Modeling for prediction of sawing force based on the maximum undeformed chip distribution in the granite sawing

Sawing by circular saw blades is predominant in the mechanized treatment of natural stone. The predictive sawing force is crucial for optimizing, controlling, and monitoring the sawing process. In this work, a novel model for predicting sawing force, which is based on the distribution of the undeformed chip thickness at the sawing contact arc, was proposed. Based on the analysis of the undeformed chip geometry of the circular saw blade, the segment surface was divided into the front end and rear end according to the contact pattern between the segments and granite, and the models of the maximum undeformed chip thickness of diamond particles on the front end and rear end were established. Results showed that the new proposed force model fully considered the distribution of undeformed chips compared with the current model. The novel model has higher prediction accuracy, the mean maximum absolute error is within 3.77%, and the maximum absolute error is within 7.83%. Through theoretical analysis, the ratio of the maximum undeformed chip thickness of the front segment to the rear segment is 1.67, which can fully explain the wear non-uniformity of the segment. The proposed model is of great significance to the optimization of the saw blade structure and process parameters.

Total number of cutting edges per unit area of the front and rear segment C g Total number of diamond particles on the rear segment ρ, η Proportion of particles of the front and rear segments involved in sawing, 2/3 Δ Mean distance between the front row particles and the segment front (μm) l w Slot width (mm) b Chip width (mm) b 1 , b i Chip width of diamond particles of the different locations (mm) r Ratio of chip width and thickness l s Segment length (mm) N Number of segments in the sawing arc ε, ε 1max , ε 2max Cutting angle (rad) B Segment width (mm) B g Ratio of the segment width to the particle diameter F tg , F ng Tangential force and normal forces of one segment (N) F n , F t Tangential force and normal force of the saw blade (N) f t Tangential force of the diamond particle (N) f t1 , f ti Normal forces of the particles of the front and rear segment (N) f n Normal force of the diamond particle (N) f n1 , f ni Normal forces of the particles of the front and rear segment (N) m, n Force distribution coefficients F x , F y , F z Force signals in three directions (N) F act, i , F model, i Measured and fitting force (N) RMSE Root means square error |E| mean Mean absolute error |E| max Maximum absolute error

