Study on wear simulation of diamond abrasive tool for rotary ultrasonic grinding

The aim of this study is to explore the wear mechanism of diamond grinding tools in rotary ultrasonic grinding process and complete the simulation study of micro wear state of diamond grinding tools in different processing stages. Primarily, the morphology of single diamond abrasive grains is analyzed and studied, and a three-dimensional irregular abrasive grain simulation is carried out according to the morphology characteristics of diamond abrasive grains. Then, the center distance and spatial distribution law between diamond grits are obtained by calculation, and the discrete three-dimensional morphology simulation of diamond abrasive end face is completed based on the virtual lattice theory. Through the rotary ultrasonic grinding experiment, the wear form of diamond grits and the mathematical model of the wear amount of diamond grits changing with the removal amount of workpiece material are determined. The wear simulation of diamond abrasive tool with different workpiece removal is completed by using the mathematical model obtained from experiments. Last section, the number of abrasive particles in the simulation model and the wear amount of the abrasive tool are compared and analyzed with the corresponding experimental results. The above research provides a new research method for exploring the wear mechanism of diamond abrasive tools.


Introduction
Diamond is a key raw material for the preparation of highquality abrasives with its high hardness, good wear resistance, long service life, and excellent thermal conductivity [1]. Abrasive machine is mainly composed of abrasive particles, binder, and pores three parts, therefor the abrasive particles, binder, and pores can also be called three elements of abrasive tool [2]. The main components of the grinding tools used in ultrasonic grinding are diamond particles, and the binders are mainly bronze, ceramic, and iron [3]. The diamond grinding tool has high working efficiency, less grinding tool wear, high grinding surface precision, and good finish. The use of diamond abrasive tools for grinding can largely avoid defects such as macroscopic cracks and surface burns on the workpiece to be processed. Diamond grinding tools in the alloy, ceramics, concrete, fiber-reinforced materials and refractory materials, and many other materials in the grinding process have a great demand [4]. However, the wear of diamond abrasive tools will inevitably occur during the machining process. The wear of abrasive tools will lead to the deterioration of the surface quality of the workpiece and adversely affect the life of the abrasive tools [5,6].
There have been many studies on the wear of diamond abrasive tools and the construction of abrasive tool models at home and abroad. Yuru Wu et al. [7] used abrasive belt grinding physical simulator to conduct diamond abrasive belt grinding experiments. It is found that there are four wear forms of diamond abrasive belt: grain flattening, grain falling off, dependent blocking, and adhesive blocking. Kun Zhou et al. [8] studied the relationship between material removal behavior and abrasive wear during diamond abrasive belt grinding of Cf/SiC ceramic matrix composites. Studies have shown that the diamond abrasive particles on the abrasive belt wear in the form of shedding, decomposition fracture, micro-adhesion, and abrasive clogging. Qinghong Wan et al. [9] explored the wear mechanism of single diamond grains in flexible wire titanium alloys. The intrinsic wear of diamond grains at high temperature was further studied by Raman scattering analysis, chemical thermodynamic calculation, and X-ray photoelectron spectroscopy. The experimental results show that the physical wear caused by high scratching force is the main wear mode at low temperature. In the study of wear behavior of diamond abrasive tools, Wei Shiliang et al. [10] found that the main wear form of diamond abrasive tools is abrasive wear.
