A novel investigation into the edge effect reduction of 4340 steel disc through induction hardening process using magnetic flux concentrators

Induction hardening serves as one of the best mass production processes used recently due to its ability to quickly generate high-intensity heat in a well-defined location of the part. Numerous advantages of this method make it a reliable technique to produce a thin martensite layer on the part surface that has compressive residual stresses. In this regard, the presented study is devoted to investigating utilizing induction heating for surface hardening of AISI 4340 steel disc. The purpose is to evaluate the performance of magnetic flux concentrators and the effects of the induction process parameter on the case-depth and edge effect in the surface hardening of the disc. Once the proper range of parameters is defined, Taguchi experimentation planning is used to frame comprehensive experimentation with the minimum possible trial. Then, the case-depth of discs is evaluated on their cross-sections (edge and middle plane) through hardness profile measurement of samples using a micro-indentation hardness machine. The results are then statistically analyzed using analysis of variance (ANOVA) and response surface methodology (RSM) to determine the best combination of parameters to achieve maximum case-depth yet minimum edge effect. The goodness-of-fit regression models are then developed to predict the case-depth profile as a function of machine parameters based on linear regression utilizing case-depth responses in the edge and middle planes of discs. Results imply that maximum case-depth with minimum edge effect can be produced by using the highest heating time along with the average amplitude of the power, axial gap, and radial gap. This study gives a good exploration of case-depths optimized by setting up process parameters when a magnetic flux concentrator is utilized; thus, a guideline to reduce discs edge effect in induction surface hardening application is given.


Introduction
The importance of induction heat treatment and its numerous applications have been increased over the last decades. One of the most sought-after usages of this phenomenon is the hardening of metals [1,2]. Multiple advantages of induction heat treatment make it superior to traditional methods: capability to produce high power density in a shorter time that results in saving time, being high energy efficient, limiting the risks of distortion and deformation during conventional heat treatments [3]. It is a fast and clean-energy process and repeatable in terms of treating parts with the same quality. Also, the ability of fast heat generation in well-defined points of the specimen and eliminating other undesired-treated parts, being environmental-friendly along with other mentioned characteristics, makes this method well-received for mass production and industrial purposes [4]. The ability to generate localized heat during the heating process in hardening processes is one of the most important industrial applications of induction heating employing improving wear and fatigue resistance of the parts such as bearings, shafts, and disc-shaped parts like gears [5,6].
Parts geometry, surface topology, chemical composition, and induction heating processing parameters are crucial elements in induction surface hardening [7]. This study aims at investigating the effects of induction machine parameters on parts with plain geometry like a disc. While the result of this simplification can be generalized to gears [8], considering simpler geometries like disc rather than gear can eliminate some redundant parameters which are related to gear-shaped parts and geometrical complications (Fig. 1).
Similar to conventional hardening methods, the induction hardening process is based on increasing the temperature up to the austenitic area, then cooling the part rapidly. As a result, the martensite phase, a very hard steel structure, is formed. However, the hardenability of the metal alloy plays a key role in the final percentage of the martensite phase. In this study, AISI 4340 steel, which is known for its heat treatable properties, is used to achieve high strength, good fatigue resistance, and proper atmospheric corrosion resistance behaviors. Referring to the main goal of the study, obtaining a hard surface on 4340 steel discs, creating maximum martensite in surface layers is intended since the formation of any pearlite or bainite phase causes a drop in the hardness of the material. To eliminate producing pearlite or bainite formation, rapid quenching of a uniform austenite layer is essential [9]. A desirable form of austenite layer highly depends on induction process parameters that could maximize the austenite phase formation. The subsequent step of quenching is done by a water shower to create a uniform martensitic layer on the parts' surface. According to these facts, a typical heat treatment temperature for AISI 4340 alloy steel parts is within a range of 790 to 915°C (850°C in our study) to create a uniform austenite layer, and a rapid quenching to produce the maximum possible percentage of martensite layer [10][11][12][13].
