On the mechanics and material removal mechanisms of vibration-assisted cutting of unidirectional fibre-reinforced polymer composites

International Journal of Machine Tools & Manufacture 80-81 (2014) 1–10

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On the mechanics and material removal mechanisms of vibration-assisted cutting of unidirectional fibre-reinforced polymer composites Weixing Xu, L.C. Zhang n School of Mechanical and Manufacturing Engineering, The University of New South Wales, NSW 2052, Australia

art ic l e i nf o

a b s t r a c t

Article history: Received 13 November 2013 Received in revised form 18 February 2014 Accepted 18 February 2014 Available online 2 March 2014

This paper aims to reveal the material removal mechanisms and the mechanics behind the vibrationassisted cutting (VAC) of unidirectional fibre reinforced polymer (FRP) composites. Through a comprehensive analysis by integrating the core factors of the VAC, including fibre orientation and deformation, fibre–matrix interface, tool–fibre contact and tool–workpiece contact, a reliable mechanics model was successfully developed for predicting the cutting forces of the process. Relevant experiments conducted showed that the model has captured the mechanics and the major deformation mechanisms in cutting FRP composites, and that the application of ultrasonic vibration in either the cutting or normal direction can significantly decrease cutting forces, minimise fibre deformation, facilitate favourable fibre fracture at the cutting interface, and largely improve the quality of a machined surface. When the vibrations are applied to both the cutting and normal directions, the elliptic vibration trajectory of the tool tip can bring about an optimal cutting process. There exists a critical depth of cut, beyond which the fibre–matrix debonding depth is no longer influenced by the vibration applied on the tool tip. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Fibre-reinforced polymer composites Vibration-assisted cutting Cutting mechanics Material removal mechanism Fibre–matrix debonding Fibre fracture

1. Introduction Fibre-reinforced polymer (FRP) composites have been widely used in advanced structural applications due to their high strength and stiffness to weight ratio. However, the machining of FRP composite products is difficult because of the significant difference in the mechanical properties of the fibres and matrix. As a result, a machined FRP composite usually contains various damages, such as fibre pull-out, fibre fragmentation, matrix cracking, fibre–matrix debonding and delamination [1–4]. To date, most experimental investigations on the machining of FRP composites are on the following issues: effect of fibre or matrix types [5,6], influence of fibre volume fraction and orientations [2,7], role of tool materials and geometries [2,8], contribution of the depth of cut [9], and selection of processing parameters [10–12]. These studies, however, are limited to the traditional machining methods, such as turning, milling and drilling, and are still facing the poor surface integrity problems highlighted above. To accomplish a high quality surface of FRP composites, the common understanding is that grinding is more appropriate [13–15], because the instant depth of cut

n

Corresponding author. Tel.: þ 61 2 9385 6078 fax: þ61 2 9385 7316. E-mail addresses: [email protected] (W. Xu), [email protected] (L.C. Zhang). http://dx.doi.org/10.1016/j.ijmachtools.2014.02.004 0890-6955 & 2014 Elsevier Ltd. All rights reserved.

of a single cutting edge in grinding is much smaller than the diameter of a fibre [2]. Nevertheless, grinding is inefficient in many cases. On the other hand, vibration-assisted cutting, which adds a displacement of micro-scale amplitude at an ultrasonic frequency to the tip motion of a cutting tool, has been experimentally evidenced to be an effective method to cut many single phase materials such as metals and ceramics [16–21]. In order to machine FRP composites effectively, the authors have developed a vibration-assisted cutting technique [22–24]. This technique applies ultrasonic vibrations to the tool tip and based on the directions of the vibrations, the tool tip trajectories can be controlled to follow, e.g., an elliptic path named as an elliptic vibration-assisted (EVA) cutting. Their investigation has shown that the EVA can significantly reduce cutting forces and subsurface damages in a workpiece even by using a simple cutting tool. The effects of vibrations in cutting and normal directions on machining characteristics have also been studied using the novel vibrator developed [23]. They found that the vibration in the cutting direction is more effective in reducing the cutting force, but that normal to the cutting direction facilitates the chip removal. When the vibration is applied to both the directions in an EVA cutting, the cutting force can be greatly reduced, the surface integrity of an FRP workpiece can be much improved, and the tool life can be largely extended. However, the mechanics and mechanisms of the EVA in the material removal process are still unclear. This has significantly

2

W. Xu, L.C. Zhang / International Journal of Machine Tools & Manufacture 80-81 (2014) 1–10

where the tool vibrates in the normal direction only (i.e., a ¼0 but ba0); and (3) EVA cutting (i.e., a a0 and ba0) as described above. The trajectories of the tool for these types of vibrationassisted cutting and traditional cutting are shown in Table 1. In order to establish a mechanics model for investigating the cutting processes, consider a single fibre as a representative unit with the following characteristics: (1) the width of the workpiece is the same as that of the cutting tool, and is equal to the diameter of the fibre; (2) the fibre experiences elastic deformation before its breakage; (3) fibre fracture takes place when the maximum tensile stress exceeds its tensile strength; (4) the contact of tool–fibre and tool–workpiece surface follows the Hertz contact theory. For convenience, the positive directions of the forces and deformations are along the positive x- and z-directions as defined in Fig. 1. 2.2. Fibre deformation

Fig. 1. A schematic illustration of an elliptic vibration-assisted cutting of an FRP composite.

hindered the optimisation and practical application of the EVA technique. The objective of this paper is to remove the above barrier through a detailed mechanics analysis to understand the science behind the EVA cutting of unidirectional FRP composites and thus to establish the essential fundamentals. The most important factors, such as cutting forces, fibre deformation, fibre fragmentation, and fibre– matrix debonding, will be comprehensively integrated in modelling. Relevant experiments will be carried out to examine the model established.