Introduction
Diamond circular saw blades with excellent cutting performance have been widely applied in brittle materials processing such as stone, structural engineering materials, and ceramics [1][2][3][4]. The diamond particles [5] are embedded into the circular saw blade segments to squeeze and crush the workpiece to remove the material. Due to the promotion of diamond crystals for stone processing, studies on product performance improvement [6], energy consumption [7], tool life by optimizing the saw blade structure [8], and process parameters [9] during stone sawing have become vitally important. It is important to understand the sawing mechanism as undeformed chips are produced during diamond removal, scholars have done lots of research on it at present. For example, Konstanty et al. [10] proposed a theoretical model for sawing natural stones using diamond tools. His work quantifies chips generated and removal processes to optimize tool composition and sawing process parameters. Tönshoff et al. [11] explained the sawing mechanism by which a diamond removes material and observed the wear characteristics of the diamond during stone sawing, noting that a detailed knowledge of sawing principles and wear characteristics are crucial. Polini et al. [12] reviewed the influence of sawing conditions on sawing force and energy. The relational models of force and equivalent chip thickness, specific energy, or material removal rate were proposed and then tests with different process conditions were performed.
Besides undeformed chip studies, force and energy consumption are important for mechanistic studies. Several pieces of research in the related literature focus on the sawing force and energy consumption when cutting hard and brittle materials with a diamond circular saw blade. Warren et al. [13] evaluated the equations that determine the number of active cutting points during ceramic grinding and then discussed the force per abrasive grain and its relationship to the average chip cross-sectional area. Malkin et al. [14] studied the removal mechanism and energy conversion during ceramic grinding. Grinding debris by SEM observation showed that brittle fracture dominates in the material removal. The energy consumed by brittle fracture accounted for only a small part of the total energy. Turchetta et al. [15] studied the effect of sawing conditions on sawing force and energy concerning the idealized chip shape during granite sawing. Xu et al. [16,17] reported forces and energy during granite sawing with circular saw blades and explained that the normal force per particle is almost proportional to the chip thickness calculated, and the tangential force component calculated is significantly different from the horizontal force components measured. These studies have fully studied the cutting mechanism of circular saw blades, but the wear analysis of individual segments is not perfect. Dong et al. [18] discussed the wear characteristics of different parts of a segment of the saw blade during granite sawing by experiments and pointed out that the wear non-uniformity is due to different forces. Besides, the sawing process by the saw blade is similar to the grinding, and many scholars have done a lot of research on diamonds' load characteristics and wear characteristics. Due to the complex processing environment during stone sawing, such as cooling fluid, stone properties, sawing method, sawing parameters, and segment materials, the wear forms of the segments and diamonds are varied. Ersoy et al. [19] discussed the effect of a series of factors on the wear of segments of saw blades and analyzed the wear characteristics of diamonds during sawing different hard rocks. Aydin et al. [20] analyzed the wear performance of saw blades during granite processing and established wear estimation models. In Jerro's work [21], the chipping geometries have been mathematically defined and derived through kinematic analysis. By understanding the kinematics of the sawing process, sawing forces can be lowered and even optimized. Later, Xu et al. [22,23] also analyzed the load and wear characteristics of high-speed grinding ceramic materials with brazed diamond wheels. However, most of these studies are conducted through statistical analysis and observational methods rather than theoretical models. Therefore, one of the difficulties in applying these studies is the need to do a lot of experiments to establish an accurate model to solve the application problem.
Unfortunately, many studies have not considered the force and wear non-uniformity of the front segment and rear segment in the process analysis of a circular saw blade. Huang et al. [24] established a theoretical model of sawing force during sawing hard granite and carried out experiments to solve the coefficients in the model. Yu et al. [25] had also done a lot of work to study the specific energy characteristics of diamond abrasives during granite sawing, and the influence of parameters such as abrasive type, granite properties, lubrication conditions, and diamond tool processing conditions on the performance of diamond tools. Wang et al. [26] carried out a study of diamond load and wear characteristics in combination saws based on chip geometry during granite sawing. Thus, a clear representation of the sawing force is very important to analyze the sawing process.
To solve the problem that the description of sawing force is not very perfect, based on theoretical analysis and experimental solution, this paper proposed a new sawing force model and analyzed the wear non-uniformity of the segment during granite sawing (Fig. 1). By analyzing the distribution of undeformed chip thickness of the diamond particles of the front segment and rear segment, the load model of the segment has been further derived. To solve the load model during sawing, the contact arc length and the maximum undeformed chip thickness were further deduced. In addition, the wear non-uniformity [18] of the particles of the front segment and rear segment has been analyzed theoretically and experimentally in this work.

Analysis of undeformed chip characteristics
The segments of the circular saw blades are brazed with diamonds. The material is removed by diamond impact and grinding [27] during granite sawing. In this work, to explore the sawing mechanism and establish sawing force characteristics, the sawing geometry is mathematically analyzed based on the segment morphology and workpiece motion. The undeformed chip geometry can be further derived, such as contact arc length and maximum undeformed chip thickness.

Contact arc length
The undeformed chip geometric characteristics of sawing are shown in Fig. 2. The circular saw blade [28] with a diameter of d s rotates at a rotation speed n and moves relative to the workpiece at a feed speed v f . The cutting depth of the circular saw blade to a granite workpiece is a p . In the actual Fig. 1 Analysis route of the novel sawing force model sawing granite process, the a p is very small, and the sawing speed v s is far greater than the feed speed v f , so the sawing arc in Fig. 2 is exaggerated. The X-Y coordinate system is established at point D' as the origin. When the circular saw blade rotates through the angle θ', the horizontal and vertical movements of the sawing edges [29] (diamond particles) at the origin along the cycloid trajectory are expressed as According to grinding technology [30] and sawing path in Fig. 2, the contact arc length l k can be simplified as It can be seen from Eq. (2) that the contact arc length increases with the increase of the cutting depth a p , the saw blade diameter d s , and the feed speed v f and decreases with the increase of sawing speed v s during sawing granite with a circular saw blade.