Despite the brilliant advantages of induction hardening, the process yet involves some challenges that lead to undesirable results. The major issue in induction heat treatment is the edge effect due to the disproportionate density of magnetic flux in different areas of a part [4,14]. This variation causes nonuniform generated temperatures over the surface of parts. Subsequently, an uneven formation of martensite phases will occur on the surface of the specimen following the quenching may lead to undesirable metallographic structure. These physical phenomena arising from magnetic flux characteristics coupled with coil geometry limitation provide a varying temperature distribution in radial and axial directions. In the case of high-frequency induction hardening of a small disc with a single-shot method, two facts could be concluded to justify the non-uniform temperature distribution: First, the temperature gradient in the radial direction due to the more proximity in the outer layer of the disc to the induction coil compared with the distant layer to the coil toward the center of the disc's geometry. The second reason for nonuniform temperature distribution comes from the distortion of magnetic field at coil and bending of the induced current in the edge regions of the workpiece, namely, the edge effect phenomenon. This results in a temperature gradient in the axial direction wherein the generated heat profile between the middle plate and side plates of a disc is not uniform as shown in Fig. 2. Thermal gradient because of the edge effect is negligible for big parts, though this is not applicable for small parts. The smaller the workpiece, the more serious challenge in the quality of the final product [15][16][17]. Due to the complexity of this physical phenomenon, there is no definite solution to eliminate it; however, a great deal of researches has been dedicated to reducing its consequences as much as possible. One efficient solution is to utilize magnetic flux concentrators. Axial/radial gaps between magnetic flux concentrators and coil/disc are two effective parameters in the edge effect that are included in this study. This study investigates the effect of the concentrators position (radial and axial gap) as well as the induction heating process parameters including machine power, frequency, and heating time on the edge effect aiming at tuning them to minimize the edge effect. Figure 3 demonstrates a schematic presentation of the position of the coil, concentrator, and workpiece.  [20] Since the discovery of the induction phenomenon in 1831, plenty of efforts have been hailed on detecting all effective elements and improving the induction machine component to make this process more productive and user-friendly. In the last few decades, the development of computer technology has brought a great deal of facilitation in this field alongside experimental investigation. Indeed, numerical computer simulation currently plays a fundamental role in the development of this complex and multi-physical process. Investigations on residual stress profiles produced by induction surface heating and quenching [18] and numerical simulation to find a relation between induction process parameters and phase transition in surface hardening [19] can be cited as examples.
Furthermore, looking at academic researchers on induction heating and heat-treating shows applying computer modeling along with experimental has replaced the word "usefulness" with the word "necessity." Studies such as the formation of surface stresses in forming rolls hardened by induction hardening are analyzed by FEM simulation [21]. In another research, case hardening of gray and ductile iron was studied by induction hardening [22]. In a practical approach, the kinetics of structural transformation of roll steels was examined [23]. Dutka et al. [24] utilized computers along with experiments to predict phase transformation during the induction process applied for cutting tools. Also, the functionality of spray cooling was tested for induction hardening of the gearwheels [25]. Multiphysics modeling was executed to make experiments faster and more accurate for induction hardening of the ring gear [26]. Furthermore, initial microstructure and heating rate were investigated for induction hardening of 5150 steel [27]. In more recent studies, induction heating was performed to enhance the fatigue strength of AISI E52100 bearing steel [28]. Experiment analyses are also applied to study distortion of cold-drawn and induction-hardened components [29]. Jeong et al. [30] tried to predict the deformation of curve shape steel plates in high-frequency induction heating. The eddy-current method is used to define the case-depth of cast iron hardened by induction [31]. Estimation of stress and distortion during the induction case hardening tube was performed by Nemkov et al. [32], and another experimental approach has been done to investigate the numerical modeling of the induction process [33]. To improve fatigue strength and avoid cracks in stress and strain during induction hardening, numerical simulations were performed by Ivanov et al. [34]. The effects of initial microstructure of 4140 steel on the fatigue behavior after induction hardening were also explored [35], and multiple investigations applied finite element analysis (FEA) to determine effective process parameters and to find ways for microstructural enhancement of the final induction-hardened product [36][37][38]. Utilizing ELTA software induction heat treatment of steel is simulated [39] wherein a combination of numerical and experimental investigations is applied for local induction hardening of large-diameter gear rolling [40]. This research team has recently utilized both experimental and FEA simultaneously to enhance induction hardening process applied to gears. 2D simulation is used to observe the effect of changing in the geometry of the component on hardness profile [41], and 3D simulation is applied to improve the case-depth of induction-hardened helical gears [42]. In another research using 2D models, the effects of induction heating process parameters on hardness profiles of 4340 steel bearing shoulders were studied [43]. The effect of frequency on the hardness profile of spline shaft heat-treated by induction is also examined [6]. Another study was focused on the effects of induction machine parameters on the casedepths of 4340 spur gear with a 2D model [44], and explore how machine parameters and geometrical factors are effective on the hardness profile of 4340 steel disc hardened by induction hardening [45]. Furthermore, surface methodology and artificial neural network modeling were exploited to observe the effects of flux concentrators on changing of edge effect in spur gear treated by induction [46]. Barglik et al. [47] recently performed a numerical study on dual frequency hardening process of a AISI 300M gear. They introduce a mathematical model of the process by the help of CCT and TTA diagrams. In 2019, Baldan et al. [48] investigated the induction heating process from another point of view, implementing multifidelity optimization approach to reduce process time. In early 2019, Li et al. [49] tried to optimize the induction heating temperature of a 55CrMo steel ball screw specimen and used the martensite fraction value as a main criterion of their study.
