A comprehensive analysis of cutting forces during routing of multilayer carbon fiber-reinforced polymer laminates

Abstract

Analyzing cutting forces during detouring of carbon fiber-reinforced polymer laminates at high cutting speeds is problematic as the recorded signal can be distorted due to resonance of the measuring system. In addition, excessive tool wear can render signal interpretation difficult. In the present study, a fully controlled experimental protocol is used to investigate the instantaneous cutting forces when milling carbon fiber-reinforced polymer laminates in a bid to avoid signal distortion and limit the tool wear effect. A polycrystalline diamond tool was selected for the experiments in order to limit the effect of tool wear on the recorded signals. The fiber orientation influences principally the cutting force amplitude, which varies nonlinearly with the feed. Based on this experimental data, a high-order mechanistic force model in terms of feed per tooth was proposed to predict the cutting forces. The tooth-to-tooth run-out was measured and included in the model, and the model was validated for different feeds, speeds, and number of plies. A good consistency between simulated and measured forces was observed. For the proposed model, the estimation error was approximately ±12.5%.

Keywords

Detouring carbon fiber-reinforced polymer multilayer composite materials polycrystalline diamond

Introduction

The machining process involves cutting forces that influence process stability, part quality, the cutting temperature, and tool conditions. While the cutting force modeling of isotropic materials may have become well established over the last decades Koenigsberger,¹ it is still only partially understood in the case of orthotropic materials such as carbon fiber-reinforced plastics (CFRP). The nonhomogeneity of CFRP directly affects the cutting forces, which can also be significantly modified by excessive tool wear. All these factors make it difficult to interpret cutting force signals. Nevertheless, since the 1980s, many researchers have carried out intensive work in this field, in an attempt to understand the cutting forces of CFRP materials.

Using high-speed-steel tools, Koplev² and Koplev et al.³ examined the milling of unidirectional carbon fiber–epoxy composite, perpendicular as well as parallel to the fiber orientation. He proved that the horizontal cutting forces (parallel to the fiber direction) are determined by the relief angle and tool wear. Konig et al.⁴ performed routing tests of unidirectional CFRP laminates, using diamond-coated carbide and polycrystalline diamond (PCD) tools. The authors demonstrated that the highest forces are achieved under compressive load of the fiber at an angle of 45°. The lowest cutting force occurred when the cut was parallel to the fiber orientation, since in this case, the matrix, rather than the fibers, was cut. Kaneeda and Masayuki⁵ also demonstrated that the milling forces are significantly affected by the fiber angle. The latter has a greater effect on the cutter tangential than on the cutter normal force. Unlike Konig et al.,⁴ Kaneeda and Masayuki⁵ found that the fiber angle between 15° and 45° offered the minimum cutting force for every rake angle during routing. Kaneeda et al.⁶ later studied the effect of the tool edge roundness and relief angle on the cutting forces of CFRP laminates and found that relief angle values did not have significant effects on the tangential force. Wang et al.⁷ studied the mechanisms of orthogonal milling of graphite/epoxy composites, with cutting speeds ranging from 4 to 14 m/min. The cutting forces when machining CFRP laminates with different fiber orientations were presented. The authors observed different fluctuations of the instantaneous force profile, which they attributed to the fiber orientations. Wang et al.⁷ found that the lowest resultant cutting force was obtained when trimming 0° material. Arola et al.⁸ used the same experimental setup as Wang et al.,⁷ and found that cutting forces varied with different tool rake and clearance angles. Colligan and Ramulu⁹ investigated the cutting forces when using diamond abrasive cutters for routing; they demonstrated that as a general rule, cutting forces increase with the material removal rate and found normal force to be generally about 60% of the thrust forces. It can be noted that most studies available until the end of the ninetie 1990s were interested in the observations of the CFRP machining processes. The initial analyses were accompanied by attempts to explain the phenomena occurring during the deformation mechanisms. These attempts allowed a thorough understanding of the CFRP machining processes and the development of the early models. Zhang et al.¹⁰ proposed an analytical model for the prediction of the average cutting forces in the orthogonal cutting of unidirectional CFRP when the fiber orientation varies from 0° to 90°. Rao et al.¹¹ simulated the cutting forces of orthogonal machining of unidirectional CFRP. The authors analyzed the cutting force for different fiber orientations, depths of cut, and tool rake angles. Rao et al.¹¹ proved that the cutting force amplitude mainly varies with fiber orientation and depth of cut. Lopèz de Lacalle et al.¹² carried out an extensive study on tool performance during trimming of CFRP laminates and proposed an exponential relationship of the radial and tangential cutting forces as a function of feed.

