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Abstract

Rotary ultrasonic machining is an excellent method for processing hard brittle materials and has been investigated extensively via theoretical and experimental studies. However, there are few researches about rotary ultrasonic machining of ceramic matrix composite materials. Material removal mode presents both ductile-mode and brittle-fracture when machining hard brittle materials. The ratio of ductile-mode and brittle-fracture has a significant impact on processing status and processing quality. In this article, we developed a cutting force model for rotary ultrasonic face machining of ceramic matrix composite materials. Based on this model, we investigated the effects of cutting parameters on ductile-mode percentage of ceramic matrix composite materials. The cutting depth, feedrate and spindle speed are assumed to be three main factors that affect the ductile-mode percentage in rotary ultrasonic face machining. The primary aim of this article is to verify the correctness of the assumption using a three-variable three-level orthogonal experiment array. The experimental results agreed well with the assumption. The main factors deciding the ductile-mode percentage were found to be cutting depth and feedrate.

Keywords

Rotary ultrasonic face machining ceramic matrix composite cutting force cutting parameters ductile-mode percentage

Introduction

Ceramic matrix composite (CMC) materials are attractive for wide application owing to their superior properties, such as high specific strength, high specific rigidity, high-temperature strength and high wear resistance, such as C/SiC composites. The modulus of elasticity is above 60 GPa. The compressive strength is close to 30 MPa, and the density is only 2000 kg/m³. CMC materials have been increasingly commercialized and used in the aerospace during the last decades. However, CMC materials are nonhomogeneous, anisotropic and extremely hard and thus are difficult to machine, which challenges the manufacturing industry. The main difficulties in machining of CMC materials are low efficiency, high processing cost and unsatisfactory processing quality.¹ Rotary ultrasonic machining (RUM) has achieved good results compared to conventional machining.^2–4

In RUM, there exist two modes of material removal: ductile-mode and brittle-fracture mode.^5,6 The brittle-fracture mode is explained well by indentation theory, and the material is removed by brittle-fracture. While the material is removed without cracks by ductile-mode, material removal mode significantly affects surface quality and removal rate. The material removal mechanism has been assumed to be brittle-fracture for a long time because a majority of materials processed by RUM are brittle.^7–9 After the plastic flow in RUM of ceramics was first observed,¹⁰ a mathematical model was presented to predict the cutting force in ultrasonic drilling based on the ductile-mode mechanism.⁵ The ductile percentage in machining of ceramics increases as the spindle speed and the cutting depth increase and also increases as the vibration amplitude decreases.¹¹ Experiments show that grit size (the diamond size on grinding wheel), amplitude, spindle speed and especially static pressure decide the ductile-mode removal mode.¹² A cutting force model was established on the basis of ductile-mode removal mechanism in ultrasonic vibration–assisted grinding (UVAG) of titanium, and the relation between cutting force and cutting parameters was studied.¹³ The effects of grain size, concentration and hardness of the cutting tool on cutting force and tool wear were studied in ultrasonic slot machining of silicon carbide matrix composite.¹⁴ However, the existing literatures focus on ceramic, glass, sapphire and titanium but rarely on rotary ultrasonic face machining (RUFM) of CMC materials or the material removal mode, especially the effects of cutting parameters on the percentage of ductile-mode. Since the angle of impact is random, the ductile-mode and brittle-fracture occur simultaneously.¹⁵ The material removal mode changes from ductile-mode to brittle-fracture with increased cutting depth.^16,17 RUM presents two material removal modes in machining. Therefore, it is necessary to study the effects of cutting parameters on the proportion of material removal mode.

In this article, we developed a mechanistic model to predict relationships between cutting force and input variables in RUFM of CMC materials based on indentation fracture mechanics. In this model, the main factors deciding the ductile percentage were proved to be spindle speed (S), cutting depth (a_p) and feedrate (f). This conclusion was verified using multi-factor orthogonal test.

Development of cutting force model

Establishment of the model

This article focuses on RUFM. According to Pei et al.,¹⁸ RUFM should be a combined process of ultrasonic machining and diamond grinding, including three material removal mechanisms: hammering, abrasion and extraction (Figure 1). Therefore, a conic tool was used as a cutting tool (Figure 2). Vibration is in the longitudinal direction.

Figure 1.

The three material removal mechanisms: (a) hammering, (b) abrasion and (c) extraction.

Figure 2.

Conic tool: (a) Tool Design; (b) Milling tool.

