The study proposed an important micro-specific coefficient based on the mathematical modeling of micro-cutting resistance to predict the mechanic conditions at cutter-edge radius. For the steady-state chip formation in the micro-cutting process, the differential angle is usually constant, and the plowing angle and rake angle are relative to the tool-edge radius, cutting resultant force, plowing resistance, surface roughness, and shearing resistance on the tool-workpiece. The optimal process included a cut of depth of 0.001 mm, cutting length of 0.003 mm, cutter-edge temperature of 38°C, and an edge radius of 0.0005 mm on workpiece Al-7075; the optimal cutting force in x-axis was 0.0005 N (Avg.) and the optimal cutting force in y-axis was 0.00028 N (Avg.) for better surface roughness Ra = 0.16. The higher temperature was 42.16°C on the workpiece and tool HSS, and the maximum strain rate occurred on the chip shearing zone was 9.33E6 (/s), which obeyed the generalized cutting criterion by numerical analysis. While the micro-specific coefficient is close to 1, the plowing zone will increase friction, stress, resistance, and even cutting excited vibration, resulting in discontinuous chipping. Besides, the process developed the micro-MDOF cutting dynamics model and applied a fractal equation to simulate the micro-cutting process. The validation can be proved as the derived theory agreed well with the simulation in the micro-cutting process.
- specific coefficient
- fractal equation
- steady-state chip formation
- micro-MDOF cutting dynamics
Micro-cutting process for depth of cut and feed is so small that it is very hard to observe under a microscope. Due to the size effect of the cutter-edge radius for larger influence on plowing and shearing zones, it results in increased friction heat, cutting force and, furthermore, specific energy. The study proposed the micro-resistance model of plowing and shearing in the quasi-state cutting process by analytic geometry. Using the fractal equation can help us to build the relations of chip fractal geometry and specific energy. The famous scholar Mandelbrot gave the set  is a compact set (Figure 1), where resisted of complex numbers c, and he studied space of quadratic polynomials and proposed the function
1.1 Paper review
Cao et al.  found that the cutting edge radius affects the microscale in the cutting process at a smaller uncut chip thickness by altering the effective rake angle, enhancing the plowing effect and the work affecting the material deformation process, expanding and widening the plastic deformation zone, and causing higher energy dissipation due to increased tool-chip contact length. The author wants to expand the work to develop the theoretical model by advanced mathematics. On the other hand, the study of Cao et al.  unestablished the micro-specific coefficient to investigate the relations of factors on plowing, shearing, cutter-edge radius, cutting heat, and stain rate. To extend the study of Cao et al. , the fractal equation proposed in Wu [3, 4, 5] can be applied on the analysis of variations in chip shapes and undeformed chip formation areas of specific energy obtained. Wu  considered the vertical force
2. Theoretical modeling
The process developed the micro-MDOF cutting dynamics model and micro-fractal equation to simulate the micro-cutting process. Through fractal mathematics, the results presented optimal geometric parameters for micro-cutting simulation for tool HSS and workpiece Al-7075 as shown in Figures 4–6. The study found that the micro-specific coefficient applied in the micromachining process is to explain the influence of shearing and plowing at cutter-edge radius.
2.1 Establishment of the specific coefficient for plowing and shearing resistance in the quasi-state micro-cutting process
The quasi-state micro-cutting process is a dynamical balance process for cutting resistance and cutting force. The three points C, D, and E are aligned as the same points from Figure 6(a). Firstly, the free body diagram (FBD) for plowing and shearing resistance in the quasi-state micro-cutting process should be established as Eqs. (1) and (2):
For the orthogonal cutting process,
Due to the size effect with considering the cutter-edge radius r in micromachining process, the differential-cutter angle is relative to two factors: plowing angle and rake angle . For the steady-state chip formation in the micro-cutting process, the rake angle is usually a constant, but the plowing angle is relative to the tool-edge precision and surface roughness on the workpiece. For the condition of resistance in the micro-cutting process, cutting resistance on points F, B, and C has a large influence on the steady-state chip formation process. Arc of cutter-edge radius
While or close to 1, called plowing and shearing coupling, large plowing resistance and vibration on the workpiece surface will occur because the plowing zone at B point is very obvious from Eq. (8). The condition should cause an increase in surface roughness:
where means the specific coefficient of micro-cutting.
