A novel method by microwave cutting ceramics based on thermal crack and trajectory control

Herein, the microwave thermal crack method is innovatively employed in cutting Al2O3 ceramics. Different from the conventional cutting technology, the thermal controlled fracture method is progressive and environmentally friendly, which adopts tensile stress to peel off the brittle material into two parts. The heat source induces tensile stress as thermal stress. Additionally, based on the Fourier heat transfer equation and thermo-elasticity, the physical model of microwave thermal crack cutting Al2O3 ceramics is established and calculated. The thickness of the graphite coating, the width of the graphite coating, and the prefabricated crack in the process of microwave thermal crack cutting Al2O3 were studied. The effect of graphite coating width on crack trajectory was also investigated by simulation combined with the test. From the result, it can be seen that the narrower the width of graphite coating, the weaker the processing ability and the better the trajectory control ability. The microscope is adopted to evaluate the surface and cross-section morphologies detailly. This study elaborates briefly on the interaction mechanism of microwave thermal crack cutting Al2O3 ceramic surface and provides practical guidance for aerospace industries applications.


Introduction
In the past few decades, hard and brittle materials such as ceramics have been attracting considerable attention in various fields, such as aerospace, automotive, petrochemical, and other industrial fields [1][2][3][4][5]. Al 2 O 3 ceramic comprehensive performance is superior, which is one of the most widely used advanced ceramic materials. Depending on a host of traits including but not limited to excellent hardness, brittleness, and high melting point, ceramic material cutting processing is more difficult than other materials, and in the process of cutting, it facile to produce damage, micro-cracks, and other defects [6][7][8]. Seen from prior studies are traditional ceramic cutting methods, such as mechanical cutting, laser melting cutting, plasma cutting, water jet cutting, electrical discharge machining (EDM), and water-guided laser [9][10][11][12][13][14][15]. The cost of mechanical cutting methods is low, but it will trigger microcrack defects. Although the laser melting cutting method is more efficient, it will produce melting layer damage. Plasma cutting and water jet cutting are fast, but the cutting surface is easily caused damage. EDM has higher cutting precision, but still leaves discharge traces on the cutting surface, and the processing efficiency is low. Although water-guided laser could reduce the thermal damage layer of laser melting cutting, it still cannot eliminate the surface machining defects. These machining defects reduce the edge strength of the actual workpiece and limit the effective use of the high-strength advantage of ceramics, which is one of the key problems restricting the application of ceramics in high-strength demand conditions. There so far exist extensive studies to provide constructive hints for solving the above concern through the thermal stress cutting method. In 1971, Lumley et al. [16] pioneered a thermal stress cutting method, and the heat source is employed to heat the workpiece to generate thermal stress so that the prefabricated crack can be controlled to expand. In the process of crack growth, there is no material removal and no macroscopic damage on the fracture surface, which enhances the edge quality of the workpiece and machining efficiency. In 2002, Ueda T and Yamada K et al. [17] employed a pulsed laser to conduct thermal crack cutting tests on silicon wafers and explored the relationship among laser power, heat source moving speed, heating temperature, and roughness. The experimental results show that if the laser power is too large, the roughness of the workpiece will increase, and the laser power required for silicon wafer fracture will also enhance with the moving speed of the heat source. In 2003, Chwanhuei T et al. [18] studied the crack growth mechanism of CO 2 laser cutting alumina ceramics and used the finite element method to simulate and calculate the thermal stress in the process of thermal crack cutting, and the change of thermal stress distribution in the process of thermal crack cutting could explain the crack growth process. In 2006, Shalupaev S et al. [19] proposed to adopt two laser beams of different wavelengths to thermal crack cutting brittle nonmetallic materials. Through experimental research, it was concluded that the double beam could effectively control the crack propagation track and improve the cutting quality. In 2008, Yukio M et al. [20] explored some questions in the process of glass cutting by the thermal cracking method. It is considered that the crack propagation behavior is controlled by the intensity factor of the maximum tangential stress, and the strength factors of the edge defects with different sizes are calculated by finite element simulation. In 2009, Motomura F et al. [21,22] simulated the process of CO 2 laser cutting glass and studied the influence of constraints, surface heat transfer, material thickness, and other factors on laser heating temperature and glass fracture strength factor. The research results showed that the stronger the surface heat dissipation capacity, the closer the crack tip was to the center of the heat source. It is pointed out that the distribution of the stress field during crack propagation has little correlation with material thickness. In the same year, Salman N et al. [23] found that there was trajectory deviation in the crack initiation and termination sections of laser thermal crack cut glass. They adopted a microscope to observe the sectional morphology of crack initiation and termination sections and employed finite element software to simulate and calculate the stress field in the crack propagation process. The deviation mechanism of the trajectory is explained by analyzing the results of the stress field. In 2011, Shalupaev S et al. [24] carried out modeling and simulation analysis on laser beams with an ellipse, normal circle, semi-ring shape, and their combined shape. They also utilized coolant to cool the heated area and analyzed the influence of heat source shape on cutting trajectory and the change of stress area after cooling. In 2019, Dudutis J et al. [25] conducted an elliptical laser cutting glass test, explored the roughness of the section at various cutting speeds, and obtained the relationship between cutting speed and roughness.
To the authors' knowledge, there are quite few, limited studies or investigations about microwave thermal crack cutting ceramic surfaces. Thus, it is meaningful to investigate the research field as well as filling these gaps. Comprehensively, innovative microwave thermal crack techniques are proposed to cut non-transparent brittle materials such as ceramics in this study. It is also a research challenge in microwave cutting ceramics as microwave heat source generation technology and its modeling, microwave thermal cracking processability, mechanism, and control method of migration propagation in microwave thermal cracking cutting process. Particularly, the uniqueness and novelty of this study is employed microwave as the energy source to heat non-transparent ceramic materials and realizes the thermal crack cutting of non-transparent ceramic materials. Furthermore, the other novelty of this study is that based on the Fourier heat transfer equation and thermo-elasticity, the physical model of microwave thermal cracking cutting Al 2 O 3 ceramics is established and calculated. The influence mechanism of graphite coating width on thermal crack cutting is obtained from the analysis of temperature field simulation results. Afterwards, a comparison test was carried out, and prefabricated cracks with different widths were prepared by different technological methods for processing test. Controlling the width of the prefabricated crack could effectively reduce the offset of the initial crack. The narrower the width of graphite coating, the weaker the processing ability and the better the trajectory control ability.

