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Modeling and verifying of sawing force in ultrasonic vibration assisted diamond wire sawing (UAWS) based on impact load
International Journal of Mechanical Sciences 164 (2019) 105161
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Modeling and verifying of sawing force in ultrasonic vibration assisted diamond wire sawing (UAWS) based on impact load Yan Wang a,∗, De-Lin Li a, Zi-Jun Ding b,∗, Jian-Guo Liu a, Rui Wang a a b
School of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
a r t i c l e Keywords: Monocrystalline Si Impact load Ultrasonic vibration Diamond wire saw Sawing force Modeling
i n f o
a b s t r a c t Ultrasonic vibration assisted diamond wire sawing (UAWS) is an effective sawing process for hard and brittle materials such as monocrystalline SiC and Si. Compared with the conventional diamond wire sawing (CWS), both of sawing force and workpiece surface quality are improved by UAWS greatly, but the principle of improvement is still unclear. In order to reveal the mechanism of sawing force reduction in UAWS, this paper presents a theoretical model for sawing force in UAWS based on the theory of impact load. Firstly, the transverse vibration model of the diamond wire saw in UAWS was established based on the transverse vibration theory of continual system. Secondly, the impact load model of single abrasive was established based on the vibration model. The validity of this impact load model was verified by the finite element simulation. Thirdly, the sawing force caused by multi abrasives was derived based on the distribution of abrasives on the surface of wire saw. Finally, the verification experiments were conducted on the monocrystalline Si workpiece in various groups of processing conditions. The average error between the experimental and theoretical results of sawing force is 7.50%, which verifies the validity of the theoretical model. The measured results also indicate that the workpiece surface roughness of UAWS are 4.3%–29.7% lower than that of CWS.
1. Introduction The monocrystalline Si is widely applied in the semiconductor industry. High quality Si wafers are the critical bases of many industries, such as chips, solar panels, and precision optoelectronic devices [1,2]]. Wire sawing technology is a key process in the manufacturing of Si wafers. The efficiency and quality of subsequent finishing process such as grinding and polishing are directly affected by the sawing quality. Wire sawing also accounts for a large proportion of costs, because it is very time consuming [3–5]. In order to improve the efficiency and quality of the sawing process, the ultrasonic vibration assisted fixed diamond abrasive wire sawing (UAWS) has been proposed. It is verified by a lot of experiments that the sawing force is significantly reduced by ultrasonic vibration [6,7]. However, there is no detailed theoretical model that can be used to reveal the mechanism of the significant reduction on the sawing force. Since the sawing efficiency and sawing quality are significantly influenced by the sawing force, it’s necessary to establish a detailed theoretical model of the sawing force in UAWS. The sawing force model in CWS has been studied by many scholars. Huang et al. [8] have conducted an experimental research on the sawing force in sawing sapphire and concluded that the tangential sawing
∗
force is affected by the crystal structure, wire speed and feed rate. Kim et al. [9] have conducted a multi wire sawing experiment on sapphire and concluded that there is a linear relationship between the tangential sawing force and the material removal rate. These two papers focus on the tangential force in CWS, however, the impact load in UAWS is along the normal direction, which is the most different between UAWS and CWS. Ge et al. [10] have conducted a scratching experiment on monocrystalline Si with the single diamond abrasive. During the experiment, a uniformly increased normal force from 0 mN to 50 mN was applied, and a transition of the material removal mode from the plastic to brittle is observed during the increasing of the pressure. The critical value of the transition is found to be 26 mN. However, the transition with a normal impact load has not been discussed. Wang et al. [11] have proposed a theoretical force model of scratching monocrystalline SiC in which the influences of the input variables on the scratching force were discussed. It was found that the theoretical and experimental results are in good agreement through the verification experiments. Wang et al. [12] have also proposed a numerical prediction method for sawing force based on the combination of plastic removal model and brittle removal model. The sawing force formula was obtained by using the new prediction method. However, due to the
Corresponding authors. E-mail addresses: [email protected] (Y. Wang), [email protected] (Z.-J. Ding).
https://doi.org/10.1016/j.ijmecsci.2019.105161 Received 24 May 2019; Received in revised form 11 September 2019; Accepted 12 September 2019 Available online 13 September 2019 0020-7403/© 2019 Elsevier Ltd. All rights reserved.
Y. Wang, D.-L. Li and Z.-J. Ding et al.
impact load, this theory could not be used to explain the significant reduction on the sawing force in UAWS. Chung et al. [13] have established a model for simulating diamond abrasive distribution in CWS and concluded that the material removal rate is decreased as the distribution density is increased. Liedke and Kuna [14] have proposed a macroscopic mechanical model of wire saw that based on experimental results in CWS. The influences of important process parameters were discussed in the model, such as the wire speed, feed rate, wire tension, workpiece size and the wire length. However, the influence of the impact load was not discussed in these two papers. In the UAWS, the abrasives are contacted against the workpiece in an intermittent mode, which is significantly different from the continuous mode in the CWS. The abrasives are forced to collide with the workpiece at high frequency, as a result, causing a significant impact load, which is the essential difference between the UAWS and the CWS. In this paper, a novel sawing force model for UAWS based on impact load is presented. Firstly, the transverse vibration model of the diamond wire saw in UAWS was established based on the transverse vibration theory of continual system. Secondly, the impact load model of single abrasive was established based on the vibration model. The validity of this impact load model was verified by the finite element simulation. Thirdly, the general sawing force caused by multi abrasives was derived based on the distribution of abrasives on the surface of the wire saw. Finally, the verification experiments were conducted on the monocrystalline Si workpiece in various groups of processing conditions. The average error between the experimental and theoretical results of sawing force is 7.50%. The theoretical result is in good agreement with experimental observations. The measured results also indicate that the workpiece surface roughness of UAWS are 4.3%∼29.7% lower than that of CWS.
International Journal of Mechanical Sciences 164 (2019) 105161
Fig. 2. Vibration model of the wire saw.
During the UAWS, the wire saw is considered as an axially moving elastic continuum. The forced vibration of the axially moving wire saw can be simplified into the vibration of an axially moving string with the transverse excitation, because the diameter and the flexural modulus of the wire saw is very small. Fig. 2 shows a schematic view of the vibration model of the wire saw. Where vs is the axially velocity of the wire saw, Ls is the span of the guide wheels, FTS is the tension within the wire saw, o is the position of ultrasonic guide wheel, o’ is the position of assistant guide wheel. The transverse ultrasonic vibration along the z axis is applied by the ultrasonic guide wheel o, with the amplitude A, and frequency f. At the time t, the transverse displacement at any position x on the string can be denoted by z, which is a function about the abscissa x and the time t, z = z(x, t). 2.1. The transverse vibration equation of the fixed string
2. The dynamic equation of wire saw Fig. 1 shows a schematic view of UAWS. The diamond wire saw is convolved between the wire drum roller and several working wheels. The workpiece is a cylindrical monocrystalline Si. The coordinate system oxyz of machine tool is established as shown in Fig. 1. During the process, the wire saw is fully tensioned by the tension wheels with a pair of constant forces FT which are applied by the air cylinders of the machine tool. The wire saw is moved reciprocally at a linear velocity vs . At the same time, the wire drum roller and all working wheels are feed downward together along the negative direction of the z axis at a same feed rate of vw . The workpiece is rotated around the y axis at a rotational speed of nw . Ultrasonic vibration is applied along the z axis through the ultrasonic guide wheel with the amplitude of A and frequency of f.
The equilibrium equation of systems shown in Fig. 1 can be derived according to theory of transverse vibration of string, as shown in Eq. (1). 𝜌𝑠
( ) 𝜕2 𝑧 𝜕 𝜕𝑧 𝐹 = 𝑓 (𝑥, 𝑡) (0 < 𝑥 < 𝐿) − 𝜕𝑥 TS 𝜕𝑥 𝜕 𝑡2
(1)
Where 𝜌s is the density of the wire saw, f (x, t) is the excitation of the vibration system. When the f (x, t) = 0, the free vibration differential equation is obtained, as shown in Eq. (2). 𝜕2 𝑧 𝜕2 𝑧 = 𝑐𝑠2 𝜕 𝑡2 𝜕 𝑥2
(2)
Where cs is the propagation velocity of the elastic wave along the chord, as shown in Eq. (3). √ 𝐹𝑇 𝑆 c𝑠 = (3) 𝜌𝑠 2.2. Transverse vibration equation of the axially moving string The relationship between the motion coordinate system and the fixed coordinate system is shown in Eq. (4) according to the relative movement theory [15]: ( ) 𝜕 𝑑 𝜕 = + 𝑣𝑠 (4) 𝑑𝑡 𝜕𝑡 𝜕𝑥 The differential equation of the string vibration in the motion coordinate system can be converted into the differential equation of the fixed coordinate system according to Eq. (4). Wherein, the left side of Eq. (3) can be expressed as Eq. (5). [ ( )] 𝜕𝑧 𝜕2 𝑧 𝜕 𝜕𝑧 𝜕2 𝑧 𝜕2 𝑧 = = + 𝑣s + 2𝑣s (5) 𝜕𝑡 𝜕𝑡 𝜕𝑥 𝜕 𝑥𝜕 𝑡 𝜕 𝑡2 𝜕 𝑡2 The right side of the Eq. (3) can be expressed as Eq. (6) according to Eq. (4):
Fig. 1. Schematic view of the UAWS.
