations [9–12], and austenitization for furnace heating [13]. However, there is limited research on the application of CA methods to predict microstructure evolution for rapid heating. The detailed description about CA method can be found in [12,13].
B.J. Yang et al. / Materials Science and Engineering A 527 (2010) 2978–2984
This paper will present a CA model for the austenitization of hypoeutectic steels during rapid induction heating. In the current model, the phase transformations of pearlite and proeutectoid ferrite to austenite are modeled simultaneously to obtain realistic information about austenitization, and is more accurate in predicting microstructure and carbon distribution before quenching than a model without consideration of the transformation of pearlite to austenite. To capture the dynamic variation in microstructure, grain size and carbon content through the modeling, the shift of start temperature (Ac1 ) of austenite formation is taken into account. In order to simulate a real induction hardening process, various time–temperature curves are applied to the corresponding locations (surface and various depths below the surface) in an induction heating part to simulate the non-uniform rates present. The approach of obtaining these time–temperature curves will be described in Section 3. The basic method will be represented, and the model validation and application are addressed thereafter. This model is being used in industrial practice to optimize the process and support a new process development. The model is also being incorporated into a more comprehensive simulation tool developed to predict microstructural evolution during quenching [14].
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Fig. 1. Definition of a neighborhood (a) and schematic illustration of the square lattice in the CA model (b). Symbols are as follows: p—pearlite, ␥—austenite, ␣/␥—interface between ferrite and austenite.
model due to fast transformation from pearlite to austenite. The nucleation probability is assumed to be the same for every cell in the pearlite region. The probability of nucleation at the interface between pearlite and ferrite is assumed to be seven times greater than in the internal pearlite region due to a difference in free energy. The nucleation rate for each cell, dN/dt, is proposed as
2. Model description
dN = ˛T˙ × Nd × fP × A dt
2.1. Start temperature of the austenitization
where N is the number of accumulated nuclei in an individual cell, T˙ is the heating rate, and ˛ is a constant that depends on the colony size of pearlite and the interlamellar spacing in pearlite colonies. For example, ˛ is taken as 0.45 for normalized steel, and 0.4 for annealed steel. Nd is the nucleation density (1 × 109 to 5 × 109 /m2 from coarse pearlite to fine pearlite), fP is the fraction of pearlite in the computational domain or subdomain in question and A is the area of the computational domain or subdomain. In any time interval, the number of nuclei is calculated in A and the nuclei are randomly assigned to the cells in question. Using the Monte Carlo method, a random number C is generated and compared with N. Once C < N, a random cell in the domain or subdomain is selected for nucleation from among the pearlite cells or pearlite/ferrite interface cells. Each new austenite grain is randomly assigned a crystallographic orientation as soon as it is formed. The crystallographic orientation for each newly generated austenite nucleus is assumed to be randomly selected from an integer in the range of 1–125. This process is continued for N times in the domain or subdomain to place the N nuclei into the cells. Two kinds of growth transformations are taken into account, namely pearlite to austenite and proeutectoid ferrite to austenite. Because of their different microstructure characteristics, two different growth models were developed. For the transformation of pearlite to austenite, the growth of any austenite grain is driven by a decrease of the local free energy and interfacial energy in the pearlite region. The energy barrier is considered to account for the variation of free energy of a cell transition from state 1 to state 2, as shown in Fig. 2. The probability of a cell transition from state 1 to state 2 is
For numerical simulation of induction heating, the accuracy of critical austenite transformation temperatures is crucial. The start temperature for the austenite transformation, Ac1 , depends strongly on three essential parameters—heating rate, initial microstructure and chemical composition. Ferrite-austenite transformations at high heating rates are characterized by shifted transformation temperatures, as was shown by Orlich’s curves [1] and expressed by an equation based upon a constant rate [15]. Unfortunately, this equation is only valid for constant heating rates. In fact, the variation of the temperature with time is nonlinear with induction heating, especially in a region of phase transformations since the absorption of heat, and heating rates (T˙ ) are not constant at any position in a workpiece during rapid heating. The critical austenitic transformation temperatures can be determined by experiments or Orlich’s curves [1]. For example, based on Orlich’s curves for SAE1070 steel, the following expression can be obtained: When T˙ < 150 ◦ C/s Ac1 = 1.2983 log3 (T˙ ) + 3.922 log2 (T˙ ) + 12.262 log(T˙ ) + 734.8 When T˙ ≥ 150 ◦ C/s Ac1 = 793 ◦ C
(1)
Not only is Ac1 used to determine the start temperature for the transformation of pearlite to austenite, Ac1 is also used for calculating the driving force for this transformation. The heating rate must be estimated based on the temperature history at the workpiece. In the current model, the heating rate at each position is calculated with an average over the range of 720–800 ◦ C. This averaged value is used to determine both Ac1 (from Eq. (1)) and the nucleation and growth rates for austenitization. 2.2. Austenitic nucleation and growth A continuous nucleation law is used in the pearlite region. The definition of a neighborhood in the CA method and the cell lattice are illustrated in Fig. 1. The interface cells are set up to represent the diffusion transformation between austenite and ferrite. However, the interface between austenite and pearlite is ignored in the
P = exp
−Gt RT
(2)
(3)
where Gt is the energy barrier, R is the universal gas constant, and T is the absolute temperature. The energy barrier, G*, is calculated considering the latent heat of the transformation from pearlite to austenite and the variation of the interfacial energy (G) from the initial configuration to the final configuration. A similar method to evaluate Gt was used in solidification modeling by Zhu and Smith [16]. The effect of carbon diffusion on the grain growth of austenite in pearlite is ignored due to the short diffusion distance in pearlite. If the change in total energy is reduced, the new configuration may
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Fig. 2. Variation of free energy for a cell transition from state 1 to state 2.