Theoretical modeling and analysis of microwave thermal crack cutting process
Initially, a tiny crack is prefabricated at the edge of the absorbing workpiece. Microwave is used to generate heating heat source in front of the workpiece crack. The heat source is scanned along the predetermined trajectory, and the nonuniform temperature field and corresponding thermal stress field are generated during the scanning process. The stress field produces stress concentration at the crack tip. When the stress intensity factor at the crack tip is greater than or equal to the fracture toughness of the material, the crack will propagate along the crack and thus realize the cutting of the workpiece. The physical process of microwave thermal crack cutting can be roughly divided into three main stages: the heating stage, thermal stress stage, and crack propagation stage. The main physical process of each stage is shown in Fig. 1. (a) Heating stage: the material absorbs the microwave and converts the electromagnetic energy in the microwave into heat energy. The temperature of the material rises rapidly. Meanwhile, the heat is transferred from the heating center area to the surrounding area, and the temperature of the surrounding area gradually rises. This stage mainly involves the relevant theories of heat transfer. (b) Thermal stress stage: the area heated by the microwave heat source heats up rapidly. After the heat source leaves the area, the temperature drops rapidly due to heat convection and heat radiation. On the path scanned by the heat source, a temperature gradient is formed between the area at the end of heating and the area under heating, and thermal stress is generated inside the material. This stage mainly involves the related theory of thermo-elasticity. (c) Crack growth stage: compressive stress is generated in the area heated by the heat source, and tensile stress is generated in the area rapidly cooling after heating by the heat source. When the tensile stress value is greater than the fracture limit of the material, the material will fracture and the crack will expand along the scanning track of the heat source correspondingly. This stage mainly involves the theory of fracture mechanics.