2. Mechanics modelling 2.1. The principle Fig. 1 illustrates the principle of an EVA cutting process, in which the cutting tool feeds at a feed rate, v, while it vibrates elliptically at an ultrasonic frequency with a micro-scale amplitude in the xz-plane. The feed rate is smaller than the maximum vibration speed in x-direction, such that an intermittent cutting is generated in each vibration cycle of the tool. In a cycle, cutting takes place only when the tip wedges into the workpiece at time instant tb, and finishes when the tangential cutting direction is parallel to the fibre orientation at time instant te. Thus, if a fibre/ matrix breakage happens during this process, chips form and are pulled out by the tool. To facilitate the breakage of the fibres and matrix, the cutting distance within a cycle of the tool vibration, Δ, is set to be smaller than the fibre diameter, D, so as to improve the surface quality as explored in the previous work [23]. Assume that a and b are the vibration amplitudes in x- and z-directions, respectively, f is the vibration frequency, ψ is the phase difference, re is the radius of the cutting edge, ap is the set depth of cut, and δ is the bouncing back of the workpiece in the contact zone of the tool with the finished surface. The relative elliptic motions of the tool in x-direction, x(t), and z-direction, z(t), can be described by the equations shown in Table 1. Based on the motion of the tool vibration relative to the feed direction, one can define three types of vibration-assisted cutting: (1) cutting-directional vibration-assisted (CDVA) cutting where the tool vibrates in the cutting direction only (i.e., a a0 but b¼0); (2) normal-directional vibration-assisted (NDVA) cutting

Consider a straight fibre in a composite supported on its entire length by the rest of the composite which is elastic. As illustrated in Fig. 1, the fibre is subjected to a horizontal force acting in the principal plane of the symmetrical cross section. The fibre thus deflects, but the bulk composite behind the fibre applies a distributed reaction to resist the fibre deflection. The reaction force can be described by Winkler's foundation model [25], as illustrated in Fig. 2. For simplicity but without losing the generality, the part of the composite that supports the fibre can be treated as an equivalent homogeneous material (EHM) whose elastic properties (e.g., modulus) can be worked out by the mixed rule of a composite [26]. Let pm be the intensity of the reaction force at a fibre cross-section with deflection x; then pm ¼ xkm, where km is the modulus of the foundation EHM, and can be obtained by using Biot's formula [27,28]: " #0:108 0:95Em D4 E m km ¼ ð1Þ ð1  ν2m Þ Ef I f ð1  ν2m Þ where Em and νm are the transverse Young's modulus and Poisson's ratio of EHM; Ef and If ( ¼πD4/64) are the transverse Young's modulus and moment of inertia of the fibre cross-section. During the cutting, part of the fibre–matrix interface can suffer from debonding of depth h, as illustrated in Fig. 2. Then, the rest of the fibre–matrix interface will have a bonding force to constrain the deflection of the fibre caused by the cutting force. Assuming that pb is the intensity of the bonding force in the un-debonding area, pb ¼ xkb, where kb is the equivalent modulus of the fibre– matrix bonding. At the onset point of the debonding E, pb should be equal to the fibre–matrix bonding strength sb. Consider an infinitesimal element of length dz of the fibre (see Fig. 2). The element equilibrium gives rise to ( ∑Q ¼ 0 Q  ðQ þ dQ Þ þpm dz þ pb dz ¼ 0 ð2Þ ∑M ¼ 0 dM  ðQ þ dQ Þdz  ðkm þ kb Þdx2 ¼ 0 where Q is the shear force and M is the bending moment. The substitution of Eq. (2) into the well-known beam bending theory [29], d2x/dz2 ¼  M/EfIf, brings about 4

Ef I f

d x þ ðkm þ kb Þx ¼ 0: dz4

ð3Þ

The general solution of Eq. (3) is x ¼ eλz ðC 1 cos λz þ C 2 sin λzÞ þ e  λz ðC 3 cos λz þC 4 sinλzÞ

ð4aÞ

or x ¼ cosh λzðB1 cos λz þ B2 sin λzÞ þ sinh λzðB3 cos λz þB4 sin λzÞ ð4bÞ

W. Xu, L.C. Zhang / International Journal of Machine Tools & Manufacture 80-81 (2014) 1–10

3

Table 1 Cutting types. Cutting types

Trace of tool

Displacement of tool (

Traditional cutting

(

CDVA cutting

(

NDVA cutting

(

EVA cutting

xðtÞ ¼ vt zðtÞ ¼ ap  r e xðtÞ ¼ vt þ a sin ð2πf tÞ zðtÞ ¼ ap  r e xðtÞ ¼ vt zðtÞ ¼ ap  r e  b þ b sin ð2πf t þ φÞ xðtÞ ¼ vt þ a sin ð2πf tÞ zðtÞ ¼ ap  r e  b þ b sin ð2πf t þ φÞ

F 3 ðλzÞ ¼ sinh λz sin λz; F 4 ðλzÞ ¼ ðcosh λz sin λz  sinh λz cos λzÞ=2 Thus, the cutting force FAx denoted in Fig. 2 is the sum of the reaction forces on parts (1) to (3) of the fibre described above, i.e., Z ap  r e Z ap þ h  δ Z þ1 F Ax ¼ km x1 dz þ km x2 dz þ kmb x3 dz ð6Þ ap  re

0

Fig. 2. Deformation of a fibre during cutting.

where λ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ðkm þ kb Þ=4Ef I f

and C1, C2, C3, C4, B1, B2, B3, B4 are constants of integration. Note that cosh(λz)¼(eλz þe  λz)/2 and sinh(λz)¼(eλz  e  λz)/2. It is clear from the above analysis that a fibre in a composite, when interacting with a cutting tool, has varying supporting conditions along its axis. Because of this, the fibre deflection can be divided into three sections as illustrated in Fig. 2: (1) the part of the fibre above the tool–fibre contact point A, i.e., z oap  re; (2) that of the fibre between points A and E, i.e., ap re r z rap þh  δ; and (3) that of the fibre below point E, i.e., z 4ap þh  δ. Correspondingly, the deflections of these parts of the fibre can be obtained below:

In calculating the FAx in Eqs. (6) and (13) parameters, B1, B2, B3, B4, B5, B6, B7, B8, C1, C2, C3, C4 and h, need to be determined by the boundary conditions of the fibre and the transition conditions across the parts of the fibre sections with different supporting constraints, when the expressions in each of the terms in Eq. (6) are obtained, as to be derived below. The general expressions of the slope of the deflection, ki (i¼1, 2, 3), bending moment, Mi (i¼1, 2, 3), and shear force, Qi (i¼ 1, 2, 3), can be derived straightforwardly. ( )  2B1 F 4 ðλm zÞ þ B2 ½F 1 ðλm zÞ þ F 3 ðλm zÞ dx1 ¼ λm k1 ¼ ð7aÞ þ B3 ½F 1 ðλm zÞ F 3 ðλm zÞ þ 2B4 F 2 ðλm zÞ dz ( k2 ¼ λm ( k3 ¼ λmb

F 1 ðλzÞ ¼ cosh λz cos λz; F 2 ðλzÞ ¼ ðcosh λz sin λz þ sinh λz cos λzÞ=2

 2B5 F 4 ðλm zÞ þ B6 ½F 1 ðλm zÞ þ F 3 ðλm zÞ

) ð7bÞ

þ B7 ½F 1 ðλm zÞ  F 3 ðλm zÞ þ 2B8 F 2 ðλm zÞ

)

C 1 ½F 1 ðλmb zÞ  F 3 ðλmb zÞ  2F 4 ðλmb zÞ þ C 2 ½F 1 ðλmb zÞ þ 2F 2 ðλmb zÞ þ F 3 ðλmb zÞ  C 3 ½F 1 ðλmb zÞ  F 3 ðλmb zÞþ 2F 4 ðλmb zÞ þ C 4 ½F 1 ðλmb zÞ  2F 2 ðλmb zÞ þ F 3 ðλmb zÞ

ð7cÞ 2

M 1 ¼ Ef I f

d x1 ¼ 2Ef I f λ2m dz2

(

B1 F 3 ðλm zÞ  B2 ½F 2 ðλm zÞ  F 4 ðλm zÞ

)

þ B3 ½F 2 ðλm zÞ þ F 4 ðλm zÞ  B4 F 1 ðλm zÞ ð7dÞ

( M 2 ¼ 2Ef I f λ2m

8 x1 ¼ B1 F 1 ðλm zÞ þ B2 ½F 2 ðλm zÞ þF 4 ðλm zÞ þ B3 ½F 2 ðλm zÞ  F 4 ðλm zÞþ B4 F 3 ðλm zÞ > > > > < x2 ¼ B5 F 1 ðλm zÞ þ B6 ½F 2 ðλm zÞ þF 4 ðλm zÞ þ B7 ½F 2 ðλm zÞ  F 4 ðλm zÞþ B8 F 3 ðλm zÞ x3 ¼ C 1 ½F 1 ðλmb zÞ þF 2 ðλmb zÞ  F 4 ðλmb zÞ þ C 2 ½F 2 ðλmb zÞ þ F 3 ðλmb zÞ þ F 4 ðλmb zÞ > > > > : þ C 3 ½F 1 ðλmb zÞ  F 2 ðλmb zÞ þ F 4 ðλmb zÞ þ C 4 ½F 2 ðλmb zÞ  F 3 ðλmb zÞ þ F 4 ðλmb zÞ

where B5, B6, B7, B8 are constants of integration, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λm ¼ 4 km =4Ef I f ; λmb ¼ 4 ðkm þ kb Þ=4Ef I f

ap þ h  δ

B5 F 3 ðλm zÞ  B6 ½F 2 ðλm zÞ F 4 ðλm zÞ þ B7 ½F 2 ðλm zÞ þ F 4 ðλm zÞ B8 F 1 ðλm zÞ

) ð7eÞ

z o ap  r e ap  r e r z r ap þ h  δ

ð5Þ

z 4 ap þ h  δ

M 3 ¼ 2Ef I f λ2mb ( ) C 1 ½F 2 ðλmb zÞ þ F 3 ðλmb zÞþ F 4 ðλmb zÞ C 2 ½F 1 ðλmb zÞ þF 2 ðλmb zÞ  F 4 ðλmb zÞ  C 3 ½F 2 ðλmb zÞ F 3 ðλmb zÞ þ F 4 ðλmb zÞ þC 4 ½F 1 ðλmb zÞ F 2 ðλmb zÞ þ F 4 ðλmb zÞ

ð7fÞ

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W. Xu, L.C. Zhang / International Journal of Machine Tools & Manufacture 80-81 (2014) 1–10 3

d x1 3 dz  ¼ 2Ef I f λ3m 2B1 F 2 ðλm zÞ  B2 ½F 1 ðλm zÞ

cutting process without an assisted vibration, or in a CDVA cutting. In these cases, the fibre deformation caused tool–fibre motion is very small, and can be neglected.

Q 1 ¼  Ef I f

 F 3 ðλm zÞ þ B3 ½F 1 ðλm zÞ þ F 3 ðλm zÞ þ2B4 F 4 ðλm zÞ



 Q 2 ¼ 2Ef I f λ3m 2B5 F 2 ðλm zÞ  B6 ½F 1 ðλm zÞ  F 3 ðλm zÞ þ B7 ½F 1 ðλm zÞ  þ F 3 ðλm zÞ þ 2B8 F 4 ðλm zÞ

ð7gÞ 2.3. Fibre fracture ð7hÞ

Q 3 ¼ 2Ef I f λ3mb ( ) C 1 ½F 1 ðλmb zÞ þ2F 2 ðλmb zÞ þ F 3 ðλmb zÞ C 2 ½F 1 ðλmb zÞ F 3 ðλmb zÞ  2F 4 ðλmb zÞ  C 3 ½F 1 ðλmb zÞ 2F 2 ðλmb zÞ þ F 3 ðλmb zÞ C 4 ½F 1 ðλmb zÞ F 3 ðλmb zÞ þ 2F 4 ðλmb zÞ