𝑐𝑠2
) 𝜕2 𝑧 𝜕2 𝑧 ( 2 = 𝑐s − 𝑣s 2 𝜕 𝑥2 𝜕 𝑥2
(6)
Y. Wang, D.-L. Li and Z.-J. Ding et al.
International Journal of Mechanical Sciences 164 (2019) 105161
The transverse vibration Eq. (7) of the axially moving string can be obtained by combining the Eqs. (5) and (6). 𝜕2 𝑧 𝜕2 𝑧 𝜕2 𝑧 + 2𝑣𝑠 − 𝑐𝑠2 − 𝑣𝑠 2 =0 (7) 𝜕𝑥 𝜕𝑡 𝜕 𝑡2 𝜕 𝑥2 The boundary conditions of the model in Fig. 2 can be expressed as Eq. (8). { 𝑧(0, 𝑡) = 𝐴𝑐𝑜𝑠𝜔𝑡 ( ) (8) 𝑧 𝐿s , 𝑡 = 0 Where 𝜔 is the angular frequency of ultrasonic vibration, 𝜔 = 2𝜋f. According to the general method of solving differential equations of undamped vibration, the vibration response of the center point of wire saw can be obtained, as shown in Eq. (9). ( ) 𝐿𝑠 1 𝑧0.5𝐿𝑠 , 𝑡 = (9) ( ) 𝐴cos 𝜔𝑡 − 𝜔 ′ 𝜔𝐿𝑠 2𝑣 2cos 2𝑐 ′
Where v’ is the average effective velocity of wire saw, c’ is the average effective wave velocity, as shown in Eqs. (10) and (11). ( 2 ) 𝑐 − 𝑣2𝑠 𝐹 𝑣′ = 𝑠 = 𝑇 𝑆 − 𝑣𝑠 (10) 𝑣𝑠 𝜌𝑠 𝑣 𝑠 ( 2 ) 𝑐 − 𝑣2𝑠 𝐹 𝑆 − 𝜌𝑠 𝑣 𝑠 2 𝑐′ = 𝑠 = 𝑇√ (11) 𝑐𝑠 𝐹 𝑇 𝑆 𝜌𝑠 As can be seen from Eq. (9), the vibration response of the center point of wire saw is a sinusoidal motion. Since the diamond abrasives are fixed on the wire saw, it can be considered that the abrasives vibrate along with the wire saw in the same amplitude, period and phase. Therefore, the vibration law of the abrasives can also be represented by the Eq. (7). 3. Modeling of sawing force The wire sawing model is shown in Fig. 3. Where vs is the reciprocating velocity of the wire saw, vw is the feed rate, nw is the rotation speed of workpiece, d is the diameter of workpiece, r is the radius of wire saw, Fn is the total normal sawing force of the wire saw, and Ft is the total tangential sawing force of the wire saw. Since the total sawing force of the wire saw is a resultant force of the sawing forces of a large amount of abrasives, there is a necessary to establish a sawing force model of single abrasive at first. 3.1. Sawing force of single abrasive in conventional diamond wire sawing Although the force of the abrasive is very complicated, it can be simplified into two parts: chip deformation force and friction force [16].
Fig. 4. Scratching model of single abrasive in CWS.
It is assumed that the sawing force of a single abrasives is consist of four components, as shown in Fig. 4, where Fnga is the normal sawing force caused by chip deformation, Fngb is the tangential sawing force caused by chip deformation, Ftga is the normal sawing force caused by friction, Ftgb is the tangential sawing force caused by friction. The sawing force of a single abrasive at the angle 𝜃 g can be expressed as the following four components [16]: 𝐹ng = 𝐹nga + 𝐹ngb = 𝜎D 𝑆1 + 𝜎C 𝑆2 𝜋 𝐹tg = 𝐹tga + 𝐹tgb = 𝐹 + 𝜇C 𝐹ngb 4tan𝛽g nga
(12) (13)
Where Fng is the normal resultant force of an abrasive, Fnga is the normal component force caused by chip deforming, Fngb is the normal force caused by friction, 𝜎 D is the contact stress between chip and abrasive, S1 is the average cross sectional area of a chip, 𝜎 C is the average contact pressure between the abrasive and the workpiece, S2 is the vertical projection area of an abrasive pressed into the workpiece, Ftg is the tangential resultant force of an abrasive, Ftga is the tangential component force caused by chip deforming, Ftgb is the tangential component force caused by friction, 𝛽 g is half of the top angle of abrasive, 𝜇 C is the coefficient of friction between abrasives and workpiece. The formula of S1 and S2 can be obtained by analyzing the geometrical relationship in Fig. 4, as shown in Eqs. (14) and (15). 1 ℎ 𝑏 = ℎg 2 tan𝛽g 2 g g ( )2 𝑏g )2 1 1 ( 𝑆2 = 𝜋 = 𝜋 ℎg tan𝛽g 2 2 2 𝑆1 =
(14) (15)
Where hg is the cutting depth of the abrasive and bg is the cutting width of the abrasive as shown in Fig. 4. The normal and tangential sawing force of the abrasive can be obtained by substituting Eqs. (14) and (15) into Eqs. (12) and (13), as shown in Eqs. (16) and (17) ( ) 1 𝐹ng = 𝐹nga + 𝐹ngb = tan𝛽g 𝜎D + 𝜎C 𝜋tan𝛽g ℎg 2 = 𝑘ng ℎg 2 2 [ ( )2 ] 2 1 1 𝐹tg = 𝐹tga + 𝐹tgb = 𝜋 𝜎D + 𝜇C 𝜎C tan𝛽g ℎg = 𝑘tg ℎg 2 2 2
Fig. 3. Schematic of wire sawing process.
(16) (17)
Where kng and ktg are the equivalent stiffness coefficient of the workpiece. The sawing force Fng and Ftg of each abrasive is different at different position on the surface of wire saw, because the cutting depth hg of abrasive at the different position 𝜃 g is different. The relationship between the cutting depth hg and the position 𝜃 g can be deduced according
Y. Wang, D.-L. Li and Z.-J. Ding et al.
International Journal of Mechanical Sciences 164 (2019) 105161
Therefore, the impact load Fng-d can be expressed as Eq. (24). √ √[ ] √ ( ) 2 √ 𝐿 −𝐴𝜔 √ ( ) sin 𝜔𝑡 − 𝜔 𝑠′ √ 2𝑣 √ 2 cos 𝜔2𝐿𝑐 ′𝑠 𝐹ng−𝑑 = 𝐾d 𝐹ng = √ 𝐹ng 𝑔 ℎg
The sawing force model of single abrasive in the UAWS can be derived by introducing the impact load model into the sawing force of CWS, as shown in Eq (25). { 𝐹uen = Ω(𝑡)ave 𝐹ng−𝑑 = Ω(𝑡)ave 𝐾d 𝐹ng (25) 𝐹uet = Ω(𝑡)ave 𝐹tg
Fig. 5. The image of function Ω(t).