Heat source model for microwave heating setup
The essence of microwave heating ceramic materials is that the dielectric loss occurs in the high-frequency electromagnetic field, and the electromagnetic energy of the microwave is converted into heat energy so that the ceramic materials are heated up quickly. According to the related theories of dielectric physics, the loss rate of medium per unit volume (i.e., thermal power density) inside the material P v is: Herein, P v and E stand for the power density of the microwave heat source for bulk heating and effective value of average electric field strength. f and tan are the microwave frequency and material dielectric loss tangent, respectively. 0 and mean the vacuum dielectric constant and relative dielectric constant of the material.
Since reflection and loss will occur during microwave transmission and coupling, the following relation between microwave output and input power is assumed: Herein, P in and P out stand for the microwave input power and output power. e and z are the microwave power coupling efficiency and system impedance, respectively. E in and E out mean the input microwave field electric field intensity and output microwave field electric field intensity. S in and S out signify the input microwave field cross-sectional area and output microwave field cross-sectional area.
The relation between input power and output electric field strength is: The relationship between heat source power density and microwave input power is: The heat source equation can be expressed as: Herein, P s and z s stand for the equivalent surface heat source power density and coating thickness. s and P v are the heat source conversion rate and power density of heat source for bulk heating, respectively.

Basic theory analysis of microwave thermal crack cutting process
After microwave loading, the ceramic workpiece heats up under the action of microwave, while the Al 2 O 3 non-absorbing material absorbs the microwave and generates heat through the absorbing layer coated on its surface. In the process of surface heating Al 2 O 3 , heat is transferred from the ceramic surface to its interior after the heat generation is absorbed by the waveabsorbing layer. According to Fourier heat transfer law and energy conservation law, the heat transfer process inside the ceramic in the Cartesian coordinate system can be expressed as: Herein, ρ and c stand for the material density and specific heat. λ and Q are the coefficient of heat conduction and heat production per unit volume, respectively.
The thermal convection and radiation equations between the surface of ceramic materials and the environment are listed as follows: Herein, h and T 0 stand for the coefficient of natural heat convection and ambient air temperature. n and σ are the emissivity of material and Boltzmann's constant, respectively.
The thermal stress generation process of ceramics mainly involves thermo-elasticity, and the equilibrium differential equation of thermal stress can be expressed as: The physical equation of thermal stress can be obtained: The condition of crack propagation in fracture mode is: Herein, K I and r stand for the stress intensity factor and tensile stress at crack. c and l r are the stress threshold for crack growth and crack length, respectively.
The stress threshold condition for crack propagation is:

Simulation analysis of influence of coating width on temperature field
Using the finite element model of microwave thermal crack cutting process established above, the surface heat source equation is employed to set the loading area of surface heat source, limited the surface heat source within the width range of 2 ~ 5 mm, simulated the width change of graphite coating layer, set the microwave power as 800 W, and the heat source speed as 1 mm/s. The processing parameters are determined by a large number of tests. Under this condition, simulation analysis and calculation are conducted. The calculated results of the obtained temperature field are shown in Fig. 2, and the temperature variation curves of the center position of the heat source perpendicular to the moving direction of the heat source under various coating widths are shown in Fig. 3. The analysis of the temperature field results shows that the wider the coating width, the larger the microwave absorption area of the coating, and the more microwave energy can be absorbed in the same time. Therefore, the wider the coating width, the higher the center temperature of the heat source, and the temperature difference between the center area of the heat source and the surrounding area as well as the thermal stress is also large, which can improve the processing ability of the thermal cracking method. Aside from this, the narrower the width of the coating, the more concentrated the high-temperature area is, so the more concentrated the tensile stress area is. When the energy of the coating is enough to ensure the stable crack propagation, the trajectory deviation is limited to a smaller range, so the degree of trajectory deviation will be smaller.