Fig. 3. Temperature histories at various locations in radial direction of a SAE1070 steel bar. r22 mm stands for 22 mm from center of bar.
be accepted from a thermodynamic point of view. The probability of accepting a new configuration depends on how P approaches unity. Once Gt is less than or equal to zero, set P equal to one; otherwise, the transformation from pearlite to austenite depends upon the transition probability calculated from Eq. (3). For the phase transformation of proeutectoid ferrite to austenite, the growth velocity of austenite in the interface cells can be calculated by Hillert [17]:
v = MF
(4)
where M is the effective mobility of the interface and F is the chemical driving force for interface motion. The effective mobility for interface migration is affected by the morphology of the local interface, chemical composition, temperature, coherency of the two phases, pinning effects, and the presence of stress or solute drag and can be described as in Loginova et al. [11]:
Q
M = M0 exp −
(5)
RT
where M0 is the pre-experimental factor (0.5 m mol J−1 s−1 in the present model, no pinning), Q is the activation energy for boundary migration (147 kJ mol−1 ), R is the universal gas constant (8.314 J/mol K), and T is the absolute temperature (K). When the mechanical driving force is neglected, F can be obtained from the work of Svoboda et al. [2]:
F = (1 − X ˛ )(˛ Fe − Fe )
(6)
where X˛ is the mole fraction of carbon atoms in ferrite at the inter and Fe are the chemical potentials of iron atoms face, and ˛ Fe in ferrite and austenite, respectively. Since X˛ approaches zero in ferrite, Eq. (6) can be simplified into:
F = ˛ Fe − Fe
(7)
The chemical potential of iron in austenite and ferrite are calculated based on Ref. [18] when using the regular sublattice model. As indicated by several researchers [10,12,15], this growth model leads to a non-equilibrium condition at the interface, or massive austenite growth without partition of any alloying element. Interstitial carbon has a much higher mobility than other alloying elements such as manganese, chromium and molybdenum. Accordingly, paraequilibrium is assumed for this type of constrained local equilibrium with reference to carbon. The competitive growth of grains of the same phase results in grain coarsening; carbon diffusion takes place during the entire process. The carbon diffusion in
the domain can be described as follows: ∂CE = ∇ (DE ∇ (C )) ∂t
(8)
where the equivalent concentration (CE ) and diffusion coefficient (DE ) in the interface cell are defined as CE = C f + C˛ (1 − f ) = C [f + k(1 − f )] DE = D f + kD˛ (1 − f )
(9) (10)
where D˛ and D are the carbon diffusion coefficients in ferrite and austenite, C˛ and C are the carbon concentrations in ␣ and ␥ phases, and k is a partition coefficient. A similar definition was given by Dong and Lee [19]. 3. Simulated results and validation The steel bars were rapidly heated to 1036 ◦ C on the surface with a stationary induction coil, and then the power was turned off. The samples were held for 0.5 s in the air, and then sprayed with water to cool to room temperature. The developed model is clearly sensitive to temperature variations, both spatially and temporally. To obtain the temperature histories at each location on the cross-section of samples, the surface temperature history was first recorded with a referred digital camcorder, and checked with surface reading crayons that melt at a specific temperature, and then the temperature histories for other locations below the surface of the round bar sample were generated by using one-dimensional induction simulation tool based on the recorded time–temperature data. Fig. 3 shows the temperature histories generated by the induction simulation tool in a SAE1070 steel bar under the above conditions. The generated temperature curves were used as given conditions to model the microstructure evolution for the same induction heating. Fig. 4 illustrates the simulated final prior austenite grain size profile from the surface to 4 mm below the surface for the temperature histories shown in Fig. 3. The austenite grain size in the material was determined using the method in ASTM E112 [20]. Different colors indicate different crystallographic orientations of austenite grains. Since the bar is subjected to various heating rates at different locations, each location has a different history for austenitization. Fewer nuclei appeared at the low heating rate region, as described in Eq. (2); for the case in Fig. 4, a low heating rate exists at locations of 3 mm or more below the surface. Thus only a few austenite nuclei occur during heating, and austenite growth dominates the final grain size in the region with a low heating rate.
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Fig. 4. Final profile of prior austenite grain size from surface in the material.
Experiments [21] have shown that the prior austenite grain size for pearlitic samples depends on the heating rate. The amount of austenite formed depends significantly on the number of nuclei and growth rate before the temperature starts to decrease. As anticipated, the finest grain size occurs on subsurface rather than at the surface in Fig. 4. No austenite forms in the regions where the local peak temperature is below a local Ac1 . The simulated microstructure and corresponding carbon concentration at two successive times are shown in Fig. 5. Because of the rapid phase transformation of pearlite to austenite, the pearlite region becomes austenite in less than 1 s once Ac1 is reached. The small amount of proeutectoid ferrite in SAE1070 steel is completely transformed when the surface of the sample reaches 880 ◦ C and grain coarsening occurs before quenching. The carbon content in the austenitized area is quite inhomogeneous, espe-
cially in the region where the proeutectoid ferrite existed before heating (see Fig. 5(d)). However, the carbon concentration in the austenite gradually becomes homogeneous due to carbon diffusion. If the carbon concentration is not uniform before quenching, the microstructure will not be uniform after quenching. The simulation shows that austenite grain size depends on both local peak temperature and austenite boundary migration. Thus, any pinning effects on the boundary will impact the austenite grain size, and can be considered in Eq. (5). Grain coarsening begins to occur before the microstructure is fully transformed to austenite, and becomes more visible with increasing temperature. Since the model herein takes into account the important role of interfacial energy in microstructural evolution, the grain morphologies shown by the model are reasonable and the dihedral angles between austenite grain boundaries at
Fig. 5. Microstructure evolution and corresponding carbon concentration in the area from the surface to 200 m below the surface. The top two images show austenite grain size at 830 ◦ C (Fig. 5(a)) and 1000 ◦ C (Fig. 5(b)). The lower two images show corresponding carbon concentrations (wt%) at each temperature.