From integration, we obtain
To obtain the function of the plowing at B in micromachining process,
Equation (12) is an important result for plowing and shearing influence of micro-cutting under the size effect. While is close to 1, the plowing zone will increase friction, stress, resistance, and even cutting excited-vibration or chipping, resulting in discontinuous chipping. The way to improve is to design the rake angle and cutter-edge radius or raise the tool precision and material.
Although denoted the resistance function in the micromachining process, the function can be expanded to investigate the relations of factors combining micro-cutting force, shear stress, strain, and strain rate.
From the view of the micro-cutting process, the definition of micro-specific coefficient is , and the definition of average micro-specific coefficient is ; if , Eq. (8) can be rewritten as follows:
The result still expresses that the function existed.
2.2 Generalized chip load of micro-cutter by the fractal differential equation
According to sinusoidal multi-cutters, the cutters’ interference ratio of cutter arrangement can be written as Eq. (14):
means (n + 1)th cutter,
denotes the length of unit wave, n means cutter number of period, b means cutter width, ϑn+1 and
means the banding angle of the unit cutter. The equivalent cutting depth of the cutting interference can be presented as Eq. (15):
where means the initial cutting depth and means the transient cutting depth. Considering the cutting depth and cutting width, Eq. (16) can be obtained if the frictional energy dissipated:
Equation (19) will delete the overlap ratio if the sinusoidal multi-cutters become a single cutter. On the other hand,
2.3 Micro-MDOF cutting dynamics equation
Assume the initial condition as
Substitute into Eq. (22),
For orthogonal conditions,
Equation (32) can be expanded by Taylor series expansion or numerical analysis.
In order to obtain the temperature variation of the cutter edge and chip surface, where the Johnson-Cook equation is used as Eq. (33), the temperature conduction between tool and workpiece is assumed. The study does not consider the cutting temperature conducting into the air:
where means the melting point temperature, means the environmental temperature,
3. Results and discussion
The study by micro-MDOF cutting dynamics simulated micro-cutting process according to micro-resistance equations (Eqs. (7) and (12)) at cutter-edge radius in order to validate plowing resistance increasing than shearing, resulted in cutting temperature raising occurred at plowing zone. Simultaneously, the parameter setup of the rake angle at 5, 10, and 15° is presented to observe the relations of plowing zone, heat rate, von Mises stress, cutting force, and specific coefficients. The study did not consider the affection of the tool-HSS elastic deformation, and it needs to focus on the chip formation and shear deformation zone by the size effect. The size effect in micro-cutting process is relative to the theoretical model . The plowing effect at B point is greater than plowing at C and shearing at F if the micro-specific coefficient is close to 1. Hence, another important factor is differential-cutter angle relative to undeformed chip thickness if the micro-specific coefficient belongs to a reasonable range to avoid shearing-plowing coupled effect, such as the range , where is similar to the vertical feed on the workpiece or digging process. The variation of differential-cutter angle will affect the cutting force because the variation of the resultant force comes from the two component forces at x-direction and at y-direction. The resultant force vectors will affect the variation of undeformed chip thickness according to the theoretical model and demonstration in Figures 4–6. Therefore, the micro-cutting theory by the size effect is different from the traditional plastic theory. In the micro-cutting process, the tool and workpiece are usually symmetric in the z-direction. In brief, the model can be simplified from three-dimensional to two-dimensional in order to reduce the calculation as shown in Table 1 and Figure 8. The influence of the factor affecting the variation of chip thickness comes from the variation of the resultant force at differential-cutter angle. If the radius of micro-cutter is larger, the resistance at x-direction will become larger due to the contacted areas for arc DE and arc BC increasing at the tool-workpiece interface, and furthermore, the friction and resistance also increase, resulting in the differential-cutter angle varying with increasing resultant force, cutting force increasing with increasing resultant force if the cutting velocity was constant in the x-direction, and finally, the chip thickness varying with variation in the cutting force. From the observation of Figures 5 and 6, the resultant force indeed varied with the differential-cutter angle. Hence, the micro-specific coefficient varied with the differential-cutter angle, and it was an important factor for the production precision in the machining process.