Temperature test and simulation result verification
In the process of microwave pyrolysis temperature measurement, the thermocouple of the temperature measurement solder joint is placed in the temperature measurement position. The ground wire is connected with the thermocouple wire screen winding, and the tape should be fixed firmly, as shown in Fig. 4a. In this test, the temperature of five points will be measured in turn on the heat source track. The temperature measuring points are arranged as shown in Fig. 4b. To avoid high-temperature ablation caused by the close distance between the tip of the coaxial inner core and the thermocouple temperature measuring line, the temperature measuring points are arranged on the lower surface of the workpiece. After the thermocouple line is fixed, place the workpiece on the mobile platform, as exhibited in Fig. 4c. The microwave input power is adjusted to 800 W, the heat source   Fig. 6.
From the comparison of temperature curves between the above temperature measurement results and the simulation results, it can be seen that the two curves are close to each other in terms of temperature value, and the temperature variation trend of each temperature measurement point is basically the same. Considering the air convection caused by workpiece movement in the actual microwave heating temperature measurement process, the influence of adhesive tape on the heat dissipation of the temperature measurement probe, and the measurement error of the thermocouple temperature measurement equipment itself, it can be considered that the simulation results are close to the actual situation, and the finite element simulation model is trustworthy. The simulation results of the temperature field and stress field of the finite element model

Experimental setup diagram
In this study, a waveguide coaxially coupled microwave thermal crack cutting device will be used to conduct experiments, which includes a microwave source controller, magnetron, excitation waveguide, circulator, rectangular waveguide, resonator, coaxial core, coaxial jacket, short-circuit piston, and other main components, as shown in Fig. 7. The working principle of the system is as follows: the microwave source controller excites the magnetron to generate microwave field inside the waveguide, and the circulator transmits the microwave to the resonator in one direction. The position of the short-circuit piston at the outlet of the resonator is adjusted to make the microwave resonant, and the coaxial core is located at the position of the highest electric field strength. The resonant microwave energy coupling is carried out in the coaxial inner position and transmitted to the inner core end, where a high-intensity focused microwave field is generated.

Preparation of graphite coating
The preparation of graphite coating is a key problem. In this experiment, 8000 mesh high purity graphite powder with a diameter of 1.6 μm is used. The graphite powder is dissolved into absolute ethanol to make a graphite-absolute ethanol solution. The solution is sprayed onto the workpiece surface utilizing a spray gun to form a graphite coating. To control the width of the graphite coating layer, adhesive tape is employed to cover both sides of the predetermined coating layer. After spraying, the adhesive tape is uncovered to leave the predetermined width of the coating layer. In order to effectively control the thickness of the graphite coating layer, the mass percentage of the graphite-absolute ethanol solution prepared is controlled at 10%. In order to ensure the uniform thickness of each spraying, control the nozzle 100 mm from the workpiece during spraying. The spray head moves to reciprocate uniformly, and the coating thickness is controlled by the number of reciprocating spraying. The air pressure of the pneumatic pump is controlled at 15 bar to ensure the slow and uniform injection of the solution.