to the distribution of abrasives on the surface of wire saw [17,18], as shown in Eq. (18). √ 𝑣w ⋅ cos𝜃g ( ) ℎ g 𝜃g = (18) 𝑃valid ⋅ 𝜌saw 𝑣s ⋅ tan𝛽g Where Pvalid is the probability of effective abrasives and 𝜌saw is the distribution density of abrasives on the surface of wire saw. 3.2. The sawing force model of single abrasive in the UAWS based on impact load
Where t0 is the time when the abrasive is impacted with the workpiece, t1 is the time when abrasive is separated from the workpiece. t2 is the starting time of the next vibration period, T is the period of ultrasonic vibration, n = 0, 1, 2… The function Ω(t) is an impulse function, as shown in Fig. 5. When t0 ≤t ≤ t1 , Ω(t)=1, the wire saw is contacted against the workpiece. When t1 ≤t ≤ t2 , Ω(t)=0, the wire saw is separated from the workpiece. The function Ω(t) is expanded by Fourier series as shown in Eq. (20), where the tq =tl -t0 . 𝑡q 𝑇
+
∞ 2∑1 sin 𝜋 𝑛=1 𝑛
(
𝑡q 𝑛 𝑇
) 𝜋 𝑐𝑜𝑠𝑛𝜔𝑡
(20)
The average of the function Ω(t) in an ultrasonic vibration period is derived by Eq. (21). Ω(𝑡)ave =
𝑡q 1 𝑇 ∫ Ω(𝑡)𝑑𝑡 = 𝑇 0 𝑇
(21)
The impact load in UAWS is greater than that in CWS, because there is a unique impact caused by abrasives on the workpiece. According to the impact load theory of material mechanics [19], the impact load Fhg-d can be expressed as the static load Fen multiplied by the dynamic load coefficient Kd , as shown in Eq. (22) √ 𝑣𝑐 2 𝐹ng−𝑑 = 𝐾d 𝐹ng = 𝐹 (22) 𝑔 Δst ng Where g is the acceleration of gravity, Δst is the deformation caused by static load Fng , and it is assumed to be the cutting depth hg . vc is the impact velocity, which can be obtained by the derivation of Eq. (9), as shown in Eq. (23). ( ) d𝑧0.5𝐿𝑠 , 𝑡 𝐿𝑠 −𝐴𝜔 𝑣𝑐 = = (23) ( ) sin 𝜔𝑡 − 𝜔 ′ 𝜔𝐿𝑠 d𝑡 2𝑣 2 cos 2𝑐 ′
3.3. The total sawing force model of wire saw in the UAWS The normal sawing force of a single abrasive can be decomposed into two components that along the y direction and the z direction. The component force along the y direction can be canceled by another abrasive located on the symmetric location of the wire saw. The total normal sawing force is the sum of the component forces along the z direction of sawing force of each single abrasive. The component force along the z direction of a single abrasive is: 𝐹uenx = 𝐹uen 𝑐𝑜𝑠𝜃g
According to the vibration response expressed by Eq. (9), it can be found that the continuous contact between wire saw and workpiece is converted into intermittent contact due to the ultrasonic vibration. A function Ω(t) is used to represent this intermittent contact, as shown in Eq. (19): { 1 𝑡0 + 𝑛𝑇 ≤ 𝑡 ≤ 𝑡1 + 𝑛𝑇 Ω(𝑡) = (19) 0 𝑡1 + 𝑛𝑇 ≤ 𝑡 ≤ 𝑡2 + (𝑛 + 1)𝑇
Ω(𝑡) =
(24)
(26)
During the sawing process, the workpiece is only in contact with the lower half of the wire saw. The sawing forces of an abrasive at the angle 𝜃 g are shown in Eq. (27). { d𝐹un = 𝐹uenx 𝑃valid 𝑐 𝑙𝑐 𝑟𝑑 𝜃g (27) d𝐹ut = 𝐹uet 𝑃valid 𝑐 𝑙𝑐 𝑟𝑑 𝜃g Where Pvalid is the probability of effective abrasives, c is the areal density of abrasives, which is related on the distribution of abrasives on the surface of wire saw, and lc is the contact length between wire saw and workpiece. The total sawing forces of UAWS are derived by integrating the Eq. (20) on [-𝜋/2, 𝜋/2], as shown in Eq. (28). 𝜋
⎧ 2 𝑑 𝐹un ⎪𝐹un = 2 ∫ 0 ⎪ ⎨ 𝜋 ⎪ 2 𝑑 𝐹ut ⎪𝐹ut = 2 ∫0 ⎩
(28)
4. The verification of theoretical sawing force model of single abrasive The theoretical sawing force model of single abrasive was verified by applying the finite element analysis (FEA) software ABAQUS. Firstly, a finite element model of the abrasive is established. The geometrical shape of the abrasive is simplified into a conical shape with a diameter of 𝛽 g = 60°, as shown in Fig. 6. During the simulation, the abrasive is defined as a rigid body based on the following reason. First, the Mohs hardness of diamond is 10 while the Mohs hardness of monocrystalline Si is 6–7 [20], so it’s reasonable to consider that the diamond is cut in the Si without deformation. Additionally, though the wire saw is
Fig. 6. The simplified geometric model of the abrasive.
Y. Wang, D.-L. Li and Z.-J. Ding et al.
International Journal of Mechanical Sciences 164 (2019) 105161
Table 1 Values of factors in the experiment.
Value Value Value Value
1 2 3 4
vs (m/s)
vw (mm/min)
nw (r/min)
2 4 6 8
0.5 1.0 1.5 2.0
5 10 15 20
Table 2 Experimental results of CWS. Fig. 7. FEA model of workpiece.
Tests
vs (m/s)
vw (mm/min)
nw (r/min)
Fn (mN)
Ft (mN)
Ra(𝜇m)
1 2 3 4 5 6 7 8 9 10
2 4 6 8 4 4 4 4 4 4
1.0 1.0 1.0 1.0 0.5 1.5 2.0 1.0 1.0 1.0
10 10 10 10 10 10 10 5 15 20
11.03 7.92 5.21 4.23 5.83 10.10 13.22 9.07 6.77 6.68
5.85 3.89 2.68 1.70 2.89 5.38 6.45 3.82 3.79 3.68
0.91 0.69 0.61 0.48 0.55 1.01 1.35 0.70 0.62 0.62
Table 3 Experimental results of UAWS.
Fig. 8. Assembled FEA model.
very thin, the diameter of it is about 0.35 mm, which is very big compared with the diameter of abrasives (40 𝜇m). Finally, though the impact load in the UAWS is very big compared with the normal force in CWS, the magnitudes of it is tens of mN, which is very small to a wire saw that was tensioned by 110 N tension force. Secondly, the finite element model of the workpiece is established by using the C3D8R unit of the Explicit cell library. The meshed model of workpiece is shown in Fig. 7. The boundary of the workpiece is: fixed the bottom and four broadsides, as shown in Fig. 7. The finite element model of the abrasive and the workpiece are assembled in the ABAQUS software, as shown in Fig. 8. The load of abrasive is: a constant velocity of vs along the x direction of the workpiece; a simple harmonic motion with the amplitude of A and frequency of f along the direction that is 𝜃 g degrade along the y axis in the oyz plane. The vs =2 m/s in every group of simulation. In every simulation for UAWS, the A = 10 𝜇m, f = 20 kHz. In every simulation of CWS, the A = 0 𝜇m, f = 0 kHz. The position angle 𝜃 g of the abrasive is taken to 0°, 30°, 60°, 85° The cutting depth hg is calculated according to the formula (18). The simulation results are shown in Fig. 9: It can be observed from the above figures that under the influence of ultrasonic vibration, the cutting depth of the abrasives is changed constantly. The trajectory of abrasive in the UAWS is different from that in the CWS. The waveforms of normal sawing force are shown in Fig. 10: The average values of the waveforms are extracted. Fig. 11 shows the comparison between the simulated theoretical results. The comparison results show that the sawing forces in the UAWS are lower than those in the CWS under the same working conditions. Additionally, the average error between the theoretical sawing force and simulation sawing force is 9.3%. Thus, the validity of the sawing force model of the single abrasive in UAWS is verified by the finite element simulation.
Tests
vs (m/s)
vw (mm/min)
nw (r/min)
Fn (mN)
Ft (mN)
Ra (𝜇m)
1 2 3 4 5 6 7 8 9 10
2 4 6 8 4 4 4 4 4 4
1.0 1.0 1.0 1.0 0.5 1.5 2.0 1.0 1.0 1.0
10 10 10 10 10 10 10 5 15 20
7.83 5.97 3.66 2.90 2.64 8.95 12.64 8.53 4.44 4.45
3.85 2.22 1.02 0.71 1.06 3.76 6.30 2.40 2.09 1.98
0.75 0.51 0.46 0.44 0.42 0.86 1.34 0.64 0.43 0.41
experimental setup is shown in Figs. 12 and 13. The working frequency of the ultrasonic generator and horn is 19.8–20.2 kHz, and the output amplitude is 10 𝜇m. The dynamometer was a Kistler three-component measuring platform type 9257B, and the charge amplifier was a Kistler dual mode amplifier model 5070A. The diameter of the wire saw is 0.35 mm, and the tension force within it was 110 N. The micrograph of the diamond wire saw is shown in Fig. 14. The sawing forces in two directions, Fn and Ft , were measured. Fn is in the normal direction, Ft is in the tangential direction, as shown in Fig. 3. The monocrystalline Si workpiece was a 36 mm × 80 mm cylinder. The surface topography and roughness on the machined surface were measured along the feed direction. The instruments used were MITUTOYO SJ-201 profile meter, KEYENCE VXH6000 optical microscope and FEI Quanta200F scanning electron microscope (SEM). Table 1 shows the matrix for the experiment, as well as the high and low settings for process variables. Three most important factors were selected as variables in the experiment: the axial velocity of the wire saw vs (m/s), the feed rate of the workpiece vw (mm/min), and the rotational speed of the workpiece nw (r/min). Three output variables were observed: sawing force, surface roughness and surface topography.
