Effect of crack propagation on surface formation mechanism and surface morphology evaluation of longitudinal–torsional composite ultrasonic mill grinding of Si3N4

Crack propagation is critical in determining the surface forming process and machined surface quality of hard brittle materials. However, there is still a lack of research on this subject. In this work, the effect of crack propagation on the surface formation mechanism and surface morphology of silicon nitride ceramics was investigated via longitudinal–torsional composite ultrasonic-assisted mill grinding, and a reconstruction model of the machined surface morphology considering crack propagation was proposed. This model was quantitatively characterized and evaluated by the average roughness Sa and the kurtosis Sku. It was found that the simulation results considering crack expansion are in good agreement with the experimental results. The average relative errors in the average roughness and kurtosis were found to be 7.95% and 9.46%, respectively. A primary effect analysis was performed to understand the influence of the process parameters on the machined surface morphology. It was found that ultrasonic vibrations lead to changes in the shear angle and shear velocity of abrasive grains, thereby changing the machined surface morphology. The results presented here provide a practical method for predicting and controlling the machined surface quality during precision machining of ceramic materials.


Introduction
Silicon nitride ceramics have excellent physical and chemical properties, including good wear resistance, high-temperature resistance, and corrosion resistance. They have a broad potential for applications in the aerospace, machinery manufacturing, medical, and healthcare fields. However, the development and application of these ceramics are seriously hampered by many problems, such as their low machining efficiency, poor machining quality, and high susceptibility to damage [1][2][3]. Numerous studies have indicated that ultrasonic-assisted milling is one of the most effective machining methods to efficiently process ceramic materials with limited damage induced to the samples [4][5][6]. Nevertheless, research on the surface formation mechanism and the evaluation of the machined surface quality of ceramic materials subject to ultrasonic-assisted milling is still lacking, which restricts the engineering application of ceramic materials.
Extensive studies on the surface formation mechanism of materials processed via ultrasonic-assisted grinding have been conducted [7][8][9]. Li et al. [10] investigated the surface formation mechanism of ceramic-based composites using ultrasonic-assisted scratching tests. They found that ultrasonic vibration effectively increased the critical cutting depth of the brittle-plastic transition and improved the surface integrity of the material; these findings were obtained through the analysis of the scratching force, friction factor, and scratch morphology. Sun et al. [11] also found that ultrasonic vibrations effectively increased the critical depth of the brittle-plastic transition, enhanced the percentage of plastic area removal, and consequently improved the surface quality of the machined sample.
Heike et al. [12] found that brittle removal was the primary removal mode of a material under the influence of ultrasonic vibrations in ultrasonic-assisted scratching tests.
In order to further investigate the effect of the interaction between multiple abrasive grains on the surface formation mechanism, double scratch tests have been undertaken [13][14][15]. Using such techniques, Yang et al. [16] showed that the interaction between the scratches was inversely proportional to the distance between them. When the scratch separation distance was small, the cracks produced by the two abrasive grains interacted; in this manner, the material between the scratches was removed. On the other hand, when the scratch separation distance was large, the cracks did not interact with each other, and no material was removed in the region between the scratches. Gu et al. [17] investigated the effect of the scratch distance on the material removal rate. They found that the material removal rate between adjacent scratches became maximum when the scratch separation distance reached a critical value. Qiu et al. [18] also demonstrated that the material removal between two scratches depended on the interaction of radial and lateral cracks. In a previous study [19], it was also found that ultrasonic vibrations promoted the expansion of lateral cracks, which led to the joining of lateral cracks in adjacent scratches, and the material between the scratches was removed due to the connection of the cracks that originated from adjacent scratches. This material removal has a significant impact on the machined surface morphology.