Experimental study on the influence of graphite coating thickness on thermal crack cutting
Nine points are chosen to measure the thickness of graphite coating and the measurement line as shown in Fig. 8. The surface of the graphite coating layer is not smooth, but there are different degrees of ups and downs, but its overall thickness is still present positive correlation with the number spray. Spray control numbers can be depended on to control the average thickness of the graphite coating layer. Across the thickness of the coating material to heat the equivalent heat source power density of the microwave cracking method has an important influence. The thicker the coating material, coating layer of microwave absorption efficiency is higher, producing higher thermal efficiency, which directly affects the ceramic material of temperature and thermal stress size. Thus, the surface heating processing capability of the microwave thermal cutting method is also affected by coating thickness. To explore the effect of graphite coating thickness on machining ability, a control variable test is carried out with graphite coating thickness as the independent variable, and whether the workpiece cracked completely  during machining is taken as the observation result. The test record is exhibited in Table 1.
As can be seen from the above table, under the conditions of graphite coating thickness of 2 mm, the ceramic thickness of 2 mm, microwave power of 1400 W, and heat source moving speed of 1 mm/s, the first two tests fail to cut the ceramic workpiece. When the graphite coating thickness increases to 200 μm, the ceramic workpiece cracks successfully. Increasing the thickness of the graphite coating could crack the workpiece, which is difficult to crack. When the thickness of the ceramic workpiece is reduced to 1 mm, the width of the graphite coating increased to 4 mm, and the cracking thickness condition is declined to 150 μm. It can be indicated that the thicker the ceramic workpiece is, the thicker the graphite coating is required for cutting. According to the above experimental phenomena, the law of influence of graphite coating thickness on machining ability can be summarized: the thicker the graphite coating is, the stronger the machining ability of surface heating microwave thermal cracking cutting is, and the thicker the ceramic workpiece could be processed. Meanwhile, it can also be discovered in the test record table, when the width of graphite coating increases from 4 to 6 mm, the thickness of cracking graphite coating decreased from 200 to 150 μm, so it is speculated that the thickness of graphite coating also has a certain impact on the processing ability.
The graphite coating thickness and crack track morphology of the workpiece were observed in Fig. 9. It is found that the graphite coating thickness obtained by graphite spraying is not uniform. At different positions of the graphite coating, its thickness fluctuates greatly, and the overall surface topography of the coating is uneven, as shown in Fig. 10.
According to the heat source density from formula 7, the graphite coating thickness has a direct impact on the heating temperature. From the perspective of thermal stress theory, the irregular variation of graphite coating thickness causes the uneven variation of graphite coating temperature, and the thermal stress is generated by the temperature gradient as well as the local temperature unstable. In this concentrated heating area, its temperature exists inconsistent, and after, the heat source over the thickness of the defect area of the cooling process will have great influence for coating thickness. In this regard, the coating thicker district cooling speed slower than the thickness of the defect area, so the thickness of defect area will produce more faster temperature gradient. Due to the uneven distribution of thickness defects on the surface of graphite coating, it affects the direction of thermal stress in this region after the heat source moves, so the original crack growth track will shift due to the change of local thermal stress. Notably, the largest temperature gradient still comes from the high-temperature region of heat source heating and the crack to be expanded region of cooling. Therefore, the overall direction of the crack can still  Fig. 9 Morphology of graphite coating under microscope: a surface morphology of graphite coating; b cross-section morphology of graphite coating be consistent with the moving track of the heat source, but small deflection always occurs during the expansion of the track, which can be observed obviously under the microscope, as shown in Fig. 11.

