Phase transformation and deformation of the high-frequency induction brazed grinding wheel based on multi-field coupling

With the characteristic of the high bonding strength to matrix, good sharpness, and large chip-storage spaces, the brazed super abrasive grinding wheels have superiorities in the machining of difficult-to-machine materials. However, thermal deformation is caused by the high temperature during the brazing process, leading to the accuracy of the brazed grinding wheel being degraded greatly. By means of local heating, high-frequency induction brazing can reduce the thermal deformation of the wheel. Aiming at the thermal deformation mechanism of the induction brazed wheel, a numerical simulation model of thermal-stress-phase multi-field coupling was established considering the temperature-dependent physical properties of the material. The simulation result indicated that the phase transformation occurred near the work surface of the wheel substrate. The depth of phase transformation layer decreased from 6.0 mm to 2.9 mm with the scanning speed increasing from 0.5 mm/s to 2.0 mm/s. The microstructure of the phase transformation layer mainly consisted of ferrite, pearlite, and bainite after brazing. An appropriate scanning speed was more important for the high accuracy of the wheel substrate during the induction brazing, since it had remarkable influence on the stress and deformation than brazing temperature. The experimental results of the microstructure morphology and deformation proved that the numerical simulation model was correct with 10.4% error.


Introduction
As a precision machining method, grinding has drawn attention to many researchers [1,2]. During the grinding process, the grinding wheels play the crucial role in obtaining a highquality ground surface. A lot of studies were conducted on fabrication and application of grinding wheels to improving machining ability [3,4]. Conventional grinding wheels primarily include electroplated wheels, sintered wheels, and resinoid bonded wheels [5]. However, the abrasive grains on these wheels are bonded mechanically, which have low holding strength and low chip-storage space. This can result in the early shedding of grits and burning on the ground surface during the grinding procedure [6,7].
In the recent decades, the active brazing technology was proposed to fabricate the super abrasive (diamond and cBN) grinding wheels [8]. During the active brazing, a reaction layer generated at grain-alloy interface contributes the excellent properties of the brazed wheels, such as high bonding strength to matrix, good sharpness, large chip-storage spaces, and long servicing cycle compared to the conventional ones. Researchers such as Ghosh et al. [9] found that a mixture of TiN and TiB 2 is generated within the reaction layer at cBN-alloy interface of the brazed cBN wheel, leading to effective wetting and higher joint strength. It is proved that the brazed super abrasive grinding wheels are very suitable for grinding of difficult-to-machine materials such as titanium, superalloys, and engineering ceramics [10]. However, it is noticed that the wheel substrate is heated to the high temperature during the brazing procedure. Such an approach is not applicable if the wheel substrate is formed by plastically deforming metal sheet, because relief of residual stress during brazing can cause thermal deformation of the wheel substrate. This leads to the accuracy of the brazed grinding wheel being degraded greatly. As known to us, the geometric accuracy of the grinding wheel plays a key role on obtaining a high-quality ground surface. Consequently, it is necessary to control the thermal deformation of the wheel substrate in the brazing process.
High-frequency induction brazing (HFIB) technology is one of the commonly used methods in brazing of diamond or cBN wheels [11]. Many investigations illustrated that the grinding wheels fabricated with HFIB had excellent performance in the grinding of different-to-machine materials [12,13]. Besides that, with the characteristics of having rapid heating rate and local heating, the thermal deformation of brazed wheel can be reduced. Li et al. [14] developed a newly high-frequency induction brazing method. A ϕ400 mm brazed cBN grinding wheel was fabricated with the thermal deformation of the wheel substrate less than 16 μm. The grinding tests proved that the ground surface quality was better than that brazed in furnace. Unfortunately, investigations on the thermal deformation mechanism of the brazed wheel are seldom reported yet.
As a common problem, thermal deformation of steel in the heat treatment process has attracted much attention of researcher till now [15,16]. Besides the temperatureinduced deformation, phase transformation occurred in cooling process which can change the microstructure of the matrix, leading to the residual stress and deformation increasing. In recent decades, numerical simulation is widely used for investigating thermal deformation mechanism usually, considering that the heat treatment process is multi-field coupling, including temperature field, stress-strain field, and microstructure field coupling with each other. This leads to the thermal deformation of the steel materials becoming a nonlinear problem. Choi et al. [17] established a highfrequency induction hardening analysis method for AISI 1045 specimen based on electromagnetic-thermal translation analysis. A good agreement between experimental and simulation results for the heating and cooling temperatures was found, as well as the hardening depth and hardened area. Herrejón-Escutia et al. [18] developed a model to predict the formation of austenite and dilatometric behavior during continuous heating in AISI 1045 steel. Phase transformation was taken into account to increase the predicted accuracy of the model. Li et al. [19] established a thermo-elastic-plastic multi-field coupling model for the laser hardening process of the disk laser 40Cr gear steel. Perform the numerical calculations on the transient temperature, martensite phase fraction, and the transformation-induced plasticity stress evolution of the 40Cr laser hardening process. An essential theoretical basis was provided for effective prediction on quenching residual stress and optimizing industrial production parameters in this study. Esfahani [20] developed a finite element algorithm by two-way coupling of microstructural and thermal fields to investigate the microstructure evolution and stress state during continuous cooling of 1045 steel gear parts. A reasonable harmony between simulated and experimental results was observed. They found that simultaneous and separate transformations occurring between the tooth and root region have a determining role in the sign of stress for internal gear parts. Zhang et al. [21] used numerical simulation and experimental methods to investigate the transformation plasticity on the welding residual stress and distortion for AF1410 steel. They pointed out that the transformation plasticity decreased the magnitude of stress variation. According to the above researches, numerical simulation can not only make up for the deficiencies of experimental methods effectively but also provides an essential theoretical basis for multi-field coupled evolution mechanism of the heat treatment process. Gao et al. [22] built an austenite kinetics model of AISI 1045 steel for spot continual induction hardening (SCIH) process. The effect of alternating magnetic field on austenite transformation process in the case of rapid heating was investigated. The experiment results proved that the austenite transformation kinetics model developed for SCIH process was valid.
In this study, the thermal-stress-phase multi-field coupling model is established considering the temperaturedependent physical properties of the material. Phase transformation and deformation of the high-frequency induction brazed grinding wheel are investigated.

