Wear Model of Silicon Nitride Ceramic Balls in Three-Body Coupling Grinding Mode

: Silicon nitride ceramic balls are the key basic components of bearings in major equipment. Their key performance indices are accuracy and batch consistency. A grinding method with the most appropriate comprehensive performance is the basis and guarantee for optimizing these performance indices. In this study, an accurate wear model was established to predict the material removal rates (MRRs) of grinding methods and improve the dynamic grinding control ability of machinery during grinding, thus enabling the mass production of high-grade silicon nitride ceramic balls. A comprehensive analysis of various grinding and polishing methods revealed that the factors affecting sphericity were mainly manifested by the increase in ball sliding and the improvement in MRRs. More over, the three-body coupling grinding mode was considered as the grinding mode that was most applicable to silicon nitride ceramic balls. The upper disk served as an external nonlinear load, and the combination of the rotating speeds of the inner and outer disks of the lower grinding disk could actively control the ball’s angle of rotation. This three-body wear mode can fully envelop the grinding trajectory and ensure uniform grinding. The traditional two-body wear model was unsuitable for three-body coupling grinding. A wear model based on three-body wear was established to predict MRRs and understand the principle of material removal in the grinding of precision spheres. Theoretical analysis and experimental veriﬁcation revealed that the MRRs of silicon nitride ceramic balls during wear are not only related to the process parameters of external load and speed but also to the physical properties and geometric parameters of balls, abrasives, and processing machinery. The wear model results of silicon nitride ceramic balls in the three-body coupling grinding mode can be obtained stably on the basis of the established wear model by removing adverse effects and adopting optimized processing parameters, thus verifying the correctness of the theoretical and simulation analyses.


Introduction
Precision spheres are the core and basic components of roundness measuring instruments, gyroscopes, precision bearings, ball screws, ball guide rails, and precision measuring instruments, which are in great demand and widely used in precision machinery, the petrochemical industry, military defense, aerospace applications, and other fields. Due to its high hardness, wear resistance, low density and thermal expansion coefficient, excellent high-temperature resistance, non-magnetic quality, corrosion resistance, high thermal conductivity, self-moistening, and excellent thermal shock resistance and oxidation resistance, silicon nitride (Si 3 N 4 ) is the preferred material for bearing balls at high temperatures, high speeds, high precision and in a special environment. it is widely used in wind turbines, high speed precision spindles, high temperature engines, and other fields [1][2][3][4][5][6][7]. The precision and batch consistency of spheres greatly affect the performance of functional precision parts [8][9][10][11]. The demand for precision spheres is increasing in industrial applications, such as precision instruments and precision machine tools, and complete sets of batch processing technology and equipment for high-efficiency precision spheres are urgently needed.
All developed countries are working hard to study new methods and processes to achieve new breakthroughs for further elevating ultra-precision machining technology. The 14th Five-Year Plan for China's machinery industry also proposes considerably improving the design and technical level of major technical equipment and markedly enhancing the international competitiveness of products. In particular, the bottleneck of key parts that restrict the development of high-end equipment should be broken. The development of precision balls, which are the key parts of supporting components in high-end equipment, is particularly important. In recent years, the market demand for precision balls has increased at a rate of 15% and reached approximately80 billion in 2021.
The batch processing of precision balls mainly follows the traditional steel ball processing method (V-groove processing method), during which the spin angle of the balls is constant [12]. The processing orientation is changed through the random movement of the circulation slot and is followed by repeated circular processing [7], which leads to low processing accuracy and poor batch consistency. Usually, precision balls are picked individually from batch-processed balls; this situation has been a difficulty perplexing the domestic and foreign fields of precision balling processing [10,11]. Moreover, production has a low automation degree and its process is affected by human factors [13][14][15][16][17][18][19].
On the basis of the National Natural Science Foundation project "Basic Research on New Grinding Method of Precision Ball" and "Basic Research on Advanced Ceramic High-efficiency Precision Machining Technology", the project team originally proposed a three-body coupling grinding method that could realize the ultra-precision grinding of balls and ensure the high precision, batch diameter variation, and batch consistency of precision ball machining in accordance with the balling principle, as shown in Figure 1.  The three-body coupling grinding method aims to control the change in the rotation angle ([−90 • , 90 • ]) of the ball blank actively by controlling the combination of the rotation speeds of the inner and outer grinding disks, such that the grinding trajectory covers the whole spherical surface evenly, achieving uniformity of the grinding trajectory on the machined spherical surface [20], effectively correcting the spherical error of the spherical surface from the balling principle, and improving the machining accuracy and batch consistency of the whole spherical surface. In previous research, the curves of inner disk rotation speed (Ω C ) and outer disk rotation speed (Ω B ) in the lower grinding disk were selected, as shown in Figure 2, and the grinding trajectory of the spherical surface inthe three-body coupling grinding mode was obtained via simulation, as illustrated in Figure 3a. The mode when the rotation speeds of the inner disk and the outer disk of the lower grinding disk are always the same is equivalent to the traditional V-groove grinding mode. The simulated grinding trajectory is shown in Figure 3b. The grinding trajectory of the three-body coupling grinding mode is all over the spherical surface, whereas that ofthe traditional V-groove grinding mode presents a girdle pattern. The related work completed by the project has proven the feasibility of the three-body coupling grinding method to a considerable extent [20][21][22][23][24][25][26]. Appl. Sci. 2023, 13, x FOR PEER REVIEW 4 of 23   In practice, however, numerous factors affect grinding accuracy. These factors include changes in processing conditions or the structural errors of grinding equipment, such that the ball does not move in the way predicted by theory during movement, which affects machining accuracy. In order to solve these problems, the factors that were more important to the grinding uniformity and material removal in actual processing were chosen and theoretically analyzed individually starting with the grinding uniformity and material removal amount that affected the balling accuracy, and their influencing laws on grinding uniformity and material removal were deduced to further ensure that precision ball processing in the three-body coupling grinding mode could obtain high accuracy and batch consistency stably and efficiently. In practice, however, numerous factors affect grinding accuracy. These factors include changes in processing conditions or the structural errors of grinding equipment, such that the ball does not move in the way predicted by theory during movement, which affects machining accuracy. In order to solve these problems, the factors that were more important to the grinding uniformity and material removal in actual processing were chosen and theoretically analyzed individually starting with the grinding uniformity and material removal amount that affected the balling accuracy, and their influencing laws on grinding uniformity and material removal were deduced to further ensure that precision ball processing in the three-body coupling grinding mode could obtain high accuracy and batch consistency stably and efficiently.