It can be seen that previous studies were mainly concerned with the average cutting forces. To understand the mechanisms at play during the trimming of CFRP composites, a more thorough investigation of the instantaneous cutting forces is needed. Over the last 5 years, some authors have looked at the modeling of the instantaneous cutting forces in order to provide greater insights into the cutting process. Sheikh-Ahmad et al.¹³ proposed a comprehensive model for cutting force modeling during the orthogonal machining of unidirectional laminates, which was based on the specific cutting energy coefficients identified though orthogonal cutting tests and the proposed model was validated for straight flute tools. More recently, Kalla et al.¹⁴ used the coefficients identified by Sheikh-Ahmad et al.¹³ to extend the model for non-null helix angles. They used the orthogonal-to-oblique transformation combined with a first-order mechanistic model to develop their cutting force model. As proven by Lopèz de Lacalle et al.,¹² the nonlinear variation of the cutting forces with the feed is obvious. It would thus appear that a high-order mechanistic model for assessing instantaneous cutting forces is adequate for modeling cutting forces when trimming CFRP.

In the present paper, a high-order mechanistic model was used to account for material nonlinearities. Many precautions were taken in the experimental protocol to eliminate all sources of nonlinearities, such as tool wear, dynamic instability, major processing defaults, and more.

This article is intended primarily to present a comprehensive analysis of the instantaneous cutting forces during the CFRP laminate trimming process. The proposed analysis was aimed at retrieving and eliminating sources of nonlinearities in cutting forces, such as tool wear, signal distortion, dynamic instability, and more. It is followed by the development of an empirical model for the cutting forces using a high mechanistic order model. To achieve these goals, the article is organized as follows: section Cutting tests methodology presents the experimental protocol. In section Cutting force analysis and modeling, the experimental results are first presented and analyzed, and then the cutting forces are modeled and the proposed model validated. This is followed by conclusions and prospective future works.

Cutting tests methodology

Experimental setup

The laminates for machining tests were prepared in a controlled aeronautical environment using preimpregnated materials. The stack, with a 64% fiber volume fraction was autoclave cured. The considered fiber orientation provided quasi-isotropic properties of the laminates. The layer orientation and laminate thickness are summarized in Table 1.

Table 1.

Properties and configuration of machined CFRP laminates

	Thickness (mm; in)	Volume fraction (%)	Process	Total number of plies	Average ply thickness (mm; in)	Ply orientation
P1	3.47 (0.1368)	64	Autoclave curing	24	0.144 (0.0057)	4/8/8/4
P2	4.63 (0.1824)	64	Autoclave curing	32	0.144 (0.0057)	6/10/10/6
P3	5.79 (0.228)	64	Autoclave curing	40	0.144 (0.0057)	8/12/12/8

CFRP: carbon fiber-reinforced polymer.

The presence of major processing defaults can constitute the first source of cutting force signal distortion. If major defaults are present, it will be difficult to interpret or model the cutting forces. Thus, prior to machining, the square-shaped laminates were inspected using ultrasound in order to verify the absence of major processing defaults. Figure 1 presents an example of the inspection result before the part is machined. This figure presents an inspected sample of each thickness.

Figure 1.

Ultrasound testing of carbon fiber-reinforced polymer (CFRP) laminates prior to machining. (a) thickness =3.47 mm (b) thickness =4.69 mm (c) thickness =5.79 mm.

It can be observed from Figure 1 that no major defects were present. It also shows that samples were free of porosity, which means that the processing of the laminates was of good quality and complied with rigorous aerospace industry standards for composites. This first control allows machining tests to be conducted without any doubts as to whether they can be affected by the presence of major processing defaults.

Once inspected, the laminates were predrilled for tightening on a machining fixture, as shown in Figure 2, where the experimental setup is presented. The predrilling was necessary both for screwing the laminate to the fixture and for facilitating a smooth entrance of the cutter in the laminate, when detouring each coupon using different cutting conditions.

Figure 2.

Experimental setup for dry trimming of carbon fiber-reinforced polymer (CFRP) laminates with different cutting parameters.