Description of chip thickness distribution
The granite is removed in the form of chips during the sawing process. Since the chips are in the form of (1) powder, for the convenience of study, they are regarded as undeformed chips [22]. The shape and distribution of the undeformed chips reflect the force on the segments of the circular saw blade. Therefore, many scholars had carried out a series of researches on stone processing, but most scholars assumed that the diamond particles on a segment of the circular saw blade tool are uniformly distributed, and the diamond's force is approximately the same. By comparing the force models in reference [15], the differences in the force models detailed are shown in this work. The first-row diamond particles that participate in sawing granite along the segment of the sawing arc are defined as the particles of the front segment [18]. The diamond particles immediately following the particles of the front segment are defined as particles of the rear segment. As shown in Fig. 3(c), particle 1 is the particle of the front segment, and particle i is the particle of the rear segment [31]. According to grinding theory, the diamond particles on the segment are defined as the cutting edges, and the maximum thickness of undeformed chips removed by the cutting edge is expressed as the maximum undeformed chip thickness. Malkin [14,30] and Xu [17] established the h m model from the diamond particle sawing path:

Fig. 2 Schematic diagram of sawing arc area
where h m (μm) is the maximum undeformed chip thickness, v s (mm/s) is the sawing speed of a single particle, v s = πd s n, n is the rotation speed of the circular saw blade, C i is the total number of cutting edges per unit area, that is, the effective diamond particles participating in cutting, r is the ratio of chip width and thickness, where r = b/h m , b is the chip width. During the sawing process, it is assumed that the diamond particles are approximately equally distributed on the segment surface. As shown in Fig. 3 (red particles), the maximum undeformed chip thickness of diamond particles of the front segment is solved as follows. The number of effective cutting edges involved in the sawing of the diamond particles per unit area of the front segment is defined as C a , where C a = 2ρ/(l w + Δ). 2 is the two rows, that is, the last row particles of the first segment and the first row particles of the second segment. ρ is defined as the proportion of the diamond particles of the front segment involved in sawing to the total number of particles on the front segment. Δ is defined as the mean distance between the front row diamond particles and the segment front ( Fig. 3(c)), and l w is the slot width. The maximum undeformed chip thickness of the diamond particles of the front segment is defined as h m1 . Replace C i with C a in Eq. (3), the equation can be expressed as Based on Eq. (4), the maximum undeformed chip thickness of the diamond particle of the front segment fully considers the existence of the slot, and the larger the width of the slot, the larger the maximum undeformed chip thickness.
As shown in Fig. 3 (blue particles), the maximum undeformed chip thickness of diamond particles of the rear 1∕2 segment is solved as follows. The number of effective cutting edges involved in the sawing of the diamond particles per unit area of the rear segment is defined as C b , where C b = C g η/ (B (l s -△)), C g is the total number of diamond particles on the rear segment, and η is the proportion of diamond particles of the rear segment involved in sawing to the total number of particles on the rear segment. l s is the segment length, and B is the segment width. The maximum undeformed chip thickness for the diamond particles on the rear segment is defined as h mi . Replace C i with C b in Eq. (3), the maximum undeformed chip thickness h mi of the diamond particles on the rear segment can be expressed as The cutting edges are assumed to be uniformly distributed on the segment surface based on the idealization of the circular saw blade segment. The certain error with the actual result is unavoidable.
The normal and tangential forces of the particles of the front segment (red particles) are f n1 and f t1 , and that of the particles (blue particles) of the rear segment are f ni and f ti . The undeformed chip morphology is shown in Fig. 3 1 and b i are the undeformed chip width of the red and blue particles, respectively.
For circular saw blades, the sawing force acting on a single diamond has an exponential power relationship with the undeformed chip thickness [12,15,16]. The relationship between the normal force f n and the tangential force f t and the maximum undeformed chip thickness h m is expressed as where m t and m n are parameters not only related to the properties of the workpiece material itself but also the sawing parameters, the larger the value, the greater the single-particle sawing force. n t and n n are parameters related to the worn shape of diamond particles [18]. When the maximum undeformed chip thickness is close to 0, it means that the single-particle sawing force does not change with the change of cutting depth a p , without the size effect.

