Open access peer-reviewed chapter

# Study on Specific Coefficient in Micromachining Process

By Sung-Hua Wu

Submitted: October 5th 2018Reviewed: November 9th 2018Published: November 20th 2019

DOI: 10.5772/intechopen.82472

## Abstract

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.

### Keywords

• specific coefficient
• fractal equation
• micro-cutting
• micro-MDOF cutting dynamics

## 1. Introduction

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 [1] is a compact set (Figure 1), where resisted of complex numbers c, and he studied space of quadratic polynomials and proposed the function fc (z)existed non-diverged and iterated from z equals zero. The fractal properties have self-similarity, hyperbolic components, and local connectivity from a topological space x. The author observed the properties of self-similarity of fractal existing on undeformed chip geometry through a series of cross-section projections on a sinusoidal wave. The undeformed chip geometry through a series of cross-section projection on a sinusoidal wave by mathematical modeling belongs to a kind of Koch curve transformation, where the Koch curve belongs to a kind of triangle fractal as shown in Figure 2(d). The fractal similarity can be applied in microanalysis of chip formation through enlarging the contour of the chip topograph as shown in Figure 2(c). From another view as shown in Figure 2(f), the fractal geometry is similar to a series of projections of sinusoidal multi-cutter as a rectangle (Figure 3(d)). Through analytic geometry, the fractal differential equation can be established to understand chip shapes, chip load, tool geometry, and specific energy. From the view under a microscope, the chip formation is similar (Figure 2(d)(f)) in different views of fractal geometry. Hence, the fractal topograph of Figure 2 is very important to establish a generalized mathematical model by the analysis of chip fractal geometry. By Fractal mathematical method, the calculation can be achieved less hardware and higher precisely results as Figure 1(b). Due to the excellent calculation and algorithm, material processing or cutting dynamics can be analyzed by specific energy obtained from undeformed chip formation (Figures 2 and 3). Besides, the theoretical modeling should consider the surface condition of workpiece because the surface roughness and tool size are enlarged in micro-dimension. So, the surface conditions need to be expressed as shown in Figure 4. Hence, the micro-MDOF cutting dynamics model can be developed as Eqs. (22)(32).

## 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 46. The study found that the micro-specific coefficient K̂tapplied 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):

x+:mdẋt+dRs,Ecosπ2ϕdRs,FsinβdRp,AcosϕdRp,BdRp,Csinθ=0E1
y+:mdḟtdRs,Esinπ2ϕdRs,FcosβdRp,Asinϕ+dRp,Ccosθ=0E2

Thus, Eqs. (3) and (4) can be obtained simultaneously as follows:

For the orthogonal cutting process,

dRs,EdRp,AmdẋtdRs,FsinβdRp,Csinθ=dRp,Bmdẋt=dRs,Fsinβ+dRp,CsinθdRp,BE5

Then,

mdḟtdRs,Fcosβ+dRp,Ccosθ=0E6

Hence,

tanγ=dẋtdḟt=dRs,Fsinβ+dRp,CsinθdRp,BdRs,FcosβdRp,CcosθE7

Due to the size effect with considering the cutter-edge radius r in micromachining process, the differential-cutter angle γis relative to two factors: plowing angle θand rake angle β. For the steady-state chip formation in the micro-cutting process, the rake angle βis usually a constant, but the plowing angle θis 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 FBCof cutter-edge radius rabout 2π3ris an important factor, which is needed to consider tool stiffness, geometric design, and material properties in the machining process.

While dẋtdḟt=1or close to 1, called plowing and shearing coupling, large plowing resistance Rp,Band 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:

dẋtdḟt=K̂r=dRs,Fsinβ+dRp,CsinθdRp,BdRs,FcosβdRp,Ccosθ=1E8

where K̂rmeans the specific coefficient of micro-cutting.

Furthermore,

dRs,Fsinβ+dRp,CsinθdRp,B=dRs,FcosβdRp,CcosθE9

Going on,

dRs,Fsinβcosβ+dRp,Csinθ+cosθ=dRp,BE10

From integration, we obtain

dRp,B=sinβcosβdRs,F+sinθ+cosθdRp,CE11

To obtain the function of the plowing at B in micromachining process,

Rp,B=sinβcosβdRs,F+sinθ+cosθdRp,C+CE12

Equation (12) is an important result for plowing and shearing influence of micro-cutting under the size effect. While dẋtdḟt=K̂ris 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 Rp,B=Rp,BRs,FRp,Cβθdenoted the resistance function in the micromachining process, the function can be expanded to investigate the relations of factors Rp,B,Rs,F,Rp,C,β,θcombining micro-cutting force, shear stress, strain, and strain rate.

From the view of the micro-cutting process, the definition of micro-specific coefficient is K̂r=dẋdḟ, and the definition of average micro-specific coefficient is K̂r,avg=dẋ2dẋ1dḟ2dḟ1=slope; if K̂r=0, Eq. (8) can be rewritten as follows:

dRs,Fsinβ+dRp,CsinθdRp,B=0dRp,B=dRs,Fsinβ+dRp,CsinθRp,B=sinβdRs,F+sinθdRp,C+CE13

The result still expresses that the function Rp,B=Rp,BRs,FRp,Cβθexisted.

### 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):

αn+1=1λnb(tanϑn+1tanϑn)E14

where

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):

E15

where d1means the initial cutting depth and dimeans the transient cutting depth. Considering the cutting depth and cutting width, Eq. (16) can be obtained if the frictional energy dissipated:

E16

Aimeans the undeformed chip shapes, which is also similar to chip fractal geometry areas Piin the steady-state chip formation process:

E17

Simultaneously,

tanϑ=Rsini×2πn/tpE18

where Rmeans amplitude of sinusoidal-set cutters, tpmeans cutter pitch, and ϑdenotes the bending angle of sinusoidal multi-cutters.

