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Abstract

A numerical-analytical approach was utilized to construct a predictive model of cutting force for machining unidirectional carbon fiber reinforced polymer (UD-CFRP) laminates. The force coefficients in the model, which include friction angle, shear plane angle, shear strength, and rebound height, can be characterized by the fiber orientations ranging from 0° to 180°. The accuracy of the model was confirmed through experimental verification. The results indicate good agreement between the predicted and experimental values, with relative errors below 14.8%, except for the thrust force at 90°. Additionally, the study examined the influence of rake angle and flank angle on the cutting force, revealing two critical points in the predicted cutting force curve throughout the fiber orientation. These turning points shifted with changes in the rake angle. For instance, the value of the first critical point changes from 60° to 45° when the rake angle range shifts from [0°, 5°] to [10°, 15°]. This indicates that a larger rake angle facilitated an earlier transformation of chip formation mode, leading to a decrease in cutting force. Furthermore, the cutting force decreased as the rake angle increased between the two turning points. The impact of the flank angle on the cutting force was determined to be minimal, and the turning points’ positions remained consistent as the flank angle increased.

Keywords

Unidirectional carbon fiber reinforced polymer force coefficient fiber orientation cutting force numerical-analytical approach

Introduction

Carbon fiber reinforced polymer (CFRP) composites are widely used in the aerospace industry as primary structural components because of their superior properties. While CFRP parts are often fabricated to near-net shape, post-machining is necessary to ensure that the components meet assembly requirements. The material removal mechanism of CFRP is different from that of metal due to its anisotropy and non-homogeneity, which is largely influenced by the fiber orientation angle.^1–3 Many scholars have made substantial efforts to understand the material removal mechanism in CFRP machining, including studying chip formation, visualizing machining damage, and predicting cutting force using models.^4–6

Numerous studies have been conducted to explore the cutting process of unidirectional CFRP (UD-CFRP) using finite element methods and analytical modeling. Three main finite element cutting models have been proposed to investigate the variation of cutting force and material failure mode, i.e., the microscopic model,^7–9 macroscopic model,^10–12 and macro-micro model.^13–15 These models have shown acceptable predictions of the principal cutting force, but the thrust force still needs improvement. Additionally, responses such as temperature distribution, delamination, surface defects, and subsurface damage have been investigated at various fiber orientations. Analytical methods have also been used to develop the quantitative relationship between the cutting forces and the process parameters in orthogonal cutting of UD-CFRP, including macro-mechanical and micro-mechanical models.^16–18 These existing analytical models focus on force prediction by pre-defining the chip formation modes from experimental observations at different fiber orientation ranges, and the macroscopic and microscopic properties of CFRP, cutting edge rounding and spring back effect of the fibers along the flank face were taken into account.^19–22 However, some of empirical parameters need to be determined for the micro-scale cutting force model.^23–27 Despite significant improvements in the process analysis of cutting CFRP, the cutting mechanics in relation to cutting conditions remain complex and require further investigation.

This paper presents a predictive model of cutting force for machining UD-CFRP using a numerical-analytical method. The force coefficients associated with the cutting force model are determined through function modeling using the available data from orthogonal cutting experiments and simulations. The relationship between the force coefficients and fiber orientation is discussed to enhance understanding of their impact on the cutting force. Experimental measurements of the cutting force at different fiber orientations are performed, and the influence of rake angle and flank angle on the cutting force is further investigated using the proposed model.

Analytical modelling of UD-CFRP

To discuss the cutting forces correlated to the friction behavior and the spring back effect, a simplified cutting model is proposed based on Zhang’s model.¹⁸ There are two deformation regions considered in the machining of UD-CFRP, i.e., the chip region and the rebound region, as shown in Figure 1. The interacting force between the rake face of the tool and the chip in the chip region is taken as the rakeface force, which is represented by $F_{R}$ . The component forces that correlate to the rakeface force $F_{R}$ are given by

F_{c 1} = F_{R} \cos (β - α)

(1)

F_{t 1} = F_{R} \sin (β - α)

