Modeling and investigation of minimum chip thickness for silicon carbide during quasi-intermittent vibration–assisted swing cutting

In micromachining, the quasi-intermittent vibration—assisted swing cutting technology alleviates the residual height problem of elliptical vibration–assisted cutting (EVC) and inherits its intermittent machining characteristics. The minimum chip thickness has a significant impact on cutting forces, tool wear, and process stability when working with difficult-to-machine materials. This study thoroughly examines the impact of cutting parameters and tool parameters on the quality of the workpiece during machining in order to better understand the time-varying characteristics of the quasi-intermittent vibration–assisted swing cutting (QVASC) machining process and the size effect on micro-cutting of silicon carbide crystal. This paper created the minimum chip thickness prediction model suited to QVASC machining process. The effects of variables such as cutting velocity and tool inclination on the minimum chip thickness were discussed as well as the scribing tests that were conducted on such challenging materials as silicon carbide. The research findings shown that during the machining process, the critical undeformed chip thickness of silicon carbide decreased continuously as the cutting velocity sequentially increased (1.51 mm/min, 1.88 mm/min, 2.26 mm/min); under the down inclination angle (0–10°), the critical undeformed chip thickness also continuously decreased. These results confirm that cutting too fast reduces the instantaneous undeformed chip thickness, which is not conducive to ductile removal.


Introduction
Quasi-intermittent vibration-assisted swing cutting (QVASC) technology is one of the ultra-precision cutting methods [1,2]. It realizes a new vibration-assisted cutting method by applying vibration to the flexible mechanism of the device to guide the tool swing and cutting according to the expected trajectory. QVASC has machining accuracy comparable to elliptical vibration cutting (EVC) for difficultto-machine materials and finds a suitable treatment method for the residual height. EVC exerts vibration in the direction of the feed and the depth of cutting, which has a considerable contribution to reducing cutting force, extending tool life and improving surface quality. At the same time, it also has disadvantages such as coupling, large error, and uneven surface residual height. QVASC enables the tool to oscillate at a constant velocity around the center of the tool nose arc by improving the device of EVC. The intermittent contact between the tool and the workpiece reduces tool wear and energy consumption during the process of swing cutting.
The ultra-precision cutting has the ability to achieve everincreasing machining accuracy, However, it is also accompanied by problems of machining quality and tool wear. It is necessary to choose the appropriate parameter conditions to avoid such problems. Then, ensure a reasonable chip evacuation. Brittle-ductile cutting mode conversion is an important phenomenon in ultra-precision machining of brittle materials. We need to analyze the factors affecting chip formation, which can be used to explore the reasonable value of cutting dosage. The relationship between the depth of undeformed chip thickness and the cutting threshold determines 1 3 the outcome of chip formation during machining. Here, the cutting threshold is referred to as the minimum cutting thickness. The parameters affecting the minimum cutting thickness are related to the radius of the cutting edge.
There are many theoretical and numerical studies about this issue over the number of experimental studies. In the Bifano et al.'s [3] study, it was found that when the minimum cutting thickness was greater than the maximum uncut chip thickness set in the experiment, the workpiece material underwent elastic and plastic deformation: delamination. Stephenson et al. [4] showed that the ductility of the workpiece material can be achieved when the radius of cutting edge is slightly larger than the thickness of the uncut chip (the ratio of the two values is greater than one), and the surface quality of the processed workpiece can be increased, and cracks and damage can be reduced. Malekian et al. [5] defined the stagnation point A to determine the critical segmentation point of the interaction area between the tool and the workpiece. The horizontal line where the stagnation point is located is the critical line for the occurrence of elastic and plastic deformation and normal chip evacuation. Kawalec et al. [6] obtained the formula h min = kr through research, in which the minimum chip thickness ( h min ) depends on the edge radius ( r n ) and the friction coefficient between the tool and the workpiece (k) (the k range belongs to 0.1 to 1). Yuan et al. [7] proposed that the minimum cutting thickness is not only related to the cutting-edge radius and the friction coefficient of the contact surface, but also related to the force ratio F y ∕F x . Grzesik et al. established a model for predicting the minimum chip thickness through molecular mechanics theory, and then studied the phenomenon of local material separation and fragment formation [8,9].
Later, some experts studied the relationship between the minimum cutting thickness ( h min ) and the stagnation angle ( ); the research of Son et al. calculated that = 45 • ; the minimum chip thickness is maintained between 0.2 and 0.35 [10,11]. When L'vov calculates = 37.6 • , the minimum cutting thickness is 0.29 [12]. According to the study of aluminum alloy Yuan et al., the minimum cutting thickness is locked at 0.25 ~ 0.32 [7]. The study of scratch grooves can be observed by micro-scale or nano-scale scratch experiments: There is a cutting thickness that allows the chip to be generated instantaneously in the plowed condition when the chip depth does not change. This is the critical depth of cutting by definition. Puttick et al. [13] determined the existence of this critical depth of cut by crack mechanics analysis, and when it is below this value, material removal without cracks after machining occurs on the surface of the material. Bifano et al. [3] elaborated on a model of plasticity and brittleness excess, which expresses the thickness of the undeformed chip by controlling the material properties of plastic deformation and brittle fracture, and found that the cutting velocity will affect the transformation phenomenon. Later Schinker and Doll [14] determined to quantify the cutting velocity. This high-temperature velocity promotes the viscosity of the glass, and the micro-cutting of glass fragments can be achieved. Siva et al. [15] proposed a model for predicting the undeformed chip thickness in ductile-brittleplastic transition. It can be viewed on the above literature that material removal must be accompanied by a fracture process. Therefore, the brittle-plastic boundary process of the machining process must be accompanied by the energy generated by brittle fracture, and the heat generated includes the energy consumption of brittle fracture of the material and contact. There are some experimental and theoretical basis for this line of research, but there are relatively few scholarly studies on QVASC.
In this paper, a predictive model is established to determine the chip thickness of an undeformed cut by determining the cutting energy associated with the brittle-plastic transition boundary region of a brittle material in QVASC process. The time-varying characteristics of the QVASC machining method are introduced into the model. Also, a new Johnson-Holmquist-2-based model of the QVASC friction coefficient for difficult-to-machine materials like SiC is available from the authors' research. After that, this study verified the model by groove experiments, and finally, the effect of different cutting conditions on the standardized minimum chip thickness of SiC material was investigated by analyzing and applying the model.