Figure 2 shows the experimental setup, where the CFRP laminate (#1) was screwed on an aluminum back plating system (#2) using 49 screws and a torque wrench. Detail A in Figure 2 shows a sample of a machined test coupon. Each side of the squared sample was machined under specific cutting conditions. Different combinations of cutting parameters were tested. The subassembly (laminate and back plate) was tightened to a three-axis dynamometer (#3) with four screws, using a torque wrench. In Figure 2, the item #3 represents a three-axis dynamometer-type Kistler 9255B, which was connected to the charge amplifiers-type Kistler 5010.The latter are not represented in Figure 2. The charge amplifiers generated five output signals that were transmitted to a data-acquisition card (type DT-9836). The sampling frequency was set to 48 kHz/channel, for a recording time of 16 s. An universal serial bus (USB) communication protocol was used to interpret the output signals of the data-acquisition card with a Matlab-based signal-processing program. The measurement system was calibrated statically and dynamically on an MTS universal testing machine. Both static and dynamic calibration allowed an estimation of drift errors and their correction, if they exceeded the acceptable limit of ±10 mN/s for the three directions. No filter was applied on the force signals in order to allow the detection of the slightest chatter signals. The cutting force signals were exported to Matlab for further analysis. All machining tests were realized under dry conditions using the same tool geometry, with the selected tool being a PCD end mill with two straight flutes having a 10° rake angle and a 5° clearance angle. In addition, for all cutting tests, the feed direction was considered parallel to the fibers oriented at 0°. The analysis performed on the force signals obtained for different cutting parameters and different ply numbers is presented in the following sections.

Measurement system validation

The use of PCD tools during the detouring operation implies high cutting speeds. The measurement of cutting forces at high speeds presented the first challenge to address in the present study. In fact, the passing bandwidth of the measurement system had to be identified adequately to ensure that the recorded cutting force signals were not distorted Astakhov.¹⁵ Thus, the dynamic response of the clamped dynamometer on the machine table was measured, and the dynamic characterization of the measurement system was performed for two different configurations:

The dynamometer is clamped on the machine-tool table and

The dynamometer is clamped on the machine-tool table and the fixture is tightened on the dynamometer.

Impact tests were performed for three measurement directions for the two cited configurations. The results obtained for the impact tests are presented in Figure 3.

Figure 3.

FRF of the fixed dynamometer on the machine tool table for two different configurations. (a) Imaginary part of the FRF in X direction (b) Imaginary part of the frequency response function (FRF) in Y direction (c) Imaginary part of the FRF in Z direction.

Figure 3(a) and (b) shows the superposition of the two frequency response functions (FRFs) with fixture and without fixture. It can be observed that at frequencies lower than 2 kHz, the superposition of the two FRFs (with and without fixture) is perfect. This may be due to the neglected effect of the fixture mass and clamping in the two directions, x and y. Some discrepancy is observed at frequency higher than 2 kHz for both direction x and y. This is most probably due to the four screws used for clamping the fixture. These screws act in both directions x and y and could change the dynamic response of the assembly. The dynamic response of the dynamometer in the z-direction is affected by the added aluminum plate (fixture). The added mass of the aluminum plate acts mainly in the z-direction, which may explain the modified FRF in the z-direction. The impact test of the fixed dynamometer plus the aluminum fixture demonstrates that the cutting force measurement setup has a passing bandwidth less than 1 kHz. To avoid signal distortion, the highest tooth passing frequency should therefore not be higher than 1 kHz. It was thus set at 900 Hz, which corresponds to a spindle speed of 450 Hz for a two-tooth cutter (27,000 r/min). It should be underlined that this precaution will ensure a nonresonance condition at the tooth-passing frequency. However, resonance may still occur at other harmonics of either spindle speed or tooth-passing frequency. The question was what could happen in the case where a natural frequency of the measurement system coincides with one of the harmonics? Such a case will be discussed in the following paragraph.

A sequence of cutting spindle speeds was selected, considering that they should be lower than 27,000 r/min, which corresponds to a cutting speed of 800 m/min for a 9.525 mm (3/8 inch) cutter. The following sequence of cutting speeds was proposed: 800, 650, 500, 400, 300, and 200 m/min. The harmonics of the spindle speeds and the tooth-passing frequency were superposed on the graphic of FRF in the x-direction. An example of superposition for the cutting speed of 500 m/min is shown in Figure 4.

Figure 4.

Chart of different spindle speeds (o symbols) and their harmonics superposed on the measuring system FRF in the x-direction (cutting speed 500 m/min).

It can be observed in Figure 4 that one of the measurement system natural frequencies in the x-direction (1671.6 Hz) coincides with the third harmonic of the tooth-passing frequency for the fourth cutting speed (500 m/min corresponds to a spindle speed of 278.6 Hz and a tooth-passing frequency of 557.2 Hz whose third harmonic is 1671.6 Hz). The cutting force obtained for this case is plotted in Figure 5.