It is difficult to characterize and evaluate the machined surface morphology due to the influence of crack propagation on the formation process of the machined surface. The effect of the process parameters on the machined surface morphology has been investigated. Wang et al. [20] performed an experimental study on quartz glass and discussed the feasibility of the longitudinal-torsional composite rotary ultrasonic-assisted machining of brittle materials. The experiment showed that the introduction of ultrasonic vibrations can effectively reduce the surface roughness and improve the surface finish of the workpiece. Wang et al. [21] explored the influence of ultrasonic vibrations on the surface morphology of composite materials via ultrasonic-assisted grinding experiments. The results of this work indicated that the surface morphology was affected by ultrasonic vibrations, and the surface roughness was reduced when the ultrasound amplitude and frequency were small. Zhang et al. [22] also found that ultrasonic vibrations effectively reduced the machined surface roughness by 9% compared with conventional grinding. Ping et al. [23,24] discovered that increasing the grinding wheel speed improved the surface quality and reduced the machined surface roughness. Wang et al. [25] concluded that the surface quality was primarily determined by brittle fractures, grooves, and pits on the processed material surface.
Evaluating the surface morphology during production is a time-consuming and laborious approach. Performing a digital reconstruction and characterization of the machined surface morphology is more efficient and convenient. Jiang et al. [26] designed a digital grinding wheel model, established a three-dimensional (3D) morphological model of the machined surface by integrating the trajectory of the abrasive grains and the number of abrasive grains, and verified the accuracy of the model through experiments. Yan et al. [27,28] proposed a method to calculate the residual material height on the surface of nano-zirconia ceramics via two-dimensional (2D) ultrasonic vibration-assisted processing. In this method, a 3D roughness prediction model was proposed, and experiments were conducted to verify the validity of the model. Zhang et al. [29] built a model to predict the 3D morphology of machined surfaces for the longitudinal-torsional composite ultrasonic machining of silicon nitride ceramics. This model was experimentally verified to have an error range in the 3D surface roughness of 0-14.07%. Wang et al. [30] introduced a method to predict the 3D morphology of elliptical ultrasonicassisted grinding surfaces. This method studied the material removal mechanism during elliptical ultrasonic-assisted machining using various ultrasonic vibration parameters. Ding et al. [31] developed a predictive model for the machined surface morphology and investigated the effect of the morphology of the grinding wheel on the machined surface roughness. Their results showed that the primary factors affecting the surface roughness include the number of abrasive grains as well as the average value and standard deviation of the undeformed chip thickness. Li et al. [32] developed a simulation model for the grinding process of the surface of single-crystal silicon considering the plastic removal mode of the material. This simulation model was found to predict the proportions of removed material based on the brittle mode and the plasticity mode as well as the roughness of the machined surface and the resulting cutting marks on the machined surface.
The material brittle-mode removal induced by crack propagation during the ultrasonic-assisted processing of ceramic materials significantly influences the surface formation process and the microscopic morphology of the resulting machined surface. However, the effect of crack propagation is not considered in current surface morphology prediction models.
In this study, focusing on the above problems, the following topics were investigated: Firstly, a kinematic analysis of the longitudinal-torsional composite ultrasonicassisted mill-grinding process was undertaken to establish the kinematic equations of the grinding process. Secondly, the influence of crack propagation on the formation of the machined surface was studied. Thirdly, a digital model of the surface morphology of the grinding wheel was established via the construction of a grain model and the definition of relevant parameters of the abrasive grains. Fourthly, a crack propagation model and algorithm that predict the machined surface contour are proposed to evaluate the 3D surface morphology. Finally, the model was experimentally verified, and the influence of ultrasonic vibrations and other process parameters on the machined surface morphology was analyzed.