Based on indentation fracture theory,^19,20 there is a deformation zone in the contact region (Figure 3). In this mechanism model, we assume that the maximum penetration depth (the maximum depth of the diamond abrasive grains penetrate into the workpiece during the ultrasonic vibration) does not reach the critical level and only consider the plastic deformation zone, ignoring the median crack and lateral crack. The material removal mode is rigid ductile-mode.

Figure 3.

Crack generation and plastic deformation zone in brittle material.

The maximum penetration depth was used as an intermediate parameter to establish the relationships between the input parameters (spindle speed, feedrate and cutting depth) and the output parameter (cutting force). There are three assumptions or simplifications:

Material removal mode is rigid ductile-mode;

All the diamond abrasive particles are of the same size;

The diamond abrasive particles are rigid octahedrons.

Relationship between maximum penetration depth and cutting force

Normal force (F_n, perpendicular to the machined surface) can be expressed as²¹

F_{n} = σ_{y} \cdot B

(1)

where $σ_{y}$ is the compression strength of the material and B is the projection of contact area on the workpiece (Figure 4).

Figure 4.

Relation between normal force and contact area.

According to the geometric relationship, the abrasive particle which is a regular octahedron can be expressed as follows

\frac{w}{S_{a} \cdot \cos β} = \frac{L}{S_{a}}

(2)

where w is the maximum penetration depth, S_a is the side length of the regular octahedron, β is the semi-angle and L is the side length at w (Figure 5).

Figure 5.

Octahedron abrasive particle.

From equation (2), then

B = L^{2} = \frac{w^{2}}{\cos^{2} β}

(3)

Substituting equation (3) into equation (1), then

F_{n} = \frac{σ_{y} \cdot w^{2}}{\cos^{2} β}

(4)

The relation between Z and f can be obtained (Figure 6)

Z = A \sin (2 π ft)

(5)

Figure 6.

Relation between effective contact time Δt and maximum penetration depth (w).

where Z is the trajectory of the diamond abrasive grains, A is the amplitude, f is the frequency and t is time.

According to equation (5) and Figure 6, effective contact time Δt can be expressed as follows

Δ t = \frac{1}{π f} [\frac{π}{2} - \arcsin (1 - \frac{w}{A})]

(6)

Based on the energy conservation theorem, then

I = \int_{cycle} F_{m} dt \approx F_{m} Δ t

(7)

I = \frac{F_{r}}{f}

(8)

where I is the impulse, F_m is the maximum impact force, Δt is the effective contact time during which an abrasive penetrates into the workpiece, cycle is a vibration cycle of the diamond abrasive grains and F_r is the cutting force caused by single diamond abrasive

F_{r} = Δ t \cdot f \cdot F_{m}

(9)

Figure 7 illustrates the geometrical relationship of the cutting force F, which can be expressed as follows

F = N_{α} \cdot F_{r} = N_{α} \cdot \frac{F_{n}}{\cos θ}

(10)

Figure 7.

Relationship of cutting force.

where F is the cutting force caused by all the active diamond abrasives, and it is the resultant force of F_n and F_t; F_n is the normal force and F_t is the tangential force.

Substituting equation (6) into equation (9), then

F_{r} = \frac{1}{π} [\frac{π}{2} - \arcsin (1 - \frac{w}{A})] \cdot F_{m}

(11)

Substituting equation (11) into equation (10), then

F = \frac{N_{α}}{π \cos θ} [\frac{π}{2} - \arcsin (1 - \frac{w}{A})] \cdot F_{n}

(12)

where N_α is the summary number of active abrasives during machining.

Based on the definition of abrasive concentration, N_α can be computed as follows⁹

N_{α} = C_{1} \cdot \frac{C_{α}^{2 / 3}}{S_{a}^{2}} \cdot A_{0}

(13)

where C_α is the concentration of the abrasive, A₀ is the contact area of the conic tool and workpiece and C₁ is a constant number, C₁ = 0.030414.

Figure 8 shows the relationship of contact area and geometry of conic tool. Whether the cutting width b_D is larger or smaller than R₁, the A₀ can be calculated as follows

\begin{matrix} A_{0} = & [R_{1} \cdot \arccos \frac{R_{1} - b_{D}}{R_{1}} + (R_{1} + a_{p} \cdot \cot θ) \\ \cdot \arccos \frac{(R_{1} + a_{p} \cdot \cot θ) - b_{D}}{(R_{1} + a_{p} \cdot \cot θ)}] \cdot \frac{a_{p}}{2} \end{matrix}

(14)

Figure 8.