Study on the influence of coating width on thermal crack cutting
The surface heating method of coated absorbing material generates heat through the joint action of microwave on coated material to form an equivalent heat source. When the width of the coating is larger than the diameter of the microwave energy beam, the effective energy density region of the equivalent heat source is determined by the diameter of the energy beam. As the cladding width is narrower than the diameter of a microwave energy beam, the effective energy density of equivalent heat source area will be limited within the scope of the width of the coating layer. The narrower the coating layer, viz., less the effective energy density area, the coating is used to transfer heat to the ceramic workpiece. It also makes the heated area more concentrated and affects the thermal stress distribution inside the workpiece. Therefore, the width of the coating layer is also a parameter that has an important influence on the trajectory expansion of surface heating by microwave thermal crack cutting. The width of coating layer has a certain influence on the processing ability of microwave thermal crack cutting and the control of trajectory offset, so the process test research is carried out, as displayed in Table 2. According to the experimental results achieved in the test record table, under the same conditions of microwave power, heat source moving speed, graphite coating thickness, and ceramic thickness, the workpiece cannot crack when the graphite coating width is 1 mm and 2 mm, but the workpiece cracks successfully when the graphite coating width increases to 4 mm and 6 mm. This result indicates that the width of graphite coating enhances    1400  1  150  2  6  Yes  1400  1  150  2  4  Yes  1400  1  150  2  2  No  1400  1  150  2  1  No  1400  1  200  2  2  Yes  1400  1  200  2  1  No  1400  1  170  1 1 Yes the processing ability of microwave thermal crack cutting. In graphite coating, thickness increased from 150 to 200 microns, the graphite cannot crack a width of 1 mm, but the successful cracking width increases to 2 mm. It can be seen that the increase in the thickness of the graphite may reduce the demand for graphite width. When graphite coating thickness increases, use a narrower width of graphite to achieve the same processing ability. The crack generally expands in the track of the hightemperature region of the heat source, which reduces width of the graphite coating and the range of high-temperature region of the heat source. Therefore, the crack propagation will be carried out in a smaller range, and the purpose of crack trajectory propagation control can be achieved. To verify this conclusion, relevant process tests are carried out. Under the condition of controlling the single variable of graphite coating width, the processing trajectories under different graphite widths are achieved. The overall situation of crack trajectories was illustrated in Fig. 12, and the comparison of trajectory offsets was exhibited in Fig. 13.
Combined with the overall trajectory diagram and trajectory offset comparison diagram, it can be seen that when the graphite coating width is 2 mm and 3 mm, the starting and ending segments of the crack are relatively straight without obvious offset, but the middle segment of the crack has appeared offset phenomenon. By comparing the machining effect of coating width of 4 mm and 6 mm, it can be illuminated that there is obvious  trace deviation at the beginning and end of the crack, but the machining effect of the middle segment is much straighter than that of the coating width of 2 mm and 3 mm. Thus, it can be indicated that the narrow graphite width can control the trajectory deviation of the crack initiation and termination section, which is consistent with the prior simulation analysis. However, when the crack expands to the middle part of the workpiece, the wider graphite coating can absorb more energy, so that the crack has enough thermal stress to drive the crack to expand stably.
Universally, considering the influence of graphite width on crack migration and machining ability, if the graphite width is too wide, the control ability of crack trajectory will decrease and the maximum crack offset will enhance. Conversely, graphite width is too narrow, microwave absorption ability is weak, and processing ability is poor, correspondingly. When the graphite width increases to 4 ~ 6 mm, the maximum temperature of the heat source center enhances with the increase of the graphite width, viz., as the graphite width is greater than 6 mm, the effect of the width on the processing capacity is weakened. Therefore, the graphite width of 4 ~ 6 mm can effectively control the crack trajectory offset and has a higher processing ability.