Figure 5.

Cutting force fast Fourier transform (FFT) analysis demonstrates the presence of amplification at measuring system resonance. (a) Instantaneous cutting force (b) FFT of the cutting force singal.

The instantaneous cutting force in the x-direction and its fast Fourier transform (FFT) are presented in Figure 5(a) and (b), respectively. It can be observed that the normalized amplitude of the third harmonic of the tooth-passing frequency (1671.6 Hz) is relatively higher than the second harmonic (1114.6 Hz). This amplification increase in the amplitude is most probably due to the coincidence of this harmonic with the natural frequency of the measurement system in the x-direction. The same analysis was repeated for different spindle speeds for the three directions, x, y, and z, and showed that the signals, for which there exists a coincidence between a resonance of the measurement system and one of the spindle speed harmonics, were not completely distorted. The distortion can be noted by the presence of energy concentration at a single frequency or of a high-amplitude modulation. While such phenomena may have been absent for the tested conditions, the amplification of one harmonic (as shown in Figure 5) was repeatedly observed, times especially for spindle speeds higher than 13,500 r/min. The amplification observed can cause an amplitude increase of the cutting force signal and probably a profile alteration. Analysis and modeling of cutting forces are difficult in the presence of such amplification, and so, using a conservative approach, the cutting speed sequence was limited to a maximum of 13,500 r/min. All recorded signals were verified for an absence of distortion through FFT analysis. The cutting force analysis is detailed in the following section.

Cutting force analysis and modeling

Cutting force analysis

The proposed experimental setup was designed to simulate a detouring operation during the finishing of composite parts. The analysis of cutting forces during such an operation is helpful for understanding causes of possible damage. The recorded cutting forces in x-, y-, and z-directions are plotted on the same graph as shown in Figure 6.

Figure 6.

Typical cutting force signal using a polycrystalline diamond (PCD) cutter (two teeth, β = 0°, α = 5°, γ = 10°). Machined part is a carbon fiber-reinforced polymer (CFRP) laminate, 24-ply 4/8/8/4.

Figure 6 sorts out the similarity of the cutting forces for the three directions. This similarity indicates the presence of three different zones in the cutting force signals. To more clearly indicate the different zones, the component in the x-direction was enlarged and presented in Figure 7.

Figure 7.

The correspondence between the cutter position in the laminate and the recorded cutting force in x-direction.

Figure 7 shows the different positions of the cutter and the cutting force signal in the x-direction. The first period (Zone I) corresponds to the engagement of the tool in the part. This zone is characterized by a partial engagement of the cutter in the part. The cutter engages in the work piece, bringing a high amount of energy to the machined part and a large part of this energy is instantaneously dissipated in the part, the rest is dissipated through the deformation of the machining system and the heat dissipated in the tool and the environment. During the steady state, the cutter advances smoothly in the part. Once the cutter is fully engaged, the cutting forces reach the second zone (Zone II), the steady state period, which is characterized by a repetitive cutting force profile from one revolution to another and by a nearly constant force amplitude.

In the last zone (Zone III), the cutting forces start to fluctuate and reach their peak values. This is the period when the cutter leaves the part, and here, the decreasing thickness of the machined section leads to a high vibration level, which causes an increase in the cutting force level. Once the thin nonsupported final part of the plate is removed, the cutting force starts to decrease slightly. This is most probably due to the decrease of chip thickness. The cutting force ultimately decreases to zero when there is no material left to remove.

The modeling of cutting forces is generally considered in the steady state period. Thus, observation was focused on the variation of the cutting forces when the tool is fully engaged. It can be seen that the cutting forces exhibit no progressive increase during the steady state period. The progressive cutting force increase is a symptom of tool wear, as confirmed by the results presented in Figure 8. The plotted cutting force signal presented in Figure 8 was obtained during the full slotting of CFRP laminate using a carbide cutter having the same geometry as the PCD cutter.

Figure 8.

The influence of tool wear on the cutting forces when detouring carbon fiber-reinforced polymer (CFRP) laminates (24-ply 4/8/8/4) using a carbide cutter (two teeth, β = 0°, α = 5°, γ = 10°).

Between the beginning and the end of the operation, the recorded cutting force signal presented in Figure 8 shows an increase of an order of magnitude of 300%. A lengthy recording for more than 30 s was considered in order to clearly show the effect of tool wear on the profile of the cutting force signal. The presence of excessive tool wear can be detected by an increase in cutting force amplitude^16–17, during the steady state period, which is not the case for recorded signals when PCD cutters were used.