Calculation of contact area (A₀).

where R₁ is the smaller radius of the conic tool, R₂ is the larger radius and a_p is the cutting depth.

Substituting equation (4) into equation (12), the relationship between the maximum penetration depth and the cutting force can be expressed as follows

F = \frac{N_{α} \cdot σ_{y}}{π \cdot \cos θ \cdot \cos^{2} β} [\frac{π}{2} - \arcsin (1 - \frac{w}{A})] \cdot w^{2}

(15)

Relationship between maximum penetration depth and cutting parameters

Figure 9 illustrates the material removal volume as the abrasive particle penetrates into the workpiece. The penetration depth increases from 0 to w first and then decreases to 0 within $Δ t$ , and the side length at w is L. Accordingly, the theoretical material removal volume V₀ during one penetration period is nearly equal to the volume of the pentahedron ABCDE and can be expressed as follows⁹

V_{0} = V_{ABCDE} = \frac{1}{3} \cdot L \cdot w \cdot L_{s}

(16)

Figure 9.

Calculation of material removal volume.

where L_s is the length when the abrasive particle moves within one period $Δ t$ . L_s can be calculated as follows

L_{s} = \frac{2 π SR}{60} \cdot Δ t

(17)

where S is the spindle speed (r/min) and R is the distance from the abrasive particle to the conic tool’s center (mm).

Substituting equations (3) and (17) into equation (16), then

\begin{matrix} V_{0} = & \frac{1}{3} \cdot \frac{w^{2}}{\cos β} \cdot \frac{2 π SR}{60} \cdot Δ t = \frac{1}{90 f} \cdot \frac{w^{2} \cdot S \cdot R}{\cos β} \\ \cdot [\frac{π}{2} - \arcsin (1 - \frac{w}{A})] \end{matrix}

(18)

The material removal volume (V) within one penetration period is nearly equal to the volume of theoretical material removal volume (V₀). We assume V and V₀ are in linear proportion, then⁹

V = k \cdot V_{0}

(19)

where k is a constant and can be obtained from cutting force experiments.

MRR_a is the material removal rate of single diamond abrasive. Since V is the material removal volume caused by single diamond abrasive in one vibration, MRR_a can be expressed as follows

{MRR}_{a} = f \cdot V

(20)

Material removal rate MRR_T is the summary of the material removed by all the effective abrasive particles during one period and can be expressed as follows

\begin{matrix} {MRR}_{T} = & N_{α} \cdot {MRR}_{a} = N_{α} \cdot f \cdot V = N_{α} \cdot k \cdot \frac{1}{90} \\ \cdot \frac{w^{2} \cdot S \cdot R}{\cos β} \cdot [\frac{π}{2} - \arcsin (1 - \frac{w}{A})] \end{matrix}

(21)

For simplification, average radius $(R_{1} + R_{2}) / 2$ is used instead of R

\begin{matrix} {MRR}_{T} = & \frac{k}{90} \cdot \frac{w^{2} \cdot S \cdot N_{α}}{\cos β} \cdot \frac{R_{1} + R_{2}}{2} \\ \cdot [\frac{π}{2} - \arcsin (1 - \frac{w}{A})] \end{matrix}

(22)

MRR_T can also be expressed as the volume swept by the conic tool during one period

{MRR}_{T} = f_{r} \cdot A_{0} \cdot \sin θ

(23)

By solving both equations (22) and (23), the relationship between maximum penetration depth and cutting parameters can be obtained as follows

\begin{matrix} \frac{C_{1} \cdot k}{180} & \cdot \frac{C_{α}^{2 / 3}}{\cos β \cdot S_{a}^{2}} \cdot w^{2} \cdot S \cdot (R_{1} + R_{2}) \\ \cdot [\frac{π}{2} - \arcsin (1 - \frac{w}{A})] = f_{r} \cdot \sin θ \end{matrix}

(24)

Cutting force model

By solving both equations (14) and (24), the relationship between cutting force and cutting parameters can be obtained as follows