CHEN et al. [11] set the abrasive particles on the abrasive tool as a spherical state, and the abrasive particles were randomly irregularly distributed, and the abrasive wear was simulated. Chakrabarti et al. [12] established the abrasive model by establishing the abrasive grain as a cone and making the abrasive grains regularly arranged on the abrasive tool. Hou et al. [13] established abrasive grains of different sizes and distributed them on the abrasive. Nguyen et al. [14] used scientific detection methods to obtain the abrasive particle parameters of the abrasive tool, and used the inverted thinking to reverse the calculation, and finally established the abrasive tool model. M. Binder et al. [15] proposed an advanced simulation method. The method combines the finite element simulation of chip formation with user-defined subroutines, and completes the tool wear simulation through program expansion. M. Binder et al. [16] studied coated tools in turning and calibrated a modified Usui tool wear model for coatings and substrates. Combined with threedimensional simulation, the wear process of local friction and wear performance change of tool matrix was simulated. Deng L et al. [17] numerically studied tool wear under stamping-hardening contact conditions and established the Archard wear model by finite element software analysis. Yu Liu et al. [18] simulated the tool wear of titanium alloy small hole by finite element method in the study of EDM titanium alloy small hole. Abrasive tool wear model is established. There is a close relationship between surface removal of hard brittle materials and tool wear. Chen Li et al. [19] studied the deformation and removal mechanism of gallium nitride single crystal during ultra-precision machining. The plastic deformation mechanism of gallium nitride crystals induced by ultra-precision machining was simulated by cross-sectional transmission electron microscopy and MD. Chen Li et al. [20] studied the deformation mechanism of YAG crystal ultra-precision grinding. A theoretical model of grinding force prediction is established, which is composed of the influence factors such as strain rate and random distribution of abrasive radius. Chen Li et al. [21] established a theoretical model of normal grinding force based on the nano-scratch test of variable cutting depth and the grain trajectory of ultrasonic vibration-assisted grinding (UVAG).
The construction of diamond abrasive particles and overall abrasive tool model is a necessary condition for wear prediction of diamond abrasive tools. At present, there are few studies on tool wear simulation in grinding process. Some scholars have also carried out research on abrasive wear and abrasive model construction in the grinding process, but few scholars have combined the two. Most scholars are still limited to single grain simulation in grinding simulation. There are few studies on the simulation of the overall abrasive tool and the wear morphology of the abrasive tool. Therefore, in order to further explore the diamond abrasive wear mechanism, the service life of abrasive should be improved. Based on the previous research on abrasive wear and abrasive model construction, the paper analyzes and studies the construction of diamond abrasive simulation model and the simulation of abrasive wear process.

Single abrasive particle simulation
In the process of rotary ultrasonic grinding, each diamond grit is a miniature grinding tool, which is the most critical component of diamond grinding tools. The three-dimensional model of single diamond grit should be studied before the three-dimensional modeling of diamond abrasive. Diamond in nature is composed of elemental carbon atoms. According to its structure, diamond is divided into equiaxed tetrahedral, hexagonal cube, and hexagonal system. The internal crystal structure of diamond is usually arranged in the form of tetrahedral atoms, and the carbon atoms are connected tightly and firmly [22].
By observing the real morphology of diamond abrasives, the morphology of diamond abrasives with different particle sizes has strong consistency and only differs in size. The larger the diamond particle size, the smaller the abrasive wear. The diamond abrasive tool with particle size number of D126 is selected for simulation modeling, and the modeling methods of abrasive tools with other particle sizes are consistent. The diamond abrasive grains on the surface of the diamond abrasive tools itself have an irregular shape, and during the rotary ultrasonic grinding process, the diamond abrasive grains will also wear and break with the continuous grinding. Therefore, no matter in the preparation process or use, the basic morphology of diamond abrasive grains cannot be determined and can only be randomly generated. In order to simulate diamond grits more realistically, the morphology of diamond grits in diamond tools was observed by scanning electron microscope. As shown in Fig. 1, it can be seen from the figure that the shape of diamond abrasive particles varies greatly, showing an irregular polyhedron shape. There are many different acute angles in diamond grains, and the distribution between diamond grains is obviously random. From the analysis of Fig. 1, it can be seen that the following four conditions should be met in the process of diamond grit simulation modeling: 1. The number of abrasive grains on the surface of the topography of the abrasive tool needs to be random. 2. The angle of the abrasive grains needs to be kept random. 3. The size of the numerical modulus of the abrasive grains needs to be kept random. 4. The arrangement position of the abrasive grains needs to be kept random.