6. Results and discussions 5. Experimental setup The validity of the total sawing force model of the wire saw is verified by the experiment. which were carried out using a conventional wire sawing machine tool assisted with an ultrasonic device, and the
Investigations were carried out into the comparison of CWS and UAWS. The effect of ultrasonic vibration on the sawing force and surface roughness were studied. The experimental results of CWS and UAWS are shown in Tables 2 and 3 respectively.
Y. Wang, D.-L. Li and Z.-J. Ding et al.
International Journal of Mechanical Sciences 164 (2019) 105161
Fig. 9. Simulation results of CWS and UAWS with different position angle 𝜃 g.
Y. Wang, D.-L. Li and Z.-J. Ding et al.
International Journal of Mechanical Sciences 164 (2019) 105161
Fig. 10. Waveforms of normal forces of CWS and UAWS with different position angle 𝜃 g.
6.1. Verification of the total sawing force model of wire saw
Fig. 11. Comparison between the simulated and theoretical results of normal force.
The validity of the total sawing force model of wire saw is verified by comparing the theoretical and experimental sawing force, as shown in Figs. 15–17. It can be seen that the trends of theoretical influences of input variables agree well with the trends determined experimentally, and the average error between the experimental and theoretical results of sawing force is 7.50%. Thus, the experimental results verified the theoretical model very well. In addition, it can also be seen from Figs. 15–17 that the sawing force decreases as the axial velocity of wire saw vs and workpiece rotation speed nw increase. The sawing force further increases as the feed rate vw increases. The sawing force in UAWS is smaller than that in CWS due to the ultrasonic vibration. The mechanism is that though the peak value of the impact load is quite high, its duration is pretty short. As a result, the impulse caused by impact load is lower than the impulse
Fig. 12. Experimental devices of ultrasonic vibration assisted wire sawing Si.
Y. Wang, D.-L. Li and Z.-J. Ding et al.
International Journal of Mechanical Sciences 164 (2019) 105161
Fig. 13. Measuring devices of experiments.
Fig. 14. The micrograph of the diamond wire saw measured by SEM.
caused by constant load, which means that the sawing force in UAWS is smaller than CWS. 6.2. Analysis of surface roughness results and workpiece topography results As Fig. 18 shows, the trends of influences of sawing forces on surface roughness were determined experimentally. It can be seen that the surface roughness Ra increases as the normal sawing force Fn and tangential
sawing force Ft increase. Thus, the lower value of surface roughness can be obtained with lower sawing force. As the sawing force in UAWS is smaller than that in CWS due to the ultrasonic vibration, it can be inferred that the lower value of surface roughness can be obtained with ultrasonic vibration. Fig. 19 shows the 3D surface topography of workpiece measured by using the KEYENCE VXH6000 optical microscope with the amplitude of 2000. It can be seen that the surface topography obtained by UAWS is better than that obtained by CWS. The size of pits on the surface of workpiece obtained by UAWS is smaller than that obtained by CWS. The mechanism is that the duration time of the contact between abrasives and workpiece is very short due to the high frequency of the ultrasonic vibration, as a result, the length of brittle fracture crack decreased. As the length of the brittle fracture crack decreases, the size of pits decreases Fig. 20 shows the surface measurements of workpieces in CWS and UAWS by using FEI Quanta200F scanning electron microscope (SEM) with the amplitude of 4000. It can be seen that the edges of pits in Fig. 20(a) are sharper than that in Fig. 20(b). According to this result, it can be recognized that the brittle removal mode is reduced by ultrasonic vibration. The removal of workpiece surface material is more uniform in UAWS. The big and sharp pits in CWS are difficult to be removed during the subsequent finishing processes. However, the small and smooth pits in UAWS can be easily removed. Therefore, UAWS is an efficient, precise and economical sawing process compared with CWS.
Fig. 15. Relation between sawing force (Fn , Ft ) and axial velocity of wire saw (vs ).
Y. Wang, D.-L. Li and Z.-J. Ding et al.
International Journal of Mechanical Sciences 164 (2019) 105161
Fig. 16. Relation between sawing force (Fn , Ft ) and feed rate (vw ).
Fig. 17. Relation between sawing force (Fn , Ft ) and workpiece rotation speed (nw ).
Fig. 18. Relation between surface roughness (Ra) and sawing force (Fn , Ft ).
Fig. 19. Comparing of workpiece surface topography (vs = 4 m/s, vw = 2 mm/min, nw = 10r/min).
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International Journal of Mechanical Sciences 164 (2019) 105161
Fig. 20. Comparing of workpiece surface measured by SEM (vs = 4 m/s, vw = 2 mm/min, nw = 10r/min).
7. Conclusions In this paper, a model of sawing force in UAWS based on the impact load is presented. The validity of the proposed model has been confirmed by simulation and experiments. The following conclusions can be drawn: 1. A novel theoretical model for sawing force in UAWS based on the theory of impact load is presented in this paper. Firstly, the transverse vibration model of the diamond wire saw in UAWS was established based on the transverse vibration theory of continual system. Secondly, the impact load model of single abrasive was established based on the vibration model. Finally, the sawing force caused by multi abrasives was derived based on the distribution of abrasives on the surface of wire saw. According to the presented theory, the impact load is the main reason for the reduction of sawing force in UAWS. 2. The theory about the sawing force of single abrasive was verified by FEA simulation. According to the theoretical results and simulation results, the average error between the theoretical and simulated value is 9.3%. The theory about the total sawing force was verified by experiments. According to the theoretical results and experimental results, the average error between theoretical and experimental results is 9.8%. The validity of presented theoretical model in this paper is verified. 3. The value of surface roughness in UAWS is lower than that in CWS. Compared with CWS, the pits on workpiece surface in UAWS are smaller and smoother. It can be recognized that surface quality in UAWS is better than that in CWS. References [1] Hao W. Wire sawing technology: a state-of-the-art review. Precis Eng 2016;43(12):1–9. [2] Kumar A, Melkote SN. Diamond wire sawing of solar Si workpieces: a sustainable manufacturing alternative to loose abrasive slurry sawing. Proc Manuf 2018;21:549–66.