Induction brazing of super abrasive
grinding wheel procedure Figure 1 shows the induction brazing of super abrasive grinding wheel procedure. The grinding wheel is made up of ANSI 1045 steel, with the diameter of ϕ400 mm and width of 10 mm. Super abrasive (diamond or cBN) grains are arranged orderly on the wheel substrate firstly. The brazing alloy powder (Ag-Cu-Ti alloy) is spread on the grains. An induction coil consisted of a 3 mm × 3 mm rectangle copper coil and magnetizer is fixed above the wheel with the gap about 2 mm. During the induction brazing, a highfrequency alternating current is running in the copper pipe of the induction coil to heat the surface of the wheel locally. A molten pool on the wheel forms when the temperature reaching the brazing temperature about 950 °C. A transparent shield with 40 L/min argon flow is used to protect the brazing zone from oxidation. The wheel rotates slowly at the speed of 1 mm/s, making the molten pool move around the wheel. The brazing procedure is finished when the wheel takes one turn. All the induction coil and brazing area are shielded in the argon atmosphere to protect the brazing area from oxidation.

Simplified model of the heat source in induction brazing
The principle of induction heating is to generate an eddy current in a workpiece, which is then heated by the Joule heat generated by the workpiece's resistance. A typical numerical simulation is performed using the magnetic field coupled to transient thermal field [23]. When the initial temperature is known, the distribution of the eddy current in the workpiece can be calculated by electromagnetic analysis. This value is then used to compute the heat generated by the Joule effect. The temperature field of the workpiece can be calculated using the eddy current distribution as the input of thermal analysis. The thermal analysis ends when a stable node temperature is obtained.
Since the material properties of the workpiece are temperature dependent, it is necessary to frequently redivide the grid based on the coil position when performing moving induction heating simulation, which results in a long computation time and low efficiency. Due to the skin effect in the induction brazing, the heat generated by high-frequency alternating magnetic field is concentrated on the surface of the wheel substrate as shown in Fig. 2. Consequently, the heat flux in the surface of the wheel substrate can be simplified and assumed to follow a Gaussian function: [24] where q(r) represents the heat flux at a distance r from the center of the projection of the copper coil, r is the coordinate along the vertical direction of the copper coil, q m is the maximum heat flux, and r H represents the effective width of the heat ling. Considering that the heat source is determined by the distribution of the edge current, as a result, the r H is equal to the half-width of the projection of the copper coil to the wheel substrate [25], i.e., 1.5 mm in this study.
Considering that the heat source has linear distribution along the axial direction of the wheel, the total power in the wheel substrate is equal to the effective power Q of the induction coil, i.e., where L w is the width of heated surface of the wheel substrate and equals 10 mm as shown in Fig. 2a. Thus, the heat flux can be written as As a result, Equation (4) is the heat flux corresponding to the induction coil. The temperature field of the wheel substrate can be directly calculated with an acceptable accuracy. The electromagnetic calculation and grid redivision during coil movement are unnecessary. The numerical simulation efficiency can increase remarkably.