Comprehensive Performance Analysis of Precision Ball Grinding Methods at Home and Abroad
Many scholars have proposed numerous new processing methods in accordance with the characteristics of ball processing to reduce the cost and improve the accuracy, efficiency, and consistency of ball processing. These methods include the rotation angle actively controlled grinding method, three-body coupling grinding method, and magnetic fluid grinding method (MFP) [20][21][22][23][24][25][26].

Traditional V-Groove Grinding Method
The processing of various spheres mainly follows the V-groove grinding method for steel ball processing. The mechanism of this method is presented in Figure 4a. In this grinding method, the value of the rotation angle only depends on the diameters of the ball and the groove of the lower grinding disk and is independent of the rotation speed of the grinding disk; it remains almost unchanged during machining because it is very small [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. The ball can only perform the grinding movement with constant relative orientation (the angle between the rotation axis and revolution axis of the ball is constant), Appl. Sci. 2023, 13, 5796 4 of 16 and the contact points between the ball and the grinding disk form a set of grinding trajectories on the surface of the ball with the rotation axis of the ball as the center.
The processing of various spheres mainly follows the V-groove grinding method for steel ball processing. The mechanism of this method is presented in Figure 4a. In this grinding method, the value of the rotation angle only depends on the diameters of the ball and the groove of the lower grinding disk and is independent of the rotation speed of the grinding disk; it remains almost unchanged during machining because it is very small [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. The ball can only perform the grinding movement with constant relative orientation (the angle between the rotation axis and revolution axis of the ball is constant), and the contact points between the ball and the grinding disk form a set of grinding trajectories on the surface of the ball with the rotation axis of the ball as the center.

Rotation Angle Actively Controlled Grinding Mode
Kurobe et al. from Kanazawa University, Japan, proposed a rotation angle actively controlled grinding method (also known as the coaxial three-disk grinding method, as shown in Figure 5) [30][31][32][33][34][35][36][37][38][39][40]. In this method, the lower grinding disk in the V-groove grinding method is separated at the V-groove, such that the whole mechanism consists of three independently rotatable grinding disks. The rotation angle of the ball blank is adjusted by controlling the rotation speed changes of the three grinding disks. In this grinding mode, the rotation angle is not only related to geometric parameters, such as the diameter of the grinding disk and the included angle of the groove of the lower disk, but also to the rotation speed of the grinding disk. By adjusting the rotation speed, the value can be taken in the whole range of [−90°, 90°], and the ball blank can execute a grinding movement with variable relative orientation. The grinding trajectory is a spatial spherical curve ( Figure 6) with the rotation axis of the ball blank as its axis, which can cover the whole surface of the ball blank. The grinding disk grinds the ball blank in a scattered manner, which is beneficial to the uniform and efficient grinding of the ball blank surface. The experimental results show that this machining method can obtain good machining accuracy and efficiency; however, its application in production is limited by the complex mechanism of the grinding device.