[3] Pei ZJ, Kassir S, Bhagavat Milind, Fisher GR. An experimental investigation into soft-pad grinding of wire-sawn Si workpieces. Int J Mach Tools Manuf 2004;44(2):299–306. [4] Wang P, Xiao S, Jia R, Sun H, Dai X, Su G, et al. 18.88%-efficient multi-crystalline Si solar cells by combining cu-catalyzed chemical etching and post-treatment process. Solar Energy 2018;169:153–8. [5] Cao F, Chen K, Zhang J, Ye X, Li J, Zou S, et al. Next-generation multi-crystalline Si solar cells: diamond-wire sawing, nano-texture and high efficiency. Solar Energy Mater Solar Cells 2015;141:132–8. [6] Shujuan LI, Aofei T, Yan LI, Li L. Influence of diamond wiresaw excited by transverse ultrasonic vibration cutting force and critical cutting depth of hard and brittle materials. J Mech Eng 2016;52(3):187–96. [7] Li S, Wan B, Landers RG. Surface roughness optimization in processing sic monocrystal workpieces by wire saw machining with ultrasonic vibration. Proc Inst Mech Eng Part B J Eng Manuf 2013;228(5):725–39. [8] Huang H, Li X, Xu X. An experimental research on the force and energy during the sapphire sawing using reciprocating electroplated diamond wire saw. J Manuf Sci Eng 2017;139(12):1210111–15. [9] Kim H, Kim D, Kim C, Jeong H. Multi-wire sawing of sapphire crystals with reciprocating motion of electroplated diamond wires. CIRP Ann Manuf Technol 2013;62(1):335–8. [10] Ge M, Zhu H, Huang C, Liu A, Bi W. Investigation on critical crack-free cutting depth for single crystal Si slicing with fixed abrasive wire saw based on the scratching machining experiments. Mater Sci Semicond Process 2018;74:261–6. [11] Wang P, Ge P, Li Z, et al. A scratching force model of diamond abrasive particles in wire sawing of single crystal SiC. Mater Sci Semicond Process 2017;68:21–9. [12] Wang P, Ge P, Gao Y, et al. Prediction of sawing force for single-crystal Si carbide with fixed abrasive diamond wire saw. Mater Sci Semicond Process 2017;63:25–32. [13] Chung C, Tsay GD, Tsai MH. Distribution of diamond grains in fixed abrasive wire sawing process. Int J Adv Manuf Technol 2014;73(9):1485–94. [14] Liedke T, Kuna M. A macroscopic mechanical model of the wire sawing process. Int J Mach Tools Manuf 2011;51(9):711–20. [15] Chen LQ, Zhang W, Zu JW. Nonlinear dynamics for transverse motion of axially moving strings. Chaos Solitons Fractals 2009;40(1):78–90. [16] Hedrih K. Transversal forced vibrations of an axially moving sandwich belt system. Arch Appl Mech 2008;78(9):725–35. [17] Li S, Liu Y, Hou X, Gao X. Analysis and modeling cutting force for sic monocrystal wafer processing. J Mech Eng 2015;51(23):189–204. [18] Cai O, Careddu N, Mereu M, Mulas I. The influence of operating parameters on the total productivity of diamond wire in cutting granite. IDR Ind Diam Rev 2007;45:25–32 3/07 page 2016. [19] Timoshenko SP, Gere j m. Mechanics of materials. New York: Van Nostrand Reinhold; 1984. p. 248–58. [20] Kumar A, Andrii K, Vanessa P, Evgeniy P, Melkote Shreyes N. Ductile mode behavior of silicon during scribing by spherical abrasive particles. In: Proceedings of the third CIRP conference on surface integrity, 45. CIRP CSI; 2016. p. 147–50.
Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Modeling and verifying of sawing force in ultrasonic vibration assisted diamond wire sawing (UAWS) based on impact load Yan Wang a,∗, De-Lin Li a, Zi-Jun Ding b,∗, Jian-Guo Liu a, Rui Wang a a b
School of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
a r t i c l e Keywords: Monocrystalline Si Impact load Ultrasonic vibration Diamond wire saw Sawing force Modeling
i n f o
a b s t r a c t Ultrasonic vibration assisted diamond wire sawing (UAWS) is an effective sawing process for hard and brittle materials such as monocrystalline SiC and Si. Compared with the conventional diamond wire sawing (CWS), both of sawing force and workpiece surface quality are improved by UAWS greatly, but the principle of improvement is still unclear. In order to reveal the mechanism of sawing force reduction in UAWS, this paper presents a theoretical model for sawing force in UAWS based on the theory of impact load. Firstly, the transverse vibration model of the diamond wire saw in UAWS was established based on the transverse vibration theory of continual system. Secondly, the impact load model of single abrasive was established based on the vibration model. The validity of this impact load model was verified by the finite element simulation. Thirdly, the sawing force caused by multi abrasives was derived based on the distribution of abrasives on the surface of wire saw. Finally, the verification experiments were conducted on the monocrystalline Si workpiece in various groups of processing conditions. The average error between the experimental and theoretical results of sawing force is 7.50%, which verifies the validity of the theoretical model. The measured results also indicate that the workpiece surface roughness of UAWS are 4.3%–29.7% lower than that of CWS.
1. Introduction The monocrystalline Si is widely applied in the semiconductor industry. High quality Si wafers are the critical bases of many industries, such as chips, solar panels, and precision optoelectronic devices [1,2]]. Wire sawing technology is a key process in the manufacturing of Si wafers. The efficiency and quality of subsequent finishing process such as grinding and polishing are directly affected by the sawing quality. Wire sawing also accounts for a large proportion of costs, because it is very time consuming [3–5]. In order to improve the efficiency and quality of the sawing process, the ultrasonic vibration assisted fixed diamond abrasive wire sawing (UAWS) has been proposed. It is verified by a lot of experiments that the sawing force is significantly reduced by ultrasonic vibration [6,7]. However, there is no detailed theoretical model that can be used to reveal the mechanism of the significant reduction on the sawing force. Since the sawing efficiency and sawing quality are significantly influenced by the sawing force, it’s necessary to establish a detailed theoretical model of the sawing force in UAWS. The sawing force model in CWS has been studied by many scholars. Huang et al. [8] have conducted an experimental research on the sawing force in sawing sapphire and concluded that the tangential sawing
∗
force is affected by the crystal structure, wire speed and feed rate. Kim et al. [9] have conducted a multi wire sawing experiment on sapphire and concluded that there is a linear relationship between the tangential sawing force and the material removal rate. These two papers focus on the tangential force in CWS, however, the impact load in UAWS is along the normal direction, which is the most different between UAWS and CWS. Ge et al. [10] have conducted a scratching experiment on monocrystalline Si with the single diamond abrasive. During the experiment, a uniformly increased normal force from 0 mN to 50 mN was applied, and a transition of the material removal mode from the plastic to brittle is observed during the increasing of the pressure. The critical value of the transition is found to be 26 mN. However, the transition with a normal impact load has not been discussed. Wang et al. [11] have proposed a theoretical force model of scratching monocrystalline SiC in which the influences of the input variables on the scratching force were discussed. It was found that the theoretical and experimental results are in good agreement through the verification experiments. Wang et al. [12] have also proposed a numerical prediction method for sawing force based on the combination of plastic removal model and brittle removal model. The sawing force formula was obtained by using the new prediction method. However, due to the
Corresponding authors. E-mail addresses: [email protected] (Y. Wang), [email protected] (Z.-J. Ding).
https://doi.org/10.1016/j.ijmecsci.2019.105161 Received 24 May 2019; Received in revised form 11 September 2019; Accepted 12 September 2019 Available online 13 September 2019 0020-7403/© 2019 Elsevier Ltd. All rights reserved.
Y. Wang, D.-L. Li and Z.-J. Ding et al.
impact load, this theory could not be used to explain the significant reduction on the sawing force in UAWS. Chung et al. [13] have established a model for simulating diamond abrasive distribution in CWS and concluded that the material removal rate is decreased as the distribution density is increased. Liedke and Kuna [14] have proposed a macroscopic mechanical model of wire saw that based on experimental results in CWS. The influences of important process parameters were discussed in the model, such as the wire speed, feed rate, wire tension, workpiece size and the wire length. However, the influence of the impact load was not discussed in these two papers. In the UAWS, the abrasives are contacted against the workpiece in an intermittent mode, which is significantly different from the continuous mode in the CWS. The abrasives are forced to collide with the workpiece at high frequency, as a result, causing a significant impact load, which is the essential difference between the UAWS and the CWS. In this paper, a novel sawing force model for UAWS based on impact load is presented. Firstly, the transverse vibration model of the diamond wire saw in UAWS was established based on the transverse vibration theory of continual system. Secondly, the impact load model of single abrasive was established based on the vibration model. The validity of this impact load model was verified by the finite element simulation. Thirdly, the general sawing force caused by multi abrasives was derived based on the distribution of abrasives on the surface of the wire saw. Finally, the verification experiments were conducted on the monocrystalline Si workpiece in various groups of processing conditions. The average error between the experimental and theoretical results of sawing force is 7.50%. The theoretical result is in good agreement with experimental observations. The measured results also indicate that the workpiece surface roughness of UAWS are 4.3%∼29.7% lower than that of CWS.
International Journal of Mechanical Sciences 164 (2019) 105161
Fig. 2. Vibration model of the wire saw.