Mechanism of multi-field coupling in induction brazing process
With the characteristics of rapid heating and rapid cooling, the induction brazing is a complex process involving multifield coupling such as electromagnetics field, temperature field, stress-strain field, and microstructure field coupling with each other. Figure 3 illustrates the mechanism of multifield coupling in the induction brazing process, ignoring the electromagnetic field, considering that the heat source in induction brazing is simplified for better calculation efficiency. The most crucial coupling is between temperature and microstructure field, where the thermal coefficients continuously change as microstructure changes, because the constructed phase shows different thermal behavior when the phase transformation occurs. Moreover, the immediate result of phase transformation is releasing latent heat, but transformational strains will also be initiated in the domain [20]. The phase transformation process is often accompanied by the change of the material volume, which dominates the evolution process of the internal stress of the material. The austenite decomposed during the brazing and cooling process, resulting in transformation-induced plastic behavior. The stress leads to the nucleation rate of the microstructure, which in turn affects the phase transformation rate. The evolution of the temperature field will produce thermal deformation and thermal stress, affecting the material stress field distribution. At the same time, the material will generate plastic deformation heat during the plastic deformation process, which will affect the temperature field.

Finite element modeling
Generally speaking, the eddy current only generates at the surface of the heated area of the wheel substrate, resulting that the heat produced in the induction brazing only exists on the wheel substrate, while the heat transfers from the wheel substrate to the super abrasive and brazing alloy layer, which makes the brazing alloy melt. Therefore, the super abrasive and brazing alloy layer have no influence on the temperature distribution during induction brazing. Additionally, the thickness of super abrasive and brazing alloy layer on the brazed wheel is less than 0.5 mm [1], which is far less than the diameter of the wheel substrate. As a result, the super abrasive and brazing alloy layer are ignored in the geometry model of the finite element analysis for better calculation efficiency, since they do not have much influence on the simulation results. The geometry dimension and meshing of the grinding wheel are illustrated in Fig. 4. The diameter of the wheel is ϕ400 mm, with center bore of ϕ127 mm. The width of the working excircle is 10 mm. Owing to the simplified model of the heat source in induction brazing, a moving heat source is loaded on the surface of the wheel substrate instead of the induction coil in the finite element model.
The meshing has very remarkable effect on the simulation results. The mesh size is controlled by a seed in the Stress/strain dependence of transformation software. A refined seed is set up in the outer of the wheel which is near the heat source, to ensure the calculation accuracy. The hexahedral elements are adopted in this area, in order to avoid not converging, while a proper big seed is set up in the rest area of the wheel which is relatively far away from the heat source, in order to guarantee the calculation efficiency. The tetrahedral elements are adopted in this area. It is shown in Fig. 4 that the mesh size of the hexahedral elements is 1 × 5 × 3 mm. The total number of mesh is 50,424. The quality of meshing is checked by the software and meets the needs of calculation. The material of the grinding wheel substrate is 1045 steel, which has temperature-dependent physical property. The involved physical properties of 1045 steel are thermal conductivity, thermal strain, enthalpy, specific heat capacity, elasticity modulus, etc. The physical properties of 1045 steel are the same as those in literature [20].

Boundary condition and loading
The heat exchanges between the wheel and the surrounding environment with heat convection and radiation. The heat convection follows the Newton cooling equation: where q a is the heat exchange energy density of the wheel and the surrounding environment; h a is the convective heat transfer coefficient and is equal to 1.5 × 10 −5 W/(mm 2 ·°C); T s is the temperature of the wheel; T a is the temperature of the surrounding environment which is equal to 20 °C.
The radiation heat transfer follows the Stefan-Boltzmann law: where q r is the transferred heat by radiation; ε is the emissivity and is equal to 0.8; σ is the Stefan-Boltzmann constant, which is equal to 5.67 × 10 −8 W/(m 2 ·K 4 ).
The moving band heat source is loaded on the surface of the working excircle instead the rotational motion of the grinding wheel during the brazing process. The heat source model is according to Eq. (4). The parameters in the numerical simulation software include the power of the heat source Q and the scanning speed v of the heat source as shown in Table 1.