Before using the simulation software to establish a diamond abrasive grain simulation model, we also need to determine the size of the diamond abrasive grains. Its abrasive particle size can be represented by [r min , r max ]. The upper limit of its radius is expressed as r max , and the lower limit is expressed as r min . When creating irregular diamonds, their size and volume need to be controlled within this standard. Under the premise of satisfying the above conditions, the mathematical modeling of diamond abrasive particles is realized, and the specific steps are as follows [23]: 1. Create a regular hexahedron of size L a . 2. Create a sphere of radius r in the XYZ coordinate system and make it tangent to the regular hexahedron. Specifically, as shown in Fig. 2. 3. A point P is randomly selected on the surface of the initial sphere, and a temporary coordinate system is established on the point P, so that the X ′ Y ′ plane is tangent to the outer surface of the sphere and intersects with the regular hexahedron. 4. Use any random plane obtained in the previous stage to cut the regular hexahedron. Simultaneously, when the cutting is completed, the entity formed by the cutting plane is removed, and the entity with the negative normal line is retained. Therefore, the value range of the diamond grain size is between [2r, L a ].
Taking the D126 grain size diamond as an example, the grain size of the D126 grain diamond is between 106~125 μm, then the side length L of the regular hexahedron is 125 μm, and the diameter of the initial sphere is 106 μm.
In the process of cutting the ball to establish the threedimensional shape of diamond abrasive particles, it is necessary to control the number of random cutting and mark the cutting surface with different colors. The above steps 3 and 4 are repeated, so that the model established according to the surface of the sphere gradually tends to be a polygonal entity with an irregular shape, thereby obtaining a random diamond abrasive particle model. The posture of the sphere after 50, 75, 100, 125, 150, 175, 200, 225 cuts is shown in Fig. 3.
From the cutting process of the sphere in Fig. 3, it can be found that the abrasive particles show a sharper shape when the initial cutting times are less (50, 75 cutting times). The morphology of diamond grains can not well reflect the characteristics of diamond grains. With the increase of cutting times, the morphology of diamond abrasive grains gradually approached an irregular polyhedron. At this time, the established abrasive grain morphology began to match the qualified diamond abrasive grain morphology (100, 125, 150 cuttings). When the number in cuttings continued to increase, the shape of the abrasive grains gradually approached a sphere. Concomitantly, the shape of the abrasive grains began to become distorted, and the state required by the qualified abrasive grain model (175 times, 200 times, and 225 times of cutting) could not be achieved. Through the above abrasive particle modeling  and simulation analysis, considering that the simulated abrasive particles should be as close as possible to the characteristic morphology of diamond abrasive particles, and the abrasive particle morphology should not be too close to a sphere, 100-150 cutting times are selected. The mathematical model of the number of times is the best area for the simulation model of diamond abrasive grains.

Arrangement law of diamond abrasive grains based on virtual lattice method
The virtual lattice theory is to constrain the positional relationship of each abrasive grain when establishing a diamond abrasive tool. The virtual lattice method is to divide a certain range of space into cubes of uniform size, and confine each abrasive particle to a cube of the same size. In each cube, the position of diamond abrasive grains can be adjusted by the parameters of the XYZ-axis of the center coordinate, so as to achieve the purpose of random position transformation [24]. The virtual lattice theory method not only satisfies the basic conditions of random distribution of diamond abrasive grains but also ensures that there will be no overlapping die phenomenon between abrasive grains. The specific implementation method of the virtual lattice method is as follows: 1. Using the XYZ coordinate system, random abrasive grains are established, M ij is used to represent each abrasive grain, and the original abrasive grain size is L a . 2. Repeatedly establish random abrasive grains and constrain them into the lattice. The density of diamond abrasive grains can be adjusted by adjusting the size of the confinement frame (lattice). The size of the lattice can be determined by The following formula 1 represents: The constant D represents a constant related to the density of diamond abrasive tools. La represents the original wear particle size, L represents the lattice size.
3. Through the XYZ coordinate coefficient value of the diamond abrasive grain model, it can change the position randomly to realize the random distribution of the abrasive grain position, thereby generating a new central coordinate system. The new coordinate can be expressed by the following formula 2: In the formula: a is the horizontal number of divided boxes; b is the vertical number of divided boxes; D 1 , D 2 , and D 3 are coordinate randomization parameters.
As shown in Fig. 4, the random arrangement is represented by two-dimensional abrasive particle modeling. In order to make the modeling representative, the diamond particle size of the diamond abrasive used in the experiment is  In the formula: ρ D is the average particle size of diamond abrasive grains, S D is the number of diamond abrasive grains within a certain range selected, m is the number of times of the selected calculation area, and S P is the actual area selected in a certain range.