Single-grain trajectory
The key principles behind the longitudinal-torsional composite ultrasonic-assisted mill-grinding process are shown in Fig. 1(a). The motion of the grinding wheel is described by a rotational motion, feed motion, axial ultrasonic vibrational, and circumferential ultrasonic vibration. In order to simplify the analysis presented here, the motion of the grinding wheel is decomposed into axial and torsional ultrasonic-assisted mill-grinding components. Figure 1(b) shows the trajectory of a single grain during processing; the trajectory equation of a single grain during the axial ultrasonic-assisted mill-grinding process can be written as follows: where R w is the distance from the grain to the grinding wheel center, n is the grinding wheel speed, v f is the feed speed, A is the axial amplitude of the ultrasonic vibration, f is the frequency of the ultrasonic vibration, and t is the processing time.
(1)  where a is the torsional ultrasound amplitude.
By coupling the longitudinal and torsional ultrasonicassisted mill-grinding motions, the trajectory of the grinding grain in the longitudinal-torsional composite ultrasonicassisted mill-grinding process can be obtained, as shown in Fig. 1(d). The equation describing this trajectory can be written as: The cutting speed of the abrasive grain at any time t is obtained by taking the derivative of Eq. (3); this yields the following expression:

Surface formation process
From Fig. 1(d), it can be seen that when the abrasive grain scratches the surface of the workpiece, part of the material is removed, and a scratch is left on the workpiece surface. After all the effective abrasive grains on the grinding wheel have scratched the surface of the workpiece, numerous scratches intersect each other and form a new surface morphology. It is known from indentation fracture mechanics that when the abrasive grains contact the workpiece, a crack system is formed within the workpiece material. The cracks connect with each other during propagation, causing another type of material removal. The two types of material removal processes mentioned above jointly determine the morphology of the machined surface. The formation process of the longitudinal-torsional composite ultrasonicassisted mill-grinding surface is shown in Fig. 2. Figure 3 shows how material removal occurs due to the intersection of lateral cracks. The material removal area after the workpiece is scratched by an abrasive grain k is shown in Fig. 3 (indicated by the symbol " I"). At the same time, median and lateral cracks expand below the abrasive grain scratch. Similar material removal and crack propagation processes occur when the abrasive grain k + 1 scratches the workpiece surface. When the distance between adjacent abrasive grains is less than a critical value, the lateral cracks generated by two abrasive grains give rise to crack bridging. This crack bridging phenomenon leads to material removal in the area between the two scratches, as shown in Fig. 3 (indicated by the symbol " II"). Based on the above analysis, it can be concluded that type-II material removal occurs when the length of the lateral cracks and the distance between adjacent abrasive grains meet specific conditions. Therefore, when studying the formation mechanism of the machined surface, these conditions should be first established.
According to indentation fracture mechanics, the lateral crack length C L can be written as [33]: where ψ is the half-top angle of the abrasive grain, and K ID is the dynamic fracture toughness of the work material. In ultrasonic vibration-assisted machining, K ID ≈ 0.3 K IC [34], where K IC is the fracture toughness of the material, H V is the Vickers hardness of the material, E is Young's modulus, v is Poisson's ratio, F k is the instantaneous scratching force of the kth abrasive grain, and C is a constant associated with the shape of the abrasive grain (C = 0.226) [35,36].

Fig. 3 Schematic illustrating the material removal process
The instantaneous scratching force F k of the abrasive grain k is related to the grinding depth according to [37]: where h(t) is the grinding depth at time t.
The relationship between the lateral crack and the scratch depth can be obtained by combining Eqs. (5) and (6) as: When the center distance between two adjacent abrasive grains is less than two times the lateral crack length (d k ≤ 2C L ) [19], type-II material removal occurs. Thus, Eq. (7) can be written as:

Grinding wheel model
The surface morphology of the grinding wheel is also a critical factor affecting the machined surface of the workpiece. Three parameters are required to describe the surface morphology model of the grinding wheel: (1) the number of effective abrasive grains K, (2) the characteristic parameters of the abrasive grains (R: distribution diameter; θ: initial phase angle), and (3) the profile dimensional characteristic parameters of the abrasive grains (h: long axis of the abrasive grain; r: short axis of the abrasive grain).
From a statistics point of view, the number of effective abrasive grains on the grinding wheel surface can be calculated using the following equation [38]: where N v is the number of grains per unit wheel volume, S w is the area of end face of the grinding wheel, d g is the diameter of the abrasive grain, a p is the cutting depth, δ = d g max − d g min is the distribution range of the abrasive grain diameter, and N is the tissue number of the grinding wheel.
Generally, abrasive grains are randomly dispersed on the surface of the grinding wheel, and the shape and dimension of different abrasive grains are also varied. In order to accurately reflect the random distribution characteristics of abrasive grains, the initial phase angle θ and the distribution diameter R are defined to describe the location of the abrasive grains on the grinding wheel surface (where 0 ≤ R ≤ 5000 μm and 0 ≤ θ ≤ 2π), as shown in Fig. 4(a). The distribution radius R and the initial phase angle θ of K abrasive grains are randomly generated and denoted as {R K } and {θ K }, respectively. θ k and R k are the initial phase angle and distribution radius of the kth abrasive grain, respectively. The distribution model of the abrasive grains on the grinding wheel surface is thus obtained. The morphology of the residual scratches on the surface of the workpiece is closely related to the outer edge profile of the abrasive grains. In this study, the outer edge profile of the abrasive grain is assumed to be semi-elliptical, as shown in Fig. 4(b), to optimize the calculation accuracy and efficiency. The long-axis h k and short-axis r k of the grinding grain profile are: where r k is the short axis of the kth abrasive grain (μm), h k is the long axis of the kth abrasive grain (μm), r g and h g are constants related to the grain dimensions of the grinding wheel, and ξ k and ζ k are the deviations of the kth grain in the shortand long-axis directions, respectively (they follow a normal distribution). Two arrays of length K are randomly generated; they are denoted as {ξ K } and {ζ K }. Thus, the long-axis and short-axis values of the K grinding grains can be denoted as A local coordinate system o′x′y′z′ is established to construct the grain profile equation. The maximal profile of the kth abrasive grain is discretized into M segments, with the interval between two adjacent discrete points being Δφ and the index of the discrete points being m = 1, 2, 3, …, M, as shown in Fig. 4(b).
In the local coordinate system, the coordinates of the mth discrete point (x′ k_m , y′ k_m , z′ k_m ) on the abrasive grain contour at t = 0 can be written as:

Surface profile reconstruction algorithm
A surface profile reconstruction algorithm is proposed to predict the machined surface morphology of the workpiece. The algorithm was divided into three steps: Firstly, the coordinate value of each discrete point on the grain profile in the global coordinate system was calculated using a coordinate transformation. Secondly, the abrasive grain profile was combined with the motion trajectory of the abrasive grain to calculate the 3D spatial coordinates of the abrasive grain during the cutting process. Finally, the minimal value of each mesh point corresponding to the z-axis was calculated.
(10) where r k and h k are the lengths of the short and long axes (μm, r k ∈ {r K } h g ∈ {h K }), respectively, m is the index of the discrete points on the grain outline, Δφ is the interval angle between the discrete points, and θ k is the initial phase angle of the kth abrasive grain (θ k ∈ {θ K }).
(2) Trajectory of the discrete point in space: As shown in Fig. 6(a), the original coordinate point was set on the workpiece surface. The location of the grinding wheel was adjusted to ensure that the z-direction coordinate z g of the lowest point of the abrasive grain profile is A at t = 0. The projection coordinates of any discrete point on the z-axis can be described as:where r max is the maximal height of the abrasive grain.
Similarly, when the cutting depth is a p , the height of any discrete point on the abrasive grain profile can be written as: During the mill-grinding process, the workpiece is cut by the maximal outline of the abrasive grain. Therefore, it is necessary to ensure that the grain outline is perpendicular to the cutting speed direction when reconstructing the machined surface morphology, as shown in Fig. 6(b). Considering the presence of ultrasonic vibrations and the rotational motion of the tool, the phase angle θ k (t) of the profile of the kth abrasive grain at any time t can be expressed as: where n is the speed of the grinding wheel, a is the torsional amplitude, f is the ultrasonic frequency, and t is the cutting time.
During cutting, the motion trajectory of each discrete point on grain outline can be written as: By combining Eqs. (3), (12), and (14)-(16), the trajectory of the abrasive grain contour can be expressed as: Then, the 3D morphology SP of the workpiece surface can be expressed as:

Surface morphology with crack propagation
The above analysis of the formation mechanism of the machined surface shows that ceramic materials are mainly removed in a brittle pattern in longitudinal-torsional composite ultrasonic-assisted mill grinding. This means that crack propagation has a crucial influence on the machined surface morphology. The following is a detailed analysis of the reconstruction method of the machined surface morphology considering crack propagation.
From Eq. (8), it can be seen that the interaction between cracks is influenced by the separation distance between the grains and the length of the lateral cracks. Therefore, the first step is to determine whether the separation distance between two adjacent abrasive grains during the movement of all abrasive grains satisfies the condition of type-II material removal. Figure 8(a) shows how the separation distance between adjacent grains is calculated. Suppose that at time t = 0, the initial phase angles of abrasive grain k and abrasive grain k + 1 are θ k and θ k + 1 , respectively. Then, at time t = t k + 1 , the phase angle of abrasive grain k + 1 reaches k . The time it takes for the phase angle of the k + 1 abrasive grain to change from θ k + 1 to θ k can be expressed as: By substituting Eq. (20) and the cutting depth a p of the abrasive grain into Eq. (8), it can be determined whether the material removal due to crack propagation occurs at this moment.
During the actual machining process, the propagation of lateral cracks is subjected to a certain degree of randomness. For the ease of calculation, it is assumed that the lateral crack extends outward from the lowest point of the abrasive grain profile, and the angle between its propagation direction and the horizontal plane direction is denoted as γ, which can be obtained experimentally.  We can obtain the coordinates of the lowest point on the z-axis of abrasive grain k + 1 and abrasive grain k, (x 1 k + 1_3π/4 , y 1 k + 1_3π/4 , z 1 k + 1_3π/4 ) and (x 1 k_3π/4 , y 1 k_3π/4 , z 1 k_3π/4 ), respectively, in the time range in which the conditions are met. In this way, the coordinates of the first crack intersection point L 1 k+1 (x 1 k + 1_L , y 1 k + 1_L , z 1 k + 1_L ) of the lateral cracks of abrasive grain k + 1 and abrasive grain k are calculated as: After determining the coordinates of the crack initiation and intersection points, the three points are connected by a straight line to approximate the morphology of the machined surface caused by crack propagation, as shown in Fig. 8(b). Similarly, at time t k + 1 + Δt, the coordinates of the three key points of the crack model can be written as (x 2 k_3π/4 , y 2 k_3π/4 , z 2 k_3π/4 ), (x 2 k + 1_3π/4 , y 2 k + 1_3π/4 , z 2 k + 1_3π/4 ), and (x 2 k + 1_L , y 2 k + 1_L , z 2 k + 1_L ). Finally, the crack model was calculated using a MAT-LAB-compiled program for each time step Δt within the processing time t, and the spatial coordinates x′, y′, and z′ through which the cracks pass are denoted as {X′},{Y′}, and{Z′}, respectively.
Using {XX′}{YY′}{ZZ′} to find the height z′ ij corresponding to the grid point (x i , y i ), the height of any point of the 3D morphology of the workpiece surface considering crack propagation can be expressed as: Then, the 3D morphology SP a of the machined surface of the workpiece considering crack propagation can be expressed in matrix form as: The machined surface morphology of the workpiece described by Eq. (23) can be solved using the algorithmic process shown in Fig. 9.

Experimental scheme
In this work, longitudinal-torsional composite ultrasonicassisted mill-grinding experiments were conducted on silicon nitride ceramics to verify the validity of the above reconstruction model. This experiment is characterized by four input variables: ultrasonic amplitude, feed rate, depth of cut, and spindle speed. Each input variable has four possible values. The average roughness (S a ) and kurtosis (S ku ) were used as response variables, and the standard orthogonal table L 16 4 5 was used for the experimental design, as shown in Table 1.
The workpiece specimen used in the test was a hot-pressed silicon nitride ceramic produced by Kexing Special Ceramics. The workpiece size was 20 × 20 × 10 mm 3 , and the performance parameters of the workpiece are shown in Table 2.

Experimental equipment
The machine tool used in the grinding test was a precision CNC milling machine (XK7124, China) with a repetitive positioning accuracy of ± 2 μm and a maximal spindle speed of 8000 r/min. The ultrasonic device was installed on the machine tool spindle, and the ultrasonic vibrations were a combination of ultrasonic vibrations along the spindle axial and in the circumferential direction, with the ultrasound vibration frequency ranging from 0 to 50 kHz and the ultrasound amplitude ranging from 10 to 20 μm. The tool was a metal-based sintered diamond wheel with a diameter of 10 mm, a granularity of 80#, and an abrasive concentration of 100%. A precision clamp was mounted on the machining platform of the precision CNC milling machine as a fixture, and the workpiece was then fixed using the fixture, as shown in Fig. 10.
After the experiment, the workpiece was placed in anhydrous ethanol and ultrasonically cleaned for 5 min. Subsequently, the 3D morphology-related parameters and surface morphology features of the machined surface were extracted using an ultra-deep field digital microscope (Smartzoom 5). In order to ensure the accuracy of the measured parameters, five sampling areas were randomly selected on the machined surface for each test, and the average of the measurement results of the five areas was taken as the final measurement result of the 3D surface roughness of the machined surfaces.