F = k \cdot \frac{f_{r} \cdot \tan θ \cdot σ_{y} \cdot 180}{π \cdot \cos β \cdot S \cdot (R_{1} + R_{2})} \cdot [R_{1} \cdot a r c c o s \frac{R_{1} - b_{D}}{R_{1}} + (R_{1} + a_{p} \cdot c o t θ) \cdot a r c c o s \frac{(R_{1} + a_{p} \cdot c o t θ) - b_{D}}{(R_{1} + a_{p} \cdot c o t θ)}] \cdot \frac{a_{p}}{2}

(25)

The following conclusions can be obtained from equation (25): the cutting force was improved with the increase in the cutting depth (Figure 10(a)) and feedrate (Figure 10(b)) but decreased with the increase in the spindle speed (Figure 10(c)). It can be assumed from the existing experiment that the percentage of ductile-mode increased with the decrease in the cutting force.¹¹ Thus, the ductile-mode percentage increases with the increase in the spindle speed but decreases with the increase in the cutting depth and feedrate.

Figure 10.

Influence of cutting parameters on cutting force (simulation): (a) cutting depth, (b) feedrate and (c) spindle speed.

Experimental verification

The cutting experiments are carried out, and the main purpose of cutting experiments is to verify the effects of cutting parameters on ductile-mode percentage.

Experimental setup and conditions

The experimental apparatus is schematically illustrated in Figure 11. The experiments are performed on a DMG ultrasonic linear 20. The experimental setup is composed of three parts: ultrasonic vibration system, conic tool and 5-axis milling machine. The ultrasonic vibration system consists of ultrasonic spindle and an ultrasonic generator.

Figure 11.

Experimental apparatus: (a) Ultrasonic machining system; (b) RUFM of C/SiC on Ultrasonic liner 20.

The workpiece is made of C/SiC composite. The mechanical properties are shown in Table 1. The sample is manufactured according to the following process:

Table 1.

The mechanical properties of C/SiC.

Density (ρ)	2000 kg/m³
Porosity (P)	17%–20%
Tensile strength (σt)	≥40 MPa
Surface shear strength (σc)	≥10 MPa
Compression strength (σy)	590 MPa

Two-dimensional carbon fiber weave → vapor deposition (2–3 weeks) → liquid phase deposition and carbonization (4–6 weeks) → the middle temperature purification process → rough machining → liquid phase deposition (about 2 weeks) and carbonization → high-temperature purification process → siliconized (2–4 weeks) → precision machining → finished.

The properties of the conic tool are shown in Table 2. The experimental design is shown in Table 4. The experiments involve three groups of input parameters (spindle speed, feedrate and cutting depth). The cutting parameter is designed by orthogonal experiment array with three factors at three levels. In order to analyze the effect of errors on the experimental results, there is a blank column as error.

Table 2.

The properties of the conic tool.

Abrasive	Diamond
Bond type	Metal-bond
Grain size	70/80
Concentration	Cα = 100
Slope angle	Θ = 15°

Measurement of ductile-mode percentage

The surface topography of the workpiece was photographed by an Xtl-2400 microscope (Shanghai Caikon Instrument Co., Ltd.) at zoom multiple of 300×. Under the material removal mechanism of ductile-mode, the fibers were clear and continuous (Figure 12, white region), but under the brittle-fracture mode, the fibers were broken with some pits on the surface (Figure 12, black region). The ductile-mode region is magnified using scanning electron microscope (SEM; SSX-550, SHIMADZU), shown in Figure 13. The brittle-fracture mode region is shown in Figure 14. The proportion of white areas is measured by measurement software supporting the Xtl-2400 microscope. The cutting force is measured by dynamometer (9257B, Kistler). The results of the machined surface and the ductile percentage and cutting force are shown in Table 3.

Figure 12.

Machined surfaces with ductile-mode percentage (80%).

Figure 13.

Ductile-mode region (1200×).

Figure 14.

Brittle-fracture region (1200×).

Table 3.

Machined surface and the ductile percentage.

	1	2	3
Machined surfaces
Ductile percentage	82	68	54
	4	5	6
Machined surfaces
Ductile percentage	72	78	65
	7	8	9
Machined surfaces
Ductile percentage	80	66	78

Experimental results and discussion

The experimental results are listed in Table 4. The results indicate that the factors significantly affecting the cutting force were the cutting depth and feedrate. The effect of experimental error is only 2 and can be ignored.

Table 4.

Intuitionistic analysis.