Rotation Angle Actively Controlled Grinding Mode
Kurobe et al. from Kanazawa University, Japan, proposed a rotation angle actively controlled grinding method (also known as the coaxial three-disk grinding method, as shown in Figure 5) [30][31][32][33][34][35][36][37][38][39][40]. In this method, the lower grinding disk in the V-groove grinding method is separated at the V-groove, such that the whole mechanism consists of three independently rotatable grinding disks. The rotation angle of the ball blank is adjusted by controlling the rotation speed changes of the three grinding disks. In this grinding mode, the rotation angle θ is not only related to geometric parameters, such as the diameter of the grinding disk and the included angle of the groove of the lower disk, but also to the rotation speed of the grinding disk. By adjusting the rotation speed, the value can be taken in the whole range of [−90 • , 90 • ], and the ball blank can execute a grinding movement with variable relative orientation. The grinding trajectory is a spatial spherical curve ( Figure 6) with the rotation axis of the ball blank as its axis, which can cover the whole surface of the ball blank. The grinding disk grinds the ball blank in a scattered manner, which is beneficial to the uniform and efficient grinding of the ball blank surface. The experimental results show that this machining method can obtain good machining accuracy and efficiency; however, its application in production is limited by the complex mechanism of the grinding device.

Three-Body Coupling Grinding Mode
Although the rotation angle actively controlled grinding method can adjust the rotation angle of the ball to achieve a uniform machining trajectory by controlling the rotation speed changes of three independent rotating grinding disks, its structure is complicated. To reduce the complexity of the mechanism and the power source of the grinding

