Design of the austempering heat treatment of a ductile iron differential case aided by computer simulation

Austempered ductile iron is frequently employed in the fabrication of automobile parts due to its good mechanical properties, and its simplicity and low cost of the manufacturing process. The heat treatment design is an im-portant stage, in which parameters such as temperatures and durations are chosen according to the required microstructure, part dimension, and initial characteristics of ductile iron. A coupled thermo-mechanical-metallurgical model is employed in this work to facilitate the heat treatment design of a ductile iron differential case. The thermal-mechanical model is solved by using a general-purpose software, and the metallurgical model is implemented into it by means of user-defined subroutines. To perform the austempering heat treatment simulation of the part, experimental results were required to complete some basic parameters of the model. In order to select appropriated values of thermal cycle parameters, the numerical model was employed to analyze the influence of austenitizing temperature, and austempering time and temperature in the microstructure, dimensional change of the part, cooling rate, and minimum required time of the heat treatment to obtain a full ausferritic matrix. A good performance of the computational tool was found by comparison of numerical and experimental results.


INTRODUCTION
Austempered Ductile Iron (ADI) is a Fe-C-Si alloy that is frequently used in powertrain and suspension systems of automobiles due to its good mechanical properties, such as 900 1600 MPa  tensile strength, 650 1300 MPa  yield strength, 1 9 %  elongation, 20 100 J  impact energy, and 269 512 HBW  hardness (ASTM A 897M-03, 2003). There are also advantages in the manufacturing process to obtain parts having complex shapes (Gundlach and Janowak, 1983;Sorelmetal, 1998;Putman and Thomson, 2003;Mendez et al., 2015).
In different types of automobile applications, ADI has been employed to replace cast steel (such as in a spring support), forged steel (ring gear, timing gear, pinion, crankshaft), and as-cast ductile iron (crankshaft), because this material allowed weight reduction and cost savings (Sorelmetal, 1998).
The ADI microstructure is formed by graphite nodules embedded into an ausferritic matrix. The ausferritic matrix is composed by sheaves and austenite blocks, and each sheaf is formed by ferrite platelets and austenite films (Sorelmetal, 1998). Parts with this microstructure are commonly obtained by a three-step austempering heat treatment, which is performed in the best case scenario after the casting and machining processes are completed. In the first step of this heat treatment the part is heated and kept at the austenitizing temperature ( T  ).
During the second step, the part is cooled and kept at the austempering temperature ( A T ). Finally, in the third step the part is cooled to the ambient temperature (Fraś et al., 2012). The described thermal cycle is characterized by the temperature and duration of the first ( t  ) and second ( A t ) steps (Sorelmetal, 1998).
The heat treatment temperatures vary in the range of 850 950 C T C      and 250 450 A C T C     , and the chosen values depend on the desired final microstructure (Bosnjak et al., 2001;Yescas, 2003), which is closely related to the required values of mechanical properties. Appropriate values of t  and A t have to be chosen to obtain ADI parts with a full ausferritic matrix, which depend on some characteristics such as the initial microstructure and chemical composition of ductile iron, and the geometry of the parts (Moncada and Sikora, 1996;Sosa et al., 2004;Sosa et al., 2009;Fraś et al., 2012).
The design process of heat treatment, which allows selecting the thermal cycle parameters, can be aided by numerical simulation, in order to reduce the number and complexity of experimental tests. Computational tools have been successfully employed in different problems of the automotive industry (Kokkula et al., 2006;Lee et al., 2016;Kong et al., 2016). However, this is not a simple problem and to perform an appropriate representation of the phenomena involved in the heat treatment, a coupled thermo-mechanical-metallurgical model is required. The thermal model should allow computation of the evolution of temperature in the part taking into account the latent heat at phase change, the mechanical model is required to compute the evolution of deformation due to temperature and phase changes, whereas the metallurgical model should predict the evolution of phases during the heat treatment development.