The absence of tool wear for the PCD cutter presents an advantage for the modeling of cutting forces. Nevertheless, other classic tooling problems could be present, especially tool run-out. To verify the significant presence of cutter run-out, a zoom on the steady state period of cutting force was realized and presented in Figure 9.

Figure 9.

Cutting force profile over four cutter revolutions when milling a 24-ply laminate with polycrystalline diamond (PCD) cutter (two teeth, β = 0°, α = 5°, γ = 10°) at 300 m/min and 0.0508 mm/tooth.

Figure 9 presents the cutting force in the x-, y-, and z-directions during four revolutions of the cutter. The force profile for each passing tooth of the cutting tool can clearly be identified in Figure 9. Each peak of the force profile presents the passage of a tooth. For a given direction (x-direction, for example), the cutting force starts with a quasi-null value when the tool is engaged (0°), and then increases until it reaches a maximum then decreases until the tooth separates from the work piece, leaving room for the second tooth. The second tooth follows a similar profile as the first. The maximum force reached by tooth number 2 is slightly lower than that reached by tooth number 1, with the difference being approximately 13% for the x-direction and 15% for the y-direction. The difference observed is most probably due to tool run-out. The tool run-out was measured and included in the modeling of the instantaneous cutting force.¹⁸

Cutting force modeling

The radial cutting force (F_r) and the tangential cutting force (F_t) were calculated from the cutting force in the feed direction (F_x) and the cutting force in the normal direction to the feed (F_y), using the following transformation:

(1)

where w is the spindle speed in rd/s and t is the time in seconds. The cutting force amplitude is variable and directly dependent on the instantaneous chip thickness (h), which is calculated using the simplified form of the chip thickness equation:¹⁹

h (j) = {\begin{matrix} f \cdot | \sin (φ) | + R (j), if φ_{start} \leq φ \leq φ_{exit} 0 if not \end{matrix}

(2)

where f is the feed per tooth (mm/tooth) and ϕ is the rotation angle of the cutting tool, which ranges between the starting angle (ϕ_start) and the exit angle (ϕ_exit). For a full slotting operation, ϕ_start and ϕ_exit are 0 and π, respectively. R(j) is tooth-to-tooth run-out of the jth tooth of the cutter.

The differential radial, tangential, and axial cutting forces (noted respectively, dF_t, dF_r, and dF_a) acting on the cutter can be derived from the equilibrium diagram of the cutting forces:

(3)

where ϕ is the instantaneous immersion angle. It should be noted that there may be more than one tooth cutting simultaneously. The maximum number of teeth cutting simultaneously can be estimated using the following equation:¹

(4)

in which m is raised to its nearest whole number, and where:

ϕ _t = angle between two consecutive teeth of the cutter = 2π/N;

N = total number of teeth in the cutter;

ϕ _b = effective angle between the leading and trailing edges;

ϕ _s = tooth engagement angle; and

R = radius of the cutter.

For the present study, the number of teeth cutting simultaneously is always one (m = 1). The elementary cutting forces acting on the cutter in the radial and tangential directions are expressed using the mechanistic equations as follows:

{\begin{matrix} d F_{t} = [K_{tc} \cdot h + K_{te}] \cdot d z d F_{r} = [K_{rc} \cdot h + K_{re}] \cdot d z \end{matrix}

(5)

where dF_t and dF_r are the differential tangential and radial forces (N), dz is the differential axial depth of cut (mm), and h is the instantaneous chip thickness (mm). K_te and K_re are the tangential and the radial edge force coefficients (N/mm), and K_tc and K_rc are the tangential and radial shearing coefficients (N/mm²). The edge and shearing coefficients can be extracted using different methods: material mechanical properties, orthogonal cutting tests, or average cutting forces, with the most popular being the average cutting force extraction method. This method is based on the experimental estimation of the average cutting force. Many authors have mentioned the dependency of average cutting forces on feed and speed.¹⁷ For isotropic materials, the average cutting forces vary linearly with the feed. Figure 10 illustrates such a variation when milling an aluminum alloy (AA6061-T6) using an uncoated carbide tool. It can be observed that for different cutting speeds, the correlation coefficients exceed 95% for x- and y-directions, while they range between 75 and 98% for the axial direction.

Figure 10.