Brittle-plastic transition analysis
It is an important direction to explore the brittle-plastic transition of materials in the process of chip formation, as shown in Fig. 1. The minimum cutting thickness is an important value for confirming the quality of the machined surface. When the uncut chip thickness (h) is greater than the minimum cutting thickness ( h min ), normal cutting will occur and chips will be generated, as shown in Fig. 1a; When the undeformed chip thickness (h) is less than the minimum cutting thickness ( h min ), chip generation cannot be carried out, and elastic and plastic deformation will deteriorate the quality of the machined surface and cause damage such as cracks, as shown in Fig. 1b.
The scribing experiment will produce a groove on the processed surface, which can also be called a groove experiment. Before that, it needs to establish a model through theoretical principles. For the study of the critical value of brittle-plastic transition, the work is based on the Kragelskii-Drujuanov equation in this chapter. In the study of Liu et al. [17] established a model for predicting the minimum cutting thickness and explained that the ratio of the minimum cutting thickness of the tool is affected by the shear strength and flow stress. Since the target materials studied in this paper are hard and brittle materials that are difficult to process, such as SiC and CaF. Therefore, the flow stress is considered to be determined by Johnson-Holmquist model-2 [18,19] in this paper, including the following situations, as shown in Eqs. (1) and (2): where * i and * f are the normalized intact equivalent stress and the normalized fracture equivalent stress, and the material constant is A, B, C, M, N, P * = P∕P HEL , here is the standard pressure. P is the actual pressure, and P HEL is the pressure at HEL. T * = T∕T HEL , where A is the standard maximum tensile hydrostatic pressure; T is the limit value of the material to withstand the tensile hydrostatic pressure; dimensionless strain rate is the actual strain rate, where L m (J/kg) is the latent heat associated with melting, (kg/m 3 ) is the density of the material, and T m is the melting point of the target metal.