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Fig. 6. Hardness profile along the radial line (HV-1 kg load-15 s).
triple junctions are evolving to the thermodynamic equilibrium value of 120◦ . Furthermore, the observation of simulation process and simulated results suggest that the rate at which proeutectoid ferrite transforms to austenite depends more on the volume of individual ferrite regions than on the total volume fraction of proeutectoid ferrite in the starting microstructure. Since normalized hypoeutectoid steels generally have smaller regions of proeutectoid ferrite and more pearlite colonies or smaller laminar spacing in pearlite than annealed steels of similar composition, this suggests that normalized steels can be austenitized more rapidly than annealed steels. The current model is able to capture the microstructure evolution and carbon diffusion during induction heating, thus it can be used to design an expected induction heating process. Induction hardening experiments have been carried out for SAE1070 steel to validate the developed model and simulated results. The as-received 50 mm round bars of SAE1070 steel were annealed before starting the induction heating. The 152 mm long bars were rapidly heated to 1036 ◦ C under the conditions of 10 kHz and 42 kW and the power was turned off. The sample was held for 0.5 s in air, then sprayed with water to quench to room temperature. The samples were rotated at 800 rpm during cooling. The hardness profile and a typical light micrograph of the prior austenite grains are shown in Figs. 6 and 7. The prior austenite grain profile was measured along the radial direction on the crosssection of the sample, as shown in Fig. 8. The hardness profile reveals the transition zone from a fully martensitic microstructure (600–900 HV) to a pearlitic one (200 HV). Material at depth of 2 mm
Fig. 7. Prior austenite grains of a SAE1070 steel sample.
Fig. 8. Austenite grain profiles from simulation and experiment in SAE1070 steel bars.
and greater from the surface was subjected to a low cooling rate, then the prior austenite transformed to pearlite during quench. One can observe in Fig. 8 that the variation of grain size occurs below the surface of the sample, and the case depth or hardened layer is about 2.3 mm thick under the given induction cycle (Fig. 3). The prior austenite grains varied from the surface to the position at which the temperature reached Ac1 (3.7 mm below the surface in the present experiments), as shown in Fig. 8. The maximum grain size in the case depth was located 1.0 mm below the surface of the sample, and the minimum grain size occurred at two locations: ∼0.3 mm and about 2.3 mm below the surface. Note that the temperatures at subsurface are higher than on the surface due to the 0.5 s delay before the water quenching, as shown in Fig. 3. The result is more austenite grain growth subsurface than on the surface. The minimum grain size occurs at about 2.3 mm below the surface because the grain coarsening is not dominating the growth behavior under 1000 ◦ C. As a result, many initially nucleated grains still exist. Since the highest temperature (1036 ◦ C) occurred at the surface in this case, the grain coarsening took place. Thus, a relatively small grain size region appeared around 0.3 mm below surface. Large prior austenite grains occurred again at 3–4 mm below the surface, as shown in Fig. 8, because full pearlite was formed during quenching, which was confirmed with the experiments. The simulated grain size distribution was also imposed in Fig. 8 to validate
Fig. 9. Temperature histories at various locations in radial direction of a SAE1045 steel bar. r25 mm stands for 25 mm from center of bar.
B.J. Yang et al. / Materials Science and Engineering A 527 (2010) 2978–2984
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stant heating rate. Fig. 11 demonstrates the time needed to obtain 99% austenite under various constant heating rates for SAE1045 and SAE1070 steels. Although a very short time is needed for austenite transformation, a relatively longer time is usually needed to have a uniform chemical composition in the austenitic grains. One can observe from Fig. 5 that the carbon concentration at the former ferrite areas is very different from that at the former pearlite region even though the region is fully austenitized. The diffusion time of elements is very critical to optimize the induction hardening. 5. Summary
Fig. 10. Austenite grain profiles of simulation and experiments for a SAE1045 steel bar.
the model works. As seen in Fig. 8, the model is able to capture the distribution trend of grain sizes observed in experiments. Also, the predicted grain sizes are in good agreement with the measured data. 4. Application The current model was applied to predict the grain size profile of SAE1045 steel bars from the surface to core under certain conditions. The critical austenitic transformation temperatures for this steel are similarly determined by experiments or Orlich’s curves [1]. Based on Orlich’s curves for SAE1045 steel, the Ac1 can be expressed: When T˙ < 150 ◦ C/s Ac1 = 4.8421 log3 (T˙ ) + 3.3313 log2 (T˙ ) + 8.234 log(T˙ ) + 735.91
(11a)
When T˙ ≥ 150 ◦ C/s Ac1 = 790 ◦ C
A numerical model for microstructure evolution during the induction heating for austenitization of hypoeutectoid Fe–C steel has been developed using a cellular automaton approach (CA). The shift of start temperature (Ac1 ) of austenite formation due to rapid heating is taken into consideration and the final microstructure can be predicted, including austenite grain size and distribution, fraction of austenite, and chemical homogeneity. The present work demonstrates that the cellular automaton (CA) method is very effective for simulating the austenitization of hypoeutectic steels during rapid heating. The kinetics of austenitization during rapid induction heating are simulated by simultaneously considering continuous nucleation, grain growth and grain coarsening as a function of heating rate. The numerical experiments demonstrate that the grain size distribution depends strongly on the local temperature history. Experimental validation of prior austenite grain size and distribution along the radial direction confirms that the current model works well for a workpiece or domain with various heating rates. The simulated grain size profile and the carbon concentration distribution agree with our experiments and knowledge, respectively. The developed model has been applied to steel induction hardening and is capable of providing information about austenitic grain size and composition homogeneity prior to quenching, making it possible to design and optimize the induction heating processes; also, the model can be used as a powerful tool to help understand microstructure evolution during austenitization.