ð7iÞ The deflection of the fibre is continuous, thus the boundary conditions and transition conditions across the individual parts of the fibre sections with different supporting constraints are as follows: At the top of the fibre (z ¼0): ( M 1 jz ¼ 0 ¼ 0  ð8Þ Q 1 z ¼ 0 ¼ 0 At the tool–fibre contact point A (z ¼zA ¼ap  re): 8 ¼ vt þ a cos ð2πf tÞ x j > > > 1 z ¼ zA ¼ ap  re > > j x < 2 z ¼ zA ¼ ap  re ¼ vt þ a cos ð2πf tÞ   k1 z ¼ z ¼ ap  re ¼ k2 z ¼ z ¼ ap  re > > A A > > > : M 1 jz ¼ zA ¼ ap  re ¼ M 2 jz ¼ zA ¼ ap  re At the fibre–matrix debonding point E (z¼ zE ¼ ap þh  δ): 8 x2 jz ¼ zE ¼ ap þ h  δ ¼ sb =kb > > > > > x3 jz ¼ zE ¼ ap þ h  δ ¼ sb =kb > > >   < k2 z ¼ z ¼ ap þ h  δ ¼ k3 z ¼ z ¼ ap þ h  δ E E > >M j > 2 z ¼ zE ¼ ap þ h  δ ¼ M 3 jz ¼ zE ¼ ap þ h  δ > > >  >  > : Q 2 z ¼ z ¼ ap þ h  δ ¼ Q 3 z ¼ z ¼ ap þ h  δ E

ð9Þ

where pmax ¼3FAx/(2πcd), and c and d are the semi-axes of the contact ellipse and are determined by the following equations [31]: 8 3F KðeÞð1  ν2 Þ  EðeÞ Ax f > ¼ r1e > > e2 c3 Ef π > > < 3F ð1  ν2 Þh i Ax EðeÞ f 2 ð14Þ 2 c3 E π 2  KðeÞ ¼ D e 1  e f > > qffiffiffiffiffiffiffiffiffiffiffi > > > : e ¼ 1  d22 c

ð10Þ

where νf is Poisson's ratio of the fibre; K(e) and E(e) are the complete elliptic integrals of the first and second kinds. As in [32], the Cartesian components of the stress field (sx, sy, sz, τxy, τyz, and τzx) generated by the wedging of the cutting tool can be calculated; thus the tensile stress in the fibre is:  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sT ¼ I1 þ 2 cos ϕ I21  3I2 =3 ð15Þ

E

At the bottom of the fibre (z -þ 1): x3 jz ¼

þ1

¼0

ðC 1 ¼ C 2 ¼ 0Þ

ð11Þ

Therefore, the aforementioned 13 parameters can be obtained by solving Eqs. (8)–(11) and the cutting force FAx in Eq. (6) is determined. Cutting force FAz represents the friction between the fibre and the cutting tool tip, and thus depends on the relative motion between the cutting tool and fibre. During the NDVA and EVA cutting, because the vibration generates a relative tool–fibre motion along the fibre axis (i.e., in the z-direction), the friction can be determined by the Coulomb friction law as F AZ ¼ μA F AX

To determine the fracture of a fibre, the tensile stress distribution in the tool–fibre contact area should be understood. When the nose of a cutting tool is in contact with a fibre, it can be viewed as the contact between two circular cylinders under the normal load FAx and friction FAz, as illustrated in Fig. 3. The corresponding distribution of the normal pressure, p(y,z), and tangential traction, q(y,z), in the contact area can be described by [30]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 > < pðy; zÞ ¼ pmax 1  ðy=cÞ2  ðz=dÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð13Þ > : qðy; zÞ ¼ μA pmax 1  ðy=cÞ2  ðz=dÞ2

where I1 ¼ sx þ sy þ sz, I2 ¼ sxsy þ sysz þ szsx  τ2xy  τ2yz  τ2zx, I3 ¼ sxsysz  sxτ2yz  syτ2zx  szτ2xy þ 2τxyτyzτzx, ϕ¼arcos[(2I31  9I1I2 þ27I3)/ (I21  3I2)3/2/2]/3. Based on the fracture criterion of maximum tensile stress, the fracture of the fibre occurs when sTmax in the fibre reaches the fibre tensile strength.

2.4. Deformation of the workpiece material below the cutting edge The nose and the clearance face of a cutting tool make the workpiece below the cutting edge deform. This deformation can be viewed as that under a half cylinder indenter (Region 1) and a half wedge indenter (Region 2) [1], as illustrated in Fig. 4. Assume

ð12Þ

where μA is the friction coefficient between the tool and fibre. In contrast, there is no such a relative motion in a traditional

Fig. 3. The contact between a cutting tool and a fibre in an FRP composite.

Fig. 4. Contact of a cutting tool with the cut surface of a workpiece.

W. Xu, L.C. Zhang / International Journal of Machine Tools & Manufacture 80-81 (2014) 1–10

that the principle of superposition applies. In Region 1, by using the indentation mechanics of a circular cylinder in contact with a half-space [33], the indentation force on the tool cutting edge can be approximated by adding up half of the indentation force on the arc length 2CB, i.e., 2

1 l πEn1 D P1 ¼ 2 4r e

ð16Þ

where P1 is the indentation force perpendicular to the finished surface, En1 is the effective elastic modulus of the workpiece material in z-direction, and l is the width of the contact arc CB defined by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 > xðtÞ r r e  r 2e  ðr e  δÞ2 > > r e  ðr e  δÞ < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ¼ r  xðtÞ r e  r 2e  ðr e  δÞ2 oxðtÞ or e > e > > : 0 xðtÞ Z r e The bouncing back value δ can be obtained by using the machining-elasticity parameter K, which is defined by the ratio of the true depth of cut to the set depth of cut, i.e., δ¼ap(1  K). However, it must be pointed out that the tool tip has a sliding motion in z-direction in NDVA and EVA cutting, which makes the bouncing back time-dependent: δ0 ¼ δ  b þ b sin ð2πf t þ φÞ

ð17Þ

When the friction coefficient is μm, the frictional force f1 ¼P1μm cos β can be resolved as ( f 1x ¼ P 1 μm cos 2 β ð18Þ f 1z ¼  μm cos β sin β where β¼

8 > arctanðδ=lÞ > > < arctan½r e  > > > : 0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2e  ðr e  xðtÞÞ2 =½r e  xðtÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xðtÞ r r e  r 2e  ðr e  δÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r e  r 2e  ðr e  δÞ2 o xðtÞ o r e

F 2z ¼ P 2 ð1 þ μm sin α cos αÞ Therefore, the total cutting forces, Fx and Fz, become ( F x ¼ F Ax þ F 1x þ F 2x F z ¼ F Az þF 1z þ F 2z