Cutting temperature, strain, and strain rate models
According to the traditional classical cutting temperature model, the main heat is divided into the mechanical energy of fracture generated by the shear surface, and the heat generated by the friction between the chip and the rake surface of the tool in the chip removal direction. Considering the actual situation of ultra-precision machining, the following elements need to be discussed in this study: Because of the specific and precise nature of the QVASC method, the temperature model also needs to take into account the changes in heat and cutting temperatures caused by significant ploughing and friction near the dead metal caused by the tool edge radius [2], as well as the lateral ploughing and friction generated on the surface by the QVASC during the process of tool feed swing. The above two parts are our calculation of the mechanical energy generated by cutting through the sliding line field model. The slip linear velocity field model studied by Liu et al. [20] is used as a model, and the state of QVASC processing is analyzed as shown in the Fig. 2a [21]. Figure 2b expresses the mechanical energy of the shearing surface to produce chips and the heat energy generated by the friction between the chips and the rake surface of the tool, where c−t , s , p , and f are tool-chip contact surface, shear profile area (see Fig. 2b), dead metal plowing and friction area, and QVASC swing plowing friction area. With the occurrence of swing, the tip of the slip line field includes not only the plough area of HK and the friction interface AC and AD but also the plough area SR caused by the lateral swing of the cutting edge with the tip of the blade, as shown in Fig. 2. 1 3 In Fig. 2b, it is circled by the four colored dashed lines. The rate of mechanical energy generated by the four regions during the cutting process can be expressed as Among them, the temperature increase mainly depends on the work Q f of the swing cutting plowing, the heat Q s generated by the shearing of the material in the shear area, and the heat Q p generated near the dead metal cap through the plowing in the whole process. Then, the current temperature can be expressed as: where T 1 is the interface temperature of the workpiece, T o is the initial temperature, and ΔT 1 is the amount of temperature changes. (4) c SV c Among them, 1 , C, , and S are the ratio of cutting heat generated at the chip working interface during the cutting process, the material density and specific heat(density and specific heat of the workpiece material), and the area of the shear surface. For details, please refer to Weiner's model [22].
where R is expressed as: where V is the cutting velocity, is the shear angle, and is the thermal energy diffusivity. The multi-part work of the swing cutting plough work can be analyzed by the slip line field so that the lengths and velocities of the locally discontinuous slip line segments can be constructed, as shown in Eq. (7). However, it is necessary to obtain Q f from the slip line field model again because of the specificity of the arc tip of the diamond tool, by transforming the equivalent approximation of the contact area in the swing cutting process, as shown in Fig. 3, and the total mechanical energy of this part can be expressed as: Among them, L 0 and V 0 are the line length and the slip line velocity of each part of the slip line of the tool contact surface [20], k 0 is the shear flow stress between the workpiece and the chip at temperature T 0 , which can be expressed as follows through iteration: Subsequently, the research needs to understand the work done at the time of complete swing, and it needs to calculate the friction heat caused by the ploughing caused by the tool arc radius sweep, as shown in Fig. 3: where is the deflection angle involved in processing, the chord length in contact with the workpiece corresponds to the cutting width w mentioned above, r e is the tool nose radius, l is a single cycle tool in the cutting direction of the forward distance, and the area of the quadrilateral enclosed by the red dashed line is the area required for a single side swing. It should be noted that the quasi-intermittent nature of the machining process is mainly reflected in the diamond tool's nose end edge in the swing angle of the drive to produce a certain gap area: It is called "low loss area," where the curved middle part of the tool tip is called "high loss area," which maintains a high-frequency non-intermittent machining at the beginning of machining and it has a longer lasting consumption capacity to retard tool damage and chipping. The low loss area has a more permanent consumption capacity and delays the duration of tool damage and breakdown. Here, it is necessary to summarize the energy generated by the part of the tool that is not severely damaged during each cycle, as shown in Fig. 3. From this, the area of the vulnerable area is defined as: The dynamic friction factor of the tool surface can be obtained from the previous modeling work of this research

Fig. 3 The area swept during processing
Since the principle of processing surface is always the fracture caused by the extrusion of face and the line, here, the circular surface of transverse extrusion can be regarded as a straight line to distinguish, and the area obtained by establishing a right angle coordinate system can be expressed as follows: Subsequently, the pendulum distance can be obtained here as l and Q f can actually be expressed as: Each angular scale corresponds to the length and velocity of the corresponding slip line field. Since the contact between the tool and the chip interface is located after the machining process, the accompanying heat is not synchronized, where Q c−t can be expressed as: k 2 can be calculated by Eq. (8), is the contact length between the tool and the chip, and V c is the cutting velocity.
For the amount of temperature change of the tool and the chip, the corresponding results can be obtained by [6]: where ΔT 2 is the rising temperature of the area where the tool and the chips meet, 2 is the proportion of heat in the incoming chips, 2 is the thermal conductivity of the chips, L 2 can be calculated by Eq. (17), and is the heat diffusion at the final temperature rate; t is the thermal conductivity of the tool. A is the function of the ratio of transverse and longitudinal friction area, as in Eq. (18): It deduces the temperatures T 1 and T 2 from the previous calculation Eqs. (4)-(17) for the cutting temperature; it is important to note here that there is some implicit coupling between the cutting temperature and the effective effects through the shear flow stresses k 1 and k 2 to effective strain and strain rate as a function of effective shear strength, which in turn is a function of Liu et al. [17], where the effective shear strain and strain rate can be obtained from Eq. (19).
Whether it is related to shear strain rate and strain rate can be obtained by Eq. (20): where the shear strain and strain rate can be derived from the above slip line field as follows: In the formula, A and B are the discontinuous velocity across the shear belt. Because the coupling of standardized minimum chip thickness n , T 1 and T 2 , etc., is more complicated, there is still research value in exploring solutions.