Three-Body Coupling Grinding Mode
Although the rotation angle actively controlled grinding method can adjust the rotation angle of the ball to achieve a uniform machining trajectory by controlling the rotation speed changes of three independent rotating grinding disks, its structure is complicated. To reduce the complexity of the mechanism and the power source of the grinding equipment, the team of the Ultra-precision Machining Research Center of Zhejiang University of Technology improved the grinding mode on the basis of the active control of the rotation angle; combined it with the process control of the rotation speed change of the grinding disk; and proposed a three-body coupling grinding mode, the principle of which is shown in Figure 7 [23]. Lv et al. and Yuan et al. [20,32] analyzed the rotation angle actually controlled grinding mode and discovered that the driving of the upper grinding disk in the grinding device was redundant and that the rotation of the upper grinding disk only increased by a relative movement velocity. A new grinding method-the three-body coupling grinding device-was proposed to reduce complexity. Figure 1 provides a schematic of the grinding structure of the three-body coupling grinding method. The upper grinding disk is under motion redundancy. The grinding movement with variable relative orientation can be realized just by controlling the rotation speed change of the lower inner and outer grinding disks, such that the grinding trajectory is evenly distributed on the ball surface. The high-efficiency and high-precision grinding of the ball blanks can thus be realized. Given that the number of driving and transmission devices is reduced from three to two, the structure of the equipment can be greatly simplified, and the precision requirements of processing and assembly are relatively reduced. This effect is highly suitable for the small-batch production of high-precision ceramic balls. Lv et al. [20] manufactured silicon nitride ceramic balls with high precision (roundness: <0.05 μm, surface roughness: Ra < 5 nm)by combining key technology and newly developed processing equipment.
In 1984, Tani and Kawata [33] proposed using the MFP for facilitating the efficient grinding of precision spheres. This method has been improved by many scholars [40,41] and has greatly enhanced the machining efficiency of precision spheres. Its principle is illustrated in Figure 8. This method uses a magnetic fluid abrasive. When a magnetic field is imposed, the magnetic fluid pushes the floating disk upward and loads the ball. The high-efficiency machining of precision balls can be realized as a result of the very high shaft rotation speed (above 10,000 rpm). However, this method has limited applications due to the high cost and short service life of magnetic fluids. Childs et al. [35][36][37] developed a non-MFP method for ceramic balls (Figure 9), in which the magnetic fluid is replaced with a cheap mixture of water and glycerol, the grinding disk is replaced with a resin-bonded diamond grinding wheel, and the self-adaptive supporting force is generated by a coil spring. Experiments demonstrated that the machining efficiency of this method is equivalent to that of magnetic fluid grinding, and the material removal rate (MRR) can be kept stable during machining [20][21][22][23][24][25][26]. In this grinding mode, the upper grinding disk does not need to rotate. The relative orientation of the ball's rotation and revolution axes can be adjusted by controlling the rotation speed changes of the inner and outer disks of the lower grinding disk, such that the grinding trajectory evenly covers the ball surface, thus achieving efficient and highprecision ball grinding. Given that the number of driving and transmission devices is reduced from three to two, the structure of the equipment can be greatly simplified, and the precision requirements of processing and assembly are relatively reduced. Therefore, the rotation angle can be continuously changed within the range of [−90 • , 90 • ]. In addition, the grinding trajectory can cover the whole ball to significantly improve the processing accuracy in the mass production of precision balls. This effect is highly suitable for the small-batch production of precision balls; meets the increasing demand for precision balls; and further compensates for the defects of the traditional processing method, such as low efficiency and high costs [37].
Lv et al. and Yuan et al. [20,32] analyzed the rotation angle actually controlled grinding mode and discovered that the driving of the upper grinding disk in the grinding device was redundant and that the rotation of the upper grinding disk only increased by a relative movement velocity. A new grinding method-the three-body coupling grinding devicewas proposed to reduce complexity. Figure 1 provides a schematic of the grinding structure of the three-body coupling grinding method. The upper grinding disk is under motion redundancy. The grinding movement with variable relative orientation can be realized just by controlling the rotation speed change of the lower inner and outer grinding disks, such that the grinding trajectory is evenly distributed on the ball surface. The high-efficiency and high-precision grinding of the ball blanks can thus be realized. Given that the number of driving and transmission devices is reduced from three to two, the structure of the equipment can be greatly simplified, and the precision requirements of processing and assembly are relatively reduced. This effect is highly suitable for the small-batch production of high-precision ceramic balls. Lv et al. [20] manufactured silicon nitride ceramic balls with high precision (roundness: <0.05 µm, surface roughness: Ra < 5 nm)by combining key technology and newly developed processing equipment.
In 1984, Tani and Kawata [33] proposed using the MFP for facilitating the efficient grinding of precision spheres. This method has been improved by many scholars [40,41] and has greatly enhanced the machining efficiency of precision spheres. Its principle is illustrated in Figure 8. This method uses a magnetic fluid abrasive. When a magnetic field is imposed, the magnetic fluid pushes the floating disk upward and loads the ball. The high-efficiency machining of precision balls can be realized as a result of the very high shaft Appl. Sci. 2023, 13, 5796 6 of 16 rotation speed (above 10,000 rpm). However, this method has limited applications due to the high cost and short service life of magnetic fluids. Childs et al. [35][36][37] developed a non-MFP method for ceramic balls (Figure 9), in which the magnetic fluid is replaced with a cheap mixture of water and glycerol, the grinding disk is replaced with a resinbonded diamond grinding wheel, and the self-adaptive supporting force is generated by a coil spring. Experiments demonstrated that the machining efficiency of this method is equivalent to that of magnetic fluid grinding, and the material removal rate (MRR) can be kept stable during machining [20][21][22][23][24][25][26].   The above analysis illustrates that the three-body coupling grinding method has obvious comprehensive advantages in terms of machining accuracy, machining efficiency, and mechanical structure (Table 1).  The above analysis illustrates that the three-body coupling grinding method has obvious comprehensive advantages in terms of machining accuracy, machining efficiency, and mechanical structure (Table 1).

Analysis of the Wear Performance of Precision Spheres
The ball surface grinding trajectory equation of the traditional V-groove grinding method was derived using ball kinematics and developed into a grinding trajectory equation suitable for other grinding methods [42]. This equation has considerable guiding importance for the research and analysis of grinding methods.

Material Removal Mechanism of Precision Spheres
Ball grinding is a process of removing materials by using abrasive particles to decrease ball diameter, improve sphericity, and reduce surface roughness. Any cross-section of an ideal sphere is a perfect circle, which is the starting point of the design of ball grinding methods. The grinding tool moves relative to the ball ( Figure 10) such that the grinding trajectory is evenly distributed on the surface of the ball blank. This situation is the geometric condition for grinding materials into a ball. A physical condition for balling is that the amount of material removed from the surface of the ball can vary with spherical deviation or can be expressed as the following two points [20][21][22][23]: (1) Equal probability of cutting: The cutting probability of every point on each spherical surface is the same.
(2) Size selectivity: In machining, large balls are ground, whereas small ones are not ground or are ground minimally. Grinding is performed along the direction of the long axis and is not performed or is lessened along the direction of the short axis. method was derived using ball kinematics and developed into a grinding trajectory equation suitable for other grinding methods [42]. This equation has considerable guiding importance for the research and analysis of grinding methods.