(2)

where

F_{c 1}

is the component of the rakeface force in the horizontal direction,

F_{t 1}

is the component of the rakeface force in the vertical direction,

β

is the friction angle acting on the rakeface and

α

is the rake angle of the tool. The compressive force generated by the spring back of CFRP is called the flankface force

F_{N}

in the rebound region, which acts in the normal direction to the machined surface. It can be seen from Figure 1 that the friction force

F_{f}

parallel to the clearance surface is produced by the flankface force. The friction force is determined by the expression of

μ F_{N} \cos γ

with the friction coefficient

μ

and flank angle

γ

. The component forces in the rebound region are then described by

F_{c 2} = μ F_{N} \cos^{2} γ

(3)

F_{t 2} = F_{N} (1 + μ \sin γ \cos γ)

(4)

where

F_{c 2}

is the horizontal component of the flankface force,

F_{t 2}

is the vertical component of the flankface force acts on the flank surface.

Figure 1.

Characteristic of component forces in the analytical cutting model.

Therefore, the total forces of $F_{c}$ and $F_{t}$ can be calculated by summing the component forces from these two deformation regions, as follows:

F_{c} = F_{c 1} + F_{c 2} = F_{R} \cos (β - α) + μ F_{N} \cos^{2} γ

(5)

F_{t} = F_{t 1} + F_{t 2} = F_{R} \sin (β - α) + F_{N} (1 + μ \sin γ \cos γ)

(6)

where

F_{c}

is the principal cutting force acts along the horizontal direction,

F_{t}

is the thrust force acts in the nominal direction of the machined workpiece. By using the inverse solution of equations (5) and (6), the rakeface force

F_{R}

and the flankface force

F_{N}

can be obtained by

F_{R} = \frac{F_{c} (1 + μ \sin γ \cos γ) - μ F_{t} \cos^{2} γ}{\cos (β - α) (1 + μ \sin γ \cos γ) - μ \cos^{2} γ \sin (β - α)}

(7)

F_{N} = \frac{F_{c} \sin (β - α) - F_{t} \cos (β - α)}{μ \cos^{2} γ \sin (β - α) - (1 + μ \sin γ \cos γ) \cos (β - α)}

(8)

On the other hand, the rakeface force $F_{R}$ can be further resolved into a component force in the shear plane, which is given by

F_{s} = F_{R} \cos (ϕ + β - α)

(9)

where

F_{s}

is the shear force,

ϕ

is the shear plane angle, which acts as the force coefficient, needs to be determined. The shear force can also be calculated by multiplying the area of the shear plane by the shear strength of the CFRP material, as follows:

F_{s} = τ (θ) A_{s} = τ (θ) \frac{A_{c}}{\sin ϕ} = τ (θ) \frac{a_{c} a_{w}}{\sin ϕ}

(10)

where

τ (θ)

is the shear strength of UD-CFRP, which can be considered as a coefficient and correlated with the fiber orientation angle

θ

A_{s}

is the area of the shear plane,

A_{c}

is the area of the undeformed chip,

a_{c}

and

a_{w}

are the depth of cut and the thickness of cut, respectively. Furthermore, the flankface force

F_{N}

can be calculated by

F_{N} = \frac{1}{2} δ (θ) a_{w} E^{*}

(11)

where

δ (θ)

is the rebound height which correlated with the fiber orientation,

E^{*}

is the effective elastic modulus for the rebound region of the workpiece material.¹⁸ This is defined by the equation of

\frac{1}{E^{*}} = \frac{1 - υ_{t}^{2}}{E_{t}} + \frac{1 - υ_{c}^{2}}{E_{c}}

(12)

where

υ_{t}

and

E_{t}

are Poisson’s ratio and elastic modulus of the cutting tool,

υ_{c}

and

E_{c}

are Poisson’s ratio and transverse elastic modulus of UD-CFRP.

