Issue 
Mechanics & Industry
Volume 21, Number 6, 2020



Article Number  610  
Number of page(s)  12  
DOI  https://doi.org/10.1051/meca/2020077  
Published online  11 November 2020 
Regular Article
Mechanism of irregular crackpropagation in thermal controlled fracture of ceramics induced by microwave
School of mechatronics engineering, Harbin Institute of Technology, Harbin 150001, PR China
^{*} email: zcytougao@163.com
Received:
18
March
2020
Accepted:
28
September
2020
Microwave cutting glass and ceramics based on thermal controlled fracture method has gained much attention recently for its advantages in lower energyconsumption and higher efficiency than conventional processing method. However, the irregular crackpropagation is problematic in this procedure, which hinders the industrial application of this advanced technology. In this study, the irregular crackpropagation is summarized as the unstable propagation in the initial stage, the deviated propagation in the middle stage, and the nonpenetrating propagation in the end segment based on experimental work. Method for predicting the unstable propagation in the initial stage has been developed by combining analytical models with thermalfracture simulation. Experimental results show good agreement with the prediction results, and the relative deviation between them can be <5% in cutting of some ceramics. The mechanism of deviated propagation and the nonpenetrating propagation have been revealed by simulation and theoretical analysis. Since this study provides effective methods to predict unstable crackpropagation in the initial stage and understand the irregular propagation mechanism in the whole crackpropagation stage in microwave cutting ceramics, it is of great significance to the industrial application of thermal controlled fracture method for cutting ceramic materials using microwave.
Key words: Crackpropagation / thermal controlled fracture method / microwave / ceramics / unstable propagation
© AFM, EDP Sciences 2020
1 Introduction
The application of advanced ceramic has become more and more widely and it is an indispensable material in hightech intensive fields such as aerospace. Most of the occasions that ceramics are competent mainly relay on its unique properties such as high hardness, high melting point and low thermal expansion. However, these excellent properties make the cutting procedure difficult. In conventional cutting technic, materialremoval process is almost ineluctable in both mechanical mode and thermal mode [1–3]. Generally, materialremoval process would cause high energyconsumption, redundant materialwaste and environmental pollution, and induce poor surface integrity and low processing precision [1–5]. Moreover, materialremoval process by mechanical force would encounter serious tool wear as well as high processing cost.
Unlike these conventional cutting technic, thermal controlled fracture method (TCFM) is a green and pollutionfree cutting mode, which uses tensile stress to separate the brittle material into two parts. The tensile stress is usually thermal stress induced by a heat source. Because this process only needs to overcome the surface energy between the new sections, it causes no materialremoval. The TCFM was invented by Lumley et al in 1969, and was mainly used to treat glass sheets [6]. At the early stage, CO_{2} laser with wavelength of 10.6 µm was used as heat source to induce thermal stress, which can only be absorbed on the surface of the glass and would not cutoff glass thoroughly. This always needs some subsequent breaking process. Then, some scholars found that laser with wavelength of 1064 nm could penetrate a certain thickness of glass to form a fullbody cutting mode, and the priority of this cutting mode has been demonstrated [7–9].
During the development of half a century, scholars in this community have carried out many experimental and theoretical research on monocrystalline silicon, polycrystalline silicon, glass/silicon twolayer bonding materials and some ceramics [10–15]. The main difference between these materials and glass is that they can't form a fullbody cutting mode with the commonly used laser wavelength. During laser cutting silicon sheets (usually <0.5 mm) process, it is generally considered that laser produces a surface heat source with a negligible laserabsorption depth [12–15]. In other words, laser is absorbed at the surface of the silicon wafer, and the heat in the whole thickness is mainly generated by heat transfer.