Temperature field
A typical temperature field of the numerical simulation results is illustrated in Fig. 5 with the power of 1500 W and scanning speed of 1 mm/s at the time of 420 s. It can be seen that the local heating is achieved in the outer of the wheel substrate with the maximum temperature of 972.8 °C. The temperature at most area of wheel substrate is below 100 °C.  In order to investigate the temperature distribution in the wheel substrate, a set of points A, B, C, and D are selected from the surface to the center along the radial direction of the wheel as shown in Fig. 6a. The distance of each point is 5 mm. The temperature field concentrates near the surface of the wheel substrate. Figure 6b illustrates the temperature variation of the four points with time varying. During the induction brazing process, the temperature of the four points is about the initial value before the heat source arrived. With the heat source approaching, the temperature of the four points rises quickly to reach the top. Finally, the temperature decreases fast to below 100 °C. Besides that, it can be seen from Fig. 6b that the maximum temperature of points A, B, C, and D is 972.8 °C, 639.9 °C, 312.6 °C, and 209.9 °C, respectively. It means that the heat affecting zone of the induction brazing is within about 5 mm of the surface of the wheel substrate.
The temperature is one of the key parameters in the brazing of super abrasive grinding wheel. Therefore, the relation between the temperature and the simulation parameters is very important for obtaining a suitable temperature during the simulation of induction brazing. The effect of the heat source power and scanning speed on the maximum temperature of point A is illustrated in Fig. 7. It can be seen that the maximum temperature has the nonlinear relation with the heat source power and scanning speed. Figure 8 illustrates the phase volume fractions and temperature of point A in Fig. 6a varying with time in the induction brazing process. The power of heat source is 1470 W with scanning speed of 1 mm/s. The brazing temperature is 950.8 °C. The phase transformation evolution primarily involves ferrite (F), pearlite (P), bainite (B), martensite (M), and austenite (A). It can be seen in Fig. 8 that the microstructure of point A consisted of 65% ferrite and 35% pearlite phase at room temperature before induction brazing. With the heat source approaching, the ferrite and pearlite begin  bainite, 0.19% martensite, and 0.01% austenite phase after induction brazing. The volume fraction of retained austenite is very low, which is ignored in the following analysis.

Phase transformation evolution
The influence of scanning speed on the phase volume fraction at point A after induction brazing is demonstrated in Fig. 9. It shows that with the scanning speed increasing from 0.5 mm/s to 1 mm/s, the volume fraction of pearlite and bainite slightly increases from 81.6% to 85.2% and 2.54% to 4.52%, respectively; that of ferrite decreases from 15.6% to 10.1%, while the volume fraction of martensite remains unchanged and is not affected by scanning speed in this study. Figure 10 demonstrates the phase volume fraction cloud chart of the wheel after induction brazing. According to the cloud chart, the phase transformation only occurs near the work surface of the wheel substrate. Most area of wheel substrate keeps the same microstructure as the initial state. Additionally, the microstructure on the work surface of the wheel substrate mainly consists of ferrite, pearlite, and bainite, as well as a small quantity of martensite. Figure 11a illustrates the austenite transformation layer during induction brazing. The depth of phase transformation layer D t is defined as the radial direction distance of austenite transformation layer within the wheel cross section. The influence of scanning speed on the depth of phase transformation layer with the brazing temperature keeping at 950 °C is shown in Fig. 11b. The D t decreases from 6.0 mm to 2.9 mm with the scanning speed increasing from 0.5 mm/s to 2 mm/s. In order to verify the microstructure simulation result, the metallographic specimen of the induction brazed wheel is prepared. The Hirox KH-7700 3D video microscope is used for observing the microstructure morphology. The result is displayed in Fig. 12. A mount of ferrite and pearlite can be found near the wheel surface in Fig. 12a. Besides that, the acicular bainite can be also observed. The acicular bainite is lower bainite, which formed in the undercooled austenite below the intermediate transformation temperature [26]. With high strength, hardness, and abrasive resistance, lower bainite can intensify the microstructure of the matrix and improve the mechanical property of the wheel substrate, while there are no evident martensite in the wheel surface. This is in accordance with the numerical simulation result in Fig. 10, considering that the volume fractions of martensite is less than 0.2% according to the simulation result. The microstructure below the surface 5 mm in the induction brazed wheel matrix consisted of ferrite and pearlite, which is the typical initial structure of the 1045 steel. Based on the simulation result in Fig. 11, the austenite transformation occurs near the wheel surface. The depth of phase transformation layer is 4.5 mm when scanning speed is 1 mm/s. This indicates that the microstructure in which depth is greater than 4.5 mm from the wheel surface keeps the same as the original state. Consequently, the microstructure morphology in Fig. 12 is consistent with the simulation result.