Number of segments in the sawing arc
During the granite sawing, the diamond particles are cut in from the workpiece surface, cut out from the bottom end, and the cutting angle ε [32] is from the maximum to the minimum. Because the diamond circular saw blade tools have slots, the cutting angle ε 1max of the diamond particles of the front segment (red particles in Fig. 3(a)) is greater than the cutting angle ε 2max of the diamond particles of the rear segment (brown or blue particles in Fig. 3(a)). It explains the wear non-uniformity phenomenon of the segment of circular saw blades. As shown in Fig. 4, when the cutting angle is ε 1max or ε 2max and so on, the number of segments in the sawing arc can be set as N. As shown (6) f t = m t h n t m f n = m n h n n m in Fig. 4(a), the number of segments in the sawing arc is difficult to calculate. In this work, the number of segments in the sawing arc region can be approximated as the ratio of the sawing contact arc length to the sum of the slot length l w and the segment length l s (Fig. 3(c)), as shown in Eq. (7).
Looking at Fig. 4(b) and (c), the cutting angle is not only significantly related to the processing parameters v f , v s , and a p of the circular saw blade but also to the diameter of the circular saw blade and the ratio of the slot and segment length [33]. In this work, the number of the segment in the sawing arc area is not solved exactly, through analysis, the number of the segment is related to the cut-in angle or cut-out angle.

Establishment of the novel force model
As mentioned in Sect. 2.1.2, the normal force and tangential force of a diamond particle are expressed to have an exponential relationship with the thickness of the undeformed chips. where f n1 and f ni are the normal forces of the particles of the front segment and rear segment, respectively, and f t1 and f ti are the tangential forces of the particles of the front segment and rear segment, respectively. m and n are the coefficients related to the workpiece material and diamond properties, respectively [34].
As shown in Figs. 3 and 4, the sawing force of a segment is the resultant force of the sawing forces on the particles of the front segment and rear segment. The total number of the diamond particles of the front segment is set to 1ρB g , where, B g is the ratio of the segment width B to the particle diameter. In this work, the diamond particle diameter is approximately the chip width b, where B g = B/b. The total number of the diamond particles of the rear segment is defined as C g η, usually η = 2/3 [34]. Thus, the normal force F tg and the tangential force F ng of a segment are expressed as The force in the sawing arc area, that is, the force on the circular saw blade, can be expressed by the resultant normal force F n and the resultant tangential force F t on the N segments.
Substituting the contact arc length (Eq. (2)), the normal force and tangential force on a segment (Eq. (10)),  (11) F t = NF tg F n = NF ng the undeformed chip thickness (Eqs. (4) and (5)), and the number of segments (Eq. (7)) into Eq. (11). The tangential force equation of the circular saw blade is expressed as Similarly, according to the force of the segment in the contact arc region, the normal force equation of the circular saw blade is expressed as According to Eqs. (12) and (13), it can be inferred that the normal and tangential forces experienced for the segments are exponential powers with the maximum undeformed chip thickness [16]. The normal force and tangential force are positively During the sawing process, the load analysis and prediction [35] play an important role in analyzing the circular saw blade life, the vibration noise, and the diamond damage forms. By establishing the prediction model of the size effect, it has far-reaching significance for the design and optimization of the circular saw blade. By analyzing Eqs. (12) and (13), for circular saw blade tools, increasing the diamond granularity on a segment can prolong its lifespan. Therefore, the distance between the front row diamond particles and the segment front can be reduced to increase the particle life.

Sawing tools and workpiece preparation
The sawing experiment was carried out on a circular saw blade made of 75Cr1 with a diameter of 390 mm, a center hole of 50 mm, and a thickness of 3 mm. The circular

Experiment set-up
A bridge sawing machine was used as the experimental machine. As shown in Fig. 5 (sawing experimental setup), a highly stable Kistler dynamometer (9257B) was used in sawing experiment. The measured signals were inputted into a computer by a charge amplifier (5070A) and the data acquisition system. During the experiment, the dynamometer monitors the dynamic changes of the sawing force in real time and records the force signals (F x , F y , and F z ) in three directions. The raw data of a sawing process cycle is shown in Fig. 6(a). A sawing cycle (workpiece length: 200 mm) consists of a cut-in stage, a steady sawing stage, and a cut-out stage. In the steady sawing process, the average value [37] of the force is taken as the sawing force on the circular saw blade. The smoothing and filtering methods are operated on DynoWare software in this paper. The smoothing option is moving mean. The window size is 200 the number of points to the left and right is different by one. In addition, the filtering option is low pass in this paper. The edge frequency has an upper limit of 100 Hz, a lower limit of 50 Hz, and a filter order of 2. After smoothing the raw data, as shown in Fig. 6(b), the forces in the F y and F z directions are used to calculate the normal F n and tangential forces F t on the circular saw blade.