Effect of cracks on the 3D morphology
The measurement and simulation results of the 3D morphology of the workpiece surface under different processing conditions are shown in Figs. 11 and 12. The figures indicate the differences between the measurement results and the simulation results of the two models without and with the influence of cracks. Figure 11(a), (b) and Fig. 12(a), (b) show the experimentally measured 3D morphology of the workpiece surface for different processing parameters. It can be seen from the figures that regular circular scratches are left by the abrasive grains on the workpiece, which are caused by the type-I material removal that occurs during the circular motion of the abrasive grains. From Fig. 11(a), it is found that when A = 0, the scratches on the workpiece surface are deep and narrow, and they are not significant fluctuant. By contrast, when A ≠ 0, the circular scratches on the surface of the workpiece are shallow and wide, and they are regularly interrupted due to the applied ultrasonic vibrations. At the same time, clear traces left by the removed material are visible between the scratches, indicating that the propagation of lateral cracks causes the occurrence of type-II material removal between the scratches. The above results are consistent with the conclusion of our previous studies, according to which ultrasonic vibrations promote lateral crack propagation [19]. Figure 11(c), (d) and Fig. 12(c), (d) show the simulated morphological features without considering the effect of the cracks for different machining parameters, whereas Fig. 11(e), (f) and Fig. 12(e), (f) show the corresponding simulated morphological features when considering the effect of the cracks. By comparing the two types of simulated morphological features, it can be observed that the simulations that consider the effect of the cracks result in lower residual material heights on both sides of the grooves, which are in better agreement with the experimental results (Fig. 11(a), (b) and Fig. 12(a), (b)). According to statistical calculations, the values of the 3D surface roughness determined experimentally, obtained from simulations when considering cracks, and obtained from simulations without considering cracks (see Fig. 11(b), (d), and (f), respectively) are 1.56, 1.69, and 1.77 μm, respectively. Furthermore, the values of the 3D surface roughness determined experimentally, obtained from simulations when considering cracks, and obtained from simulations without considering cracks (see Fig. 12(a), (c), and (e), respectively) are 1.79, 1.71, and 1.97 μm, respectively. Thus, from both the image characteristics and the quantitative evaluation results, it is inferred that the simulation model considering the influence of cracks has better prediction accuracy than the model that does not take into account the influence of cracks.

Evaluation of the 3D morphology
In order to further verify the validity of the above simulation model of the surface morphology, two 3D morphology characterization parameters, namely the average roughness S a and kurtosis S ku , were selected as quantitative evaluation  parameters. The experimental and simulation results of the evaluation parameters were obtained for different processing parameters, and the S a and S ku of the simulation model were calculated. The results are shown in Table 3.
As can be seen from Table 3, the results of the simulation model without considering crack expansion deviate significantly from the experimental results in predicting S a and S ku , and the corresponding maximal relative errors are 42.58% and 40.7%, respectively. By contrast, the simulation model considering crack propagation predicts S a and S ku with relative errors exceeding 10% in only a few cases, and the average relative errors of S a and S ku are only 7.95% and 9.46%, respectively. The above results indicate that the model considering crack expansion has better accuracy and stability.

Main effects analysis of S a and S ku
The results of the orthogonal experiments were analyzed to determine the main factors affecting S a and S ku , as shown in Figs. 13 and 14. Figure 13 shows that the spindle speed is the factor that affects S a the most, followed by the depth of cut, amplitude, and feed rate. S a decreases with increasing spindle speed and feed rate; S a increases with increasing amplitude and depth of cutting. Both the model considering S ku is a measure of the sharpness of the surface height distribution. It can be seen from Fig. 14 that except for the spindle speed, the three other factors cause a similar fluctuation in S ku around the mean value of 3. This means that the surface morphology at this time is characterized by a normal distribution. It can be seen that variations in these three factors do not cause large fluctuations in the morphology of the machined surface. The effect of the spindle speed is different. S ku gradually decreases with increasing spindle speed. When the spindle speed is low, the S ku value is 4.6, which implies that the workpiece surface is characterized by deeper valleys. The above results are consistent with the conclusions drawn from Fig. 12.
By comparing the deviations of the calculation results from the experimental ones for the two models in Figs. 13 and 14, it is evident that the model considering crack expansion is in better agreement with the experimental results than the model ignoring cracks. The reason for the larger error in the model ignoring cracks is that crack propagation inevitably will lead to material removal during the machining process, which changes the morphology of the machined surface, but the simulated model ignored this factor. However, the model considering crack propagation still exhibits a small error. According to the analysis, the reason is  mainly due to two factors. The first is that crack propagation is random and is affected by other grains. The proportion of crack interaction between adjacent grains was derived experimentally and considered in the simulation model. However, the affection of other grains is difficult to predict. The second factor is that crack propagation paths were simplified as straight lines during simulation. This simplification would necessarily miss some removed material, which affected the simulation results. Even so, the model considering crack propagation is able to predict the surface morphology more accurately.