Simulation modeling of UD-CFRP

Description of 3D micromechanical cutting model

There is limited research on the experimental measurement of the shear plane due to the difficulty in obtaining the integrity of the machined surface.^6,28 In this study, the shear plane angle $ϕ$ was determined through simulation cutting of UD-CFRP with four different fiber orientations. The simulations were conducted using the software ABAQUS/Explicit. The 3D micromechanical FE model with consideration of the three phases (i.e., fiber, matrix and the interphase between the fiber and matrix) of CFRP composites, as shown in Figure 2. The fiber diameter of 7 μm in the composite workpiece is arranged in a hexagonal array to simulate the impact of random distribution, and the interphase thickness is considered to be 0.5 μm based on experimental measurement.²⁵ The tool is considered a rigid body with rake and flank angles of 20° and 10°, respectively, and an edge radius of 5 μm. The tool’s constant cutting velocity is set along the y direction during the simulation. The 8-node brick element with reduced integration (C3D8R) is adopted for the model with a size of $215 \times 215 \times 50$ μm, which includes approximately 100,000 elements. All the degrees of freedom in the x, y, and z directions are constrained at the bottom surface, and the displacement of U_y is also restricted at the left side surface in the model. The fiber orientation angle $θ$ is defined relative to the cutting path in a clockwise direction. The kinematic contact definition was applied between the tool surface and workpiece using the penalty-function principle, and the self-contact condition was adopted for each phase of the composites to prevent penetration among adjacent elements during the cutting simulation. The coefficient of friction μ is taken as 1.76, 1.31, 0.8, and 1.16 for the 0°, 45°, 90°, and 135° fiber orientations, respectively, by using the formula of μ=(F_t+F_ctanα)/(F_c-F_ttanα). The principal cutting force F_c, the thrust force F_t, and the rake angle α can be measured in the experiments (see Section on “Experiment verification”). The mechanical properties of the fiber, matrix, and interphase are summarized in Table 1, in which the properties of the fiber and matrix were obtained from the supplier’s specification sheet, while the interfacial properties were derived from the model developed in previous work.²

Figure 2.

Scheme of 3D finite element cutting model for a 90° fiber orientation.

Table 1.

Properties of the constituents used in the cutting simulation of CFRP.

Constituents	Properties	Values
Carbon fiber	Longitudinal elastic modulus, $E_{f 1}$ (GPa)	294
	Transverse elastic modulus, $E_{f 2}$ (GPa)	15
	Poisson’s ratio, $υ_{f 12}$	0.3
	Tensile strength, $X_{f}$ (GPa)	5.49
	Shear strength, $S_{f}$ (GPa)	0.38
	Diameter, $d_{f}$ (μm)	7
	Fiber volume fraction, $V_{f}$ (%)	65
	Density, $ρ_{f}$ (g/cm³)	1.81
Epoxy	Elastic modulus, $E_{m}$ (GPa)	4
	Poisson’s ratio, $υ_{m}$	0.4
	Tensile strength, $X_{m}$ (MPa)	85
	Shear strength, $S_{m}$ (MPa)	70
	Yield strength, $σ_{m}^{y}$ (MPa)	27
	Fracture energy, $G_{m}^{f}$ (N/mm²)	0.01
	Density, $ρ_{m}$ (g/cm³)	1.26
Interphase	Longitudinal elastic modulus, $E_{i 1}$ (GPa)	83.9
	Transverse elastic modulus, $E_{i 2}$ (GPa)	7.03
	Poisson’s ratio, $υ_{i 12}$	0.37
	Tensile strength, $X_{i}$ (MPa)	167.5
	Shear strength, $S_{i}$ (MPa)	25
	Thickness, $t_{i}$ (μm)	0.5
	Density, $ρ_{i}$ (g/cm³)	1.41
	Fracture energy, $G_{i}^{f}$ (N/mm²)	0.05

Failure model of fiber and matrix

The carbon fiber is considered an anisotropic material, and $E_{f}$ is used to represent the anisotropic elasticity moduli of the fiber. Based on previous studies, carbon fiber will fail when the fiber stress exceeds the ultimate stress in either tension or shear immediately. Hence, the plastic deformation of the fiber is not taken into account.