The outcome of the study shows that the temperature increase is in favor of martensite formation and hardness gain.
This study aims at determining the most effective process parameter on edge effect in induction hardening of discshaped parts while flux concentrators are applied. As previously mentioned, employing disc-shaped parts in this study could helpfully eliminate some difficulties of using parts with complex geometry like gears while the result can be applied for both shapes. In this experimental effort, the ANOVA statistical method is used to model and analyze case-depth for different combinations of process parameters and location of concentrators. This method has proved its reliability in statistic approaches and shows the way to understanding the effect and measure of each multi-physical parameter and helps to reduce errors and wasting time on calculations. In this regard, the first step is to identify the most effective range of the process parameters by preliminary tests. The second step is to plan the most effective tests with the minimum number of trials employing Taguchi planning method. The last steps are respectively performing the planned tests, statistical analysis, and then discussing the results and finally, elaborating conclusion according to the evidence. These proceeds led to yield an effective, accurate, and robust statistical model that can be used to predict the case-depth of disc-shaped workpieces. This article is formed as follows: Section 2 presents the methodology of our experimental tests based on induction surface heating and its related parameters, while results of experimentation, statistical, and sensitivity analyses are reported in Section 3. In the end, Section 4 presents conclusions and ideas for future works in this field.
2 Experimental procedure

Materials and method
Induction hardening is applied to 4340 steel discs aiming at fast surface heat treatment, which is low alloy steel known for its high toughness and strength in the heat-treated condition while preserving fatigue resistance. Mentioned characteristics make it a reliable choice for making industrial disc-shaped parts. The chemical composition of AISI 4340 steel and its mechanical characteristics are noted in Tables 1 and 2 respectively.
The induction heating machine that is used for this study employs two generator sets of a medium and high frequency to produce a wide range of frequencies. The maximum range of frequency, approximately 1 MW, could be achieved by a combination of the two generators. Here is the machine procedure to drive the maximum range of frequency: first is to create maximum power of 550 kW by using audio-frequency technology with solid-state converters that operate at 10 kHz, and the second is to deliver maximum power of 450 kW by running a transistor radiofrequency generator at 200 kHz. To achieve high frequency, a thyristor radiofrequency (RF) generator operates at 200 kHz. Process controlled utilizing real-time monitoring (RTM) module to ensure the state of machine parameters during the process validating whether the actual parameters are the same as the input parameters.
A disc is positioned on a ready-made stand kit between two magnetic flux concentrators. Discs and flux concentrators are in the same material, shape, and geometry; AISI 4340 steel with an outer diameter of 104.3 mm, and a thickness of 6.5 mm. AISI 1010 steel shims are inserted between disc and concentrators to ensure distance between them (axial gap). The thickness and the outer diameter of each shim are 0.2 ± 0.05 mm and 31.75 mm respectively. The geometrical specification of copper-made coil is 140 mm for the outer diameter and 110 mm for the inner diameter with a useful section of 7 mm × 7 mm and thickness of 2 mm. Figure 3 displays a schematic position of magnetic flux concentrators, master disc, and coil. Finally, a jet of a cooling solution containing 92% water and 8% of polymers is used as coolant fluid for quenching the disc after induction heating. It should be mentioned that all disc samples have undergone a uniform quenching and tempering treatment to have approximately a uniform hardening of 45 HRC through the discs. This is to prepare an equal initial condition for all samples before starting the induction heating process. As previously mentioned, because of the variety of flux densities in different areas of the disc, different amounts of heat will be generated through the disc and can be stocked in some areas. Therefore, the area that is exposed to the more magnetic flux is considered critical that must be heated within the range of austenization (above Ac 3 ) to avoid overheating more or less than this specific temperature range. To serve this purpose, a typical guideline for the induction hardening of the AISA 4340 disc is followed.