Only a few works are found in the literature that consider coupled models for the austempering heat treatment simulation. Yoo et al. (1999) proposed a thermo-metallurgical model to simulate the in-situ austempering heat treatment. The metallurgical model allows to predict a ferritic-pearlitic-ausferritic matrix at ambient temperature. The pearlitic and ausferritic transformations are modeled by means of Avrami equation; the model is difficult to employ because the parameters of the equation need to be fitted based on a large number of experiments. In the works of Pérez Villalobos et al. (2011) and Hepp et al. (2012), the modeling of the austempering heat treatment of ADI parts was presented. Even though the models employed in these works predict the evolutions of temperature and phases in a part, they do not consider microstructural features such as the shape and size of phases and the chemical composition of phases. Moreover, information regarding model application, such as part modeling (initial and boundary conditions, domain discretization) and metallurgical model setting (employed data base or fitting technique) is rather scarce. Boccardo et al. (2017b) proposed a thermo-mechanicalmetallurgical model for the three-step austempering heat treatment, considering the reverse eutectoid, ausferritic, and martensitic transformations; the modeling involved the microscale in order to consider the microstructure features of the ductile iron, such as graphite nodule size, shape and dimension of the growing phases, heterogeneous carbon concentration within the matrix, among others. This model allows predicting a final martensiticausferritic matrix and the final dimensional change of the part. The aim of the present work is to design the austempering heat treatment of a differential case, which was originally made of an as-cast ductile iron. In order to do that, the general-purpose software ABAQUS is employed to solve the thermal and mechanical problems at a macroscale by means of the finite element method. The metallurgical problem is solved by taking into account the microscale features of the material by a novel model previously published and validated by the authors (Boccardo et al., 2017b), which is here implemented into ABAQUS and coupled with its thermo-mechanical model by means of user-defined subroutines.
The main characteristics of the thermo-mechanical-metallurgical model and the implementation of the metallurgical model into ABAQUS are described in Section 2. In Section 3 the modeling of the differential case is developed. Moreover, the strategy to fit the metallurgical model by means of simple and standard experimental tests is included. Numerical results obtained for different values of heat treatment parameters are presented and discussed in Section 4. Finally, conclusions are drawn about the computational tool employment, fitting process of the metallurgical model, and the obtained thermal cycle, dimensional changes, and microstructure.

NUMERICAL MODEL
The numerical tool used in this work is a coupled thermo-mechanical-metallurgical model. It allows computing the evolution of the microstructure as a function of the temperature and, in turn, the evolution of the temperature as a function of the phase changes. Moreover, the model predicts the evolution of deformations due to temperature and phase changes. The thermal and mechanical problems are solved at the macroscale, whereas the evolution of the microstructure is predicted by means of a model developed at microscale. The temperature, deformation, and microstructure characteristics are calculated in the whole part.  (ABAQUS, 2010). For the mechanical model, the constitutive response of the ductile iron was modeled assuming an elastoplastic behavior with isotropic hardening according to the J2-flow theory of plasticity. Furthermore, this behavior was considered to be temperature-dependent. Based on the work of Boccardo et al. (2017b), the relation between the stress  and total strain e is given by the constitutive law defined as: where C is the temperature-dependent isotropic elastic constitutive tensor, p e is the plastic strain tensor, and v e is the strain tensor associated to a volume expansion or contraction due to temperature and phase changes. The von Mises yield function is assumed to represent the plastic behavior of the ductile iron (Celentano et al., 2013): is the isotropic plastic hardening function (where A is the strength index, n is the strain hardening exponent, and p eq e is the equivalent plastic deformation), and o C is the temperature-dependent yield strength defining the initial elastic boundary.
It is assumed that temperature and phase changes generate an isotropic volume deformation at the macroscale. The volume deformation is evaluated by employing the model proposed by Christien et al. (2013), which is based in the variation of the microstructure density. The increment of the strain tensor v e  , required by the ABAQUS formulation, is evaluated as:   , and mic t  are the microstructure density at beginning of the simulation (reference state), at time Δ t t  , and at time t . Moreover, 1 is the unit tensor of second-rank.