In the past, TCFM is mainly used to treat these thin plates [16,17]. Although it has also been used to cut thick ceramics, the TCFM induced by laser is reported not competent for highquality cutting process. This is because that laser is not easy to induce fullbody cutting mode for ceramics [18,19]. Wang et al. first proposed to use microwave to treat ceramics based on TCFM [20]. They have successfully cut SiC ceramic with a thickness of 3 mm with fullbody cutting mode. However, some ceramics are difficult to absorb microwave, even though they could be completely penetrated. Then, Wang et al. used graphite to coat the surface of these ceramics to realize TCFM based on microdischarge mechanism. Although under the similar mode of surface absorption, the microdischarge makes its cutting quality better than that of laser cutting [18,19,21].
Wang et al. have found that crackpropagation path always deviates from the expected propagation path during TCFM [20–22]. This irregular crackpropagation hinders the industrial application of this advanced process. To deal with this problem, it is important to understand the mechanism of irregular crackpropagation. However, there are few reports on this issue.
In this paper, analytical models have been developed to reveal the irregular crackpropagation mechanism in microwave cutting ceramics using TCFM. Simulation of thermal fracture was conducted by general finite element simulation software to simulate the temperature and stress distribution. Combining the analytical models with the simulation work, the prediction of the maximum deviation range and the initial unstable crackpropagation length have achieved. Microwave cutting experiments with regards to glass, SiC, Al_{2}O_{3} and ZrO_{2} ceramics were conducted to verify this prediction method. Simulation work was also used to reveal the mechanism of the irregular crackpropagation in the middle stage and end segment.
2 Materials and Method
2.1 Principle of microwave cutting ceramics based on thermal controlled fracture method
The main physical process in the microwave cutting ceramics based on TCFM is shown in Figure 1. It can be divided into three procedures. Firstly, microwave heating ceramics as shown in Figure 1a: this process mainly consists of the dielectric loss of ceramic materials to microwave, the generation and conduction of heat. The incident microwave could penetrate the ceramics and be absorbed by the materials. According to absorptive capacity to microwave, it can be divided into bulk heating mode (ceramics can absorb microwave well by themselves) and surface heating mode (coating graphite on ceramics to add absorption of microwave); secondly, the generation of thermal stress as shown in Figure 1b: this process is mainly based on the theory of thermal stress. Tensile stress occurs where temperature is lower than the average value, while compressive stress occurs where temperature is above the average value. The absolute value of thermal stress is mainly determined by the temperature gradient. Finally, the crack propagation process as shown in Figure 1c: this process is mainly based on the theory of fracture mechanics of solid materials. When the initial crack on the ceramic encounters a tensile stress greater than the fracture strength of the ceramic, it will propagate. If the tensile stress can guide the crack to propagate continuously at a certain speed, the cutting process achieves.
Fig. 1 Physical processes ((a) microwave heating ceramics, (b) generating thermal stress and (c) crack propagation process) in microwave cutting ceramics based on TCFM. 
2.2 Experimental equipment, materials, and methods
Figure 2 shows the experimental apparatus of microwave cutting system. It is produced by Nanjing Huiyan Microwave System Engineering Co., Ltd of china and its maximum output power is 1.5 kW. The type used in this study is MY1500S. The system is composed of a microwave source and a cutting machine (2.45 GHz) shown in Figure 2a, microwave controller shown in Figure 2b, a xy moving plate, and a circular focusing waveguide shown in Figure 2c. The circular focusing waveguide is used to generate circular heat source. Previous studies have shown that the circular heat source is more feasible for curve cutting of glass [22].
To improve the security of the experiment, the inner wall of the machine tool shield is made of absorbing material, and four utensils containing water are placed into the cabin to absorb the leaky microwave during experiment. The material is placed on the workbench which could adjust the distance between the waveguide and workpiece.
The experimental materials include NaCa glass, SiC ceramic, Al_{2}O_{3} ceramic and ZrO_{2} ceramic. The dimensions of these materials are 100 × 100 × 1 mm.
Curve cutting experiments are carried out. The start point of scanning position on the workpiece was marked in advance for recording the relative position of the crackpropagation path conveniently. Optical microscope was used to observe the crackpropagation path after the cutting experiments. The controllable processing parameters are microwave power and scanning speed. The values of these parameters for each material are given in Table 1. Each group of experimental parameters were tested six times repeatedly.