Through scanning electron microscope photos, 18 end face areas were randomly selected, and the distribution of diamond abrasive grains was recorded. Through the diamond morphology photos, the number of diamond abrasive grains was recorded, the average value was calculated, and the average number of a single area was about 25. The area of each randomly selected surface area is 0.53 mm 2 . By calculation, the average density of the abrasive grain surface of the diamond abrasive tool is 47.17/mm 2 . According to the average density of diamond, the above virtual lattice method is used, and the diamond abrasive grain distribution model is established. The nine abrasive grains are arranged in the intersection of the square grid of the field shape, and the square side length is set to 2L, and the four small square side lengths are L, as shown in Fig. 5.
In the field-shaped grid, the spacing between diamond grits is L, and within the field-shaped area, the actual number of effective diamond grits is 4 (one complete grit, four half grits, four 1/4 abrasive grains), that is, the area of the black part in the figure. Through the field grid analysis, we can deduce 5 according to the calculation formula 4, and calculate the distance L between the actual abrasive particles.
The density of the abrasive grain surface of the diamond abrasive tools is 46.17/mm 2 . After calculation, the diamond abrasive grain spacing in the grid is 0.1471 mm. In the process of randomizing abrasive grains, it is necessary to prevent the phenomenon of diamond abrasive grains overlapping the die, so the center distance between diamond abrasive grains can be calculated by formula 6: In the formula: L ′ is the distance between the centers of the diamond abrasive grains, L is the actual distance between the abrasive grains, and D m is the average diameter of the diamond abrasive grains. According to the diamond grain size comparison table, we can know that the abrasive grain size of diamond grain size D126 selected in this paper is 106-125 μm, and the median average value is 115.5 μm. After calculation, the center distance between diamond grains is 0.2626 mm.

Discrete 3D model of abrasive tool end face
According to the virtual lattice theory, an abrasive grain model with 4 rows × 6 columns × 1 layer randomly distributed in a virtual cube grid is established to realize the mathematical modeling of diamond abrasive tools.
1. Firstly, the initial abrasive grain model is established in the rectangular range of 836 × 625 μm, and the initial arrangement model of diamond abrasive grains is established through the regularly arranged initial origin coordinate system. And generate a mathematical model of diamond abrasive grains in the initial origin coordinate system, so as to obtain the initial regular arrangement and distribution of diamond abrasive grains. At this time, the mathematical abrasive grain model and the center point of the grid are in a state of coincidence, as shown in Fig. 6. 2. By adjusting the random change of the center coordinate position of diamond grits, the random distribution of diamond grits mathematical model is realized. As shown in Fig. 7, it is a random distribution model of abrasive particles. 3. Through the establishment of a random abrasive particle model, the abrasive particles and the binder are uniformly combined for modeling. The simulation program is used to control the height of abrasive grains in the Z-axis direction to obtain the height of diamond abrasive grains. The average value of the edge height H of the diamond abrasive grains should be limited between a [0, d m /3], where d m represents the maximum value of the diamond abrasive grain size. Finally, a local model of the end face of the diamond abrasive tool is established, as shown in Fig. 8.
Adjust the observation angle of the mathematical model to observe the random distribution of diamond abrasive grains in different view directions. The shape and position of diamond grits are uncertain. Figure 9 shows the local model of the main view of the diamond abrasive tool. From the figure, it can be observed that the implant depth of the diamond Fig. 6 The initial distribution of abrasive grains is regularly arranged Fig. 7 Abrasive particle random distribution model abrasive particles in the bond is randomly distributed, and the maximum cutting edge height H is 58 μm. Figure 10 shows the partial model of the top view of the diamond abrasive tool. From the diagram, the randomness of the distribution of diamond abrasive particles on the end face of the abrasive tool can be observed, which is consistent with the distribution of abrasive particles on the surface of the real abrasive tool. Figure 11 shows the partial model of the cross section A-A in the top view of the diamond abrasive tool. The random distribution of diamond grains in the matrix can be observed from the figure. In the model, there are individual abrasive particles implanted in the matrix with a shallow depth and a sharp shape of the implanted matrix. This kind of abrasive particle is easy to fall off during processing. It can be seen that the local model structure and distribution

Ultrasonic grinding experiment platform and instruments
Rotary ultrasonic grinding experiments mainly use ordinary vertical machining centers and acoustic instruments as experimental platforms and domestic scanning electron microscopes as observation instruments. The vertical threeaxis milling machine used in this experiment is VDL-1000E produced by DMTG. The ultrasonic vibration generator and ultrasonic tool holder are CD-Tech of Tianjin University. The experimental platform and instruments are shown in Fig. 12. Taking into account the rationality of the experiment and the existing experiments, when selecting abrasive tools, choose a bronze-based diamond abrasive tool with an outer diameter of 10 mm and an inner diameter of 6 mm, which is produced by Changxing Diamond Precision Abrasives Co., Ltd.