Experimental setup
In this section, the groove experiment is set up to verify the type in this paper. The turning machine model is Precitech Nanoform 250 ultra-precision lathe, the spindle velocity is 0-3000 rpm, and the feed rate is 0-4000 mm/min. The minimum motion accuracy can be 15 nm. An aluminum rod with a diameter of 12 mm and a workpiece length of 50 mm is selected, and one end of the aluminum rod is connected to a SiC with a diameter of 12 mm and a length of 10 mm through industrial glue for processing. The device adopts the quasi-intermittent vibration-assisted swing cutting device independently developed by the research group, as shown in Fig. 4a-b. Generally, it will observe the transition area where the plow head begins to generate chips to determine if the surface of the target workpiece has undergone a brittle-plastic transition during machining on the QVASC machine. and capturing sudden changes in height to determine the minimum cutting thickness. In this study, a Zygo NewView 9000 3D optical profiler was used for the observation of the material surface, as shown in Fig. 4c. As shown in Fig. 5a, there is a transition boundary region at the cutting edge, an example of the 3D profile of the surface is measured with a white light interferometer on the turned surface, and the profile is analyzed by generating a 2D view to capture the minimum chip thickness. It is required that the scratched areas with brittle-plastic transition points at four different radial locations are captured as measurements for the analysis to calculate the average value of the workpiece end face. Finally, the normalized minimum cut thickness was determined by the Kragelskii-Drujuanov model, as shown in Fig. 5. The thermal parameters of the workpiece are shown in Table 1. In this part of the difficult-to-machine brittle materials SIC and Si, the parameters of the constitutive model JH-2 model [23] are shown in Table 1 [22]. The JH-2 model has certain applications in cutting and grinding [24]. It is chosen as the constitutive model to accurately describe the dynamic feedback of brittle-plastic difficult-to-machine materials at high strain rates and high pressure. The target of this paper is silicon carbide.
Due to the limitations of the equipment to a certain extent, the QVASC equipment is subject to a certain degree of machining instability during vertical cutting. So the work here finds and determines the brittle-plastic transition point of the material in the circular groove of the workpiece surface after cutting.
In the case of ultra-precision machining, the blunt radius of the cutting edge needs to be in the same order of magnitude as the cutting amount (depth of cutting). This section will use the indentation method to measure the blunt radius of the cutting edge of the tool. First, squeeze the test piece slightly and slowly with the tool, and then observe the extrusion marks with a white light interferometer (Zygo NewView 9000 3D Optical profiler)1.45 μm, as shown in Fig. 6.

Effect of velocity on critical undeformed chip thickness
This section explores the effect of the value of cutting velocity on the instantaneous undeformed chip thickness during the machining of QVASC, and the parameters are shown in Table 1. First of all, for different velocities, we selected 1.51 mm/min, 1.88 mm/min, and 2.26 mm/min as the deep scribing test. The surface scribing morphology and cutting width achieved by different velocities are shown in Fig. 8. Figure 7a illustrates the variation of the critical undeformed chip thickness as the cutting velocity increases for a certain range. It increases with the cutting velocity from 1.51 mm/min, 1.88 mm/min, and 2.26 mm/min. The thickness of the critical undeformed chip decreases continuously under the circumstances of changing the tool radius of circle, frequency, and depth of cutting, as shown in Fig. 7. The pattern of decline is comparable for different tools. The vibration-assisted cutting ductility production criteria estimate the crucial undeformed chip thickness under various velocity circumstances. Figure 8 displays the projected trend together with the experimental machining results. Experimental results are consistent with the expected value of chip thickness trend; the critical undeformed chip thickness value declines steadily with increasing cutting velocity. QVASC has several processing traits with EVC in its implementation. Without taking into account other variables, yet inconsistently reflecting the processing time delay, also, the quasiintermittent vibration is not unlike from regular cutting of tough-to-machine materials (taking SiC as an example). Due to this feature, any accompanying processing-related temperature rise will be minimal. QVASC's intermittent nature means it causes less of a temperature increase during cutting and has a less pronounced dominating influence on the brittle-plastic transition than EVC. With the thickness of the instantaneously undeformed chips being reduced, ductile removal is not possible at too high a cutting velocity.