During the UAWS, the wire saw is considered as an axially moving elastic continuum. The forced vibration of the axially moving wire saw can be simplified into the vibration of an axially moving string with the transverse excitation, because the diameter and the flexural modulus of the wire saw is very small. Fig. 2 shows a schematic view of the vibration model of the wire saw. Where vs is the axially velocity of the wire saw, Ls is the span of the guide wheels, FTS is the tension within the wire saw, o is the position of ultrasonic guide wheel, o’ is the position of assistant guide wheel. The transverse ultrasonic vibration along the z axis is applied by the ultrasonic guide wheel o, with the amplitude A, and frequency f. At the time t, the transverse displacement at any position x on the string can be denoted by z, which is a function about the abscissa x and the time t, z = z(x, t). 2.1. The transverse vibration equation of the fixed string
2. The dynamic equation of wire saw Fig. 1 shows a schematic view of UAWS. The diamond wire saw is convolved between the wire drum roller and several working wheels. The workpiece is a cylindrical monocrystalline Si. The coordinate system oxyz of machine tool is established as shown in Fig. 1. During the process, the wire saw is fully tensioned by the tension wheels with a pair of constant forces FT which are applied by the air cylinders of the machine tool. The wire saw is moved reciprocally at a linear velocity vs . At the same time, the wire drum roller and all working wheels are feed downward together along the negative direction of the z axis at a same feed rate of vw . The workpiece is rotated around the y axis at a rotational speed of nw . Ultrasonic vibration is applied along the z axis through the ultrasonic guide wheel with the amplitude of A and frequency of f.
The equilibrium equation of systems shown in Fig. 1 can be derived according to theory of transverse vibration of string, as shown in Eq. (1). 𝜌𝑠
( ) 𝜕2 𝑧 𝜕 𝜕𝑧 𝐹 = 𝑓 (𝑥, 𝑡) (0 < 𝑥 < 𝐿) − 𝜕𝑥 TS 𝜕𝑥 𝜕 𝑡2
(1)
Where 𝜌s is the density of the wire saw, f (x, t) is the excitation of the vibration system. When the f (x, t) = 0, the free vibration differential equation is obtained, as shown in Eq. (2). 𝜕2 𝑧 𝜕2 𝑧 = 𝑐𝑠2 𝜕 𝑡2 𝜕 𝑥2
(2)
Where cs is the propagation velocity of the elastic wave along the chord, as shown in Eq. (3). √ 𝐹𝑇 𝑆 c𝑠 = (3) 𝜌𝑠 2.2. Transverse vibration equation of the axially moving string The relationship between the motion coordinate system and the fixed coordinate system is shown in Eq. (4) according to the relative movement theory [15]: ( ) 𝜕 𝑑 𝜕 = + 𝑣𝑠 (4) 𝑑𝑡 𝜕𝑡 𝜕𝑥 The differential equation of the string vibration in the motion coordinate system can be converted into the differential equation of the fixed coordinate system according to Eq. (4). Wherein, the left side of Eq. (3) can be expressed as Eq. (5). [ ( )] 𝜕𝑧 𝜕2 𝑧 𝜕 𝜕𝑧 𝜕2 𝑧 𝜕2 𝑧 = = + 𝑣s + 2𝑣s (5) 𝜕𝑡 𝜕𝑡 𝜕𝑥 𝜕 𝑥𝜕 𝑡 𝜕 𝑡2 𝜕 𝑡2 The right side of the Eq. (3) can be expressed as Eq. (6) according to Eq. (4):
Fig. 1. Schematic view of the UAWS.
𝑐𝑠2
) 𝜕2 𝑧 𝜕2 𝑧 ( 2 = 𝑐s − 𝑣s 2 𝜕 𝑥2 𝜕 𝑥2
(6)
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International Journal of Mechanical Sciences 164 (2019) 105161
The transverse vibration Eq. (7) of the axially moving string can be obtained by combining the Eqs. (5) and (6). 𝜕2 𝑧 𝜕2 𝑧 𝜕2 𝑧 + 2𝑣𝑠 − 𝑐𝑠2 − 𝑣𝑠 2 =0 (7) 𝜕𝑥 𝜕𝑡 𝜕 𝑡2 𝜕 𝑥2 The boundary conditions of the model in Fig. 2 can be expressed as Eq. (8). { 𝑧(0, 𝑡) = 𝐴𝑐𝑜𝑠𝜔𝑡 ( ) (8) 𝑧 𝐿s , 𝑡 = 0 Where 𝜔 is the angular frequency of ultrasonic vibration, 𝜔 = 2𝜋f. According to the general method of solving differential equations of undamped vibration, the vibration response of the center point of wire saw can be obtained, as shown in Eq. (9). ( ) 𝐿𝑠 1 𝑧0.5𝐿𝑠 , 𝑡 = (9) ( ) 𝐴cos 𝜔𝑡 − 𝜔 ′ 𝜔𝐿𝑠 2𝑣 2cos 2𝑐 ′
Where v’ is the average effective velocity of wire saw, c’ is the average effective wave velocity, as shown in Eqs. (10) and (11). ( 2 ) 𝑐 − 𝑣2𝑠 𝐹 𝑣′ = 𝑠 = 𝑇 𝑆 − 𝑣𝑠 (10) 𝑣𝑠 𝜌𝑠 𝑣 𝑠 ( 2 ) 𝑐 − 𝑣2𝑠 𝐹 𝑆 − 𝜌𝑠 𝑣 𝑠 2 𝑐′ = 𝑠 = 𝑇√ (11) 𝑐𝑠 𝐹 𝑇 𝑆 𝜌𝑠 As can be seen from Eq. (9), the vibration response of the center point of wire saw is a sinusoidal motion. Since the diamond abrasives are fixed on the wire saw, it can be considered that the abrasives vibrate along with the wire saw in the same amplitude, period and phase. Therefore, the vibration law of the abrasives can also be represented by the Eq. (7). 3. Modeling of sawing force The wire sawing model is shown in Fig. 3. Where vs is the reciprocating velocity of the wire saw, vw is the feed rate, nw is the rotation speed of workpiece, d is the diameter of workpiece, r is the radius of wire saw, Fn is the total normal sawing force of the wire saw, and Ft is the total tangential sawing force of the wire saw. Since the total sawing force of the wire saw is a resultant force of the sawing forces of a large amount of abrasives, there is a necessary to establish a sawing force model of single abrasive at first. 3.1. Sawing force of single abrasive in conventional diamond wire sawing Although the force of the abrasive is very complicated, it can be simplified into two parts: chip deformation force and friction force [16].
Fig. 4. Scratching model of single abrasive in CWS.
It is assumed that the sawing force of a single abrasives is consist of four components, as shown in Fig. 4, where Fnga is the normal sawing force caused by chip deformation, Fngb is the tangential sawing force caused by chip deformation, Ftga is the normal sawing force caused by friction, Ftgb is the tangential sawing force caused by friction. The sawing force of a single abrasive at the angle 𝜃 g can be expressed as the following four components [16]: 𝐹ng = 𝐹nga + 𝐹ngb = 𝜎D 𝑆1 + 𝜎C 𝑆2 𝜋 𝐹tg = 𝐹tga + 𝐹tgb = 𝐹 + 𝜇C 𝐹ngb 4tan𝛽g nga
(12) (13)
Where Fng is the normal resultant force of an abrasive, Fnga is the normal component force caused by chip deforming, Fngb is the normal force caused by friction, 𝜎 D is the contact stress between chip and abrasive, S1 is the average cross sectional area of a chip, 𝜎 C is the average contact pressure between the abrasive and the workpiece, S2 is the vertical projection area of an abrasive pressed into the workpiece, Ftg is the tangential resultant force of an abrasive, Ftga is the tangential component force caused by chip deforming, Ftgb is the tangential component force caused by friction, 𝛽 g is half of the top angle of abrasive, 𝜇 C is the coefficient of friction between abrasives and workpiece. The formula of S1 and S2 can be obtained by analyzing the geometrical relationship in Fig. 4, as shown in Eqs. (14) and (15). 1 ℎ 𝑏 = ℎg 2 tan𝛽g 2 g g ( )2 𝑏g )2 1 1 ( 𝑆2 = 𝜋 = 𝜋 ℎg tan𝛽g 2 2 2 𝑆1 =
(14) (15)
Where hg is the cutting depth of the abrasive and bg is the cutting width of the abrasive as shown in Fig. 4. The normal and tangential sawing force of the abrasive can be obtained by substituting Eqs. (14) and (15) into Eqs. (12) and (13), as shown in Eqs. (16) and (17) ( ) 1 𝐹ng = 𝐹nga + 𝐹ngb = tan𝛽g 𝜎D + 𝜎C 𝜋tan𝛽g ℎg 2 = 𝑘ng ℎg 2 2 [ ( )2 ] 2 1 1 𝐹tg = 𝐹tga + 𝐹tgb = 𝜋 𝜎D + 𝜇C 𝜎C tan𝛽g ℎg = 𝑘tg ℎg 2 2 2
Fig. 3. Schematic of wire sawing process.