Illustration of the linear variation of average cutting forces for an isotropic material (AA6061-T6, thickness = 3 mm) using a bright carbide cutter (β = 0°, α = 5°, γ = 10°) under different cutting speeds. (a) V_c = 50 m/min (b) V_c = 75 m/min (c) V_c = 100 m/min.

The particularity of composite materials resides in the nonlinear variation of the average cutting forces with the feed. To illustrate this variation, several tests were conducted on a CFRP multilayer under different feeds and speeds. Generally, the average cutting forces are estimated over two revolutions²⁰ in the steady state period. This approach is suitable for isotropic materials where the variation of the cutting forces from one revolution to the other is negligible. For the case of nonisotropic materials, it is more suitable to consider a higher revolution number. For the present study, 10 revolutions were considered in the steady state zone to estimate the average cutting forces. Considering a high revolution number permits also to determine the corresponding bar errors as illustrated in Figure 11. The bar errors in Figure 11 were estimated at 95% confidence level.

Figure 11.

Variation of the average cutting forces with feed and speed, polycrystalline diamond (PCD) cutter (two teeth, β = 0°, γ = 10°, α = 5°), 24-ply carbon fiber-reinforced polymer (CFRP) laminates 4/8/8/4. (a) V_c = 200 m/min (b) V_c = 300 m/min.

The trend of the average cutting forces presented in Figure 11 was obtained using a polynomial curve fitting. From Figure 11, it can be observed that:

The average cutting forces vary nonlinearly with the feed per tooth.

When the cutting speed is increased, the nonlinear form is preserved (compare subplots in the column a to those in the columns b or c).

The feed has a more significant effect on the average cutting forces than does the cutting speed.

The average cutting force depends on the cutting speed. To illustrate the effect of high cutting speeds on cutting forces, some machining tests were conducted at 500, 650, and 800 m/min, and average cutting forces collected. Figure 12 summarizes the results found.

Figure 12.

Cutting speed effect on average cutting forces. (a) Cutting speed effect on average cutting force F_x (b) Cutting speed effect on average cutting force F_y.

It can be observed that at high cutting speeds, the average of the cutting forces is affected.

The effect of the cutting speed increase is obvious on the y component of the cutting forces. For the F_x component, the effect of the cutting speed is not very significant. In fact, if the cutting speed is quadrupled (from 200 to 800 m/min), the average cutting force increase is less than 5% for a range of feeds between 0.05 and 0.254 mm/tooth. This slight increase may be due to an increase in the abrasive effect induced by the augmented speed. It is known that for isotropic materials, the dynamic friction coefficient increases with increased cutting speed²¹ up to a certain limit, and then it starts decreasing (for example, threshold sliding speed is around 13 m/s for ordinary steel). In the author’s opinion, this phenomenon may occur when CFRP are machined. In the present case, the threshold of the sliding speed to have a decreasing friction coefficient is approximately equal to 650 m/min. This is because there is no significant difference in the average of cutting forces in x direction at 500 m/min and 650 m/min. The observed decreasing of the average cutting forces in x direction with the increased speed may also be due to the thermal softening of high strain rate. On the other hand, the F_y component average increases significantly with the speed. This is most probably due to three different major causes, namely, abrasive effects, shearing strain rate effects, and thermal effects. Abrasive effects can cause a slight rise in cutting forces, but the two other effects (shearing strain rate effects and thermal effects) can mask this increase. Another issue that can reduce the impact of the abrasiveness, and the other phenomena is the symmetry aspect of the profiles. In fact, the nearenti symmetry of the F_x component profile during the slotting process results in a relatively low amplitude which can reduce the effect of the abrasiveness, thermal softening, and the strain rate. In contrast, the nonsymmetry of the F_y component will result in more pronounced effect of abrasiveness, thermal softening, and the strain rate.

While the influence of the cutting speed on the shearing strain rate is well known for isotropic materials,²² to the authors’ knowledge, this effect is not well understood for composite materials. The third effect can be related to the thermal properties of the matrix and the fibers. Carbon fibers have a high conductivity, which facilitates heat flow from the tool interface to the vicinity of the surrounding epoxy matrix. This matrix has a very low transition temperature (around 80°C on average), which can cause it to easily soften, leading to reduced cutting forces. It was observed that for high feeds (higher than 0.2 mm/tooth), the increase in cutting forces led to increased cutting forces in the y-direction. The author believes that this may be explained by the relatively high quantity of materials removed, which can limit the effect of thermal softening. It is much easier to soften low quantities of CFRP with the generated heat due to friction than a high quantity of material when the feed per tooth is increased. This observation may explain the known rule of the thumb when machining CFRP: high cutting speed and low feed.