Influence of tool inclination angle on critical undeformed chip thickness
Furthermore, the effect of tool inclination angle on the critical undeformed thickness was investigated. In Fig. 5a-c depicts three different tools with tool inclination angles of − 10°, 0°, and 10°. The process using the process parameters is listed in Table 2. Figure 7 shows that the predicted value is similar to the experimental value. As seen in Fig. 5, the effect of varying the tool inclination angle and cutting velocity on the critical undeformed chip thickness is compared for various tool settings. There seems to be a trough tendency from a negative inclination of 10° to a positive inclination of 10°, with lowest point being 0°, and this trend growing from small to big tool inclination. The critical undeformed chip thickness on the left side of the trough reduces as the negative declination angle lowers, and it is difficult to produce brittle removal; however, on the right side of the trough, the instantaneous undeformed chip thickness progressively increases under the condition of gradually rising positive declination angle. The critical undeformed chip thickness reduces monotonically with increasing inclination angle, which is the equivalent conclusion necessary to meet the criterion of reducing brittleness. Three tools with varying inclination angles were chosen for the tool parameter problem, and their trends and prediction outcomes were compared; these are depicted in Fig. 9. A zero-angle inclination angle indicates that the cutting Fig. 7 The influence of different parameters on the scribing morphology. a ℧ = −10 • ; b ℧ = 0 • ; c ℧ = 10 • direction and the rake face are perpendicular to one another. Presently, the prediction data investigation is focusing on a range of 0-10° for the inclination angle. At velocities of 1.51 mm/min, 1.88 mm/min, and 2.26 mm/min, respectively, the instantaneous undeformed chip thickness drops from 1067 nm, 781 nm, and 713 nm to 1038 nm, 765 nm, and 530 nm when the tool inclination angle reduces from − 10 to 0°.

Conclusions
This paper focuses on the prediction of the minimum cutting thickness model in silicon carbide machining of quasi-intermittent vibration-assisted swing cutting, as well as effect of the tool inclination angle and cutting velocity The outcomes of this study indicate the following: 1. This paper develops a prediction model for the minimum cutting thickness of silicon carbide based on quasi-intermittent vibration-assisted swing cutting, which can be used to predict the normalized minimum cutting thickness. The model derives inferences based on the thermomechanical energy created during the cutting process. 2. In addition, as a result of the intermittent character of QVASC, the cutting heat is relatively low and the dominance of brittle-plastic transformation is reduced; so the quicker velocity is not favorable to ductile excision. The instantaneous undeformed cut thickness approaches the condition of debrittle excision when the positive inclination angle (− 10, 0, 10) is gradually increased. Thus, when the angle of inclination is increased, the critical undeformed cut thickness decreases. In the event of a steadily growing positive inclination angle, the transitory undeformed cutting thickness rises to the condition of de brittle removal (− 10, 0, 10). Thus, when the angle of incidence is increased, a smaller critical thickness for an undeformed chip is required. 3. The predictions made by the brittle-plastic transition critical undeformed chip thickness theoretical model exhibit a consistent prediction trend that is consistent with real machining trials. While the EVC direction's ductility-creation requirements may be transferred to the QVASC machining method, there are several caveats that need to be taken into account. There is discrepancy between the predicted value and the experimental value, with a portion of the error perhaps attributable to: The machined surface is affected in some way by the reverse friction caused by the tool's deflection. Since the tool wear was not carefully considered during the experiment, the actual times of use and wear of the tool were not replaced and recorded in detail. There is a paucity of testing of important parameters due to the rapid pace at which new equipment is being developed.
In addition, because of the characteristics of the QVASC machining process, the tool trajectory will exhibit twodimensional movement when difficult-to-machine materials   . 9 Influence of inclination angle on critical undeformed chip thickness are being machined. To prevent catastrophic tool failure, it is vital to have a high degree of control over the parameters chosen for the depth of cutting. This is where further indepth investigation on this topic is needed. This research demonstrates the method's usefulness for ductile removal of silicon carbide at a cutting velocity of 1.51 mm/min and a tool inclination of 10°, as well as its potential benefits in the treatment of materials that are notoriously difficult to machine.