ð21Þ

ð22Þ

2.5. Effect of vibration on the dynamic stiffness of the cutting system Because the vibration frequency of the tool tip far exceeds the natural frequency of the cutting system, the tool–workpiece interaction is not continuous, but is of multiple micro-impacts. Thus the dynamic stiffness of the cutting tool assembly on the machine can be greatly increased to T/Tc times that of a traditional cutting system under the same cutting condition [34–37], where T is the vibration period of the cutting tool and Tc (¼te  tb) is the cutting duration. Assuming that kx is the equivalent coefficient of dynamic stiffness in x-direction at a point along the fibre axis in traditional cutting, the corresponding dynamic modulus of the foundation k0 m during the cutting process can be written as 8 a ¼ 0 ðTraditional cuttingÞ < kx km =ðkx þ km Þ  0 km ¼ T T a 4 0 ðVibration  assisted cuttingÞ : T c kx km = T c kx þ km ð23Þ Similarly, the effect of vibration on the dynamic stiffness of the cutting system in z-direction can be expressed by modifying the machining-elasticity parameter. ( K b ¼ 0 ðTraditional cuttingÞ 0 ð24Þ K ¼ 1  TTc ð1  KÞ b 4 0 ðVibration  assisted cuttingÞ

xðtÞ Z r e

Therefore, the total cutting forces in Region 1 become ( F 1x ¼ P 1 μm cos 2 β F 1z ¼ P 1 ð1  μm cos β sin βÞ

3. Cutting conditions ð19Þ

Similarly, by using the contact mechanics between a wedge and a halfspace [30], the total force P2 in Region 2 can be calculated by P2 ¼

Region 2 are ( F 2x ¼ P 2 μm cos 2 α

5

1 Dδ n E 2 sin α 2

ð20Þ

where α is the clearance angle of the tool, and En2 is the effective modulus of the workpiece material in Region 2. Note that En2 must be smaller than that of the original workpiece material, because the material in this region has been damaged during the deformation experienced in Region 1. When adding the friction force between the clearance face and the workpiece material, the cutting forces in

In order to verify the mechanics model, the same conditions listed in Table 2 were applied in the simulation and experiments. The set-up used in the present experiments, including the depth of cut, tool feed rate, vibration frequency and vibration amplitudes, were the same as that in the authors' previous work [23]. A micrograin grade TiAlN/TiN coated tungsten carbide insert was used as the cutting tool, and the workpieces were prepared from a unidirectional carbon fibre-reinforced polymer (CFRP) laminates which were made by stacking up the unidirectional prepreg plies (carbon fibres impregnated with epoxy resin) with one fibre direction. The friction coefficients μA and μm were taken as 0.22 and 0.25, respectively, based on the experimental measurements available [2,23]. Table 3 outlines the material properties obtained from the literature about machining simulations [1,38–40].

Table 2 Simulation and experimental conditions. Tool & workpiece Tool material Cutting edge radius re [μm] Tool clearance angle α [1] Workpiece material Fibre orientation θ [1] Fibre diameter D [μm] Fibre volume fraction Vf [%]

Cutting conditions Tungsten carbide 5 7 Unidirectional CFRP 90 7 60

Depth of cut ap [μm] Tool feed rate v [m/min] Vibration frequency f [kHz] Vibration amplitude a [μm] Vibration amplitude b [μm] Friction coefficient μA Friction coefficient μm

4–100 1 17.43 2.07 1.67 0.2 0.25

6

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4. Results and discussion 4.1. Deformation and fracture of fibre Fig. 5 shows the variations of fibre deformation (blue lines) and fibre–matrix debonding depth (red lines) during the cutting process with ap ¼ 30 μm and v ¼1 m/min. When the ultrasonic vibration was applied, the frequency was f¼ 17.43 kHz. With the CDVA cutting, a¼ 2.07 μm and b ¼0 μm; under the NDVA cutting, a ¼0 μm and b ¼1.67 μm; while under the EVA cutting, a ¼2.07 μm and b¼1.67 μm. Fig. 5(a) illustrates the deformation of fibre in a traditional cutting process. It is clear that the fibre bends ahead of the cutting tool due to the continuous pressing of the cutting tool. The deflection increases with the advancement of the tool until the

fibre fractures at t¼248 μs. Usually, the fibre–matrix debonding depth is used to describe the degree of subsurface damage. In this case, with the increase of deflection of fibre, debonding appears at t¼23 μs for the first time, followed by the increase of the debonding depth with the penetration of the tool tip into the fibre. The maximum depth comes to h ¼14.07 μm at the end. Obviously, the traditional method leads to a large workpiece deformation, brings about a deep damage in the subsurface, and causes the material to be cut at a low efficiency. When a vibration is applied to the feeding direction, i.e., the CDVA cutting shown in Fig. 5(b), the resultant velocity is determined by the feed rate and the vibration speed. At the beginning, both the feed rate and the vibration speed are in the same direction, which leads to the earlier appearance of fibre–matrix debonding at t¼3 μs in comparison with that in the traditional

Table 3 Material properties. Fibre Transverse Young's modulus Ef [GPa] Tensile strength st [GPa] Poisson's ratio νf Shear strength ss [GPa]

EHM 15 3.5 0.2 0.38

Interface Bonding strength sb [MPa] Equivalent modulus kb [GPa/m]

Transverse Young's modulus Em [GPa] Effective elastic modulus E1* [GPa] Effective elastic modulus E2* [GPa] Poisson's ratio νm

5.6 10 3.5 0.318

Cutting system 30 115

Machining-elasticity parameter K Coefficient of dynamic stiffness in traditional cutting kx [GPa/m]