The microstructure density varies with both temperature and phase changes; and it is evaluated by means of the rule of mixtures: where ph f and ph  are the volume fraction and density of the microconstituents, respectively. The phase volume fractions are computed with the metallurgical model described in Section 2.2. The temperaturedependent phase densities are calculated as in Boccardo et al. (2017b). For the thermal problem, the heat released or absorbed by phase change is considered (Boccardo et al., 2017b). The heat transfer equation is written as follows: where T is the temperature, t is the time,  is the density, c is the specific heat capacity, Q  is the thermal power released or absorbed by phase changes, and K is the isotropic conductivity tensor of second-rank defined as 1 K k  , where k is the conductivity. The material parameters c and k are considered temperaturedependent.
In order to simulate the heat transfer between the part and the environment, a boundary condition based on Newton's law was imposed at the part/environment interface (Boccardo et al., 2017b): where conv q is the normal heat flux, h is the heat transfer coefficient at interface, and T and env T are the temperatures at both sides of the interface.
The initial condition applied to the entire domain is

Metallurgical model
The metallurgical model simulates the microstructure evolution during the three-step of the austempering heat treatment. The metallurgical model considers an initial microstructure formed by ferritic, ferritic-pearlitic, or pearlitic matrices. During the heat treatment the microstructure suffers several phase transformations.
The model considers the development of: • Stable (EITs) and metastable (EITm) reverse eutectoid transformations, in which ferrite and pearlite transform into austenite, respectively, during the heating from ambient temperature up to austenitizing temperature. • Austenite carbon homogenization (ACH) during which the austenite carbon concentration is homogenized at austenitizing temperature. • Ausferritic transformation (AT) in which austenite transforms into ferrite platelets during the cooling from austenitizing temperature down to ambient temperature. In the development of this transformation the ausferritic matrix is obtained. • Martensitic transformation (MT) in which austenite transforms into martensite during the cooling from austenitizing temperature down to ambient temperature.
The solution scheme of the metallurgical model is illustrated in Figure 1, where TEITs, TEITm, TAT, and TMT are the temperatures at which transformations EITs, EITm, AT, and MT start, respectively; additionally, Gr f and f  are the graphite and austenite volume fractions, respectively. Notice that the developments of phase changes depend on the value of temperature and the microstructure at time t . The microscale metallurgical model represents the evolution of the phases by means of representative volume elements that allow considering microstructure characteristics such as the type, shape, and size of phases that form the ductile iron. The graphite nodules are assumed as spheres which are grouped into sets according to their radii; the number of sets is denoted as nsetsg . Additionally, the pearlite colonies are considered as a laminated composite of cementite and ferrite layers and they are grouped into nsetsc sets according to their interlam- for ACH 3 where Gr r , r  , and shell r are the radii of graphite nodule, austenite that grows within ferrite, and the spherical representative volume elements employed for ACH and AT, respectively; x  , x  , and x  are the coordinates of cementite-austenite and austenite-ferrite interfaces, and the length of the unidimensional representative element respectively. The kinetics of phase transformation is computed by means of ordinary differential equations that are solved numerically with the explicit Euler method. The model employed for the ausferritic transformation simulates the nucleation of ferrite platelets by an incubation time function. This exponential function, Equation 11, has two parameters ( 1 k and 2 k ) that need to be fitted with experimental data for each chemical composition of ductile iron (Boccardo et al., 2017a).
where m G  is the maximum free energy available for nucleation at paraequilibrium condition, f  is the attempt frequency factor, R is the gas constant, 1 k and 2 k are empirical parameters to be fitted, and 3 2540 / Once the model is fitted correctly for a chemical composition, it is able to represent the evolution of ferrite platelets for other values of graphite nodule count and austempering temperature. A full description and equations of the employed metallurgical model has been presented by Boccardo et al. (2017b).

Metallurgical model implementation
The metallurgical model was implemented and coupled to the thermo-mechanical model of ABAQUS using user-defined subroutines that were written in Fortran (ABAQUS, 2010). These subroutines allow to extend the functionality of ABAQUS with modules defined by the user.
The USDFLD subroutine is used to implement the metallurgical model represented in Figure 1. Thus, the metallurgical model is solved at material points (Gauss points) of each finite element of the thermo-mechanical model. The variables related to the phase evolutions are stored into the STATEV array (solution-dependent state variables array) and it allows to pass information to HETVAL subroutine and visualize the results at the postprocess. The temperature within USDFLD is obtained by GETVRM subroutine.