Stress and deformation
Numerical simulation results of the stress after induction brazing are illustrated in Fig. 13. It can be seen in Fig. 13a that the equivalent stress mainly distributes near the surface around the wheel substrate with the maximum value of 602.5 MPa. The equivalent stress at most area of wheel substrate is less than 100 MPa. Figure 13b displays the equivalent stress distribution along the white arrow direction in the wheel substrate. It reveals the fact that the equivalent stress increases slightly to the top value and then declines fast to about 20 MPa. The maximum equivalent stress occurs in the place which is about 5 mm under the wheel surface. The phase transformation that happens near the wheel surface should contribute for this phenomenon. That is to say, during the cooling process of induction brazing, the undercooled austenite transforms to generate ferrite, pearlite, bainite, and martensite. Because the volume and density of the transformed phase are different from the matrix, a lot of mechanical stress formed in the phase transformation layer. According to the simulation result in The influence of scanning speed on the phase volume fraction after induction brazing Fig. 11, the depth is about 5 mm. As a result, the maximum equivalent stress occurs in this area. Figure 14 demonstrates the influence of temperature and scanning speed on the maximum equivalent stress and the maximum deformation. It should point out that the brazing temperature keeps at 950 °C in all the cases with the scanning speed varying. Figure 14a shows that the maximum equivalent stress changes nonsignificant with brazing temperature increasing from 850 °C to 1000 °C, while it varies significantly with the scanning speed. Besides that, the maximum equivalent stress rises with the scanning speed increasing. A similar relation can be found in that of  Fig. 14b. This conclusion indicates that an appropriate scanning speed is more important for the high accuracy of the wheel substrate during the induction brazing.
In this study, the temperature in all cases is above the complete austenitic temperature. Theoretically, the equivalent stress of the workpiece is affected by the cooling speed more than the temperature. The cooling speed depended on the scanning speed. In other words, with the scanning speed raising, the cooling speed increases, leading to generating more bainite in the workpiece. So the equivalent stress and deformation increase simultaneously. This is the main reason for the above phenomenon that the maximum equivalent stress and the maximum deformation change nonsignificant with brazing temperature, while they vary significantly with the scanning speed.
The flank of the brazed wheel as shown in Fig. 4 is measured both before and after brazing by Micro-Hite DCC coordinate measuring machine. It is difficult to obtain the exact data of the excircle dimension, which is covered with super abrasive and brazed alloys. The changes in flatness of the flank are adopted to define the deformation in the induction brazing experiment. It is measured that the changes in flatness of flank 1 and flank 2 are 20.9 μm and 11.1 μm, respectively. A comparison of the simulation value and the experiment is shown in Fig. 15. It can be concluded that the simulation value is a little smaller than that of experiment The simplification of the heat source is the main reason for this error of the simulation value. On the other hand, the super abrasive and brazing alloy layer is ignored in the geometry model of the finite element analysis for better calculation efficiency. This layer generates the additional residual stress after brazing procedure. It should be the possible reason for the deformation of the wheel in experimental measurement which is larger than that of simulation. Moreover, the brazing alloy layer has more influence on the deformation of flank 1 than that of flank 2, since the former is much closer to flank 1. As a result, the simulation error in flank 1 is bigger than that in flank 2. Anyway, all the simulation errors are still within a reasonable range. Above all, the experimental results of the microstructure morphology and deformation prove that the numerical simulation model is correct.

Conclusions
Aiming at the thermal deformation mechanism of the induction brazed wheel, the temperature field, stress field, and phase transformation evolution of the high-frequency induction brazed grinding wheel were investigated based on a multi-field coupling numerical simulation model in this study. Based on the simulation results, the following conclusions can be obtained: 1. A simplified model of the heat source in induction brazing was established based on Gaussian function. The numerical simulation results proved that this model was feasible.
2. Phase transformation only occurred near the work surface of the wheel substrate. The depth of phase transformation layer decreases from 6.0 mm to 2.9 mm with the scanning speed increasing from 0.5 mm/s to 2 mm/s. The microstructure of the phase transformation layer mainly consists of ferrite, pearlite, and bainite after brazing. 3. The maximum equivalent stress changes nonsignificant with brazing temperature, while it varies significantly with the scanning speed. Besides that, the maximum equivalent stress has positive correlation with the scanning speed. A similar conclusion can be found in that of the maximum deformation. 4. The experimental results of the microstructure morphology and deformation prove that the numerical simulation model is correct with 10.4% error.