Model solving of force model
To solve the coefficients m 1 , m i , n 1 , and n i in Eqs. (12) and (13), it is necessary to count the forces under different sawing parameters. The rotation speed is 1860 rpm. According to statistics, the diamond particle number C g on the segment in this work is approximately 36, the proportions η and ρ of effective particles are 2/3, the ratio of chip width and thickness r is 1.714 [16], the segment width B is 4.30 mm, the selected diamond diameter is about 400-500 µm, the width   . 8 The wear morphology of the diamond particles: (a) segment wear and (b) six forms of the worn diamond particles of the slot is l w , l w = πd s /z-l s . The distance Δ between the front row diamond particles and the segment front ranges from 1 to 5 mm, 1 mm is brought into the calculation in this work. Looking at Fig. 5, the angle between the force action point and the y-axis in the experiment is β, β = kθ, k is a constant, θ is the arc angle, and the force analysis [38] of the circular saw blade can be deduced according to the theoretical relationship as follows Since the cutting depth value is small, β is half of θ, θ = cos −1 (1-2a p /d s ). The forces measured in the experiments are shown in Table 1, calculating the undeformed chip thickness values for different sawing parameters. Using the sawing force under different sawing parameters to fit Eq. 15, the coefficients m n1 , m ni , n n1 , and n ni ; m t1 , m ti , n t1 , and n ti can be obtained. The linear fitting curve of F ng /F tg or F n /F t and h m1 /h mi was shown in Fig. 7. Compared with the literature [16], the slope of F ng /F tg or F n /F t shows the correctness of the measured experimental data. The slopes of h m1 /h mi show that the chip thickness of the particles of the front segment is 1.67 times larger than that of the rear segment. In this work, for R-square and adjusted R-square, the normal force F ng is well fitted, and the tangential force F tg is generally fitted with large errors on a segment. It is considered that the error sources may be saw blade deflection caused by excitation forces during machining (e.g., related to granite type, segment material, and diamond particle size), as well as geometric errors of the machining system (e.g., blade runout) [39]. Besides, there is a large error in predicting sawing force at different feed speeds [16]. Therefore, R square (R 2 ) and root means square error (RMSE, Eq. (16)) were used to evaluate the fitted performance of the proposed model, reflecting the degree to which the measured data (F act, i ) deviates from the fitting value (F model, i ). The fitted experiment results were shown in Eq. (17) and Table 2. Six combinations (v f and a p ) were randomly selected, and their normal forces were estimated. The corresponding actual normal force is experimentally measured and calculated. The error of the results is shown in Table 3. In this case, |E| mean is Fig. 9 The proportion of worn diamond particles of the front segment and rear segment 3.77%, and |E| max is no more than 7.83%. This essentially proves the prediction accuracy of the model, and the proposed models had a high accuracy compared with current models [12,16]. This is conducive to its universal application.