Equation (19) will delete the overlap ratio αi=0if the sinusoidal multi-cutters become a single cutter. On the other hand,

P=1αibd1αi1αin1+1αidi1di2+d1=bd1E19

where Pmeans the generalized chip load of the micro-cutter by the fractal differential equation; the average cutting depth d1can be obtained through a few experiments and simulation or specific energy Kt can be obtained if the cutting length bis usually constant in the micro-cutting process. Average specific energy Kt can be obtained through unit cutting areas of multi-cutters by average chip load of each chip formation as Eq. (20):

K¯t=WΔV=PLbd1L=Pbd1E20

The cross-sections Type A, Type B, and Type C of undeformed chip shapes can be derived as Eq. (21) and Figure 7:

PtypeA=K¯tbd1for TypeAPtypeB=K¯tb1tanφd1forTypeBPtypeC=12K¯tbd1forTypeCE21

### 2.3 Micro-MDOF cutting dynamics equation

A tiny cutter should have a tiny mass δm, damping c, and elastic constant kwith micro-dynamic loading Pas shown in Figure 4. It needs to construct a dynamic equation as Eq. (22):

δmx¨+cẋ+kx=pE22

Assume the initial condition as

x¯0=Φq0x¯̇0=Φq̇0E23

Substitute xiTinto Eq. (22),

δmx¨+cẋ+kx=pE24

Then,

x˜iTδmΦq¨+x˜iTcΦq̇+x˜iTkΦq=x˜iTpi=123nE25

For orthogonal conditions,

x˜jTδmx˜i=0,ijx˜jTcx˜i=0,ijx˜jTkx˜i=0,ijE26

Going on,

δmiq¨i+ciq̇i+kiqi=pi,i=123nE27

To define

δmi=x˜iTδmx˜ici=x˜iTcx˜iki=x˜iTkx˜ipi=x˜iTpE28

Hence,

q¨i+ciδmiq̇i+kiδmiqi=piδmiE29

Assume

ciδmi=2ξiωikiδmi=ωi2E30

Furthermore,

q¨i+2ξiωiq̇i+ωi2qi=piδmiE31

To solve, the micro-MDOF cutting dynamics equation can be obtained as Eqs. (31) and (32):

qi=qit=eξiωitq̇i0+ξiωiqi0ωpsinωpt+qi0cosωpt+i=123nE32

where ωp=ωi1ξi2.

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:

σeq=A+Bεn1+Clnε̇ε̇01TTroomTmTroommE33

where Tmmeans the melting point temperature, Troommeans the environmental temperature, Tmeans the workpiece temperature, Ameans the yielding stress, Bmeans the strain factor, nmeans the strain coefficient, mmeans the temperature coefficient, ε̇means the plastic strain ratio, and ε0means the stain ratio. The simulation processed the orthogonal micro-cutting by the Lagrangian finite element method and numerical analysis. Eq. (33) offers the cutting temperature distribution on the tool-workpiece.

## 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 Rp,B=Rp,BRs,FRp,Cβθ. 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 γrelative to undeformed chip thickness if the micro-specific coefficient belongs to a reasonable range to avoid shearing-plowing coupled effect, such as the range 0K̂r0.9, where K̂r=0is similar to the vertical feed on the workpiece or digging process. The variation of differential-cutter angle γwill affect the cutting force because the variation of the resultant force comes from the two component forces mdẋat x-direction and mdḟat y-direction. The resultant force vectors will affect the variation of undeformed chip thickness according to the theoretical model and demonstration in Figures 46. 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 K̂rvaried with the differential-cutter angle, and it was an important factor for the production precision in the machining process.

Material and tool geometry parameters
ToolHSS
WorkpieceAl-7075
Length*height (2D)0.005 mm*0.002 mm
Rake angle (°)51015
Clearance angle (°)10
Process parameters
Feed0.0005 mm
Room temp. (°C)20
Initial temp. of the workpiece (°C)20
Cutting depth (average chip thickness)0.001 mm
Cutting conditionDry cutting
Cutting velocity (mm/min)30
Friction coefficient0.4–0.5
Cutter conditionSingle-cutter; orthogonal cutting

### Table 1.

Simulated parameters.

### 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 θ. 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.

## 4. Conclusions

The major results have been summarized as follows:

1. 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.

2. 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.

3. While micro-specific coefficient K̂ris close to 1, the plowing zone will increase friction, stress, resistance, and even cutting excited-vibration or chipping, resulting in discontinuous chipping.

4. 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 K̂r= 0.44; the maximum cutting temperature is distributed at 35–40°C.

5. 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.

6. The study proposed the mathematical model of micro-cutting resistance to predict the conditions at cutter-edge radius.

7. 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.

8. The study developed the micro-MDOF cutting dynamics model and micro-fractal equation to simulate the micro-cutting process.

9. 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.

10. From the view of the micro-cutting process, the definition of micro-specific coefficient is K̂r=dẋdḟ, and the definition of average micro-specific coefficient is K̂r,avg=dẋ2dẋ1dḟ2dḟ1=slope.

11. 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.

## Acknowledgments

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.

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Sung-Hua Wu (November 20th 2019). Study on Specific Coefficient in Micromachining Process, Micromachining, Zdravko Stanimirović and Ivanka Stanimirović, IntechOpen, DOI: 10.5772/intechopen.82472. Available from:

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