Material Removal Mechanism of Precision Spheres
Ball grinding is a process of removing materials by using abrasive particles to decrease ball diameter, improve sphericity, and reduce surface roughness. Any cross-section of an ideal sphere is a perfect circle, which is the starting point of the design of ball grinding methods. The grinding tool moves relative to the ball ( Figure 10) such that the grinding trajectory is evenly distributed on the surface of the ball blank. This situation is the geometric condition for grinding materials into a ball. A physical condition for balling is that the amount of material removed from the surface of the ball can vary with spherical deviation or can be expressed as the following two points [20][21][22][23]: (1) Equal probability of cutting: The cutting probability of every point on each spherical surface is the same.
(2) Size selectivity: In machining, large balls are ground, whereas small ones are not ground or are ground minimally. Grinding is performed along the direction of the long axis and is not performed or is lessened along the direction of the short axis. Combining the basic balling conditions and the machining process shows that the factors that affect balling accuracy can be classified into two aspects: the uniformity of the grinding trajectory and MRR. Previous research shows that the uniformity of the grinding trajectory is more important than other factors and that MRR is subordinate [23]. Therefore, this study explored the influencing factors of grinding uniformity and their final influence on sphericity with the expectation of providing theoretical and technical support for actual production.

Status of Research on the Wear Model of Silicon Nitride Ceramic Balls
The surface energy of ceramic balls is low, and the adhesion of grinding media and abrasives is poor. It affects the machining efficiency, surface roughness, and batch diameter variation of ceramic balls. The dimensional accuracy and surface quality of ceramic balls have a very significant impact on the life of shaft bearings, and the pursuit of dimensional accuracy and surface quality will significantly increase the manufacturing cost, so it is of great significance to study high efficiency and low-cost ceramic ball grinding and polishing processing technology. A gentle process with a high removal rate is needed to minimize surface damage and achieve high polishing speeds. This process is called magnetic float polishing (MFP). It was originally developed by Tani and Carvatta Combining the basic balling conditions and the machining process shows that the factors that affect balling accuracy can be classified into two aspects: the uniformity of the grinding trajectory and MRR. Previous research shows that the uniformity of the grinding trajectory is more important than other factors and that MRR is subordinate [23]. Therefore, this study explored the influencing factors of grinding uniformity and their final influence on sphericity with the expectation of providing theoretical and technical support for actual production.

Status of Research on the Wear Model of Silicon Nitride Ceramic Balls
The surface energy of ceramic balls is low, and the adhesion of grinding media and abrasives is poor. It affects the machining efficiency, surface roughness, and batch diameter variation of ceramic balls. The dimensional accuracy and surface quality of ceramic balls have a very significant impact on the life of shaft bearings, and the pursuit of dimensional accuracy and surface quality will significantly increase the manufacturing cost, so it is of great significance to study high efficiency and low-cost ceramic ball grinding and polishing processing technology. A gentle process with a high removal rate is needed to minimize surface damage and achieve high polishing speeds. This process is called magnetic float polishing (MFP). It was originally developed by Tani and Carvatta [39], then greatly improved by Meiyuan and Kato [40], Childs et al. [43][44][45], Raghunandan et al. [41,46], and Komandruri et al. [47,48] by adding a float to generate a uniform and high polishing force. Zhang and Childs [49][50][51] developed a non-magnetic fluid grinding process for ceramic balls. In this process, ceramic balls are supported by springs instead of a magnetic force. They also discussed the grinding ratio, magnetic force cost, and non-magnetism. Chemical mechanical polishing (CMP) was also introduced. Vora et al. [52] and Uematsu et al. [53] demonstrated the feasibility of polishing silicon nitride via chemical mechanical grinding with iron oxide. Suga et al. [54], Kikuchi et al. [55], Ming J., and R. Komanduri et al. [56] studied CMP with various abrasives and discovered that chromium oxide is the most suitable abrasive for polishing silicon nitride. Bhagavatula and Komanduri [57] analyzed the principle of material removal using a chromium oxide abrasive in finishing silicon nitride ceramic balls.
Wei W. X. et al. investigated the micro-frictional wear characteristics and damage behaviors of silicon nitride ceramic balls and ball bearing steel [ that grinding speed and pressure have great influences on the removal method and the surface defects and surface quality of ceramic balls [59].
Childs et al. [44,45] also discussed ball blank movement and MRR. The volume removal rate dV/dt of the ball blank, which depends on the sliding speed U s , was established on the basis of two-body wear, and the contact load W and the hardness Hv of the ball blank were decided by the abrasive wear principle (with K, the wear coefficient, being approximately 0.06): The above model is based on the fact that the embedded abrasive is located in the shaft surface. It indicates typical two-body wear, which is only related to process parameters and the hardness of the ball blank. In this study, the wear mode was reinvestigated, and a new three-body wear model based on the three-body coupling grinding mode was established to describe the interaction among the wafer, abrasive, and tool. The model is based on the normal distribution of the abrasive particle size on the ball blank-abrasive-tool interface and the plastic contact on the ball blank-tool interface.