Explaining wear non-uniformity with the novel models
The saw blade segments used in the experiment were taken down, and 16 segments were selected at equal distances for observation, as shown in Figs. 8 and 9 (the sawing parameters are n = 1860 rpm, a p = 6 mm, and v f = 15 mm/s). The wear morphology of diamond particles of the segments was observed with a scanning electron microscope (JSM-6610LV). The morphologies of the particles of the front segment and rear segment on the 16 segments can be divided into six forms (b): fresh, whole, micro-fractured, macrofractured, polished, and pulled-out crystal [40]. Fresh crystal is an emerging diamond, polished crystal means a smooth surface on the diamond, and both show a relatively low residual height and poor sawing ability. Whole crystals and micro-fractured crystals have relatively high residual heights for better sawing capability. Pulled-out crystals indicate no sawing capability. The proportions of different worn particles of the front segment and rear segment were counted and analyzed in Fig. 9. The effective sawing morphology (whole and micro-fractured) proportions of the particles of the front segment and rear segment were 22% and 41%, respectively. The proportions of macro-fractured and pulled-out particles of the front segment and rear segment were 61% and 38%, respectively. Similar to the test results [18], the diamonds of the front segment are dominated by macro-fractured and pulled out, and the particles of the rear segment are dominated by micro-fractured and polished diamonds. It can be seen that the sawing force of the particles of the front segment is greater than that of the rear segment due to the slots, and the service life of the front segment is shorter than that of the rear segment. As shown in Fig. 7, the cause of wear non-uniformity on a segment is due to uneven force during granite sawing. By the parameters used in Sect. 4.1, it is calculated that the maximum undeformed chip thickness of the particles of the front segment is 1.67 times that of particles of the rear segment, and the ratio of h m1 and h mi is shown in Eq. (18).
1∕2 Under the constant diamond particle concentration and an effective number of particles on the segment, the theoretical ratio of f n1 and f ni is consistent with the ratio of h m1 and h mi . It can be concluded that the different normal forces particles of the front segment and rear segment are the main reason for the wear non-uniformity of a segment.
Observe the residual height of the selected 16 diamond segments by a laser confocal microscope (LSM800) (measurement method: measuring the distance from the worn diamond top by scribing the matrix surface (segment surface), Fig. 10). Although the whole form diamond particles of the front segment are less than that of the rear segment, the residual height of the diamond particles of the front segment is significantly higher than that of the particles of the rear Fig. 11 Model of crack generation and development on the workpiece during sawing segment. The diamond particles of the front segment are updated faster than that of the rear segment.

Mechanism explanation of size effects
According to grinding theory [30], the process that the diamond particle cutting the workpiece material can be considered as the shearing process of the workpiece material by the cutting edge, that is, the workpiece material is fractured along the diamond particle with the undeformed chip thickness. Figure 11(a) and (b) shows the models of the non-uniformity generation and development of workpiece cracks during sawing. This crack [41,42] is not generated by the material itself but formed during sawing. The red particle during the sawing process represents the model of the generation and development of cracks in removing the workpiece material process by the diamond particles of the front segment. Because of the slots, the particles of the front segment during the sawing process may make the crack generation and development of the workpiece periodical. Meanwhile, the blue particle during the sawing process represents the model of the generation and development of cracks in removing the workpiece material process by the diamond particles of the rear segment. From the crack on the top of the sample in Fig. 11(c), it can be seen that the presence of the slots leads to certain small periodic cracks on the workpiece surface.
This work investigated the load characteristics of diamond particles of the segment by analyzing the relationship between sawing force and the maximum undeformed chip thickness formed by the diamond particles of the front segment and rear segment. By refining the maximum undeformed chip thickness models of the particles of the front segment and rear segment, the force on the segment was related to the distribution of diamond particles and the slots. By analyzing the number of segments in the contact arc area and the relationship between the force and the two maximum undeformed chip thicknesses, the size effects [43] of the diamond particles of the front segment and the segment were presented.

Conclusion
This paper conducted a systematic study of granite sawing by circular saw blades and proposed a novel force model by analyzing the distribution of thickness of undeformed chips and geometrical morphology. The following conclusions are drawn from this work: (1) The sawing force model needs to consider the distribution of undeformed chips. Compared with current models, the proposed model fully takes into account the slots. It has higher prediction accuracy, the mean maximum absolute error is within 3.77%, and the maximum absolute error is within 7.83% (2) The application of the proposed model was verified by the experimental samples, and the wear non-uniformity between the front segment and rear segment was explained by observation. The effective diamond proportions of the front segment are less than that of the rear segment, and the wear out of diamond particles of the front segment is much faster than the rear (3) The wear nonuniformity in the front segment and rear segment can be explained theoretically by analyzing the ratio of maximum thickness of undeformed chips of the front segment and rear segment, and it is consistent with the observation results. The larger the thickness of undeformed chips of diamond particles is, the larger the force is and the easier it is to loss sawing ability