Fig. 2 Experimental apparatus of microwave cutting system consisting of (a) microwave source and cutting machine, (b) microwave controller and (c) circular focusing waveguide. 
Processing parameters in microwave cutting ceramic using TCFM.
2.3 Simulation model for cutting ceramics based on thermal controlled fracture method
The main physical process during microwave cutting are microwave absorption, heat transfer, thermal induced stress and crack propagation. The modeling process are as following:
The thermal power density P_{v} in the workpiece when loading with microwave can be given by [23]:(1)where P_{v} is thermal power density (W/m^{3}); f is microwave frequency (Hz); ε_{0} is vacuum dielectric constant (F/m); ε is relative dielectric constant of material; tanδ is dielectric loss tangent of material. E_{out} is RMS of electric field intensity for output microwave (V/m).
When heated to a steady temperature, the temperature at any point in a linear elastic half space at the initial crack can be given by:(2)where a is thermal conductivity (W/m · °C), ρ is density (kg/m3), c is specific heat capacity (J/(kg · °C), z′ is workpiece thickness (m).
According to thermal stress theory, this would produce a thermal stress field in the material. The normal stress σ_{x} along Xdirection and σ_{y} along Ydirection at any point at the initial crack caused by temperature field are given by:(3)and(4)where G is modulus of elasticity in shear, ε_{x} and ε_{y} are normal strain, β is thermal stress coefficient.
According to thermal stress theory of fracture mechanics of brittle solid materials, the critical fracture stress σ_{F} at the crack front can be given by [24]:(5)
Where c_{0} is the size of the precrack, γ is the free surface energy per unit area, E' is the equivalent modulus, which is equal to the elastic modulus E under the condition of thin specimen. When the thermal stress loaded at crack front is greater than or equal to σ_{F}, the crack system begins to expand.
The goal of analyzing the main physical process during microwave cutting is to simulate the dynamic process of crackpropagation. The crackpropagation process is discontinuous, so the conventional finite element method (FEM) can't be used to calculate the crackpropagation process directly. To resolve this problem, the extended finite element method (EFEM) is used. This method needn't to refine grid dynamically in simulation process, and it has good convergence characteristics when the mesh around the crack is subdivided to a certain extent. These simulations are implemented in commercial finite element software ABAQUS 6.141.
In the simulation model, the element type is C3D8R, and the material damage type is evolution damage. As shown in Figure 3, the grid density nearby the microwave scanning position is increased. In this way, the calculation results of the concerned areas become more accurate, and the calculation efficiency can be improved.
The physical parameters of the ceramics are shown in Table 2. The parameters are effective in room temperature, however, since the processing temperature of the TCFM is low (can be lower than 200 °C), these can also be used in the prediction model. Physical parameters given in the table include electromagnetic parameters, mechanical parameters and thermal parameters. The dielectric constants of these materials are provided by AET Corporation of Japan. The mechanical and thermal parameters of these materials are provided by Harbin Xinhui Special Ceramics Co., Ltd of china. According to the Equation (1), ε and tanδ are two material property which determines the microwave absorption and heat production capacity of materials. Among them, NaCa glass and SiC ceramic can absorb microwave of 2.45 GHz well. Al_{2}O_{3} ceramic and ZrO_{2} ceramic used in this study can hardly absorb microwave of this frequency band, so the surfaces of the radiation site are coated with graphite, which is good absorbing material for microwave. The ε and tanδ of the graphite coating are also given in the table, which are used in the cutting simulation of Al_{2}O_{3} and ZrO_{2}.
Fig. 3 Finiteelementmeshes of cutting ceramics based on TCFM. 
Physical parameters of the materials.
3 Results
3.1 Crackpropagation in initial stage
To observe the crackpropagation state, cutting experiments on glass were conducted. Figure 4 shows the crackpropagation in the initial stage during cutting glass with microwave. When the initial crack propagates, it makes a sharp blasting noise. As shown in Figure 4, the initial crackpropagation deflects form the scanning path. The distance between the starting point of crackpropagation and the heat source center is defined as the starting point deviation (Y_{1}′ ). The length of the initial unstable propagation segment in scanning direction is defined as the unstable propagation length (X_{1}′ ). It is found that Y_{1}′ is approximately 3 mm and X_{1}′ is approximately 7 mm.