To avoid useless trial and error tests, an optimized number of tests should be done; therefore, a rigid method for designing the experiments is adopted to provide the maximum amount of information from the minimum number of experiments. In this study, the Taguchi planning method is applied because of the capability of this method in executing efficient and simplified factorial results [52,53]. To apply this optimized experimental planning method, a list of effective parameters is required with a defined level of each. During the induction heating, machine power, heating time, and frequency are three input variables of the induction machine that control the generated heat and consequently the temperature amplitude on the part. It should be considered, in contrast to the power and heating time, frequency is not tunable. Other effective factors are the axial gap (the gap between each concentrator and master disc), and the radial gap (the gap between the coil and discs) as shown in Fig. 3. Accordingly, the input parameters are power, time, axial, and radial gap. The levels of input parameters and their ranges were defined by low, medium, and high levels as presented in Table 3. Regarding the abbreviation list in Table 3, in the following, P, T, AG, and RG are used instead of power, heating time, axial gap, and radial gap respectively. Taguchi experiment planning is well-known as one of the most reliable and robust planning-designers to perform a minimum efficient number of experiments. A proper range of factors by preliminary tests along with using Taguchi experimental design makes the process more robust, efficient, and with a minimum number of trials [54,55]. Indeed, performing an induction hardening within the maximum and minimum range of parameters is of critical importance to create maximal and minimal case-depth transformation. An L 9 orthogonal matrix corresponding to nine (9) experimental tests is addressed by the Taguchi planning method; consequently, each test consisted of a unique combination of parameter levels as shown in Table 4.

Micro-indentation hardness tests
Micro-indentation tests using Clemex machine are performed to measure the hardness of specimens at their cross-section. After surface-hardening process by induction heating method, discs specimens are cut into two halves. In the following, the middle plane refers to the surface that cut the thickness of disc in two equal parts parallel to the edge plane. The edge plane, in its turn, refers to the outer surfaces of the disc as depicted in Fig. 4. Micro-indentation tests are then performed in both edge and middle planes, and results are summarized in Table 5. These responses present the measured case-depths of discs at the edge and middle of discs respectively denoted as d E , d M . It worth mentioning that the defined case-depths on edge planes are calculated as the mean value of case-depths measured on the two exterior surfaces of the disc (edge planes). Doing so could avoid complications in analyses related to outliers.
The results in Table 5 imply that the maximum case-depth in both middle and edge planes occurred using test 9 parameters, while the minimum case-depth in middle and edge planes is observed in test 1. In detail, results of tests 9 and 1 are presented in Fig. 5a and b, in which the hardness profile of induction heat-treated discs is evaluated. Three distinguished regions could be defined in these figures; the first region presents a hardened surface layer due to martensitic phase formation (the line approximately parallel horizontal axis), the second region displays a hardness drop to approximately the core material hardness (the part with deep slope), and the third region in which discs tempered before the tests and preserve the initial core material hardness (approximately 45 HRC) [56]. Figure 5a demonstrates the hardness profile and obtained case-depth at edge plane (d E ) resulted respectively from test 9 and test 1, while Fig. 5b shows the hardness profile and casedepth at the middle plane (d M ).