The variables of the metallurgical model placed into STATEV array are initialized, at the beginning of the simulation, by employing the SDVINI subroutine. This subroutine also provides a check point to the input data. If problems are detected during the checking process, the calculation is stopped by STDB_ABQERR subroutine and a suitable error message is generated.
The heat flux generation Q  due to phase changes is computed by HETVAL subroutine, and it is calculated as follows (Stefanescu, 2009)

DIFFERENTIAL CASE MODELING
The austempering heat treatment of the differential case was performed in this paper by means of computer simulation. The specific part of interest is a gearbox component of front wheel drive automobiles.
The initial microstructure of the whole part is composed by graphite nodules embedded into a ferriticpearlitic matrix as shown in Figure 2. The graphite nodule count per unit of area is 2 200nod mm , and the volume fractions of graphite, ferrite, and pearlite are 0.16, 0.64, and 0.2, respectively, which were determined by image analysis of optical micrographs. The chemical composition, which is a design constraint, is 3.4C-0.8Mn-0.03Cr-0.4Cu-2.3Si-Fe (in weigh percent). In order to select appropriated values of the heat treatment parameters, several simulations were performed varying the austenitizing temperature, and austempering time and temperature.  (6) (Lacaze and Gerval, 1998;Kapturkiewicz et al., 2005;Celentano et al., 2013).

Model fit
As described in the previous section, the ausferritic transformation model has two parameters to be fitted ( 1 k and 2 k ). The fitting strategy employed requires both experimental tests and numerical simulations, and it allows the model to consider the variation of the austempering temperature when the graphite nodule count remains constant.
Fourteen cuboid samples of small dimensions ( 20 1 0 5 mm mm mm   ) were extracted of one differential case, see Figure 4, and subjected to an experimental austempering heat treatment, all of them were austenitized at 950 T C    during 1200 t s   , while the values of A T and A t were varied, as is shown in Table 3 and Figure   5, in order to characterize the kinetics of the ausferritic transformation for two different values of austempering temperature. The three steps of the heat treatment were developed into a small laboratory furnace, a molten salt bath (sodium nitrate and sodium carbonate with 1:1 ratio), and a room at ambient temperature.  Figure 6. Then, volume fractions of sheaves were measured by means of image analysis. As shown in Figure 7a for both values of austempering temperature, the microstructure presented an ausferritic-martensitic matrix at lower austempering time (small sheaves volume fraction) and a full ausferritic matrix at higher time.
Additionally, Brinell hardness measurement was performed using 10 mm steel ball and 3000 kgf . As shown in Figure 7b, the hardness decreases as austempering time increases, because sheaves volume fraction increases (or martensite volume fraction decreases).    Finally, the parameters 1 k and 2 k were obtained by fitting the model response to the experimental results.
In order to do that, the thermo-mechanical-metallurgical model was employed to simulate the heat treatment of k and 2 k . The geometry was meshed using 1950 tetrahedron elements of 4 nodes (C3D4T) and the convection boundary condition was applied at the external surfaces of the geometry. The optimal combination of these parameters minimizes the sum of the square differences between numerical and experimental results, being the employed results the normalized sheaf volume fraction ( n f ). This sum takes into account the mentioned difference of the two specimens used in the fitting process.