Using previous observations, it can be concluded that the average cutting force is a result of many effects during the machining process. These effects are:

Resistance to abrasion, which is proportional to the quantity of material removed per tooth.

Resistance to shearing, which is proportional to the surface of the removed material. The latter is proportional to the square of the feed per tooth.

Resistance to penetration, which is divided into two components, one static and the other, dynamic. The dynamic resistance to penetration is directly proportional to the feed, while the static resistance to penetration should be a constant that depends on the material and the geometry of the penetrant (in our case, the penetrant geometry is the cutter geometry).

The above properties are speed-dependent, and so, for a given cutting speed, the average cutting forces can be modeled using a third-degree polynomial equation:

{\bar{F}}_{q} = {\bar{F}}_{qc 1} f^{3} + {\bar{F}}_{qc 2} f^{2} + {\bar{F}}_{qc 1} f + {\bar{F}}_{qe}, where q = x, y or z

(6)

In this equation, each coefficient can be interpreted as one of the resistances presented above. Equation (6) generates a satisfactory correlation coefficient (over 95% for x-direction and around 75% for y-direction), which means that this relationship can be used to identify the edge and the shearing coefficients, with an estimated risk error lower than 25%.

Using the analytical expression of the average cutting force extracted from equation (6), it is possible to identify the polynomial-tangential-edge-force coefficients (K_tei, i = 1, 2, 3), the polynomial-radial-edge-force coefficients (K_rei, i = 1, 2, 3), the polynomial-tangential-edge shearing coefficients (K_tci, i = 1, 2, 3), and the polynomial-radial-edge-shearing coefficients (K_rci, i = 1, 2, 3).

{\begin{matrix} {\bar{F}}_{xe} = \frac{Na}{π} K_{re} {\bar{F}}_{xc 1} = - \frac{K_{rc 1} \cdot a \cdot N}{4} {\bar{F}}_{xc 2} = - \frac{K_{rc 2} \cdot a \cdot N}{4} {\bar{F}}_{xc 3} = - \frac{K_{rc 3} \cdot a \cdot N}{4} {\bar{F}}_{ye} = \frac{Na}{π} K_{re} {\bar{F}}_{yc 1} = \frac{K_{tc 1} \cdot a \cdot N}{4} {\bar{F}}_{yc 2} = \frac{K_{rc 2} \cdot a \cdot N}{4} {\bar{F}}_{yc 3} = \frac{K_{rc 3} \cdot a \cdot N}{4} {\bar{F}}_{ze} = \frac{N \cdot a}{2} K_{ae} {\bar{F}}_{ac 1} = \frac{K_{ac 1} \cdot a \cdot N}{π} {\bar{F}}_{ac 2} = \frac{K_{ac 2} \cdot a \cdot N}{π} {\bar{F}}_{ac 3} = \frac{K_{ac 3} \cdot a \cdot N}{π} \end{matrix}

(7)

In this equation, N denotes the tooth number of the tool, while a denotes the axial depth of cut (a = the laminate thickness in the present case). Detailed calculations leading to equation (7) are presented in Annex I.

It should be underlined that the coefficients are estimated separately for each cutting speed. Five feeds were used to identify these coefficients, with a sixth used to validate the model. First, the model was tested to predict the cutting forces for a 24-ply laminate (thickness 3.47 mm). Two cutting speeds, 200 m/min and 400 m/min, were tested at a feed of 0.2032 mm/tooth. It was found that the high mechanistic order simulation method is adequate for CFRP materials. In fact, the results obtained for the model validation show a good consistency, as presented in Figure 13 for the x- and y-directions.

Figure 13.

Validation for the high-order mechanistic model for carbon fiber-reinforced polymer (CFRP) 24-ply laminate. (a) F_x' 200 m/min, 0.2332 mm/tooth, PCD cutter, 24 plies, 4/8/8/4 (b) F_y' 200 m/min, 0.2332 mm/tooth, PCD cutter, 24 plies, 4/8/8/4 (c) F_z' 200 m/min, 0.2332 mm/tooth, PCD cutter, 24 plies, 4/8/8/4.

From Figure 13, it can be observed that the axial cutting force is negligible compared to the x and y components. Thus, the radial and tangential cutting forces were calculated and presented in Figure 14 without considering the axial component of the cutting forces.

Figure 14.