There are three main conclusions to be drawn from Table 4. First, the maximum case-depth at the edge plane happens in the maximum level of power and axial gap, and a minimum level of radial gap. Meanwhile, the minimum case-depth at edge plane occurs when P, T, AG, and RG are set at their minimum level. Second, the minimum amplitude of the case-depth difference between middle and edge planes (Δd), which is an indication of reaching a minimum edge effect, is

Statistical analysis and parameters contribution
To estimate the effectiveness degree of each variable in the experimental, outputs are analyzed utilizing the statistical method of analysis of variance (ANOVA) [57,58]. ANOVA uses an F-test to appraise the regression model. To evaluate the degree of parameters' participation in responses (case-depth at middle and edge planes), the main effect diagram for each dependent variable (specific range combination of the parameters) is first plotted. The results of ANOVA concerning the case-depth at edge and middle planes are shown in Tables 6 and 7 respectively. In these tables, the degrees of freedom (DoF), sum of squares, contribution percentage, F-value, and p-value are the terms that declare the results of the analyses. Also, to estimate the precision of the regression model, the coefficient of determination (denoted by R 2 or R-squared) is calculated. The term p-value determines statistically whether the effect of process parameters on the case-depth is significant. In other words, ANOVA calculates the p-value and compares it with the significance level (α) to determine whether the null hypothesis is rejected. In fact, the null hypothesis states that the effect of studied parameters is insignificant. The null hypothesis can be rejected when the calculated p-value is lower than or equal to the designated significance level. The defined significance level (α) for this study is 95%, which means that there is a 5% risk of considering a specific parameter as having a noticeable effect on the process when there is no real effect. Linear regression is then performed based on ANOVA results to develop predictive equations for case-depths in the edge and middle planes (d E and d M respectively) as a function of power (P), heating time (T), axial gap (AG), and radial gap (RG). Here outputs are case-depths that are termed dependent variable, and input parameters of P, T, AG, and RG are termed independent variables. The ANOVA conditions in this study are selected as a stepwise method wherein interactions between factors are included. The stepwise method aims at excluding less important factors and interactions in each step giving more importance to the effective factors. Analyses were proceeded by including the most important interactions to prevent changing the meaning of the lower order coefficients. Furthermore, involving any redundant term to the model while the related P-value is higher than 0.05 could make the model more complicated [59].
P-values for T, P, P×AG, and RG are respectively 0.001, 0.002, 0.030, and 0.060 as given in Table 6. Except for RG, all p-values are less than 0.05; hence, the null hypothesis is rejected. It can be observed that in d E , the heating time (T) and machine power (P) are the most effective parameters on the case-depth and then, P×AG with less degree of importance regarding their descending quantity of p-values. P-value of 0.030 for P×AG connotes interaction between power and the axial gap was in such a way that combining these two independent variables gave a different effect on d E . Table 7 introduces P, T, T×RG, and RG as the most effective variables in d M concerning the P-values (respectively 0.00, 0.005, 0.015, and 0.031). Results for both d E and d M prove the involving interactions as the P-values of P×AG and T×RG are respectively 0.030 and 0.015 from Tables 6 and 7. The contribution of each variable and interaction is also defined in the edge and

Main effect plot and predictive regression models
To define the correlation between each independent variable and dependent variable, main effect plots are drawn. Interpretation by plots provides us a visual trend of response variation with respect to parameter levels. This approach allows exploiting a goodness-of-fit model that describes the relationship between variables and predicting the case-depth with a combination of the input parameters. Main effect plots are drawn in Matlab TM to describe the most effective parameters affecting the response. These plots show a general trend of how one parameter affects the response allowing their effectiveness comparison separately. This is performed by verifying the slope of each variation line between two levels in which the steeper the slope, the more effective the parameter on the response. The main effect plots of the responses (d E and d R ) over the 3 levels of their parameters (P, T, RG, and AG) are shown in Fig. 6. In this figure, d E and d R are presented with respectively blue and red lines. It can be observed from Fig. 6 for d E that P and T have a significant effect on the case-depth. The bigger domain of changing in response with steeper variation lines is the evidence of this interpretation. The effect of the axial and radial gap (AG and RG) on the case-depth at edge plane is less compared to P and T. This observation is based on the fact that the slope of AG and RG variation is less than the one of P and T. Like d E , P and T are the effective parameters on d M wherein the slopes of variations are steep. The effect of AG and RG on d M can be interpreted as less than P and T since their variation slop is much smaller. Compared to AG, increasing in RG shows the opposite effect on the response as it is vivid from the negative slop from levels 1 to 3. The main effect plots are used to reveal the relationship of each independent variable on the dependent variables. The interaction between the parameters is also studied based on ANOVA as a combination of interacting variables because the effect of one variable depends on the value of the other [55]. Studying the interaction of variables indicates that P and AG (P×AG) in edge plane (d E ) and T and RG (T×RG) in middle   plane (d M ) present interactions. To this end, a predictive linear regression of case-depth, which is a function of the important input parameters and interactions, is exerted for the edge and middle plane. These predictive functions are respectively presented in Eqs. (1) and (2).