It is notable that microcracks on the left of the heat source center are about 250 µm, and microcracks on the right side are about 327 µm. As shown in Figure 4, the crack propagates from the microcrack with 327 µm.
According to the Equation (5), the larger the crack size c_{0}, the smaller the critical stress σ_{F} is required for crack initiation. The critical stress corresponding to 327 µm is about 87.4% to it of 250 µm. The tensile stress at the crack front is determined by its distance from the heat source center. Therefore, for microcracks with same distance, these of 327 µm have greater opportunities to propagate than these of 250 µm.
Fig. 4 Unstable propagation of initial crack of glass. 
3.2 Crackpropagation in middle and end stage
The crackpropagation morphology of curve cutting of glass is shown in Figure 5. The red reference line represents the scanning path. The crackpropagation path consists of an initial straightline segment, a middle curve segment and a final straightline segment. It is noteworthy that in the curve cutting segment the crack path deviates from the scanning path and inclines to the inside of the curve. There is a lag between the crack front and the heat source center in the curve cutting segment. The crack undergoes a nonpenetrating propagation (the crack do not propagate close to the outlet of scanning) in the end segment.
Fig. 5 Crackpropagation morphology of curve cutting of glass with circular microwave spot. 
4 Discussion
4.1 Prediction of the unstable crackpropagation in initial stage
The irregular crackpropagation phenomena shown in Figure 5 indicate that the realistic crackpropagation path is not along with the expected scanning path accurately. These nonideal propagation phenomena can be simplified as a diagram shown in Figure 6. It shows that the actual crack initiation point S deflects from the ideal point P in the initial propagation stage. Because of this deflection, the initial realistic crackpropagation deflects from the ideal path (the microwave scanning path) evidently. In the intermediate segment, the crack path deviates from the heat source path obviously in curve cutting site. In the end segment, the crack does not penetrate the specimen in the outlet. In order to understand the mechanism of irregular crackpropagation, models for predicting the maximum crack initiation range and the length of the initial unstable crackpropagation have been established respectively.
In the fracture mode I, the critical conditions for crack initiation can be written as [24]:(6)where K_{I} is the stress intensity factors in fracture type I, K_{Ic} is the fracture toughness of an infinite plate with unilateral crack. According to theory of fracture mechanics, under the condition of edge crack, K_{I} can be written as:(7)where σ_{r} is tensile stress at the crack front without considering of stress concentration; l_{r} is the crack length. The K_{Ic} can be the expressed by:(8)where σ_{c} is the critical stress for crack initiation (when the tensile stress at crack front is beyond σ_{c}, the crack would propagate). It can also be written as:(9)where l_{c} is the critical cracklength for crack initiation. According to Equations (8) and (9), σ_{c} and l_{c} can be written by:(10)and(11)
Equation (10) is the critical stress for crackpropagation and Equation (11) is the critical cracklength for crackpropagation. The crack initiation condition is: σ_{r} ≥ σ_{c} or l_{r} ≥ l_{c}.
Generally, there are many microcracks on the edges of ceramic and glass workpieces as shown in Figure 4. Experimental results show that the crack system tends to trigger from the microcracks with longer length. These microcracks commonly distribute randomly with different length at the edge of workpiece. Reference to Figure 6, the ideal propagation location is P and the actually propagation location is S. According to Equation (10) and the crack initiation condition, the stress conditions for crack propagating from P not S are:(12)
According to Equation (11) and the crack initiation condition, the cracklength conditions for crack propagating from P not S are:(13)
The meaning of these parameters consults the parameter description in Equations (9) and (10), just adding P or S as subscripts to represent the corresponding value in these two places shown in Figure 6.
Because of the random distribution of the microcrack at the edge of specimen, the location of crack initiation is random and can't be accurately determined. However, the region of it could be predicted theoretically by analyzing the crack initiation condition above. Figure 7 shows the principle to predict the crackpropagation region under typical stress distribution conditions.