(11b)
Two cylindrical samples of SAE1045 steel were heated up for 8 s, held for 0.5 s in air, and then quenched to room temperature. The peak temperature at the surface reached 1070 ◦ C. The corresponding temperature histories at various locations are shown in Fig. 9. Under the given conditions, the validated model was applied to predict the grain sizes. Test samples were sectioned and the grain sizes were measured. Fig. 10 shows the grain size profiles obtained from the simulation and experiment, indicating that the model can predict the grain sizes and distribution in the hardened layer and case. The model was also applied to predict the minimum time needed from Ac1 to 99% austenitic phase formation under a con-
Fig. 11. The time needed to obtain 99% austenite under various heating rates.
Acknowledgements The authors would like to acknowledge support for this work from Caterpillar, Inc., and the Dept. of Energy-Office of Heavy Vehicle and Transportation Applications, WR10648. References [1] J. Orlich, H.J. Pietrzeniuk, Atlas zur Warmebehandlung der Stahle, 4, Part 2, Verlag Stahleisen M.B.H., Dusseldorf, Germany, 1976. [2] J. Svoboda, F.D. Fischer, P. Fratzl, E. Gamsjager, N.K. Simha, Acta Mater. 49 (2001) 1249. [3] F.G. Caballero, C. Capdevila, C. Garci de Andres, ISIJ International 41 (2001) 1093–1102. [4] K. Clarke. The Effect of Heating rate and Microstructural Scale on Austenite Formation, Austenite Homogenization, and As-quenched Microstructure in Three Induction Hardenable Steels, PhD Thesis, Colorado School of Mines, 2008. [5] A. Jacot, M. Rappaz, Acta Mater. 47 (1999) 1645–1651. [6] M. Rappaz, C. Gandin, A. Jacot, C. Charbon, in: M. Cross, J. Campell (Eds.), Modeling of Casting, Welding and Advanced Solidification Processes VII, TMS, Warrendale, PA, 1995, p. 501. [7] V. Marx, F.R. Reher, G. Gottstein, Acta Mater. 47 (1999) 219–1230. [8] R.L. Goetz, V. Seetharaman, Metall. Mater. Trans. A 29A (1998) 2307–2321. [9] G.P. Krielaart, J. Sietsma, S. van der Zwaang, Mater. Sci. Eng. A237 (1997) 216–223. [10] M. Kumar, S. Sasikumar, P.K. Nair, Acta Mater. 17 (1998) 6291–6303. [11] I. Loginova, J. Odqvist, G. Amberg, J. Agren, Acta Mater. 51 (2003) 1327. [12] Y.J. Lan, D.Z. Li, Y.Y. Li, Acta Mater. 52 (2004) 1721–1729. [13] B.J. Yang, L. Chuzhoy, M.L. Johnson, Comput. Mater. Sci. 41 (2007) 186–194. [14] D.H. Sherman, B.J. Yang, A.V. Catalina, Mater. Sci. Forum 539–543 (2007) 4795–4800. [15] J. Rodel, H.J. Spies, Surf. Eng. 12 (1996) 313–318.
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P. Zhu, R.W. Smith, Acta Metall. Mater. 40 (1992) 683–692. M. Hillert, Acta Mater. 47 (1999) 4481. M. Hillert, L.I. Staffansson, Acta Chem. Scand. 24 (1970) 3618–3626. H.B. Dong, P.D. Lee, Acta Mater. 53 (2005) 659–668.
[20] ASTM E 112-96, Standard Test Method for Determining Average Grain Size, 1996. [21] K.C. Smith. A Microstructural Analysis of Rapid Austenite Formation in Induction Hardened Steels, M.S. Thesis, Colorado School of Mines, 2001.
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Simulation of steel microstructure evolution during induction heating B.J. Yang ∗ , A. Hattiangadi, W.Z. Li, G.F. Zhou, T.E. McGreevy Virtual Product Development Technology, Technical Center, Caterpillar Inc., USA
a r t i c l e
i n f o
Article history: Received 29 October 2009 Received in revised form 11 January 2010 Accepted 12 January 2010
Keywords: Induction heating Microstructure evolution Cellular automaton
a b s t r a c t An effective model and approach have been developed and validated to simulate the microstructure evolution and composition variation during induction heating. The basic method is described, and the validation and application are addressed in the paper. In this model, a real initial microstructure is used as an input to simulate austenitization, and the intrinsic chemical difference is utilized to describe the ferrite and pearlite phases. A set of temperature curves is applied to the corresponding locations in an induction heating part and the dynamic variation in microstructure, grain size and carbon content is revealed through modeling. The shift of the start temperature of austenite formation (Ac1 ) is taken into account and the correlation between Ac1 and the heating rate is presented. This model is being used in industrial practice to optimize existing processes and support new process development. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Induction heating is one of rapid heating processes, which is widely used for surface hardening to high-volume continuous heat treatment operations. Modeling is an efficient way to aid engineers in developing optimal induction hardening operations and to develop insights about thermal processes and physical transformations. The first essential issue for induction hardening is rapid heating, which could shift the critical temperatures of phase transformation, such as Ac1 and Ac3 , and consequently impact the microstructure prior to quenching. The microstructure state prior to quenching significantly affects the final microstructure and mechanical properties of surface-hardened workpieces; development of an integrated tool to simulate microstructure evolution during the induction heating for austenitization is important to achieve the desired final microstructure and properties. Heating is different from cooling because during heating both solute diffusion and the driving force for austenite formation increase with increasing temperature. On cooling, diffusion decreases whereas the driving force for transformation to martensite or pearlite/bainite increases. The start temperature for austenite formation, Ac1 , is a function of the heating rate, initial microstructure and composition, and usually shifts to higher temperature with increasing heating rates [1]. When the temperature reaches Ac1 , austenite nucleation and grain growth will occur simultaneously. Due to temperature gradients in a workpiece dur-
∗ Corresponding author at: Virtual Product Development Technology, Technical Center, Caterpillar Inc., Old Galene, Peoria, IL 61656, USA. Tel.: +1 309 578 0047; fax: +1 309 578 3322. E-mail address: yang bing [email protected] (B.J. Yang). 0921-5093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2010.01.038