Effect of ultrasonic vibrations on S a and S ku
In longitudinal-torsional composite ultrasonic-assisted mill grinding, ultrasonic vibrations change the initial velocity and shear angle of the abrasive grains when cutting into the workpiece, forming different machined surface topographies. Figure 15 shows the effect of different processing conditions on the cutting speed and shear angle of the abrasive grain. The initial velocity of the abrasive grain can be obtained from Eq. (4), and the shear angle η can be defined as follows: where v g is the initial cutting speed of the abrasive grain, v′ g is the horizontal component of the cutting speed, and η is the shear angle. Figure 15(a) shows that the initial velocity of the abrasive grain cutting into the workpiece is v g = 1.05 × 10 6 μm/s in exp. no. 2; the velocity direction is parallel to the workpiece surface, and the shear angle is η = 0°. By contrast, in exp. no. 12, owing to the effect of ultrasonic vibrations, the shear angle and the initial cutting velocity of the abrasive grain increase to η = 74.49° and v g = 1.96 × 10 6 μm/s, respectively, as shown in Fig. 15(b). The abrasive grains impact the workpiece surface at higher velocities, prompting the lateral cracks to extend forward, causing the material to be removed irregularly. This results in an increase in S a and S ku on the workpiece surface (24) = arccos v � g ∕v g (e.g., S a increases from 1.33 to 1.79 μm). Compared with exp. no. 12, in exp. no. 15, the ultrasonic amplitude and spindle speed increase simultaneously. Furthermore, the shear angle η decreases from 74.49° (exp. no. 12) to 38.64° (exp. no. 15), and the cut-in velocity v g increases from 1.96 × 10 6 (exp. no. 12) to 4.02 × 10 6 (exp. no. 15), as shown in Fig. 15(b), (c). According to the experimental results shown in Table 3, the S a of the workpiece surface decreases from 1.79 (exp. no. 12) to 1.10 μm (exp. no. 15). At the same time, ultrasonic vibrations reduce S a from 1.33 μm (exp. no. 2) to 1.10 μm (exp. no. 15).
In longitudinal-torsional composite ultrasonic-assisted mill grinding, the morphology of the machined surface is the result of the combined influence of various parameters, and the quality of the machined surface can be effectively improved through the selection of the appropriate ultrasonic amplitude, spindle speed, and feed rate.

Conclusions
In this work, the influence of crack expansion on the surface formation process during the longitudinal-torsional composite ultrasonic-assisted mill grinding of silicon nitride ceramics was investigated. A crack bridging model and a surface profile reconstruction algorithm were proposed, and a 3D morphological reconstruction model of the machined surface considering crack propagation was established and experimentally verified. Thus, the influence of different machining parameters on the surface morphology could be revealed. The main conclusions are as follows: (1) In longitudinal-torsional composite ultrasonic-assisted mill grinding of silicon nitride ceramics, the machined surface is the result of a combination of type-I material removal via abrasive grain shear and type-II material removal via lateral crack propagation.
(2) The model that takes into account crack propagation can reproduce the texture characteristics of the Fig. 15 Initial cutting speed and shear angle of the abrasive grains for different machining parameters machined surface more accurately than the model that ignores crack propagation. The average relative errors of S a and S ku for the model that considers crack propagation are 7.95% and 9.46%, respectively. (3) The surface reconstruction model considering crack expansion can accurately predict the influence of each factor on the evaluation parameters of the machined surface morphology, which is beneficial to the optimization of the process parameters in actual production. (4) Ultrasonic vibrations change the initial velocity and shear angle of the abrasive grain when the grain contacts the workpiece, thus changing the machined surface morphology. The surface quality can be effectively improved by choosing a suitable ultrasonic amplitude and an appropriate spindle speed and feed rate.