The curve L_{r} in Figure 7 represents the length variation of microcrack along the edge of specimen. The crack length at the ideal position P is defined as the smallest one. The curves of σ_{r1} and σ_{r2} correspond to two different stress distribution curves. The curves of L_{c1} and L_{c2} represent the cracklength threshold curves corresponding to σ_{r1} and σ_{r2} respectively. The intersection of L_{c1} and L_{r} is A_{1} and A_{2}; the intersection of L_{c2} and L_{r} is B_{1} and B_{2}.
According to Equation (11), the closed shadow region of L_{c1} and L_{r} is the possible crackpropagation region under the condition of σ_{r1}. Because it satisfies the crack initiation condition of l_{r} ≥ l_{c} in this region. When the cracklength threshold curve is L_{c2}, the crackpropagation region would add a distance of ΔY' that is the length between A_{1} and B_{1}. However, it is difficult to predict A_{1}, A_{2}, B_{1} and B_{2} in practice.
To simplify the prediction process of crackpropagation region, D_{1}, D_{2}, D_{3} and D_{4} are introduced to replace the A_{1}, A_{2}, B_{1} and B_{2}. The determination process and the meaning of these points are as followings:
1. The determination of D_{1}, D_{2}, D_{3} and D_{4}:
First, obtaining the fluctuation value of cracklength as ΔL_{r}; then, drawing upwards a line whose distance from the bottom of curve L_{c1} and L_{c2} is ΔL_{r}; finally, the interactions between this line and the L_{c1} and L_{c2} are D_{1}, D_{2}, D_{3} and D_{4}.
2. The meaning of D_{1}, D_{2}, D_{3} and D_{4}:
D_{1} and D_{2} are the extreme location for crack initiation corresponding to σ_{r1}, and Y_{1} is the maximum crack initiation range corresponding to σ_{r1}. Similarly, D_{3} and D_{4} are the extreme location for crack initiation to σ_{r2}, and Y_{2} is the maximum crack initiation range corresponding to σ_{r2}.
3. Its relationship with A_{1}, A_{2}, B_{1} and B_{2}:
When the lowest points of L_{c1} and L_{c2} are close to position P, D_{1}, D_{2}, D_{3} and D_{4} are approximately coincident with A_{1}, A_{2}, B_{1} and B_{2}, and the ΔY (distance between D_{1} and D_{3}) is approximately equal to ΔY'.
By introducing D_{1}, D_{2}, the step to obtain maximum crack initiation range Y_{1} is as following:
Frist, σ_{r} curve is obtained by stress calculation or simulation; then, the corresponding curve Lc can be fitted using some discrete points calculated by Equation (9). Finally, the D_{1}, D_{2} is obtained by the above methods, and the Y_{1} is achieved.
By using this method, the maximum crack initiation ranges of the four kinds of specimens are gotten and are shown in Figure 8. The σ_{r} curve and L_{c} curve are obtained by thermal stress simulation. The length change of microcrack in the edge of each workpiece is defined as Δl_{max} corresponding to Δl_{rc} in Figure 7. This value is obtained by calculating the length difference between the maximum value and the minimum value of the microcracks. Due to the different manufacturing methods of workpieces, these values for these four materials are different. After measurement and calculation, the Δl_{max} for glass is approximately 200 µm, as well as 500 µm for SiC, 400 µm for Al_{2}O_{3} ceramic and 300 µm for ZrO_{2} ceramic. As shown in Figure 8, it is found that the Y_{1} for glass is approximately 6 mm as well as 12 mm for SiC, 8 mm for Al_{2}O_{3} and 10 mm for ZrO_{2}.
Under the condition of a unilateral crack on a rectangular plate with unit thickness, if the crack length is x and an increment dx is added, according to the theory of Griffith energy release, then the released stressstrain energy dU_{ε} can be written as:(14)where σ_{f} is the tensile stress perpendicular to the crackfront (Pa), which is balanced with the external load; The generated surface energy dE_{s} (J)is:(15)
When the crack propagates, the increment of kinetic energy is:(16)
Where k ′ is integral coefficient of displacement component; v_{s} is crackpropagation velocity (m/s); ρ is the reciprocal of acoustic velocity in the medium (s/m); E_{k} is crack kinetic energy (J).
The above energy meets the following requirements:(17)
Introducing Equations (14)–(16) into Equation (17), then vs can be written as:(18)
When v_{s} is constant, the crack growth is stable. Then, σ_{f} can be written as:(19)where B_{k} is the constant of stress function, which can be written as:(20)
In actual thermalcracking process, k′ρvs is negligible (because the crackpropagation is far less than acoustic velocity in the medium), so the stress σ_{f} can be written as:(21)
Equation (21) is an ideal stress condition in which the crack velocity is equal to the heat source velocity of a constant.