ing heating, austenitization occurs at different rates at different locations, so the microstructure and the grain size distribution will be non-uniform from the surface to the core. This means an accurate evaluation of Ac1 is very important for rapid heating processes. A typical equilibrium condition for hypoeutectoid carbon steel consists of two distinctive microstructures—pearlite and proeutectoid ferrite. The mechanism and the kinetics of austenitization have been widely discussed in the literature [2–4]. The transformation occurs in two ways—pearlite transforms to austenite and proeutectoid ferrite transforms into austenite. Pearlite transforms to austenite in a relatively short time due to the short distance for carbon diffusion in pearlite colonies. Jacot and Rappaz demonstrated this using a combined model [5]. If pearlite transformation is ignored, the description of microstructure evolution during heating is incomplete. The rate of austenitization depends on the heating rate, the start temperature of austenite formation (Ac1 ), composition and initial microstructure, austenite nucleation rate, and grain growth. Although the direct observation of austenitization is difficult, modeling and simulation of microstructural evolution is possible and realistically achievable for industrial use. Computer modeling using cellular automata (CA) is an applicable and powerful way to simulate microstructural evolution during both austenitization and cooling processes. Cellular automata can often express local conditions and structural detail more efficiently than typical continuum models. In recent years, CA methods have been successfully applied to simulate microstructural evolution of dendritic structures in solidification [6], recrystallization [7,8], austenite to ferrite transformations [9–12], and austenitization for furnace heating [13]. However, there is limited research on the application of CA methods to predict microstructure evolution for rapid heating. The detailed description about CA method can be found in [12,13].
B.J. Yang et al. / Materials Science and Engineering A 527 (2010) 2978–2984
This paper will present a CA model for the austenitization of hypoeutectic steels during rapid induction heating. In the current model, the phase transformations of pearlite and proeutectoid ferrite to austenite are modeled simultaneously to obtain realistic information about austenitization, and is more accurate in predicting microstructure and carbon distribution before quenching than a model without consideration of the transformation of pearlite to austenite. To capture the dynamic variation in microstructure, grain size and carbon content through the modeling, the shift of start temperature (Ac1 ) of austenite formation is taken into account. In order to simulate a real induction hardening process, various time–temperature curves are applied to the corresponding locations (surface and various depths below the surface) in an induction heating part to simulate the non-uniform rates present. The approach of obtaining these time–temperature curves will be described in Section 3. The basic method will be represented, and the model validation and application are addressed thereafter. This model is being used in industrial practice to optimize the process and support a new process development. The model is also being incorporated into a more comprehensive simulation tool developed to predict microstructural evolution during quenching [14].
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Fig. 1. Definition of a neighborhood (a) and schematic illustration of the square lattice in the CA model (b). Symbols are as follows: p—pearlite, ␥—austenite, ␣/␥—interface between ferrite and austenite.
model due to fast transformation from pearlite to austenite. The nucleation probability is assumed to be the same for every cell in the pearlite region. The probability of nucleation at the interface between pearlite and ferrite is assumed to be seven times greater than in the internal pearlite region due to a difference in free energy. The nucleation rate for each cell, dN/dt, is proposed as
2. Model description
dN = ˛T˙ × Nd × fP × A dt
2.1. Start temperature of the austenitization
where N is the number of accumulated nuclei in an individual cell, T˙ is the heating rate, and ˛ is a constant that depends on the colony size of pearlite and the interlamellar spacing in pearlite colonies. For example, ˛ is taken as 0.45 for normalized steel, and 0.4 for annealed steel. Nd is the nucleation density (1 × 109 to 5 × 109 /m2 from coarse pearlite to fine pearlite), fP is the fraction of pearlite in the computational domain or subdomain in question and A is the area of the computational domain or subdomain. In any time interval, the number of nuclei is calculated in A and the nuclei are randomly assigned to the cells in question. Using the Monte Carlo method, a random number C is generated and compared with N. Once C < N, a random cell in the domain or subdomain is selected for nucleation from among the pearlite cells or pearlite/ferrite interface cells. Each new austenite grain is randomly assigned a crystallographic orientation as soon as it is formed. The crystallographic orientation for each newly generated austenite nucleus is assumed to be randomly selected from an integer in the range of 1–125. This process is continued for N times in the domain or subdomain to place the N nuclei into the cells. Two kinds of growth transformations are taken into account, namely pearlite to austenite and proeutectoid ferrite to austenite. Because of their different microstructure characteristics, two different growth models were developed. For the transformation of pearlite to austenite, the growth of any austenite grain is driven by a decrease of the local free energy and interfacial energy in the pearlite region. The energy barrier is considered to account for the variation of free energy of a cell transition from state 1 to state 2, as shown in Fig. 2. The probability of a cell transition from state 1 to state 2 is
For numerical simulation of induction heating, the accuracy of critical austenite transformation temperatures is crucial. The start temperature for the austenite transformation, Ac1 , depends strongly on three essential parameters—heating rate, initial microstructure and chemical composition. Ferrite-austenite transformations at high heating rates are characterized by shifted transformation temperatures, as was shown by Orlich’s curves [1] and expressed by an equation based upon a constant rate [15]. Unfortunately, this equation is only valid for constant heating rates. In fact, the variation of the temperature with time is nonlinear with induction heating, especially in a region of phase transformations since the absorption of heat, and heating rates (T˙ ) are not constant at any position in a workpiece during rapid heating. The critical austenitic transformation temperatures can be determined by experiments or Orlich’s curves [1]. For example, based on Orlich’s curves for SAE1070 steel, the following expression can be obtained: When T˙ < 150 ◦ C/s Ac1 = 1.2983 log3 (T˙ ) + 3.922 log2 (T˙ ) + 12.262 log(T˙ ) + 734.8 When T˙ ≥ 150 ◦ C/s Ac1 = 793 ◦ C