The values obtained of the parameters were  In Figure 9a the temperature evolutions at points 1 and 2 of the part, see Figure 3b, are compared. Even though these points presented the largest difference in temperature evolution, it was small. In the first step, the variation of temperature evolution caused by the heat absorption of reverse eutectoid transformation was observed in the region A. Regarding phase evolutions, Figure 9b, during the first step the austenite volume fraction increased up to its maximum value (0.925) due to both reverse eutectoid transformation and austenite carbon homogenization. The volume fraction of this phase decreased during the second step as a consequence of the ausferritic transformation, in which it was consumed by ferrite platelets. The ausferritic transformation ended at 330 A T C   , but it restarted during the cooling down to the ambient temperature, a fact that explains the changes in austenite and ferrite platelet volume fractions as is shown in the figure. According to the numerical simulation, there was no martensite formation. The predicted evolution of austenite and ferrite platelets is in agreement with the experimental observations reported by Fraś et al. (2012). The length dimensional change during the heat treatment was studied, which is defined as , where m is the value of the dimension, and the subscripts o and t stand for the initial value and the value at time t , respectively. At the end of the first step an increment of the part length was observed, Figure 9c. This dimensional change had the contribution of both temperature increment, which produced an expansion, and phase change, which produced a contraction as may be observed in region A. At the beginning of the second step the length reduction occurred due to cooling from T  down to A T , followed by an expansion caused by the austempering transformation, see region B. Finally, the length was decreased in the third step as a consequence of the cooling down to ambient temperature.   To obtain a full ausferritic matrix with a maximum ferrite platelet volume fraction, the austempering time has to be greater than 2400 s , which is the required time to end the ausferritic transformation at A T . It is worthwhile to notice that the same behavior was observed in samples employed to fit the metallurgical model. The dimensional change at the end of the heat treatment is defined as , where m is the value of the dimension, and the subscripts o and f stand for the initial and final values, respectively. As is shown in Figure   11b, dimension reductions occurred for any austempering time. The maximum reduction occurred for the maximum ferrite platelet volume fraction, which is obtained when the ausferritic transformation has been completely developed at A T . Moreover, DCD, DCT, and DCL were very close each other, because the final volume fraction is almost uniform in the whole part. The minimum required time for the first step ( min t  ) has the contribution of a) time to get the temperature at which the reverse eutectoid transformation starts and b) time to transform the initial microstructure. In the simulation it was observed that when T  increased, the first time is increased and the second one decreased. The resulting variation of min t  with T  is shown in Figure 12a, and the maximum difference was 2.6 %. On the other hand, the minimum required time for the second step (  Figure 12b, and the maximum difference was 9.7 %. Regarding the microstructure at the end of the heat treatment, the graphite volume fraction depended on the austenitizing temperature, decreasing its fraction with the increment of the temperature, as may be seen in Figure   13a. The maximum difference in the graphite volume fraction was 9.3 %. The ferrite platelet volume fraction increased with the decrement of both T  and A T , therefore the maximum volume fraction was obtained at 880 T C    -300 A T C   , and the maximum difference was 16.3 % , as is shown in Figure 13b. The variation of this phase fraction with A T was less than the expected, because a large amount of ferrite platelets was formed during the third step when the part was cooling from a high value of A T . The behavior of austenite volume fraction was opposite to the found for the ferrite platelets, because austenite was consumed by the second phase during the ausferritic transformation. The final dimensional change of diameter, thickness, and length of the part are presented in Figure 14 for different values of austenitizing and austempering temperatures. As mentioned in Section 4.1, DCD, DCT, and DCL dimensional changes were very close to each other, because the final volume fraction is almost uniform in the whole part. Moreover, the dimensional changes were highly influenced by T  , being the maximum difference of 30 % . In order to obtain a full ausferritic matrix, the eutectoid transformation has to be avoided (transformation of austenite into ferrite and pearlite). One way to do that is by subjecting the part to high cooling rate during the cooling from T  down to A T (Venugopalan, 1990). In Figure 15 the cooling rate at point 1, developed between 880 C  and 600 C  , is presented for different values of T  and A T . The analysis is presented for this part point, because it presented the lowest value of cooling rate. The cooling rate is defined as where Δt is the required time to cool the part from 880 C  to 600 C  . As may be seen, the cooling rate increased when T  increased and A T decreased, and the maximum difference was 38.6 %. The size and geometry of the differential case is a design constraint. However, in order to analyze how the heat treatment is developed in parts with different size and geometry, the numerical results of both differential case (DFC) and small sample employed to fit the model (SS) were compared. The heat treatment simulations were performed for 950 T C    and 330 A T C   . 4 % greater than for SS, because different amount of this phase was formed during the cooling from A T down to ambient temperature. As a consequence the final dimensional change (in absolute value) in SS is 3 % greater than for DFC. The minimum required time for the first step was highly influence by the size and geometry, being 1130 s and 3200 s for SS and DFC, respectively. As was mentioned in Section 4.2, min t  has the contribution of a) time to get the temperature at which the reverse eutectoid transformation starts and b) time to transform the initial microstructure. In SS these contributions were smaller than for DFC. The minimum required time for the second step was 2335 s and 2400 s for SS and DFC, respectively, indicating that it was slightly influenced by the mentioned changes. The small difference occurred because a different time was necessary to get the temperature at which the ausferritic transformation starts, due to different cooling rate for SS ( 21.05 / C s  ) and DFC ( 5.91 / C s  ) during the cooling down to A T , and that the duration of the ausferritic transformation at A T was almost the same for both parts.