0.96 1.77

Fig. 5. Fibre deformation and fibre–matrix debonding depth in (a) traditional cutting (ap ¼ 30 μm, v¼ 1 m/min), (b) CDVA cutting (ap ¼ 30 μm, v¼1 m/min, f ¼17.43 kHz, a¼ 2.07 μm, b¼ 0 μm), (c) NDVA cutting (ap ¼ 30 μm, v¼ 1 m/min, f ¼17.43 kHz, a¼ 0 μm, b ¼1.67 μm), and (d) EVA cutting (ap ¼ 30 μm, v¼ 1 m/min, f¼17.43 kHz, a¼ 2.07 μm, b ¼1.67 μm). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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method at t ¼23 μs as listed previously. The resultant velocity of these two components increases sharply to the maximum and then decreases to 0 (t¼10 μs) with the continuous increase of debonding depth. After that, the cutting tip begins to move backwards, due the vibration phase shift, and the debonding stops to propagate downwards and reaches its the largest depth in the first cutting cycle (h ¼5.13 μm). The process lasts until the commencement of the second cutting cycle at t¼59 μs, which brings about a further propagation of the debonding. Lastly, the fibre fractured in the third cutting cycle at t¼ 121 μs with the depth of debonding h¼ 10.62 μm. Clearly, the application of the vibration in the cutting direction has improved not only the surface quality (one third of the debonding depth is decreased) but also the cutting efficiency. When a vibration is applied perpendicular to the feeding direction, i.e., the NDVA cutting shown in Fig. 5(c), it is evident that the application of the vertical vibration brings about a continuous change of the resultant velocity, which changes the originally linear trajectory of the cutting tool. Thus a relevant movement of the tool along the fibre axis causes an extra friction which makes the fibre fracture earlier, reducing the debonding penetration depth. Additionally, the cutting speed downwards occupies only half of a vibration cycle, which makes the bedonding depth further shortened. It can be seen that the application of a vertical vibration can also improve the subsurface quality and increase the cutting efficiency. When an elliptical vibration is applied, as shown in Fig. 5 (d), the EVA cutting combined the merits of the CDVA and the

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advantages of the NDVA. Apparently, an EVA cutting is merely a motion synthesis of the CDVA and NDVA. However, such a synthesis has brought about extremely superior characteristics. On one hand, the application of the horizontal vibration contributes to the early emergence of debonding (same as CDVA cutting) and increases the normal force on the fibre to initiate earlier fracture of the fibre. On the other hand, the vertical vibration alters the trajectory of the tool to bring about a higher friction along the fibre to accelerate its fracture (fibre fractures at t¼ 7 μs).

Fig. 7. Influence of depth of cut on fibre–matrix debonding depth.

Fig. 6. Finished surfaces of CFRP composites machined by (a) traditional cutting (ap ¼ 30 μm, v¼1 m/min), (b) CDVA cutting (ap ¼30 μm, v ¼1 m/min, f ¼17.43 kHz, a¼ 2.07 μm, b¼ 0 μm), (c) NDVA cutting (ap ¼ 30 μm, v¼ 1 m/min, f ¼17.43 kHz, a¼ 0 μm, b ¼ 1.67 μm), and (d) EVA cutting (ap ¼ 30 μm, v¼ 1 m/min, f¼ 17.43 kHz, a ¼2.07 μm, b ¼1.67 μm).

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Meanwhile, this much shortened time span hinders the debonding propagation, making the debonding depth at only h¼ 1.9 μm which is far smaller than all the other cutting processes discussed above. Fig. 6 shows the finished CFRP composite surfaces machined by the traditional and vibration-assisted cutting methods, where the debonding zones have been highlighted by red curves. It can be

seen that the conventional cutting produced a number of large debonding zones, resulting in a deep damage to the subsurface as shown in Fig. 6(a). The vibration-assisted cutting improved the machined surface integrity to a large degree. As shown in Fig. 6 (b), the CDVA cutting (vibration in the cutting direction) accelerated the fibre fracture in the very close neighbourhood of the tool–fibre interacting zone, and therefore made the fibre–matrix

Fig. 8. Finished surfaces of CFRP composites machined by (a) traditional cutting (ap ¼100 μm, v¼ 1 m/min), and (b) EVA cutting (ap ¼100 μm, v ¼1 m/min, f ¼17.43 kHz, a¼ 2.07 μm, b ¼1.67 μm).

Fig. 9. Variation of the cutting forces and the maximum tensile stress in (a) traditional cutting (ap ¼30 μm, v¼ 1 m/min), (b) CDVA cutting (ap ¼ 30 μm, v ¼1 m/min, f¼ 17.43 kHz, a¼ 2.07 μm, b ¼ 0 μm), (c) NDVA cutting (ap ¼30 μm, v¼ 1 m/min, f¼ 17.43 kHz, a¼ 0 μm, b¼ 1.67 μm), and (d) EVA cutting (ap ¼ 30 μm, v ¼1 m/min, f ¼17.43 kHz, a¼ 2.07 μm, b ¼1.67 μm).

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Fig. 10. Influence of the depth of cut on the cutting force in (a) x-direction, and (b) z-direction (traditional cutting: v¼ 1 m/min; CDVA cutting: v¼ 1 m/min, f ¼17.43 kHz, a¼ 2.07 μm, b ¼0 μm; NDVA cutting: v¼ 1 m/min, f ¼17.43 kHz, a ¼0 μm, b¼ 1.67 μm; EVA cutting: v¼1 m/min, f ¼17.43 kHz, a¼ 2.07 μm, b ¼1.67 μm).

debonding zones much smaller. However, owing to the frequent reciprocation of the cutting tool along the cutting direction, fibres and matrix of the finished surface were debonded as well. Fig. 6 (c) demonstrates the surface quality by NDVA cutting (vibration being perpendicular to the cutting direction), which shows that the vibration application was effective in reducing the fibre–matrix debonding. In the NDVA cutting, however, the tool–fibre contact position varied instantly due to the reciprocating sliding of the cutting tool on the fibre in the vertical direction. As such, the breaking point on a fibre is unpredictable, leading to an irregular fractured surface. The best surface quality was produced by the EVA cutting, which not only made the surface much smoother, but also minimised the fibre–matrix debonding zone, as shown in Fig. 6 (d). These experimental results are in good agreement with the predictions from the model predictions presented above. Fig. 7 shows the influence of the depth of cut on the fibre– matrix debonding depth. It appears that the cutting tool vibration in either the cutting or normal direction can reduce the fibre– matrix debonding depth. As discussed in Section 2.2 on the mechanics of fibre deformation, the deflection of a fibre decreases from the tool–fibre contact position along the fibre into the subsurface depth, depending on the depth of cut. Specifically, with increasing the depth of cut till π/λ ( ¼27 μm), where 1/λ is the characteristic length of the fibre on the EHM foundation, the rate of the deflection decrease reduces. As a result, the increase of the depth of cut (till π/λ) brings about the lengthening of the debonding depth. In contrast, the tool vibration along z-direction makes the tool–fibre contact position vary instantly during a cutting process, and in turn alter the debonding depth. Fig. 8 shows the finished surfaces by traditional and EVA cutting under the depth of cut of 100 μm. Compared with the results under the depth of cut of 30 μm shown in Fig. 6, it can be seen that once the depth of cut reaches a critical value, its further increase does not influence the debonding depth.