(1)
Not only is Ac1 used to determine the start temperature for the transformation of pearlite to austenite, Ac1 is also used for calculating the driving force for this transformation. The heating rate must be estimated based on the temperature history at the workpiece. In the current model, the heating rate at each position is calculated with an average over the range of 720–800 ◦ C. This averaged value is used to determine both Ac1 (from Eq. (1)) and the nucleation and growth rates for austenitization. 2.2. Austenitic nucleation and growth A continuous nucleation law is used in the pearlite region. The definition of a neighborhood in the CA method and the cell lattice are illustrated in Fig. 1. The interface cells are set up to represent the diffusion transformation between austenite and ferrite. However, the interface between austenite and pearlite is ignored in the
P = exp
−Gt RT
(2)
(3)
where Gt is the energy barrier, R is the universal gas constant, and T is the absolute temperature. The energy barrier, G*, is calculated considering the latent heat of the transformation from pearlite to austenite and the variation of the interfacial energy (G) from the initial configuration to the final configuration. A similar method to evaluate Gt was used in solidification modeling by Zhu and Smith [16]. The effect of carbon diffusion on the grain growth of austenite in pearlite is ignored due to the short diffusion distance in pearlite. If the change in total energy is reduced, the new configuration may
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Fig. 2. Variation of free energy for a cell transition from state 1 to state 2.
Fig. 3. Temperature histories at various locations in radial direction of a SAE1070 steel bar. r22 mm stands for 22 mm from center of bar.
be accepted from a thermodynamic point of view. The probability of accepting a new configuration depends on how P approaches unity. Once Gt is less than or equal to zero, set P equal to one; otherwise, the transformation from pearlite to austenite depends upon the transition probability calculated from Eq. (3). For the phase transformation of proeutectoid ferrite to austenite, the growth velocity of austenite in the interface cells can be calculated by Hillert [17]:
v = MF
(4)
where M is the effective mobility of the interface and F is the chemical driving force for interface motion. The effective mobility for interface migration is affected by the morphology of the local interface, chemical composition, temperature, coherency of the two phases, pinning effects, and the presence of stress or solute drag and can be described as in Loginova et al. [11]:
Q
M = M0 exp −
(5)
RT
where M0 is the pre-experimental factor (0.5 m mol J−1 s−1 in the present model, no pinning), Q is the activation energy for boundary migration (147 kJ mol−1 ), R is the universal gas constant (8.314 J/mol K), and T is the absolute temperature (K). When the mechanical driving force is neglected, F can be obtained from the work of Svoboda et al. [2]:
F = (1 − X ˛ )(˛ Fe − Fe )
(6)
where X˛ is the mole fraction of carbon atoms in ferrite at the inter and Fe are the chemical potentials of iron atoms face, and ˛ Fe in ferrite and austenite, respectively. Since X˛ approaches zero in ferrite, Eq. (6) can be simplified into:
F = ˛ Fe − Fe
(7)
The chemical potential of iron in austenite and ferrite are calculated based on Ref. [18] when using the regular sublattice model. As indicated by several researchers [10,12,15], this growth model leads to a non-equilibrium condition at the interface, or massive austenite growth without partition of any alloying element. Interstitial carbon has a much higher mobility than other alloying elements such as manganese, chromium and molybdenum. Accordingly, paraequilibrium is assumed for this type of constrained local equilibrium with reference to carbon. The competitive growth of grains of the same phase results in grain coarsening; carbon diffusion takes place during the entire process. The carbon diffusion in
the domain can be described as follows: ∂CE = ∇ (DE ∇ (C )) ∂t
(8)
where the equivalent concentration (CE ) and diffusion coefficient (DE ) in the interface cell are defined as CE = C f + C˛ (1 − f ) = C [f + k(1 − f )] DE = D f + kD˛ (1 − f )
(9) (10)
where D˛ and D are the carbon diffusion coefficients in ferrite and austenite, C˛ and C are the carbon concentrations in ␣ and ␥ phases, and k is a partition coefficient. A similar definition was given by Dong and Lee [19]. 3. Simulated results and validation The steel bars were rapidly heated to 1036 ◦ C on the surface with a stationary induction coil, and then the power was turned off. The samples were held for 0.5 s in the air, and then sprayed with water to cool to room temperature. The developed model is clearly sensitive to temperature variations, both spatially and temporally. To obtain the temperature histories at each location on the cross-section of samples, the surface temperature history was first recorded with a referred digital camcorder, and checked with surface reading crayons that melt at a specific temperature, and then the temperature histories for other locations below the surface of the round bar sample were generated by using one-dimensional induction simulation tool based on the recorded time–temperature data. Fig. 3 shows the temperature histories generated by the induction simulation tool in a SAE1070 steel bar under the above conditions. The generated temperature curves were used as given conditions to model the microstructure evolution for the same induction heating. Fig. 4 illustrates the simulated final prior austenite grain size profile from the surface to 4 mm below the surface for the temperature histories shown in Fig. 3. The austenite grain size in the material was determined using the method in ASTM E112 [20]. Different colors indicate different crystallographic orientations of austenite grains. Since the bar is subjected to various heating rates at different locations, each location has a different history for austenitization. Fewer nuclei appeared at the low heating rate region, as described in Eq. (2); for the case in Fig. 4, a low heating rate exists at locations of 3 mm or more below the surface. Thus only a few austenite nuclei occur during heating, and austenite growth dominates the final grain size in the region with a low heating rate.