(16) (17)
Where kng and ktg are the equivalent stiffness coefficient of the workpiece. The sawing force Fng and Ftg of each abrasive is different at different position on the surface of wire saw, because the cutting depth hg of abrasive at the different position 𝜃 g is different. The relationship between the cutting depth hg and the position 𝜃 g can be deduced according
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International Journal of Mechanical Sciences 164 (2019) 105161
Therefore, the impact load Fng-d can be expressed as Eq. (24). √ √[ ] √ ( ) 2 √ 𝐿 −𝐴𝜔 √ ( ) sin 𝜔𝑡 − 𝜔 𝑠′ √ 2𝑣 √ 2 cos 𝜔2𝐿𝑐 ′𝑠 𝐹ng−𝑑 = 𝐾d 𝐹ng = √ 𝐹ng 𝑔 ℎg
The sawing force model of single abrasive in the UAWS can be derived by introducing the impact load model into the sawing force of CWS, as shown in Eq (25). { 𝐹uen = Ω(𝑡)ave 𝐹ng−𝑑 = Ω(𝑡)ave 𝐾d 𝐹ng (25) 𝐹uet = Ω(𝑡)ave 𝐹tg
Fig. 5. The image of function Ω(t).
to the distribution of abrasives on the surface of wire saw [17,18], as shown in Eq. (18). √ 𝑣w ⋅ cos𝜃g ( ) ℎ g 𝜃g = (18) 𝑃valid ⋅ 𝜌saw 𝑣s ⋅ tan𝛽g Where Pvalid is the probability of effective abrasives and 𝜌saw is the distribution density of abrasives on the surface of wire saw. 3.2. The sawing force model of single abrasive in the UAWS based on impact load
Where t0 is the time when the abrasive is impacted with the workpiece, t1 is the time when abrasive is separated from the workpiece. t2 is the starting time of the next vibration period, T is the period of ultrasonic vibration, n = 0, 1, 2… The function Ω(t) is an impulse function, as shown in Fig. 5. When t0 ≤t ≤ t1 , Ω(t)=1, the wire saw is contacted against the workpiece. When t1 ≤t ≤ t2 , Ω(t)=0, the wire saw is separated from the workpiece. The function Ω(t) is expanded by Fourier series as shown in Eq. (20), where the tq =tl -t0 . 𝑡q 𝑇
+
∞ 2∑1 sin 𝜋 𝑛=1 𝑛
(
𝑡q 𝑛 𝑇
) 𝜋 𝑐𝑜𝑠𝑛𝜔𝑡
(20)
The average of the function Ω(t) in an ultrasonic vibration period is derived by Eq. (21). Ω(𝑡)ave =
𝑡q 1 𝑇 ∫ Ω(𝑡)𝑑𝑡 = 𝑇 0 𝑇
(21)
The impact load in UAWS is greater than that in CWS, because there is a unique impact caused by abrasives on the workpiece. According to the impact load theory of material mechanics [19], the impact load Fhg-d can be expressed as the static load Fen multiplied by the dynamic load coefficient Kd , as shown in Eq. (22) √ 𝑣𝑐 2 𝐹ng−𝑑 = 𝐾d 𝐹ng = 𝐹 (22) 𝑔 Δst ng Where g is the acceleration of gravity, Δst is the deformation caused by static load Fng , and it is assumed to be the cutting depth hg . vc is the impact velocity, which can be obtained by the derivation of Eq. (9), as shown in Eq. (23). ( ) d𝑧0.5𝐿𝑠 , 𝑡 𝐿𝑠 −𝐴𝜔 𝑣𝑐 = = (23) ( ) sin 𝜔𝑡 − 𝜔 ′ 𝜔𝐿𝑠 d𝑡 2𝑣 2 cos 2𝑐 ′
3.3. The total sawing force model of wire saw in the UAWS The normal sawing force of a single abrasive can be decomposed into two components that along the y direction and the z direction. The component force along the y direction can be canceled by another abrasive located on the symmetric location of the wire saw. The total normal sawing force is the sum of the component forces along the z direction of sawing force of each single abrasive. The component force along the z direction of a single abrasive is: 𝐹uenx = 𝐹uen 𝑐𝑜𝑠𝜃g
According to the vibration response expressed by Eq. (9), it can be found that the continuous contact between wire saw and workpiece is converted into intermittent contact due to the ultrasonic vibration. A function Ω(t) is used to represent this intermittent contact, as shown in Eq. (19): { 1 𝑡0 + 𝑛𝑇 ≤ 𝑡 ≤ 𝑡1 + 𝑛𝑇 Ω(𝑡) = (19) 0 𝑡1 + 𝑛𝑇 ≤ 𝑡 ≤ 𝑡2 + (𝑛 + 1)𝑇
Ω(𝑡) =
(24)
(26)
During the sawing process, the workpiece is only in contact with the lower half of the wire saw. The sawing forces of an abrasive at the angle 𝜃 g are shown in Eq. (27). { d𝐹un = 𝐹uenx 𝑃valid 𝑐 𝑙𝑐 𝑟𝑑 𝜃g (27) d𝐹ut = 𝐹uet 𝑃valid 𝑐 𝑙𝑐 𝑟𝑑 𝜃g Where Pvalid is the probability of effective abrasives, c is the areal density of abrasives, which is related on the distribution of abrasives on the surface of wire saw, and lc is the contact length between wire saw and workpiece. The total sawing forces of UAWS are derived by integrating the Eq. (20) on [-𝜋/2, 𝜋/2], as shown in Eq. (28). 𝜋
⎧ 2 𝑑 𝐹un ⎪𝐹un = 2 ∫ 0 ⎪ ⎨ 𝜋 ⎪ 2 𝑑 𝐹ut ⎪𝐹ut = 2 ∫0 ⎩
(28)
4. The verification of theoretical sawing force model of single abrasive The theoretical sawing force model of single abrasive was verified by applying the finite element analysis (FEA) software ABAQUS. Firstly, a finite element model of the abrasive is established. The geometrical shape of the abrasive is simplified into a conical shape with a diameter of 𝛽 g = 60°, as shown in Fig. 6. During the simulation, the abrasive is defined as a rigid body based on the following reason. First, the Mohs hardness of diamond is 10 while the Mohs hardness of monocrystalline Si is 6–7 [20], so it’s reasonable to consider that the diamond is cut in the Si without deformation. Additionally, though the wire saw is
Fig. 6. The simplified geometric model of the abrasive.
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International Journal of Mechanical Sciences 164 (2019) 105161
Table 1 Values of factors in the experiment.
Value Value Value Value
1 2 3 4
vs (m/s)
vw (mm/min)
nw (r/min)
2 4 6 8
0.5 1.0 1.5 2.0
5 10 15 20
Table 2 Experimental results of CWS. Fig. 7. FEA model of workpiece.
Tests
vs (m/s)
vw (mm/min)
nw (r/min)
Fn (mN)
Ft (mN)
Ra(𝜇m)
1 2 3 4 5 6 7 8 9 10
2 4 6 8 4 4 4 4 4 4
1.0 1.0 1.0 1.0 0.5 1.5 2.0 1.0 1.0 1.0
10 10 10 10 10 10 10 5 15 20
11.03 7.92 5.21 4.23 5.83 10.10 13.22 9.07 6.77 6.68
5.85 3.89 2.68 1.70 2.89 5.38 6.45 3.82 3.79 3.68
0.91 0.69 0.61 0.48 0.55 1.01 1.35 0.70 0.62 0.62
Table 3 Experimental results of UAWS.
Fig. 8. Assembled FEA model.
very thin, the diameter of it is about 0.35 mm, which is very big compared with the diameter of abrasives (40 𝜇m). Finally, though the impact load in the UAWS is very big compared with the normal force in CWS, the magnitudes of it is tens of mN, which is very small to a wire saw that was tensioned by 110 N tension force. Secondly, the finite element model of the workpiece is established by using the C3D8R unit of the Explicit cell library. The meshed model of workpiece is shown in Fig. 7. The boundary of the workpiece is: fixed the bottom and four broadsides, as shown in Fig. 7. The finite element model of the abrasive and the workpiece are assembled in the ABAQUS software, as shown in Fig. 8. The load of abrasive is: a constant velocity of vs along the x direction of the workpiece; a simple harmonic motion with the amplitude of A and frequency of f along the direction that is 𝜃 g degrade along the y axis in the oyz plane. The vs =2 m/s in every group of simulation. In every simulation for UAWS, the A = 10 𝜇m, f = 20 kHz. In every simulation of CWS, the A = 0 𝜇m, f = 0 kHz. The position angle 𝜃 g of the abrasive is taken to 0°, 30°, 60°, 85° The cutting depth hg is calculated according to the formula (18). The simulation results are shown in Fig. 9: It can be observed from the above figures that under the influence of ultrasonic vibration, the cutting depth of the abrasives is changed constantly. The trajectory of abrasive in the UAWS is different from that in the CWS. The waveforms of normal sawing force are shown in Fig. 10: The average values of the waveforms are extracted. Fig. 11 shows the comparison between the simulated theoretical results. The comparison results show that the sawing forces in the UAWS are lower than those in the CWS under the same working conditions. Additionally, the average error between the theoretical sawing force and simulation sawing force is 9.3%. Thus, the validity of the sawing force model of the single abrasive in UAWS is verified by the finite element simulation.