Wear Classification
In general, microscale wear is a two-body wear process when the abrasive is embedded in the counter-face but is a three-body wear process when abrasive particles are not embedded and can roll between two surfaces [60,61]. The three-body process is defined in the wear system. For two-body wear, the first body is the wear of greatest concern, and the second body is the counter-face that moves relative to the first body and directly or indirectly makes contact with the first body such that force can be transmitted between the two bodies. In three-body wear, the first and second bodies are as mentioned above, and the third body, which is composed of any solid material, is completely separated from the first two bodies and may appear on the contact surface between the first and second bodies.
In recent research, the abrasive wear process in which particles are fixed to the counterface is described as groove abrasive wear, whereas the abrasive wear process in which abrasives can roll between two surfaces is described as rolling abrasive wear. Groove abrasive wear and rolling abrasive wear are used to replace two-body wear and three-body wear, respectively. For an effective abrasive wear process, the abrasive particles or the same area with a rough surface must remain in contact with the worn surface throughout the process. The worn surface caused by groove wear is characterized by grooves and is parallel to the sliding direction. The abrasive wear process in which the area of wear particles contacts the wear surface is constantly changing. The worn surface caused by rolling abrasive wear features severe deformation, multiple dented surfaces, and little or no directionality. The typical surfaces of the two wear processes are depicted in Figure 11.

Wear Classification
In general, microscale wear is a two-body wear process when the abrasive is embedded in the counter-face but is a three-body wear process when abrasive particles are not embedded and can roll between two surfaces [60,61]. The three-body process is defined in the wear system. For two-body wear, the first body is the wear of greatest concern, and the second body is the counter-face that moves relative to the first body and directly or indirectly makes contact with the first body such that force can be transmitted between the two bodies. In three-body wear, the first and second bodies are as mentioned above, and the third body, which is composed of any solid material, is completely separated from the first two bodies and may appear on the contact surface between the first and second bodies.
In recent research, the abrasive wear process in which particles are fixed to the counter-face is described as groove abrasive wear, whereas the abrasive wear process in which abrasives can roll between two surfaces is described as rolling abrasive wear. Groove abrasive wear and rolling abrasive wear are used to replace two-body wear and three-body wear, respectively. For an effective abrasive wear process, the abrasive particles or the same area with a rough surface must remain in contact with the worn surface throughout the process. The worn surface caused by groove wear is characterized by grooves and is parallel to the sliding direction. The abrasive wear process in which the area of wear particles contacts the wear surface is constantly changing. The worn surface caused by rolling abrasive wear features severe deformation, multiple dented surfaces, and little or no directionality. The typical surfaces of the two wear processes are depicted in Figure 11.

Conversion between Two-Body Wear and Three-Body Wear
The micro-scale wear test (also known as the indention abrasive wear test of ball blanks), in which one ball blank rotates around the sample in the presence of fine abrasive slurry, is currently the standard wear resistance test of coating film processing in the project [61] funded by the European Union. Figure 12 shows that the ball blank is sandwiched between coaxial rods and driven by a motor. A flat sample is vertically mounted on the rotating arm and loaded by the static weight suspended on the horizontal shaft against the ball. In the micro-scale wear test, the dominant wear mode is influenced by the external load, volume fraction of the abrasive in the slurry, wear materials, materials of the ball blank and sample, and surface conditions of the ball blank [60,61]. For all tested abrasives, the two-body groove wear mechanism is dominated by a high load and/or low slurry concentration. The mechanism dominated by a low load and/or high slurry concentration is a three-body process, in which abrasive particles are not embedded but are rolling between two surfaces, producing a severely deformed and multidented worn surface, and no obvious surface directionality exists. At a moderate load and/or slurry concentration, some wear marks present mixed characteristics, with a two-body groove in the center and three-body rolling around.