Study on the effect of prefabricated crack on initial crack
The most obvious part of the crack trajectory deviation in microwave thermal crack cutting is the beginning and end of the crack. In order to explore the cause of the crack initiation deviation in microwave thermal crack cutting, a microscope is employed to observe the crack initiation section of the finished workpiece under the condition of 100 times magnification, as shown in Fig. 14. In the workpiece with an obvious deviation of the initial crack, the crack deviates from the expected position from the beginning. Ideally, the crack should start from the tip of the prefabricated crack, with the direction of initiation pointing to the predetermined trajectory direction, as marked by the dashed red line. However, the actual cases, the position of the prefabricated crack is not at the crack tip, and there is a large deflection between the crack angle and the desired trajectory. Therefore, a crack has occurred, and during the crack growth, the crack will expand in the direction of the heat source moving. It is noteworthy that the crack will gradually be corrected to the predetermined trajectory, and the crack in the average trajectory will be deflected again. As a result, the initial crack will shift, as signified in Fig. 14.
The prefabricated cracks were further observed, and the results are illustrated in Fig. 15. As can be seen from the figure, the tip of the prefabricated crack presents an arc shape with a diameter of 0.5 mm, which is not an ideal sharp crack. Therefore, the crack initiation position can occur anywhere in the arc, and the crack initiation angle is also the normal direction of the crack initiation position on the arc, which has a certain angle with the direction found at the top of the arc. In this case to crack extension, the direction of the reservation will be expected to make the starting process of the maximum thermal stress direction pointing to the top of the arc completely normal direction, the heat source movement, and the thickness of the graphite material, and the average distribution of temperature field around the desired trajectory has very strict requirements, more difficult to achieve in practical operation. Therefore, it can be analyzed that the  crack top produced by using diamond wire saw to make prefabricated cracks is circular arc instead of sharp shape, so the crack initiation position and angle will inevitably shift. To solve this problem, the preparation method of prefabricated cracks can be optimized, and the deviation of crack initiation position and angle can be controlled, to control the deviation of crack initiation segment.
According to the above analysis, prefabricated cracks need to be prepared with a higher technology to obtain a narrower width of prefabricated cracks. This paper tries to use two ways to optimize the prefabricated cracks. The diameter of the diamond wire saw used is 0.5 mm, and its initial crack is shown in Fig. 16a. In order to reduce the arc diameter of the crack tip and control the initiation position of the initial crack, a diamond wire saw with a diameter of 0.1 mm was attempted to use, and the initial crack is shown in Fig. 16b. Under the conditions of a repetition rate of 50 kHz, a processing power of 10 W, a scanning speed of 100 mm/s, and pulse width of 15 ns, a prefabricated crack with a length of 2 mm was prepared at the starting position of the predetermined trajectory of alumina ceramic using a UV laser with a wavelength of 355 nm. The crack was exhibited in Fig. 16c, as can be seen from the figure. The width of the prefabricated crack prepared by UV laser is only about 1 μm, which is 500 times smaller than that of the crack prepared by diamond wire saw, and the tip of the prefabricated crack is sharp.
Prefabricated cracks are prepared under the above three conditions for comparative processing, and other process parameters are controlled. The initial crack comparison of the machining results was shown in Fig. 17, in which the blue dashed line is the ideal trajectory and the red dashed line stands for the actual trajectory. The average and maximum offset of the initial crack section are compared as observed in Fig. 18. It can be seen that when the diamond wire saw with a thickness of 0.1 mm is employed to prefabricate the initial crack, the initial crack tip is more concentrated than before. The location of the initial crack is well-controlled, but the initial angle is still offset. From the comparison of crack offset, it can also be seen that the finer the prefabrication method, the maximum and average offsets of the initial crack are significantly reduced. Therefore, the analysis of the crack initiation phenomenon has also confirmed that the appropriate prefabrication method of the initial crack can effectively avoid the crack initiation offset.

Concluding remarks
In this study, based on the Fourier heat transfer equation and thermo-elasticity, the physical model of microwave thermal crack cutting Al 2 O 3 ceramics is established. The results satisfactorily achieved the aim of cutting Al 2 O 3 ceramics by microwave thermal crack. The thickness of the graphite coating, the width of the graphite coating and the prefabricated crack in the process of microwave thermal cracking cutting Al 2 O 3 were studied. The detailed information is listed as follows: (1) The present findings are very encouraging. The result indicates that the uneven thickness of graphite coating will affect the crack propagation path, and the influence law of graphite coating thickness on the thickness of processed materials is summarized. (2) The influence of graphite coating width on the crack trajectory is studied by means of simulation combined with the test. The influence mechanism of graphite coating width on thermal crack cutting was obtained from the analysis of temperature field simulation results, and the law of processing ability and trajectory control ability is concluded that the narrower the width of graphite coating is, the weaker the processing ability is. (3) From the microscopic observation of the crack trajectory, it can be seen that the crack has shifted since the initiation of the prefabricated crack, and then, the prefabricated cracks with different widths are prepared by different technological methods for processing experiments. The experimental results elaborate that controlling the width of the prefabricated crack can effectively reduce the offset of the initial crack.