Due to the particularity of the boundary conditions of initial crack, the real stress σ_{l} in front of the crack is often greater than the ideal stress σ_{f}. This will induce the overspeed propagation of the initial crack. In theory, this unstable propagation segment tends to end when the stressstrain energy released by the crack system is equal to the surface energy produced by the newly generated surface.
Figure 9 shows the diagram of the relationship of these two stress curves and it can be used as a method to predict the overspeed propagation length. The distance between the specimen edge and the interaction of these two curves is the overspeed propagation length. It is apparent that the increase of the stress gradient in front of the crack tends to reduce the overspeed propagation length.
The prediction value of overspeed propagation length of each material is obtained and is shown in Figure 10. The intersection of σ_{l} curve and σ_{f} curve is defined as the stable point. The distance between the stable point and the workpiece edge is defined as overspeed propagation length X_{1}, which is actually the length of the initial unstable crackpropagation along Xdirection (the microwave scanning direction).
Therefore, combining the σ_{l} curve and σ_{f} curve, the X_{1} can be predicted. The σ_{f} curve can be obtained by Equation (21), while σ_{l} curve can be obtained by microwave cutting simulation. It indicates that X_{1} of ZrO_{2} ceramic is 10 mm; X_{1} of Al_{2}O_{3} ceramic is smallest, only 2 mm; X_{1} of glass and SiC ceramic are 4.5 mm and 6 mm respectively.
Figure 11 shows the starting point deviation Y_{1}′ and the unstable propagation length X_{1}′ in cutting experiments of glass, SiC, Al_{2}O_{3} and ZrO_{2}. The mean value and deviation value were calculated. Due to the different distribution of microcracks around workpiece edges and the unique mechanical and thermal properties of different materials, the mean values are different. As shown in Figure 11, it indicates that the results of Y_{1}′ and X_{1}′ fluctuates in cutting experiment of these materials.
Table 3 shows the comparation of experimental and predicted results on parameters in initial unstable propagation. It indicates that the mean values of Y_{1}′ are not beyond half of the maximum crack initiation range Y_{1}. In the case of ZrO_{2}, Y_{1}/2 does not exceed 5% of the mean value of Y_{1}′ . Combining with Figure 11, it is noteworthy that the maximum values of Y_{1}′ do not deviate too much from half of Y_{1}. What' more, the mean value of X_{1}′ does not deviate too much from the overspeed propagation length X_{1}. It indicates that the theoretical model can effectively predict the unstable propagation in the initial stage of microwave cutting ceramics based on TCFM.
Fig. 6 Sketch of irregular crackpropagation in microwave cutting experiment. 
Fig. 7 Principle of determining the crackpropagation region under typical stress distribution conditions. 
Fig. 8 Thermal stress and representative zone of crack along the cutting in edge of four kinds of ceramics. 
Fig. 9 Diagram of overspeed propagation length prediction. 
Fig. 10 Prediction results of overspeed propagation length of four kinds of ceramics. 
Fig. 11 Results of the starting point deviation length (Y_{1}′ ) and the unstable propagation length (X_{1}′ ) in the initial crack propagation stage. 
Comparation of experimental and the prediction results on parameters in initial unstable propagation.
4.2 Crackpropagation mechanism of middle and end stage
In the cutting experiments, the deviated crackpropagation appears in the middle stage of crackpropagation. In order to understand the mechanism of this phenomenon, the relationship between the tensile stress at the crack front and the propagation state of the crack was investigated. It is found that crack front does not always own the maximum tensile stress in the middle propagation segment. The crack front is sometimes at the front of the maximum tensile tress zone. This is the reason for discontinuous propagation of the intermediate crack. This is caused by the intermittentequilibrium mechanism between the strain energy at the crack front and the newly generated surface energy for the newly propagated crack.
Figure 12 shows the relationship between the strain energy release and crack propagation at the crack front. The specific explanation is as follows: as shown in Figure 12a, tensile stress field is generated and concentrated around the crack front to balance with the compressive stress induced by the heat source center. This concentrated tensile stress would increase with the increase of its distance from the heat source. When the concentrated stress is greater than the threshold of the crackpropagation stress, the strain energy releases immediately and converts into surface energy for the generating new surface. At this moment, the new crack front can propagate instantaneously as shown in Figure 12b. Then, with the moving forward of heat source, the tensile stress at the crack front will gradually concentrate to its peak to induce a next propagation as shown in Figure 12c.