Selection of heat treatment parameters
As shown in Sections 4.1 and 4.2, the austenitizing temperature, the austempering temperature, and the austempering time highly influence the final phase fraction, final dimensional change, among other variables.
In order to obtain a final microstructure composed by graphite nodules and a full ausferritic matrix, the duration of the second step is chosen to be equal to . The final dimensional change, due to the heat treatment, has to be considered in both pre and post machining processes in order to satisfy the dimensional tolerance.
The differential case was heat treated in a laboratory in order to verify the results obtained with the simulation. The durations chosen for the first and second steps were around 10% greater than the durations predicted by the model ( 3600 t s   and 2700 A t s  ), and the experimental procedure was the same that the employed for the small samples.
As was predicted by the simulation, the microstructure obtained in the experiment is uniform in the whole part. The presence of ferrite, pearlite, and martensite was not observed, as is shown in Figure 16, even in the region with the lowest cooling rate from T  to A T (point 1). Figure 16: Final microstructure of the differential case at point 1, which presents a full ausferritic matrix. Graphite nodules in black, sheaves in brown, and austenite in white.

CONCLUSIONS
The design of the austempering heat treatment of a ductile iron differential case, aided by computer simulation, was presented in this paper. The simulation was performed with a coupled thermo-mechanical-metallurgical model, which was fitted to the chemical composition of the part.
The main conclusions drawn from the work may be summarized as follows: (a) The metallurgical model was implemented into ABAQUS by means of user-defined subroutines. The pre-process, resolution, and postprocess required to perform the simulations and analyses of results were carried out in spite of the complexity of the part geometry and the large size of the employed finite element mesh. (b) The metallurgical model was successfully fitted to the alloy chemical composition by employing a standard procedure that involved experimental heat treatment, at different austempering times and temperatures, and microstructure analysis of several small samples obtained of one differential case. (c) According to the simulations, the austempering time modifies the type of final microstructure. In the differential case, ausferriticmartensitic and ausferritic matrices are obtained for short and large values of austempering times, respectively. The same behavior is observed in the samples employed to fit the metallurgical model. Additionally, the austenitizing temperature highly influences the final phase volume fraction (decrement of graphite and ferrite platelet volume fractions with a temperature increment) and final dimensional change (increment of the contraction with a temperature increment) when the ausferritic transformation was completely developed at the austempering temperature. (d) The minimum time required for the first step of the differential case heat treatment is influenced by the austenitizing temperature (decrement of this time with a temperature increment). On the other hand, when the size of the part is reduced, the minimum required time decreases. The minimum required time for the second step of the differential case heat treatment is influenced by the austenitizing and austempering temperatures in a not simple way, and it suffers a small variation when the size of the part is reduced. (e) The temperature evolution in whole part presented small differences; a fact that allowed to obtain a uniform ausferritic matrix, at the end of the heat treatment, in the whole part. (g) Amounts of ferrite and pearlite were not observed in the micrographs of the treated differential case, indicating that the cooling rate was correctly chosen. The cooling rate highly depends on the austempering temperature. When considerable amounts of ferrite and pearlite are obtained during the cooling from austenitizing temperature down to austempering temperature, the metallurgical model has to consider the development of the eutectoid transformation in order to represent adequately the phase evolution throughout the three-step austempering heat treatment.