Tests
vs (m/s)
vw (mm/min)
nw (r/min)
Fn (mN)
Ft (mN)
Ra (𝜇m)
1 2 3 4 5 6 7 8 9 10
2 4 6 8 4 4 4 4 4 4
1.0 1.0 1.0 1.0 0.5 1.5 2.0 1.0 1.0 1.0
10 10 10 10 10 10 10 5 15 20
7.83 5.97 3.66 2.90 2.64 8.95 12.64 8.53 4.44 4.45
3.85 2.22 1.02 0.71 1.06 3.76 6.30 2.40 2.09 1.98
0.75 0.51 0.46 0.44 0.42 0.86 1.34 0.64 0.43 0.41
experimental setup is shown in Figs. 12 and 13. The working frequency of the ultrasonic generator and horn is 19.8–20.2 kHz, and the output amplitude is 10 𝜇m. The dynamometer was a Kistler three-component measuring platform type 9257B, and the charge amplifier was a Kistler dual mode amplifier model 5070A. The diameter of the wire saw is 0.35 mm, and the tension force within it was 110 N. The micrograph of the diamond wire saw is shown in Fig. 14. The sawing forces in two directions, Fn and Ft , were measured. Fn is in the normal direction, Ft is in the tangential direction, as shown in Fig. 3. The monocrystalline Si workpiece was a 36 mm × 80 mm cylinder. The surface topography and roughness on the machined surface were measured along the feed direction. The instruments used were MITUTOYO SJ-201 profile meter, KEYENCE VXH6000 optical microscope and FEI Quanta200F scanning electron microscope (SEM). Table 1 shows the matrix for the experiment, as well as the high and low settings for process variables. Three most important factors were selected as variables in the experiment: the axial velocity of the wire saw vs (m/s), the feed rate of the workpiece vw (mm/min), and the rotational speed of the workpiece nw (r/min). Three output variables were observed: sawing force, surface roughness and surface topography.
6. Results and discussions 5. Experimental setup The validity of the total sawing force model of the wire saw is verified by the experiment. which were carried out using a conventional wire sawing machine tool assisted with an ultrasonic device, and the
Investigations were carried out into the comparison of CWS and UAWS. The effect of ultrasonic vibration on the sawing force and surface roughness were studied. The experimental results of CWS and UAWS are shown in Tables 2 and 3 respectively.
Y. Wang, D.-L. Li and Z.-J. Ding et al.
International Journal of Mechanical Sciences 164 (2019) 105161
Fig. 9. Simulation results of CWS and UAWS with different position angle 𝜃 g.
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International Journal of Mechanical Sciences 164 (2019) 105161
Fig. 10. Waveforms of normal forces of CWS and UAWS with different position angle 𝜃 g.
6.1. Verification of the total sawing force model of wire saw
Fig. 11. Comparison between the simulated and theoretical results of normal force.
The validity of the total sawing force model of wire saw is verified by comparing the theoretical and experimental sawing force, as shown in Figs. 15–17. It can be seen that the trends of theoretical influences of input variables agree well with the trends determined experimentally, and the average error between the experimental and theoretical results of sawing force is 7.50%. Thus, the experimental results verified the theoretical model very well. In addition, it can also be seen from Figs. 15–17 that the sawing force decreases as the axial velocity of wire saw vs and workpiece rotation speed nw increase. The sawing force further increases as the feed rate vw increases. The sawing force in UAWS is smaller than that in CWS due to the ultrasonic vibration. The mechanism is that though the peak value of the impact load is quite high, its duration is pretty short. As a result, the impulse caused by impact load is lower than the impulse
Fig. 12. Experimental devices of ultrasonic vibration assisted wire sawing Si.
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International Journal of Mechanical Sciences 164 (2019) 105161
Fig. 13. Measuring devices of experiments.
Fig. 14. The micrograph of the diamond wire saw measured by SEM.
caused by constant load, which means that the sawing force in UAWS is smaller than CWS. 6.2. Analysis of surface roughness results and workpiece topography results As Fig. 18 shows, the trends of influences of sawing forces on surface roughness were determined experimentally. It can be seen that the surface roughness Ra increases as the normal sawing force Fn and tangential
sawing force Ft increase. Thus, the lower value of surface roughness can be obtained with lower sawing force. As the sawing force in UAWS is smaller than that in CWS due to the ultrasonic vibration, it can be inferred that the lower value of surface roughness can be obtained with ultrasonic vibration. Fig. 19 shows the 3D surface topography of workpiece measured by using the KEYENCE VXH6000 optical microscope with the amplitude of 2000. It can be seen that the surface topography obtained by UAWS is better than that obtained by CWS. The size of pits on the surface of workpiece obtained by UAWS is smaller than that obtained by CWS. The mechanism is that the duration time of the contact between abrasives and workpiece is very short due to the high frequency of the ultrasonic vibration, as a result, the length of brittle fracture crack decreased. As the length of the brittle fracture crack decreases, the size of pits decreases Fig. 20 shows the surface measurements of workpieces in CWS and UAWS by using FEI Quanta200F scanning electron microscope (SEM) with the amplitude of 4000. It can be seen that the edges of pits in Fig. 20(a) are sharper than that in Fig. 20(b). According to this result, it can be recognized that the brittle removal mode is reduced by ultrasonic vibration. The removal of workpiece surface material is more uniform in UAWS. The big and sharp pits in CWS are difficult to be removed during the subsequent finishing processes. However, the small and smooth pits in UAWS can be easily removed. Therefore, UAWS is an efficient, precise and economical sawing process compared with CWS.
Fig. 15. Relation between sawing force (Fn , Ft ) and axial velocity of wire saw (vs ).
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International Journal of Mechanical Sciences 164 (2019) 105161
Fig. 16. Relation between sawing force (Fn , Ft ) and feed rate (vw ).
Fig. 17. Relation between sawing force (Fn , Ft ) and workpiece rotation speed (nw ).
Fig. 18. Relation between surface roughness (Ra) and sawing force (Fn , Ft ).
Fig. 19. Comparing of workpiece surface topography (vs = 4 m/s, vw = 2 mm/min, nw = 10r/min).
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International Journal of Mechanical Sciences 164 (2019) 105161
Fig. 20. Comparing of workpiece surface measured by SEM (vs = 4 m/s, vw = 2 mm/min, nw = 10r/min).
7. Conclusions In this paper, a model of sawing force in UAWS based on the impact load is presented. The validity of the proposed model has been confirmed by simulation and experiments. The following conclusions can be drawn: 1. A novel theoretical model for sawing force in UAWS based on the theory of impact load is presented in this paper. Firstly, the transverse vibration model of the diamond wire saw in UAWS was established based on the transverse vibration theory of continual system. Secondly, the impact load model of single abrasive was established based on the vibration model. Finally, the sawing force caused by multi abrasives was derived based on the distribution of abrasives on the surface of wire saw. According to the presented theory, the impact load is the main reason for the reduction of sawing force in UAWS. 2. The theory about the sawing force of single abrasive was verified by FEA simulation. According to the theoretical results and simulation results, the average error between the theoretical and simulated value is 9.3%. The theory about the total sawing force was verified by experiments. According to the theoretical results and experimental results, the average error between theoretical and experimental results is 9.8%. The validity of presented theoretical model in this paper is verified. 3. The value of surface roughness in UAWS is lower than that in CWS. Compared with CWS, the pits on workpiece surface in UAWS are smaller and smoother. It can be recognized that surface quality in UAWS is better than that in CWS. References [1] Hao W. Wire sawing technology: a state-of-the-art review. Precis Eng 2016;43(12):1–9. [2] Kumar A, Melkote SN. Diamond wire sawing of solar Si workpieces: a sustainable manufacturing alternative to loose abrasive slurry sawing. Proc Manuf 2018;21:549–66.
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