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Fig. 4. Final profile of prior austenite grain size from surface in the material.
Experiments [21] have shown that the prior austenite grain size for pearlitic samples depends on the heating rate. The amount of austenite formed depends significantly on the number of nuclei and growth rate before the temperature starts to decrease. As anticipated, the finest grain size occurs on subsurface rather than at the surface in Fig. 4. No austenite forms in the regions where the local peak temperature is below a local Ac1 . The simulated microstructure and corresponding carbon concentration at two successive times are shown in Fig. 5. Because of the rapid phase transformation of pearlite to austenite, the pearlite region becomes austenite in less than 1 s once Ac1 is reached. The small amount of proeutectoid ferrite in SAE1070 steel is completely transformed when the surface of the sample reaches 880 ◦ C and grain coarsening occurs before quenching. The carbon content in the austenitized area is quite inhomogeneous, espe-
cially in the region where the proeutectoid ferrite existed before heating (see Fig. 5(d)). However, the carbon concentration in the austenite gradually becomes homogeneous due to carbon diffusion. If the carbon concentration is not uniform before quenching, the microstructure will not be uniform after quenching. The simulation shows that austenite grain size depends on both local peak temperature and austenite boundary migration. Thus, any pinning effects on the boundary will impact the austenite grain size, and can be considered in Eq. (5). Grain coarsening begins to occur before the microstructure is fully transformed to austenite, and becomes more visible with increasing temperature. Since the model herein takes into account the important role of interfacial energy in microstructural evolution, the grain morphologies shown by the model are reasonable and the dihedral angles between austenite grain boundaries at
Fig. 5. Microstructure evolution and corresponding carbon concentration in the area from the surface to 200 m below the surface. The top two images show austenite grain size at 830 ◦ C (Fig. 5(a)) and 1000 ◦ C (Fig. 5(b)). The lower two images show corresponding carbon concentrations (wt%) at each temperature.
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Fig. 6. Hardness profile along the radial line (HV-1 kg load-15 s).
triple junctions are evolving to the thermodynamic equilibrium value of 120◦ . Furthermore, the observation of simulation process and simulated results suggest that the rate at which proeutectoid ferrite transforms to austenite depends more on the volume of individual ferrite regions than on the total volume fraction of proeutectoid ferrite in the starting microstructure. Since normalized hypoeutectoid steels generally have smaller regions of proeutectoid ferrite and more pearlite colonies or smaller laminar spacing in pearlite than annealed steels of similar composition, this suggests that normalized steels can be austenitized more rapidly than annealed steels. The current model is able to capture the microstructure evolution and carbon diffusion during induction heating, thus it can be used to design an expected induction heating process. Induction hardening experiments have been carried out for SAE1070 steel to validate the developed model and simulated results. The as-received 50 mm round bars of SAE1070 steel were annealed before starting the induction heating. The 152 mm long bars were rapidly heated to 1036 ◦ C under the conditions of 10 kHz and 42 kW and the power was turned off. The sample was held for 0.5 s in air, then sprayed with water to quench to room temperature. The samples were rotated at 800 rpm during cooling. The hardness profile and a typical light micrograph of the prior austenite grains are shown in Figs. 6 and 7. The prior austenite grain profile was measured along the radial direction on the crosssection of the sample, as shown in Fig. 8. The hardness profile reveals the transition zone from a fully martensitic microstructure (600–900 HV) to a pearlitic one (200 HV). Material at depth of 2 mm
Fig. 7. Prior austenite grains of a SAE1070 steel sample.
Fig. 8. Austenite grain profiles from simulation and experiment in SAE1070 steel bars.
and greater from the surface was subjected to a low cooling rate, then the prior austenite transformed to pearlite during quench. One can observe in Fig. 8 that the variation of grain size occurs below the surface of the sample, and the case depth or hardened layer is about 2.3 mm thick under the given induction cycle (Fig. 3). The prior austenite grains varied from the surface to the position at which the temperature reached Ac1 (3.7 mm below the surface in the present experiments), as shown in Fig. 8. The maximum grain size in the case depth was located 1.0 mm below the surface of the sample, and the minimum grain size occurred at two locations: ∼0.3 mm and about 2.3 mm below the surface. Note that the temperatures at subsurface are higher than on the surface due to the 0.5 s delay before the water quenching, as shown in Fig. 3. The result is more austenite grain growth subsurface than on the surface. The minimum grain size occurs at about 2.3 mm below the surface because the grain coarsening is not dominating the growth behavior under 1000 ◦ C. As a result, many initially nucleated grains still exist. Since the highest temperature (1036 ◦ C) occurred at the surface in this case, the grain coarsening took place. Thus, a relatively small grain size region appeared around 0.3 mm below surface. Large prior austenite grains occurred again at 3–4 mm below the surface, as shown in Fig. 8, because full pearlite was formed during quenching, which was confirmed with the experiments. The simulated grain size distribution was also imposed in Fig. 8 to validate
Fig. 9. Temperature histories at various locations in radial direction of a SAE1045 steel bar. r25 mm stands for 25 mm from center of bar.