The coefficient of determination (R 2 ) concerning the regression equation of d E is 97% which declares that the predictive equation represents 97% of the response behavior based on the presented factors and their interactions. The R 2 of d M is calculated 99% which means that the predictive regression equation represents almost perfectly the variation of response (d M ) in the range of experimentations. The accuracy of regression equations can also be estimated based on the residual evaluations. In this regard, the difference between predicted and measured values represents the possible regression error (residual). Using Eq. (3), the relative error of predicted models can be calculated. Our study's outcomes indicate a good agreement between experimentally measured and predicted model for case-depths at the edge plane (d E ) and the middle plane (d M ). The estimated errors of d E are less than 4% except for test 2 where it reaches approximately 12% while the error of d M is less than 4% for all tests.
The goodness of fit, which visually represents the level of agreement between the measured experiments and predicted models, is depicted in Fig. 7a and b for d E and d M case-depths respectively. Predicted values that agree exactly with measured values would fall on the red line, while the dispersion of these values (blue points) with respect to the red line represents the lack of fitness. The plots of Fig. 7 illustrate a very good agreement between the experimental values and predictive models, for which the predicted model of d M fits better than d E .

Response surfaces and sensitivity analysis
For empirical modeling, response surface methodology (RSM) is applied whereby the results are shown by contour plots. Contour plots are 3D diagrams that the response curve is projected on the plane of variables. In this regard, the response is shown by contours (color gradient and isolines) on the plane of variables. Here the variables are P, T, AG, and RG, and responses are d E and d M . The results of RSM are displayed in Fig. 8 for d E and in Fig. 9 for d M . It should be noted that the non-presented parameters in the plots are considered as a constant independent variable in their medium level of 100 kW, 0.5 s, 0.8 mm, and 2.8 mm for P, T, AG, and RG respectively. A contour plot of case-depths in edge plane (d E ) versus P and T is illustrated in Fig. 8a wherein the maximum values of case-depth (1.7 mm) in the upper right corner of the plot correspond to the highest values of both P and T. The minimum value of case-depth (0.7 mm) in this graph corresponds with the lowest values of both P and T. In other words, both T and P have an increasing effect on the case-depth value as well as approximately the same impact on the cade depth enhancement. The same trend can be observed in Fig. 8b and c for P and AG as well as T and AG, but the case-depth value is affected more by changes in P and T than AG in these plots. As it is already explained in Section 3.2 and can be observed in Fig. 8d, RG shows the opposite effect on the case-depth when it is compared with other parameters and the maximum case-depth happens at minimum RG. This is demonstrated in the contour plot of case-depths at edge plane versus AG and RG. As a conclusion, the highest case-depth can be achieved by increasing P, T, and AG whereas decreasing RG.
Based on ANOVA in Section 3.1, the most effective parameters of case-depths in middle plane (d M ) are defined as P, T, and RG in which their contour plots are presented in Fig. 9. The increase of P and T results in the case-depth increase as shown in Fig. 9a. Similar to d E , RG has an inverse effect on d M as illustrated in Fig. 9b and c. It can be concluded from Fig. 9 that case-depth in middle plane of discs is mainly affected by the P and T in which their increase would increase the case-depth.

Edge effect discussion
Considering the principle of magnetic fields, they are denser near the source of the field. Consequently, moving from the outer diameter of the disc toward the center of the disc hardness profile is reduced considerably to around the initial hardness of material as is shown in Fig. 5. Loops of induced electrical currents, namely "eddy currents" arising from changes in the magnetic field, tend to be stuck in corners that lead to non-uniform thermal distribution through the disc. Therefore, a non-uniform hardness profile and finally a different casedepth profile in edge plane versus the middle plane are formed. To make a comparison of case-depth profiles in the edge and middle planes, in this section, the middle plane is considered as the reference plane and the differences between the planes (Δd = d E − d M ) present the edge effect. The ultimate purpose of this study is to determine the best combination of process parameters that maximized the case-depth with minimum edge effect utilizing flux concentrators. To this end, the less the Δd, the less the edge effect that results in a more desirable uniform case hardening.