Wear Mode of Silicon Nitride Ceramic Balls
The wear conditions in the micro-scale wear test are similar to those in the grinding of silicon nitride ceramic balls. Therefore, the previous research conclusions are also applicable to the wear mode classification of silicon nitride ceramic balls in grinding processes.
Silicon nitride ceramic balls were ground under different conditions. Figure 13 shows the SEM surface images of ground silicon nitride ceramic balls [62]. These images illustrate that the dominant wear mode of the ball blank is the three-body (rolling wear) wear mode instead of the two-body wear mode. Studies have also shown that when fragile materials are used, plastic deformation occurs through dislocation. Therefore, a new model must be built to optimize the wear form of the old model on the silicon nitride ceramic ball. In the micro-scale wear test, the dominant wear mode is influenced by the external load, volume fraction of the abrasive in the slurry, wear materials, materials of the ball blank and sample, and surface conditions of the ball blank [60,61]. For all tested abrasives, the two-body groove wear mechanism is dominated by a high load and/or low slurry concentration. The mechanism dominated by a low load and/or high slurry concentration is a three-body process, in which abrasive particles are not embedded but are rolling between two surfaces, producing a severely deformed and multidented worn surface, and no obvious surface directionality exists. At a moderate load and/or slurry concentration, some wear marks present mixed characteristics, with a two-body groove in the center and three-body rolling around.

Wear Mode of Silicon Nitride Ceramic Balls
The wear conditions in the micro-scale wear test are similar to those in the grinding of silicon nitride ceramic balls. Therefore, the previous research conclusions are also applicable to the wear mode classification of silicon nitride ceramic balls in grinding processes.
Silicon nitride ceramic balls were ground under different conditions. Figure 13 shows the SEM surface images of ground silicon nitride ceramic balls [62]. These images illustrate that the dominant wear mode of the ball blank is the three-body (rolling wear) wear mode instead of the two-body wear mode. Studies have also shown that when fragile materials are used, plastic deformation occurs through dislocation. Therefore, a new model must be built to optimize the wear form of the old model on the silicon nitride ceramic ball.

Three-Body Wear Model of Silicon Nitride Ceramic Ball Grinding
The three-body wear of fragile materials in grinding has been previously stud [63]. The removal rate, Z, which is defined as the height of materials removed per u time, is given by the following formula: where n is the number of the indentation points on the circumference of rolling partic Pi is the normal load of each load-bearing particle; Lm,c denotes the average size load-bearing particles; Ew and Hw are the Young's modulus and hardness of the wo piece, respectively; KIC is the fracture toughness of the workpiece; and p and v are

Three-Body Wear Model of Silicon Nitride Ceramic Ball Grinding
The three-body wear of fragile materials in grinding has been previously studied [63]. The removal rate, Z, which is defined as the height of materials removed per unit time, is given by the following formula: where n is the number of the indentation points on the circumference of rolling particles; P i is the normal load of each load-bearing particle; L m,c denotes the average size of loadbearing particles; E w and H w are the Young's modulus and hardness of the workpiece, respectively; K IC is the fracture toughness of the workpiece; and p and v are the applied pressure and relative speed (sliding), respectively, which usually depend on the particle shape. For cone indentation, α = 0.01(cosψ) 7/6 (ψ is the included angle of indentation points). Therefore, the MRR of silicon nitride ceramic balls can be obtained as follows: In the Formula (3), p.A = W, where A is the interaction area between the ball blank and pad (in short, the tool surfaces, including the shaft, container, and float, are all called the pad here); W is the external load; and ρ w represents the workpiece density.
Assuming that the concentration of abrasive particles in the contact zone is directly proportional to the volume fraction of the abrasive in the slurry provided, the load P i carried by each abrasive particle ( Figure 14) can be obtained from the following formula [62]: where c is the proportionality constant, which varies with the abrasive type; C f stands for the volume fraction of abrasives in the slurry; and D is the long axis of particles (for example, the diameter of abrasive particles). Assuming that the macrodeformation of the ball blank and pad is elastic, and that the interaction zone A (Figure 15) is the upper zone when the spacing between the ball blank and pad is smaller than the diameter of abrasive particles, then A can be obtained using the following formula: where a is the radius of the Hertzian contact zone, and Rb denotes the radius of the ball blank. Hertzian contact circulates along the radius a as if it were caused by the contact of the ball blanks on the plane;  Assuming that the macrodeformation of the ball blank and pad is elastic, and that the interaction zone A (Figure 15) is the upper zone when the spacing between the ball blank and pad is smaller than the diameter of abrasive particles, then A can be obtained using the following formula: where E b , H b , K IC,b , and ρ w represent the Young's modulus, hardness, fracture toughness, and density of the ball blank, respectively.