Experimental process and result analysis
In the rotary ultrasonic grinding experiment, the diamond abrasive with diamond particle size D126 and concentration 125% was selected. The outer diameter of the abrasive tool was 10 mm, the inner diameter was 6 mm, and the total length of the tool was 84.5 mm. The diameter of the workpiece was 40 mm and the thickness was 10 mm. The rotational speed is set to 5000 r/min, the feed speed is set to 400 mm/min, and the ultrasonic vibration power is set to 90%. The processing path is reciprocating grinding motion, the depth of single processing is 0.04 mm, and the grinding width is 5 mm. For the method of removing an equal volume, the total removal depth of the workpiece is 0.4 mm. Every time 0.08 mm of workpiece material is removed, use a tool projection measuring instrument to measure the amount of wear of the tool in the axial direction relative to the initial state, as shown in Table 1.
Using the curve fitting method to construct a graph of the wear amount of the rotary ultrasonic grinding tool with the amount of workpiece material removal is shown in Fig. 13.
The curve fitting function of rotary ultrasonic grinding tool wear (T) with workpiece material removal (h) is:   Fig. 13, we can see that the wear of the diamond abrasive tool gradually increases with the increase of the workpiece material removal. When the workpiece material is removed by 0.08 mm, the abrasive tool wear amount is 0.0157 mm. The abrasive wear is relatively severe. When the workpiece material is removed by 0.16 mm, the abrasive wear is 0.0474 mm. Compared with the growing trend in the previous stage, the wear is relatively gentle at this time, and the abrasive wear enters the normal wear stage.
When the workpiece is in different material removals, there are three forms of abrasive wear related to diamond abrasive particles observed by scanning electron microscopy, namely abrasive wear, abrasive fracture, and abrasive shedding, as shown in Fig. 14.
For three kinds of wear, according to the results of ultrasonic grinding test, select three different workpiece removal amount of abrasive used for observation. The surface morphology of diamond abrasive tool was observed by scanning electron microscope, and the statistics were made according to the abrasive wear form and wear times on the surface of diamond abrasive tool. As shown in Fig. 15, the surface morphology of the diamond abrasive tool used in the observation of different workpiece removal amounts by scanning electron microscopy. The corresponding workpiece removal amount is 0.08 mm, 0.24 mm, and 0.40 mm from low to high. For the convenience of observation, the graph is binarized.
According to the experimental scheme shown above, three groups of ultrasonic grinding experiments with different workpiece removal amounts (0.08 mm, 0.24 mm, 0.40 mm) are continued. The abrasive surface used in three different workpiece removals was randomly selected for observation. The number of diamond abrasive grains in the observation range was counted, and the number of occurrences of each wear condition of the abrasive grains was recorded. The average number of abrasive particles in the same wear form under the same wear amount is taken, and the proportion of each wear form is calculated.
It can be seen from Table 2 that abrasive wear is mainly abrasive wear in rotary ultrasonic grinding process. Abrasive particle fracture and abrasive particle shedding accounted for relatively small. Abrasive breakage is accompanied by abrasive wear. During the wear process of the diamond abrasive grains, when the stress of the workpiece reacting to the diamond abrasive grains is greater than the critical stress value of the diamond abrasive grains, the diamond abrasive grains break. In the grinding process, when the grinding force is greater than the holding force of the matrix for the diamond abrasive grains, the diamond abrasive grains fall off.