	Factor	Spindle speed (r/min)	Feedrate (mm/s)	Error	Cutting depth (mm)	Ductile percentage	Cutting force (simulation)	Cutting force (measure)
Experiment	Experiment 1	2000	0.5		0.1	82	20.0	19.1
	Experiment 2	2000	1		0.2	68	80.0	79.3
	Experiment 3	2000	2		0.3	54	240.2	150.8
	Experiment 4	4000	0.5		0.3	72	30.0	30.4
	Experiment 5	4000	1		0.1	78	20.0	20.4
	Experiment 6	4000	2		0.2	65	80.0	78.4
	Experiment 7	6000	0.5		0.2	80	13.3	12.2
	Experiment 8	6000	1		0.3	66	40.0	39.2
	Experiment 9	6000	2		0.1	78	26.6	26.2
Mean	Mean 1	68.0	78.0	71.0	79.3
	Mean 2	71.7	70.7	72.7	71.0
	Mean 3	74.7	65.7	70.7	64.0
Range	Range	6.7	12.3	2.0	15.3

The measured values and simulated values (k = 2) of cutting force are almost consistent except experiment 2. This may due to the high proportion of brittle-fracture and cannot be ignored in experiment 2. Thus, the cutting force must be the synthesis of ductile-mode and brittle-fracture. In this article, the influence of ductile-mode percentage on the surface quality is the main research, and most of the simulation results agree well with the experimental results, so the cutting force caused by brittle-fracture is ignored in this article.

The effect curves indicate that the cutting force increases with the increasing cutting depth and feedrate but decreases with the rising spindle speed (Figure 15), while the ductile-mode percentage is opposite (Figure 16), which are consistent with the theoretical analysis. Thus, the theoretical model is accurate.

Figure 15.

Effect curves (cutting force): (a) spindle speed, (b) feedrate and (c) cutting depth.

Figure 16.

Effect curves (ductile percentage): (a) spindle speed, (b) feedrate and (c) cutting depth.

The significance of each factor was tested at level A = 0.05. Table 5 shows that F_ap > Ff > F (critical value) > F_S, indicating that the ductile-mode percentage was affected most significantly by the cutting depth, followed by feedrate but not significantly by the spindle speed. The reason might be that the cutting depth and feedrate should be controlled more stringently so as to further improve the ductile-mode percentage after it reaches a certain value. Since orthogonal experiment array does not select all the combinations of parameters, in this set of experiments, the combination of cutting depth and feedrate might limit the effect of spindle speed on the increase of the ductile-mode percentage. Moreover, despite the effect from the cutting force, the ductile-mode percentage is more directly affected by the maximum penetration depth. Therefore, it is necessary to establish a maximum penetration depth model for further research. Based on this model, we can analyze the effects of different cutting parameters on ductile-mode percentage. In Table 5, DEVSQ is sum of squares of deviations from the mean, and DOF is degree of freedom.

Table 5.

Variance analysis (F-test).

Factor	DEVSQ	DOF	F-ratio	F (critical value)	Significant
S	66.889	2	9.710	19.000
f	230.889	2	33.516	19.000	Yes
a_p	353.556	2	51.322	19.000	Yes
Error	6.89	2

Conclusion

We developed a cutting force model for RUFM of CMC materials based on the ductile-mode removal mechanism. Based on this model, the effects of cutting parameters on ductile-mode percentage were studied. Major conclusions are as follows:

Ductile-mode percentage decreases with the increase in the cutting depth or feedrate but increases with the increase in the spindle speed. The ductile-mode percentage is affected most significantly by cutting depth, followed by feedrate and then rotational speed.

Lower cutting depth and higher feedrate can be used in RUFM of CMC materials to obtain higher surface quality and processing efficiency.

Despite the effect from the cutting force, the ductile-mode percentage is more directly affected by the maximum penetration depth. Therefore, it is necessary to establish a maximum penetration depth model for further research.

RUFM can effectively reduce the fracture of carbon fiber and improve the surface quality (Figure 17).

Figure 17.

Machined surface: (a) conventional milling and (b) RUFM.

Footnotes

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This work was supported by National High Technology Research and Development Program of China (863 Program) (2013AA040105) and Basic Scientific Research Program of China.

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Effects of cutting parameters on ductile material removal mode percentage in rotary ultrasonic face machining

Abstract

Keywords

Introduction

Development of cutting force model

Establishment of the model

Relationship between maximum penetration depth and cutting force

Relationship between maximum penetration depth and cutting parameters

Cutting force model

Experimental verification

Experimental setup and conditions

Measurement of ductile-mode percentage

Experimental results and discussion

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References