|Material and tool geometry parameters|
|Length*height (2D)||0.005 mm*0.002 mm|
|Cutter-edge radius||0.00005 mm|
|Rake angle (°)||5||10||15|
|Clearance angle (°)||10|
|Room temp. (°C)||20|
|Initial temp. of the workpiece (°C)||20|
|Cutting depth (average chip thickness)||0.001 mm|
|Cutting condition||Dry cutting|
|Cutting velocity (mm/min)||30|
|Cutter condition||Single-cutter; orthogonal cutting|
3.1 Micro-MDOF cutting dynamic simulation
The initial conditions for the material and workpiece as shown in Table 1 and Figure 8 can be set up by H-adaptive model fine mesh raising the cutting precision. Von Mises stress can be expressed as the wear on tool flank or the cutting resistance under plastic stress and friction. Figure 9 demonstrated that the analysis of vectors on cutter-edge radius can prove that plowing at B (934 MPa) is larger than shearing at F (800 MPa) for (c) compared with (b) and (a); (a) theoretical model FBD for micro cutter-edge radius; (b) defined plowing and shearing on cutter radius, where the plowing angle is . The shearing and plowing resistance model can be established to predict the influence of tool geometry, rake angle, and plowing angle on the stress of the plowing zone and shearing zone. On the other hand, it can explain why cutting force increased, friction heat increased, and tool flank wear occurred by the size effect in the micro-cutting process. The finding results offer the importance for micro-MDOF cutting dynamics differed from the traditional cutting process. According to the simulated micro-cutting conditions of cutter-edge radius, the finding results are as follows: the maximum von Mises 934 MPa occurred on tool flank; the maximum heat rate 9.5E6 W/mm2 occurred on the plowing zone to a part of the shearing zone; the maximum strain rate 9.5E6 occurred on the plowing zone to a part of the shearing zone similar to heat rate; and the higher temperature 43.4°C on the whole cutter-edge radius and shearing-plowing zone suffering size effect (Figure 10).
3.2 The relations of rake angle, cutting force, and cutting temperature
The results are nicely reasonable for workpiece Al-7075 and tool HSS. The variation of micro-cutting force is not uniform because of the roughness increasing on the workpiece surface after micro-cutting. The result can be proved by the plowing influence of size effect according to Eq. (7). The optimal process included a cut of depth of 0.001 mm, a cutting length of 0.003 mm, and a cutter edge of 38°C on workpiece Al-7075; the optimal cutting force in x-axis was 0.0005 N (Avg.) and the optimal cutting force in y-axis was 0.00028 N (Avg.) for better surface roughness Ra = 0.16. The higher temperature was 42.16°C on workpiece and tool HSS, and the maximum strain rate that occurred on the chip shearing zone was 9.33E06 (/s). The variation of rake angle compared with cutting force Fx, Fy, and cutting temperature presented that the rake angle 15° has better longer tool life because the cutting force is smaller than others as shown in Figure 11. The results can also be expressed by the resistance at B smaller than others. The cutter with rake angle 15° has lower cutting temperature through a long-time micro-cutting process as Figure 12. The trends of different rake angles for cutting force and cutting temperature are the same. The trends are nonlinear because of the size effect. The future work can investigate the influence of plowing angle.