Therefore, to keep this cycle going, the maximum concentrated stress at the crack front should yield the tensile stress threshold for crackpropagation. If this maximum value can't reach the threshold in a segment, the tensile stress in the crack front would decrease as the heat source continues to go ahead. So, the crackpropagation would end. To avoid this phenomenon, the output power of microwave should not be less than a certain value.
Figure 13 shows a diagram of the relationship between the crack propagation path and the heat source path. When the heat source moves ahead, the crack will lag behind the heat source for a certain distance based on the intermittent propagation mechanism discussed above. As is shown in Figure 13, the crack always propagates towards the center of the heat source and keeps a distance of D_{h} from it. The actual propagation path of the crack is h_{2} and is composed of a series of segments of broken line. When the broken line is short enough, its envelope curve (dotted line in the figure) is the actual crackpropagation path. It can be seen that the actual propagation path always deviates from the actual scanning trace in curve cutting.
Figure 14 shows one of the temperature field and tensile stress field contours in the process of curvedcracking of glass. As discussed above, the actual propagation path in curve cutting propagates into the inside of preset trace. This speculation reproduces in the simulation of curve cutting glass as shown in Figure 14. From the temperature distribution contours in Figure 14a, it indicates that the maximum temperature in microwave cutting glass is about 140 °C, which is lower than the temperature value in laser cutting glass based on TCFM reported by Zhao [9]. Figure 14b shows simulation results of the stress distribution in curvedcracking of glass. As is shown in Figure 14b, the tensile stress is asymmetrically distributed along the scanning line.
According to the experimental results, the terminal crack failed to penetrate the workpiece. To reveal the reason for this phenomenon, the stress characteristic nearby crack front is analyzed when the crack propagates near the terminal point.
Figure 15 shows the simulation and sketch of the stress characteristic at the end propagation segment. As discussed above, the intermediate crack would undergo a discontinuous propagation caused by the intermittentequilibrium mechanism between the strain energy at the crack front and the newly generated surface energy. However, this relative equilibrium state would be broken when the crack propagates near the end stage.
As shown in Figure 15a, a compressive stress field σ_{k1} is generated in the heating region. Tensile stress fields σ_{p1} and σp2 would generate behind and in front of this compressive stress field to balance with it. In the middle segment, the tensile stress fields σ_{p1} and σ_{p2} are in equilibrium with the compressive stress field σ_{k1} as the heat source moving. However, this equilibrium no longer holds when the crack approaches the edge of the workpiece. As shown in Figure 15b, the distance between the center of σp2 and workpiece edge is ΔD_{k}. When the heat source is close to the edge of the workpiece (ΔD_{k} is close to zero), the front tensile stress area σ_{p2} would produce a large stress concentration. Since the compressive stress in σ_{k1} is kept near constant, the average stress in σ_{p1} should decrease to keep balance. This decreases the stress intensity factor K at the crack front. When K < K_{IC}, the crack would stop propagating and result in nonpenetrating propagation.
Fig. 12 Process of stain energy releasing and crack propagating. (a) Tensile stress field concentrating; (b) New crack front propagates; (c) A next propagation. 
Fig. 13 Mechanism of deviation between cutting trajectory and heating path. 
Fig. 14 Simulation of the deviated propagation in curvedcracking of glass and its corresponding temperature and stress distribution. (a) Temperature distribution (middle propagation). (b) Stress distribution (middle propagation). 
Fig. 15 Simulation and sketch of the stress state nearby the propagating crack at end segment. (a) Simulation of stress distribution. (b) Sketch of stress distribution. 
5 Conclusion
In conclusion, this paper has revealed the irregular propagation mechanism in the whole crackpropagation stage in microwave cutting ceramics using TCFM by experimental and theoretical study. The following conclusions can be drawn:

The phenomena of deviated crack propagation at the initial and intermediate segment and the nonpenetrating propagation at the end segment were observed through curve cutting experiments. It indicates that the maximum temperature in microwave cutting glass is about 140 °C.

The mechanism of the irregular crackpropagation was revealed by the combination of analytical model and finite element model. Through theoretical analysis, the propagation speed of the initial stage is higher than the scanning speed of microwave.

The maximum crack initiation range and the length of the overspeed propagation could be predicted by analytical and finite element models. The microwave cutting experimental results show good agreement with the prediction results, and the relative deviation between them can be <5% in cutting of some ceramics.

The crack front is sometimes in front of the maximum tensile tress. This is caused by the intermittentequilibrium mechanism between the strain energy at the crack front and the newly generated surface energy.
By effectively predicting the unstable propagation offset and understanding the mechanism of irregular propagation, this study is of great significance to avoid and reduce the offset in thermal controlled fracture method using microwave.
Acknowledgments
This research is supported by the National Science Foundation of China (Grant No. 51275118).
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Cite this article as: X. Cheng, C. Zhao, H. Wang, Y. Wang, Z. Wang, Mechanism of irregular crackpropagation in thermal controlled fracture of ceramics induced by microwave, Mechanics & Industry 21, 610 (2020)
All Tables
Comparation of experimental and the prediction results on parameters in initial unstable propagation.
All Figures
Fig. 1 Physical processes ((a) microwave heating ceramics, (b) generating thermal stress and (c) crack propagation process) in microwave cutting ceramics based on TCFM. 

In the text 
Fig. 2 Experimental apparatus of microwave cutting system consisting of (a) microwave source and cutting machine, (b) microwave controller and (c) circular focusing waveguide. 

In the text 
Fig. 3 Finiteelementmeshes of cutting ceramics based on TCFM. 

In the text 
Fig. 4 Unstable propagation of initial crack of glass. 

In the text 
Fig. 5 Crackpropagation morphology of curve cutting of glass with circular microwave spot. 

In the text 
Fig. 6 Sketch of irregular crackpropagation in microwave cutting experiment. 

In the text 
Fig. 7 Principle of determining the crackpropagation region under typical stress distribution conditions. 

In the text 
Fig. 8 Thermal stress and representative zone of crack along the cutting in edge of four kinds of ceramics. 

In the text 
Fig. 9 Diagram of overspeed propagation length prediction. 

In the text 
Fig. 10 Prediction results of overspeed propagation length of four kinds of ceramics. 

In the text 
Fig. 11 Results of the starting point deviation length (Y_{1}′ ) and the unstable propagation length (X_{1}′ ) in the initial crack propagation stage. 

In the text 
Fig. 12 Process of stain energy releasing and crack propagating. (a) Tensile stress field concentrating; (b) New crack front propagates; (c) A next propagation. 

In the text 
Fig. 13 Mechanism of deviation between cutting trajectory and heating path. 

In the text 
Fig. 14 Simulation of the deviated propagation in curvedcracking of glass and its corresponding temperature and stress distribution. (a) Temperature distribution (middle propagation). (b) Stress distribution (middle propagation). 

In the text 
Fig. 15 Simulation and sketch of the stress state nearby the propagating crack at end segment. (a) Simulation of stress distribution. (b) Sketch of stress distribution. 

In the text 
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