Validation for the high-order mechanistic model for carbon fiber-reinforced polymer (CFRP) 24-ply laminate (radial and tangential forces). (a) F_r' 200 m/min, 0.2332 mm/tooth, PCD cutter, 24 plies, 4/8/8/4 (b) F_t' 200 m/min, 0.2332 mm/tooth, PCD cutter, 24 plies, 4/8/8/4.

The results presented in Figure 14 were confirmed for another cutting speed of 400 m/min. The simulated and measured results were superposed on the same plot as shown in Figure 15.

Figure 15.

Comparison of measured and simulated forces at a cutting speed of 400 m/min. (a) F_r' 400 m/min, 0.2032 mm/tooth, 24 plies, 4/8/8/4 (b) F_t' 400 m/min, 0.2032 mm/tooth, 24 plies, 4/8/8/4.

Using the results obtained in Figure 15, it was possible to predict the influence of the number of plies (depths of cut) on the cutting forces. The new tested laminate has 32 plies, configured as follows: 6/10/10/6.

When comparing the results obtained in Figures 15 and 16, it can be observed that the cutting forces have similar profiles but different amplitudes. This means that the cutting force profile does not depend significantly on the fiber orientation, which affects only the cutting force amplitude. This may be explained by the following circumstances:

The inherent symmetry of the slotting process.

The inherent symmetry of the laminates.

The proportionality between the number of layers and their orientation.

Figure 16.

Prediction of cutting forces when the number of plies was increased to 32, configured as: 6/10/10/6 (a) F_x' 400 m/min, 0.2032 mm/tooth, PCD cutter, 32 plies, 6/10/10/6 (b) F_t' 400 m/min, 0.2032 mm/tooth, PCD cutter, 32 plies, 6/10/10/6.

This masked effect of fiber orientation was encountered in a third test using a thicker laminate with 40 plies, configured as 8/12/12/8. A good agreement between simulated and measured cutting forces is shown in Figure 17.

Figure 17.

Prediction of cutting forces when the number of plies was increased to 40, configured as 8/12/12/8. (a) F_r' 400 m/min, 0.2032 mm/tooth, 40 plies, 8/12/12/8 (b) F_t' 400 m/min, 0.2032 mm/tooth, 40 plies, 8/12/12/8 .

The normalized error estimation for the tangential and radial direction can be approximated using the following equation:

(8)

Where n denotes the number of samples in the considered portion of the cutting force signal and the subscript i indicates the ith sample of the signal. For example, the estimation errors (for the results presented in Figure 17) were ±7.4% for the radial direction and ±12.2% for the tangential direction.

Conclusions

The analysis of instantaneous cutting forces led to the conclusion that nonisotropic material properties cause a nonlinear variation of average cutting forces. The estimated average forces were useful for the determination of the average cutting force coefficients, and the latter were utilized in building a predictive model for the cutting forces for multilayer laminated CFRP. The modeling of instantaneous cutting forces was performed through a high-order mechanistic model. The model developed allowed the prediction of the cutting forces for different feeds and speeds. It was shown that the cutting force profile does not depend significantly on the fiber orientation. The validation of the proposed model showed a good consistency between the simulated and the measured forces for different speeds and feeds. In addition, when the number of plies in the stack was increased from 24 to 32, and then to 40, the model predicted the cutting forces well. The simulated results were accurate to ±12.5%, which is an acceptable result for nonisotropic materials. The model accuracy can be improved through a more accurate equation for the average cutting forces, especially for the normal direction of feed (y-direction). The estimated cutting forces can be used to predict possible damage occurring during a composite laminate detouring operation. One of the limitations of the model presented is that even if it were to be successfully applied, it would not be fully predictive. In fact, experimental tests are mandatory in order to estimate the cutting force average coefficients. A fully predictive model needs to be based on material properties, cutting conditions, and tool geometry.

Footnotes

Funding

This study was performed as part of a large collaborative research project on “Optimization of the machining processes of graphite/epoxy composites and multilayer materials.” This project was funded jointly by the National Research Council of Canada, the Quebec Research Consortium for Innovation in Aerospace (CRIAQ), MITACS Accelerate, and the participating industrial partners which are Bombardier Aerospace, Avior Integrated Products, AV&R Vision&Robotics, Delastek, and Minicut International.

Acknowledgments

We sincerely thank the Industrial Materials Institute (IMI) of the National Research Council of Canada and Mr. Harold Hébert for its scientific and technical assistance regarding the inspection of laminates using the C-scan technology.

Conflict of interest

The authors declare that they do not have any conflict of interest.

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