3.3 The simulation for multipoints and shearing and plowing
In order to express steady-state continuous chip formation, the cutting of multipoints has been simulated on the quasi-state cutting process as shown in Figure 13 and the maximum chip loading of 0.56E−3 mm2 as shown in Figure 13(d). The non-overlap ratio and cutting depth ratio for cutters can be calculated by fractal geometry of the microscope. Through the assumed d1* = 0.125 mm of fractal mathematics, the results for multipoint cutting presented optimal geometric parameters for quasi-state cutting simulation of tool HSS and workpiece_Al-7075 as shown in Figure 13: (a) non-overlap ratio for sinusoidal multi-cutters, (b) cutting depth ratio for sinusoidal multi-cutters, (c) discrete chip loading distribution, (d) continuous chip loading for multi-cutters, and (e) cutters arrangement. Figure 13(d) can be expressed as a continuous loading trend for unit period cutting similar to the trend of single cutter in the micromachining process without considering the influence of the size effect. Another simulation is to prove that the plowing and shearing coupling existed while is close to 1 by traditional tool design in the cutting process. The special tool with two cutters of larger cutter-edge radius has simulated Kr to be close to 1 under the traditional cutting as shown in Figure 14(a) and (b). The specific coefficient Kr = 1.3 showed discontinuous characteristics, where the traditional specific coefficient defined as and , means the average cutting force. Figure 14(e) shows similar frequency for
The major results have been summarized as follows:
The optimal process included a cut of depth of 0.001 mm, a cutting length of 0.003 mm, and a cutter edge of 38°C on workpiece Al-7075; the optimal cutting force in x-axis was 0.0005 N (Avg.) and the optimal cutting force in y-axis was 0.00028 N (Avg.) for better surface roughness Ra = 0.16. The higher temperature was 42.16°C on workpiece and tool HSS, and the maximum strain rate that occurred on the chip shearing zone was 9.33E06 (/s), which obeyed the generalized cutting criterion by numerical analysis.
For the steady-state chip formation in the micro-cutting process, the rake angle is usually constant, but the plowing angle is relative to the tool-edge precision and surface roughness on the workpiece.
While micro-specific coefficient is close to 1, the plowing zone will increase friction, stress, resistance, and even cutting excited-vibration or chipping, resulting in discontinuous chipping.
The variation of the rake angle will affect the cutting force and plowing zone according to Eq. (7). The average micro-cutting force Fx = 0.5 mN and Fy = 0.22 mN and micro-specific coefficient = 0.44; the maximum cutting temperature is distributed at 35–40°C.
Through fractal mathematics, the results presented optimal geometric parameters for micro-cutting simulation for tool HSS and workpiece_Al-7075: overlap ratio between cutters, average cut-depth ratio between cutters, and chip load (undeformed chip formation areas and shapes) distribution on cutter order.
The study proposed the mathematical model of micro-cutting resistance to predict the conditions at cutter-edge radius.
The average micro-cutting force Fx = 0.5 mN and Fy = 0.22 mN and specific coefficient Kr = 0.44; the maximum cutting temperature is distributed at 35–40°C. To compare the variation of rake angle, rake angle at 15° has smaller micro-cutting force, and hence, the tool design has longer tool life in the micro-cutting process.
The study developed the micro-MDOF cutting dynamics model and micro-fractal equation to simulate the micro-cutting process.
Due to suffering from the size effect of cutter-edge radius r in the micromachining process, the differential-cutter angle is relative to two factors: plowing angle and rake angle. For the steady-state chip formation in the micro-cutting process, the rake angle is usually constant, but the plowing angle is relative to the tool-edge precision and surface roughness on the workpiece.
From the view of the micro-cutting process, the definition of micro-specific coefficient is , and the definition of average micro-specific coefficient is .
To compare with micro-cutting, the traditional cutting process has the same plowing effect, but the plowing in microscope effect is more obvious as shown in Figure 15. The validation can be proven as the derived theory agreed with the simulation in the micro-cutting process.
This work was supported in part by Taiwan NSC under Grant No. MOST 105-2221-E-327-015 and the industrial plan—Development of Ultra Speed Intelligent CNC Band Saw Machine from No. 105RB07. Special thanks to Prof. Ching-Hua Wei, Prof. Chin-Tu Lu, Prof. Jung-Zen Huang, One-on-One group members and my good friend Sam Fang for supporting my study; Prof. Sheng-Jye Hwang, Prof. Ta-Hui Lin, and Prof. Rong-Shean Lee at NCKU for their support in the process of this study; and Prof. Yunn-Lin Hwang and Prof. Jeng-Haur Horng for their research cooperation at the National Formosa University.