force is used here for comparison. The results demonstrate that the traditional cutting requires the largest cutting force, followed by the NDVA and CDVA cutting processes. The EVA mode has the smallest. In the traditional cutting, the cutting forces in x-direction (Fsim  x) and the maximum tensile stress (szmax) increase with the feeding of the cutting tool until the fibre is fractured at t¼248 μs (Fig. 9(a)). During the process, two turning points are of particular interest: one corresponds to the first appearance of fibre–matrix debonding at t ¼248 μs where the variation pattern of the force in x-direction changed from exponential to linear; the other corresponds to the beginning of the shortening of the contact width l at t¼248 μs defined by Eq. (16), which brings about the drop of the cutting force in z-direction. However, the application of the vibration makes such pattern weakened. As the result of the intermittent contact between the cutting tool and the workpiece in the vibration direction, the corresponding cutting force and tensile stresses fluctuate accordingly. Nevertheless, because the vibration frequency is much higher than the natural frequency of the cutting system, the dynamic stiffness of the cutting tool is greatly improved compared with that in the traditional cutting [34–37] and thus the tensile stress increased rapidly in each cutting duration. This effect works prominently in EVA cutting, with the fibre fractured much earlier under a much smaller cutting force, as shown in Fig. 9(d). Fig. 10 compares the predicted results (lines) with the experimental measurements (scattered dots), when the depth of cut ap changes from 5 μm to 100 μm. It can be seen that the cutting forces in both the cutting and normal directions increase with the rise of the depth of cut. When the vibration is applied on the tool tips, the cutting forces become smaller, and the EVA cutting has the best performance. It can be seen that the model predictions are in excellent agreement with the experiments, indicating that the mechanics model established above has captured the major deformation mechanisms in cutting FRPs with and without ultrasonic vibration.

4.2. Cutting force Fig. 9 shows the variation of the cutting forces from both the model prediction (Fsim) and experiments (Fexp) under the same conditions, where the force values are the cutting force per unit width of cut. Due to the limitation of dynamometer Kistler 9256A1 (sampling frequency ¼100 kHz; natural frequency ¼5.5 kHz), the exact instant variation of the cutting force in a high frequency cycle (larger than 17.43 kHz) was not measurable; thus an average

5. Conclusions This paper has successfully established the mechanics of cutting unidirectional FRP composites with and without the vibration of a tool tip. The study has also revealed the major material removal mechanisms considering the effects on the fibre deformation, fibre fragmentation and fibre–matrix debonding.

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The capability of the model has been verified by experiments. The results from the simulation and experiments bring about the following major conclusions: (1) The application of tool vibration can significantly decrease the cutting forces, minimise fibre deformation, facilitate the fibre fracture at a location very close to the tool–fibre contact zone, reduce the penetration depth of fibre–matrix debonding, and hence provide a much better surface integrity of a finished surface. (2) The elliptic vibration trajectory of a tool tip (EVA mode) provides the best performance. This is because the vertical vibration always exerts an additional tensile stress to a fibre surface and the dynamic effect of the vibration component in the cutting direction at a high frequency further accelerates the fibre fracture while minimising the fibre deflection in the cutting process. (3) In general, increasing the depth of cut increases the fibre– matrix debonding depth in the subsurface of a FRP composite. However, there exists a critical depth of cut beyond which the debonding depth does not vary.

Acknowledgement The authors appreciate the Australian Research Council for its financial support to this work. This work was supported by an award under the Merit Allocation Scheme on the NCI National Facility. References [1] L.C. Zhang, H.J. Zhang, X.M. Wang, A force prediction model for cutting unidirectional fibre-reinforced plastics, Mach. Sci. Technol. 5 (2001) 293–305. [2] X.M. Wang, L.C. Zhang, An experimental investigation into the orthogonal cutting of unidirectional fibre reinforced plastics, Int. J. Mach. Tool. Manuf. 43 (2003) 1015–1022. [3] L.C. Zhang, Cutting composites: a discussion on mechanics modelling, J. Mater. Process Technol. 209 (2009) 4548–4552. [4] H.J. Zhang, W.Y. Chen, D.C. Chen, L.C. Zhang, Assessment of the exit defects in carbon fibre-reinforced plastic plates caused by drilling, Precis. Mach. Adv. Mater. 196 (2001) 43–52. [5] D.H. Wang, M. Ramulu, D. Arola, Orthogonal cutting mechanisms of graphite/ epoxy composite1. Unidirectional laminate, Int. J. Mach. Tool. Manuf. 35 (1995) 1623–1638. [6] W. Koenig, C. Wulf, P. Grass, H. Willerscheid, Machining of fibre reinforced plastics, CIRP Annals-Manufact. Technol. 34 (1985) 537–548. [7] R. Teti, Machining of composite materials, CIRP Annals-Manuf. Technol. 51 (2002) 611–634. [8] I.S. Shyha, D.K. Aspinwall, S.L. Soo, S. Bradley, Drill geometry and operating effects when cutting small diameter holes in CFRP, Int. J. Mach. Tool. Manuf. 49 (2009) 1008–1014. [9] N. Bhatnagar, N. Ramakrishnan, N.K. Naik, R. Komanduri, On the machining of fiber-reinforced plastic (Frp) composite laminates, Int. J. Mach. Tool. Manuf. 35 (1995) 701–716. [10] J.C. Rubio, A.M. Abrao, P.E. Faria, A.E. Correia, J.P. Davim, Effects of high speed in the drilling of glass fibre reinforced plastic: evaluation of the delamination factor, Int. J. Mach. Tool. Manu. 48 (2008) 715–720. [11] N.K. Jain, V.K. Jain, K. Deb, Optimization of process parameters of mechanical type advanced machining processes using genetic algorithms, Int. J. Mach. Tool. Manuf. 47 (2007) 900–919.