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stant heating rate. Fig. 11 demonstrates the time needed to obtain 99% austenite under various constant heating rates for SAE1045 and SAE1070 steels. Although a very short time is needed for austenite transformation, a relatively longer time is usually needed to have a uniform chemical composition in the austenitic grains. One can observe from Fig. 5 that the carbon concentration at the former ferrite areas is very different from that at the former pearlite region even though the region is fully austenitized. The diffusion time of elements is very critical to optimize the induction hardening. 5. Summary
Fig. 10. Austenite grain profiles of simulation and experiments for a SAE1045 steel bar.
the model works. As seen in Fig. 8, the model is able to capture the distribution trend of grain sizes observed in experiments. Also, the predicted grain sizes are in good agreement with the measured data. 4. Application The current model was applied to predict the grain size profile of SAE1045 steel bars from the surface to core under certain conditions. The critical austenitic transformation temperatures for this steel are similarly determined by experiments or Orlich’s curves [1]. Based on Orlich’s curves for SAE1045 steel, the Ac1 can be expressed: When T˙ < 150 ◦ C/s Ac1 = 4.8421 log3 (T˙ ) + 3.3313 log2 (T˙ ) + 8.234 log(T˙ ) + 735.91
(11a)
When T˙ ≥ 150 ◦ C/s Ac1 = 790 ◦ C
A numerical model for microstructure evolution during the induction heating for austenitization of hypoeutectoid Fe–C steel has been developed using a cellular automaton approach (CA). The shift of start temperature (Ac1 ) of austenite formation due to rapid heating is taken into consideration and the final microstructure can be predicted, including austenite grain size and distribution, fraction of austenite, and chemical homogeneity. The present work demonstrates that the cellular automaton (CA) method is very effective for simulating the austenitization of hypoeutectic steels during rapid heating. The kinetics of austenitization during rapid induction heating are simulated by simultaneously considering continuous nucleation, grain growth and grain coarsening as a function of heating rate. The numerical experiments demonstrate that the grain size distribution depends strongly on the local temperature history. Experimental validation of prior austenite grain size and distribution along the radial direction confirms that the current model works well for a workpiece or domain with various heating rates. The simulated grain size profile and the carbon concentration distribution agree with our experiments and knowledge, respectively. The developed model has been applied to steel induction hardening and is capable of providing information about austenitic grain size and composition homogeneity prior to quenching, making it possible to design and optimize the induction heating processes; also, the model can be used as a powerful tool to help understand microstructure evolution during austenitization.
(11b)
Two cylindrical samples of SAE1045 steel were heated up for 8 s, held for 0.5 s in air, and then quenched to room temperature. The peak temperature at the surface reached 1070 ◦ C. The corresponding temperature histories at various locations are shown in Fig. 9. Under the given conditions, the validated model was applied to predict the grain sizes. Test samples were sectioned and the grain sizes were measured. Fig. 10 shows the grain size profiles obtained from the simulation and experiment, indicating that the model can predict the grain sizes and distribution in the hardened layer and case. The model was also applied to predict the minimum time needed from Ac1 to 99% austenitic phase formation under a con-
Fig. 11. The time needed to obtain 99% austenite under various heating rates.
Acknowledgements The authors would like to acknowledge support for this work from Caterpillar, Inc., and the Dept. of Energy-Office of Heavy Vehicle and Transportation Applications, WR10648. References [1] J. Orlich, H.J. Pietrzeniuk, Atlas zur Warmebehandlung der Stahle, 4, Part 2, Verlag Stahleisen M.B.H., Dusseldorf, Germany, 1976. [2] J. Svoboda, F.D. Fischer, P. Fratzl, E. Gamsjager, N.K. Simha, Acta Mater. 49 (2001) 1249. [3] F.G. Caballero, C. Capdevila, C. Garci de Andres, ISIJ International 41 (2001) 1093–1102. [4] K. Clarke. The Effect of Heating rate and Microstructural Scale on Austenite Formation, Austenite Homogenization, and As-quenched Microstructure in Three Induction Hardenable Steels, PhD Thesis, Colorado School of Mines, 2008. [5] A. Jacot, M. Rappaz, Acta Mater. 47 (1999) 1645–1651. [6] M. Rappaz, C. Gandin, A. Jacot, C. Charbon, in: M. Cross, J. Campell (Eds.), Modeling of Casting, Welding and Advanced Solidification Processes VII, TMS, Warrendale, PA, 1995, p. 501. [7] V. Marx, F.R. Reher, G. Gottstein, Acta Mater. 47 (1999) 219–1230. [8] R.L. Goetz, V. Seetharaman, Metall. Mater. Trans. A 29A (1998) 2307–2321. [9] G.P. Krielaart, J. Sietsma, S. van der Zwaang, Mater. Sci. Eng. A237 (1997) 216–223. [10] M. Kumar, S. Sasikumar, P.K. Nair, Acta Mater. 17 (1998) 6291–6303. [11] I. Loginova, J. Odqvist, G. Amberg, J. Agren, Acta Mater. 51 (2003) 1327. [12] Y.J. Lan, D.Z. Li, Y.Y. Li, Acta Mater. 52 (2004) 1721–1729. [13] B.J. Yang, L. Chuzhoy, M.L. Johnson, Comput. Mater. Sci. 41 (2007) 186–194. [14] D.H. Sherman, B.J. Yang, A.V. Catalina, Mater. Sci. Forum 539–543 (2007) 4795–4800. [15] J. Rodel, H.J. Spies, Surf. Eng. 12 (1996) 313–318.
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[20] ASTM E 112-96, Standard Test Method for Determining Average Grain Size, 1996. [21] K.C. Smith. A Microstructural Analysis of Rapid Austenite Formation in Induction Hardened Steels, M.S. Thesis, Colorado School of Mines, 2001.