Discussion
The above formula only calculates the MRR of one contact zone. However, in the three-body coupling grinding mode, three contact zones exist between the ball blank and pad. Therefore, the total MRR of a ball blank should be the sum of the MRRs of the three contact surfaces. The MRR formula consists of three parts.

Influence of Wear
The effects of geometrical α, n, size L m,c , and concentration C f are considered in this model. The lower limit of the average size of load-bearing particles is indirectly provided as L m,c ≥ 2 L m (the average size of abrasive particles). The distribution of the abrasive particle size can satisfy the normal probability density function presented in a previous study [64]. C can be calculated via the possibilistic method: When fine abrasive particles are used, especially in the polishing of the ball blank, threebody grinding is likely transformed into two-body grinding due to the ratio of the indention height to the abrasive particle size [60,61].

Influence of the Material Properties of the Ball Blank and Pad
Stiffness, strength, and other properties of materials can affect the wear rate and friction coefficient, which is one of the main causes of wear [65][66][67]. The formula includes the size, Young's modulus, hardness, and fracture toughness of the pad. These factors are all constants for a certain material.

Influence of Process Parameters
The nonlinear correlation of the MRR with the external load of each ball blank shown in this model is related to the contact zone and abrasive indentation. The sliding speed is the most uncertain factor in the formula because the ball blank on the pad slides randomly. Childs et al. [44,45] analyzed the ball blank movement and load in the magnetic fluid grinding process in detail, as shown in Figures 16 and 17. The high removal rate is attributed to the high sliding speed between the ball blank and the driving shaft. Viscosity is the most important fluid variable to be controlled. Although high viscosity promotes a high sliding speed, excessive viscosity leads to a reduction in the removal rate.

Deficiencies of the Model
This study focused on the mechanical removal mechanism but did not discuss the chemical effect on MRR. Meanwhile, the degradation of the cutting ability of abrasive particles during ball grinding was not considered. A previous study [45] found that the removal rate would decrease with operation time. In addition, some factors that might affect the movement of the abrasive and ball blank, such as the shaft rotation speed, and the profile charts of the ball blank and pad, were ignored.
In view of these factors not being covered in this paper, we will specifically elaborate in two other papers to be published.

Deficiencies of the Model
This study focused on the mechanical removal mechanism but did not discuss the chemical effect on MRR. Meanwhile, the degradation of the cutting ability of abrasive particles during ball grinding was not considered. A previous study [45] found that the removal rate would decrease with operation time. In addition, some factors that might affect the movement of the abrasive and ball blank, such as the shaft rotation speed, and the profile charts of the ball blank and pad, were ignored.
In view of these factors not being covered in this paper, we will specifically elaborate in two other papers to be published.

Experimental Study on the Removal Form of Silicon Nitride Ceramic Spheres
The contact area of a silicon nitride ceramic ball and cast-iron disk was used in the experiment. The mechanical characteristics of the two are shown in Table 2. With different abrasive particle sizes, the relationship between wear form and load

Experimental Study on the Removal Form of Silicon Nitride Ceramic Spheres
The contact area of a silicon nitride ceramic ball and cast-iron disk was used in the experiment. The mechanical characteristics of the two are shown in Table 2. With different abrasive particle sizes, the relationship between wear form and load and abrasive particle concentration is basically consistent. That is to say, the material removal form may have little relationship with the abrasive size. That is to say, in addition to the mechanical and material characteristics of the wear device itself, the wear form mainly depends on the external load and abrasive concentration and has nothing to do with the particle size of the abrasive. Therefore, according to the ceramic wear experimental conditions set in Table 3, different loads and different concentrations of grinding fluid in Table 4 were taken as conditions to test the material removal mode and corresponding MRR.

Conclusions
In this paper, the application of a three-body coupled lapping method in the batch processing of silicon nitride ceramic balls was introduced. Then, based on the traditional two-body grinding model, the relevant parameters were analyzed, and the wear model of the three-body coupled grinding mode of silicon nitride ceramic precision balls was optimized. This model can be used to predict material removal during the grinding process of silicon nitride ceramic balls. The basic idea behind this model is MRR = Z · A · ρ b . Compared with previous modeling methods based on the two-body wear model, this model not only combines the hardness, geometry, and fracture toughness of the ball billet itself, but also incorporates the abrasive concentration, mechanical motion, and structural parameters into the formula to predict the material removal rate. Through theoretical derivation and experimental verification, the appropriate load, abrasive concentration, and other related parameters can be given according to the model control, so as to obtain the required wear mode. The elimination of other adverse factors and the optimal design of the speed of the disk are also discussed in other articles.

Conflicts of Interest:
The authors declare no conflict of interest.