[12] V. Tagliaferri, G. Caprino, A. Diterlizzi, Effect of drilling parameters on the finish and mechanical-properties of Gfrp composites, Int. J. Mach. Tool. Manuf. 30 (1990) 77–84. [13] N.S. Hu, L.C. Zhang, Some observations in grinding unidirectional carbon fibrereinforced plastics, J. Mater. Process Technol. 152 (2004) 333–338. [14] N.S. Hu, L.C. Zhang, A study on the grindability of multidirectional carbon fibre-reinforced plastics, J. Mater. Process Technol. 140 (2003) 152–156. [15] S.L. Soo, I.S. Shyha, T. Barnett, D.K. Aspinwall, W.M. Sim, Grinding performance and workpiece integrity when superabrasive edge routing carbon fibre reinforced plastic (CFRP) composites, CIRP Annals-Manufact. Technol. 61 (2012) 295–298. [16] D. Brehl, B. Brocato, M. Cemaway, T. Dow, N. Negishi, A. Sohn, Micromachining results for ductile, brittle, and polymer materials using an elliptical vibrationassisted machining system, in: Proceedings of ASPE Spring Topical Meeting on Precision Micro/Nano Scale Polymer Based Component and Device Fabrication, North Carolina State University, United States 2005, pp. 83–88. [17] D.E. Brehl, T.A. Dow, 3-D microstructure creation using elliptical vibrationassisted machining, in: Proceedings of ASPE Spring Topical Meeting on Vibration Assisted Machining Technology, North Carolina State University, United States 2007, pp. 21–26. [18] D.E. Brehl, T.A. Dow, Review of vibration-assisted machining, Precis. Eng. 32 (2008) 153–172. [19] Z.Q. Liang, X.B. Wang, Y.B. Wu, L.J. Xie, L. Jiao, W.W. Zhao, Experimental study on brittle-ductile transition in elliptical ultrasonic assisted grinding (EUAG) of monocrystal sapphire using single diamond abrasive grain, Int. J. Mach. Tool. Manuf. 71 (2013) 41–51. [20] Y. Li, Y. Wu, L. Zhou, M. Fujimoto, Vibration-assisted dry polishing of fused silica using a fixed-abrasive polisher, Int. J. Mach. Tool. Manuf. 77 (2014) 93–102. [21] Y. Wang, B. Lin, S. Wang, X. Cao, Study on the system matching of ultrasonic vibration assisted grinding for hard and brittle materials processing, Int. J. Mach. Tool. Manuf. 77 (2014) 66–73. [22] L. Zhang, W. Xu, Vibration-Assisted Cutting System and Method, Australian Patents No. 2013900905, 2013. [23] W. Xu, L.C. Zhang, Y. Wu, Elliptic vibration-assisted cutting of fibre-reinforced polymer composites: understanding the material removal mechanisms, Compos. Sci. Technol. 92 (2014) 103–111. [24] W.X. Xu, L. Zhang, Y.B. Wu, Micromechanical modelling of elliptic vibrationassisted cutting of unidirectional FRP composites, Adv. Mater. Res. 591 (2012) 531–534. [25] L.C. Zhang, Z. Lin, An analytical solution to springback of sheet metals stamped by a rigid punch and an elastic die, J. Mater. Process Technol. 63 (1997) 49–54. [26] N. Cristescu, E.-M. Craciun, E. Soós, Mechanics of Elastic Composites, Chapman & Hall/CRC, Boca Raton, Fl., 2003. [27] M. Biot, Bending of an infinite beam on an elastic foundation, J. Appl. Mech. (1937) A1–A7. [28] M. Hetényi, Beams on elastic foundation; theory with applications in the fields of civil and mechanical engineery, University of Michigan Press, Ann Arbor, Michigan, United States, 1946. [29] X.Y. Tongxi, L. Zhang, Plastic Bending: Theory and Applications, World Scientific Publishing Co Pte Ltd, Singapore, 1996. [30] K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge Cambridgeshire; New York, 1985. [31] A.I. Luré, Three-Dimensional Problems of the Theory of Elasticity, Interscience Publishers, New York, 1964. [32] M.D. Bryant, L.M. Keer, Rough contact between elastically and geometrically identical curved bodies, J. Appl. Mech.-T Asme 49 (1982) 345–352. [33] L. Zhang, Solid Mechanics for Engineers, Palgrave, 2001. [34] V.K. Astashev, V.I. Babitsky, Ultrasonic cutting as a nonlinear (vibro-impact) process, Ultrasonics 36 (1998) 89–96. [35] J.L. Fu, Z.J. Yuan, Y.X. Yao, A unified system model of cutting chatter and its transformation function, Int. J. Mach. Tool. Manuf. 29 (1989) 601–609. [36] J. Kumabe, Vibratory Cutting, Dzikke Sjuppan, Tokyo, 1979. [37] C.X. Ma, E. Shamoto, T. Moriwaki, L.J. Wang, Study of machining accuracy in ultrasonic elliptical vibration cutting, Int. J. Mach. Tool. Manuf. 44 (2004) 1305–1310. [38] G.V.G. Rao, P. Mahajan, N. Bhatnagar, Machining of UD-GFRP composites chip formation mechanism, Compos. Sci. Technol. 67 (2007) 2271–2281. [39] G.V.G. Rao, P. Mahajan, N. Bhatnagar, Micro-mechanical modeling of machining of FRP composites – cutting force analysis, Compos. Sci. Technol. 67 (2007) 579–593. [40] D. Iliescu, D. Gehin, I. Iordanoff, F. Girot, M.E. Gutierrez, A discrete element method for the simulation of CFRP cutting, Compos. Sci. Technol. 70 (2010) 73–80.