The effects of induction hardening process parameters (P, T, RG, and AG) on the case-depth profile have been already discussed separately in the middle and edge planes. Results showed that the case-depth profile is mainly affected by P and T, while other parameters including RG and AG participated less. In this study, the interaction of parameters is also considered and regarding ANOVA, only P×AG and T×RG are effective parameters in case-depth. Applying the concept of Δd, and presenting it in Fig. 10, a reducing trend in edge effect from test 1 to test 9 can be observed. The minimum edge effect is then noted in test 7 and test 5. The experiment parameters, P, T, AG, and RG for test 7, are 85 kW, 0.6 s, 1.4 mm, and 2.8 mm respectively which refers to the low-level of machine power, the maximum level of T and AG, and mid-level of RG. The process parameters of test 5 are 100 kW, 0.6 s, 0.8 mm, and 2.6 mm, respectively, for P, T, AG, and RG. They refer to the maximum level of P and T as well as the mid-level of AG and RG. It is worth mentioning that the average case-depth in test 7 is 1.30 mm while in test 5 is 1.56 mm. Although the best case-depth results in test 9, the edge effect is also considerable. It could be concluded that the best scenario, deeper casedepth, and less edge effect occurred in tests 5 and 7 where the heating time is tuned in its maximum level, the machine power, and radial gap are in low and mid-levels, and the axial gap in its maximum and mid-levels.

Conclusions
4340 steel discs are used in this study to investigate the effects of induction hardening process parameters on the case-depth and edge effect with the presence of magnetic concentrators. The process parameters, machine power (P), heating time (T), the axial gap between disc and concentrator (AG), and the radial gap between disc and coil (RG) are considered to evaluate two responses, case-depth in edge and middle of discs (d E and d M ). The Taguchi design method is employed to plan nine tests (L 9 ) in threeparameter variation levels. The micro-hardness test and consequently the case-depths are then analyzed by analysis of variance (ANOVA). Using linear regression, predictive models are then developed for d E and d M with the coefficient of determination (R 2 ) of 97% and 99% respectively. The analyses show that heating time and machine power are dominant parameters to control the case-depth in d E and d M as far as the contribution percentages of T and P for d E are respectively 43.69% and 40.31%. Taking into account the effect of parameter interactions, P×AG is contributed 7.60%. In d E variation, the contribution of T, P, RG, and T×RG in the variation of d E is 48.98%, 44.81%, 3.29%, and 2.36%. Moreover, contour plots based on response surface methodology (RSM) show that setting independent variables of P, T, and AG at maximum level and RG at minimum level would enhance the case-depths (d E and d M ). The edge effect, which presents the nonuniformity of case-depth in the edge plane with respect to the middle plane, is then analyzed in which tests 5 and 7 present the minimum edge effect. Based on the test results, maximum heating time, medium machine power, and radial gap as well as maximum axial gap help reduce the edge effect. It is worth mentioning that the average case-depth in tests 5 and 7 reaches 1.56 and 1.30 mm respectively. The results of this study can also be used in disc-shaped parts such as gears.
Although the results of this study are promising, some improvements and applications can be foreseen. Future works can study the generator frequency fluctuation according to the variation of geometrical factors (thickness, axial, and radial gaps) as an induction machine factor. Moreover, the concentration thickness can be optimized during the extensive design of experiments. To optimize surface treatment of the other parts like bevel gears, and helical gears which have different and more complex shapes, experimental test benches as well as predictive models can be developed. In this regard, applying artificial intelligence to create and enhance the models could be the next project to perform.
Author contribution The authors' contributions are as follows: Noureddine Barka and Mohamed Khalifa conceived, planned, and carried out the experiments; Mahyar Parvinzadeh, Sasan Sattarpanah Karganroudi, and Narges Omidi contributed to the interpretation of the results. Sasan Sattarpanah Karganroudi took the lead in writing the manuscript; Mahyar Parvinzadeh and Narges Omidi contributed actively in writing the manuscript; all authors provided critical feedback and helped shape the research, analysis, and manuscript. Data availability All data, material, and codes used in this paper are available. Fig. 10 Case-depth difference between middle plane and edge plane (Δd) resulted from the 9 arrays of experiments

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