Models of different wear forms of diamond abrasives
It can be seen from the above that there will be three types of wear in the process of diamond abrasive tool wear: abrasive wear, abrasive cracking, and abrasive shedding. Therefore, before establishing the wear stage morphology model of diamond abrasive tools, we should first establish three-dimensional models of three different wear forms in the wear process of abrasive tools. Then, according to the proportion of each wear form in Table 2, it is embedded in the three-dimensional model of different stages of diamond abrasive wear in the form of random distribution. Complete the morphology modeling of diamond abrasive wear stage.
In the normal wear process of diamond abrasive tools, abrasive wear is the most common form of wear. Compared with abrasive wear, abrasive cracking and abrasive shedding account for a small proportion. The process of Fig. 13 Variation of abrasive wear amount with workpiece removal continuous wear of abrasive tools can be regarded as diamond abrasive particles. So the simulation of abrasive wear can be based on the fifth-order fitting function of workpiece removal and abrasive wear (formula 1) to control the simulation of the height change of diamond abrasive particles in the Z-axis direction.
Each set of experimental cutting parameters is consistent in the process of simulating diamond abrasive wear. In order to distinguish different wear forms in the local model of the abrasive tool below, the abrasive wear part is characterized. The abrasive wear model in the process of abrasive wear is shown in Fig. 16.
In Fig. 16, the left picture is the original abrasive wear image constructed by simulation, the right picture is the characteristic treatment picture, and the light white area is the abrasive wear part.
During the wear process of diamond abrasive tools, in addition to abrasive grain wear, abrasive grain rupture is also accompanied. The abrasive fracture can be understood as the brittle fracture of abrasive grains in the process of abrasive wear. In the process of simulation, in addition to controlling the height change of abrasive particles in Z-axis direction, it is also necessary to adjust the position change of X-axis and Y-axis to simulate the abrasive particle breakage model. The abrasive fracture model is embedded into the diamond grinding tool using a random replacement command. Similarly, in order to distinguish different wear forms in the local model of the abrasive tool, the fractured part of the abrasive grains is characterized. Figure 17 shows the abrasive fracture model in the process of abrasive tool wear.
In Fig. 17, the left picture is the original abrasive grain fracture image constructed by simulation, and the right picture is the characteristic processing picture. The light white area is the abrasive grain wear part, and the black area is the abrasive grain broken part.
Abrasive wear forms also include abrasive shedding. Abrasive particle shedding accounts for a small proportion in abrasive wear. Therefore, it is difficult to observe the phenomenon of abrasive particle shedding during local observation. Part of the abrasive particles fall off because of falling off is not complete can be classified as abrasive fracture. Complete shedding of abrasive grains can be expressed as pits in the base part of the abrasive tool. The impact of abrasive particle cracking and abrasive particle shedding on abrasive tool wear is to increase the abrasive tool wear rate. Since the impact of abrasive particle fracture and shedding on the wear of abrasive is very small, so in the normal wear stage, abrasive simulation process does not consider the impact of both the amount of wear. In order to make the abrasive wear simulation model closer to the actual morphology, the abrasive particle breakage and shedding are given to the simulation program according to the proportion of experimental data. Thus, a wear model with complete surface morphology is obtained.

Modeling of abrasive topography at different wear stages
The diamond abrasive tool is in the process of continuous wear of diamond abrasive grains in the normal wear stage.
The above abrasive wear forms are substituted into the three-dimensional model of the abrasive tool according to the proportional relationship in Table 2 to construct the wear stage model of the diamond abrasive tool. It is known that the surface density of the abrasive grains of the diamond abrasive tool is 46.17/mm 2 . The surface size of the model matrix is consistent with the size of the diamond surface morphology obtained in the experiment. The length is 0.836 mm and the width is 0.625 mm. The model should include 24 abrasive grains. As shown in Fig. 18, it is a three-dimensional model of the local wear of the diamond abrasive tool when the workpiece removal amount is 0.16 mm during rotary ultrasonic grinding. In this stage, the main wear forms of abrasive tools include abrasive grain breakage and abrasive grain shedding in addition to abrasive grain wear. The abrasive particle wear and the abrasive particle crushing are consistent with the above description, and the abrasive particle shedding is characterized by pits on the abrasive matrix. Abrasive wear area is consistent with Fig. 15. Figure 19 shows the main view of the surface morphology simulation of the abrasive tool used in ultrasonic grinding when the workpiece removal is 0.08 mm. From the figure, the simulation results relative to the main view of the local model of the complete diamond grinding tool shown in Fig. 9 can be observed. When the workpiece removal is 0.08 mm, the wear of diamond abrasive is 0.0143 mm.
According to the wear statistics table of the abrasive tool end face, a three-dimensional model of local wear of  The corresponding diamond grinding tool simulation when the workpiece is removed by 0.08 mm Fig. 19 The corresponding diamond grinding tool simulation front view when the workpiece is removed by 0.08 mm diamond abrasive tool is established when the workpiece removal amount is 0.24 mm during rotary ultrasonic grinding. Compared with the previous wear, the proportion of abrasive cracking and abrasive falling off has increased, but the increase is small. The mathematical model of abrasive wear is shown in Fig. 20. Similarly, the area of abrasive wear area is the same as that in Fig. 15, which remains consistent. Figure 21 shows the main view of the surface micro-morphology simulation of the abrasive tool used when the workpiece removal amount of ultrasonic grinding is 0.24 mm. From the figure, the simulation results relative to the main view of the local model of the complete diamond grinding tool shown in Fig. 9 can be observed. When the workpiece removal is 0.08 mm, the wear of diamond abrasive is 0.0298 mm.
According to the wear statistics table of the abrasive tool end face, the three-dimensional model of the local wear of the diamond abrasive tool when the workpiece removal amount is 0.40 mm during the rotary ultrasonic grinding process. The proportion of abrasive particles breaking down has decreased, and the proportion of abrasive particles falling off has begun to rise. The mathematical model of abrasive tool wear at this stage is shown in Fig. 22. Similarly, the area of abrasive tool wear is consistent with that in Fig. 15. Figure 23 shows the main view of the simulation of the surface micro-morphology of the abrasive tool used when the workpiece removal amount of ultrasonic grinding is 0.24 mm. The simulation results of the main view of the partial model show that when the workpiece removal amount is 0.08 mm, the wear amount of the diamond abrasive tool is 0.0503 mm.

Comparison and verification
The three-dimensional micro-morphology simulation experiment of diamond abrasive wear provides a new research method for rotary ultrasonic grinding simulation. The simulation model of normal wear stage of diamond abrasive tool can replace the simulation of single diamond abrasive particle, which makes the simulation research environment of rotary ultrasonic grinding closer to the real situation. The reliability of the diamond abrasive wear simulation research method needs to be further verified. The number of abrasive particles in the abrasive model is compared with the number of abrasive particles obtained experimentally, as shown in Fig. 24. From Fig. 24, we can see that the number of abrasive particles in the wear simulation model is very close to the number of abrasive particles obtained in the experiment. The difference between the number of diamond abrasive grains obtained by simulation and experiment is 1, 2, 4 respectively, and the error is within the acceptable range.
It is difficult to verify the reliability of the simulation by the number of abrasive grains. Furthermore, the wear degree of the diamond local wear model established by this research method is compared with that of the unprocessed diamond abrasive local model, and the abrasive wear value

Conclusions
Rotary ultrasonic grinding experiments show that the main wear forms of diamond tools are abrasive wear, abrasive fracture, and abrasive shedding. Based on the experimental research, the single diamond grain model, the local model of diamond grinding tool end face, and the wear model of diamond grinding tool in rotary ultrasonic grinding experiment are established by simulation.
1. Three-dimensional model of irregular diamond grits was obtained by sphere cutting method. The diamond grit model is most consistent with the actual diamond grit morphology in the range of 100 times to 175 times of cutting. 2. Based on the virtual lattice theory, a microscopic threedimensional simulation model of diamond grinding tool end face is established, which lays a foundation for the construction of diamond grinding tool wear simulation model. 3. The mathematical model of the variation of diamond abrasive wear with the removal of workpiece material was obtained by rotary ultrasonic grinding experiment. 4. The error between the wear amount of diamond abrasive obtained by simulation and the wear amount of diamond abrasive obtained by experiment is 8.9%, 4%, and 6.1% respectively. The experimental error is small, and the simulation has certain reliability.