The matrix is modeled as an isotropic elastic-plastic material, and the elastic modulus $E_{m}$ and Poisson’s ratio $υ_{m}$ are used to characterize the elastic behavior of the matrix. The von Mises yield criterion and isotropic hardening are used to describe the plastic behavior. The damage to the matrix occurs when the stress reaches its maximum tensile or compressive stress, as expressed by

E_{m}^{d} = (1 - d_{m}) E_{m}

(13)

where

E_{m}^{d}

is the matrix modulus with damage, and

d_{m}

is the damage variable linearly varies in the range of 0 to 0.99.

Failure model of interphase

By comparing the cohesive zone model (CZM), the solid continuum element has been used to represent the interphase between the fiber and the matrix due to the lower computational time and the excessive element distortion. The interphase is defined as an anisotropic material, and $E_{i}$ is used to represent the elasticity moduli of the interphase. The damage model of the interphase is then established to describe the process of interphase failure, as shown in Figure 3. The elastic modulus varies linearly in segment AB, and plastic deformation occurs at point B when the stress exceeds the yield stress of the interphase ( $σ_{i}^{y}$ ). The damage will occur once the stress exceeds the ultimate stress ( $σ_{i}^{u}$ ) at point C, which leads to modulus degradation until complete failure (i.e., 0) in the segment CD.

Figure 3.

Failure model of the interphase.

The interphase modulus with damage ( $E_{i}^{d}$ ) can be determined by

E_{i}^{d} = (1 - d_{i}) E_{i}

(14)

where

d_{i}

is the damage variable of the interphase, which is assumed to linearly evolve by

d_{i} = \frac{L_{i}}{{\bar{u}}_{f}^{p l}} {\bar{ε}}_{i}^{p l}

(15)

where

L_{i}

is the characteristic length of the element, which can be determined by the cube root of the integration point volume of element,

{\bar{ε}}_{i}^{p l}

is the plastic energy-induced strain, and

{\bar{u}}_{f}^{p l}

is the failure displacement caused by the equivalent plastic strain, which is given by

{\bar{u}}_{f}^{p l} = \frac{2 G_{i}^{f}}{σ_{i}^{y}} = \int_{{\bar{ε}}_{0}^{p l}}^{{\bar{ε}}_{f}^{p l}} L_{i} σ_{i}^{y} d {\bar{ε}}^{p l} / σ_{i}^{y} = \int_{0}^{{\bar{u}}_{f}^{p l}} d {\bar{u}}^{p l}

(16)

where

G_{i}^{f}

is the interphase’s fracture energy.

In the simulation, the shear plane is defined as the geometrical surface where elements are deleted to produce the chip, i.e., the separation surface between chip and workpiece, as shown in Figure 4. The line of depth of cut is taken as the baseline, which is used to determine the shear plane angle with the separation surface of chip. After completing the simulation, the shear plane angle $ϕ$ was evaluated at the fiber orientations of 0°, 45°, 90°, and 135°, resulting in angles close to 66°, 50°, 34°, and 45° respectively.

Figure 4.

The shear plane in the simulation of CFRP cutting at the fiber orientation of (a) 0°, (b) 45°, (c) 90° and (d) 135°.

Force coefficients determination

Friction angle $β$

It is known that the angle between the resultant force and the normal force acting on the rake face is defined as the friction angle, $β$ , which is appear to be critically influenced by the fiber orientation in UD-CFRP machining. It can be calculated by

β = \arctan μ = \arctan (\frac{F_{c} \sin α + F_{t} \cos α}{F_{c} \sin α - F_{t} \cos α})

(17)

Based on the experimental $F_{c}$ and $F_{t}$ extracted from various sources,^5,9,17 the friction angle $β$ was determined using equation (17). And then the relationship between the friction angle and the fiber orientation $θ$ is presented in Figure 5. To describe the variation of $β$ with $θ$ at $0 ° \leq θ \leq 180 °$ , the polynomial function is proposed to predict the data of the friction angle, which is given by

β = {\begin{cases} 60.5517 + 0.01831 θ - 0.00456 θ^{2} 0 ° \leq θ < 90 ° \\ 63.6238 - 0.7724 θ + 0.00399 θ^{2} 90 ° \leq θ < 180 ° \end{cases}

(18)

Figure 5.

Variation of the friction angle with the fiber orientation.

It can be seen that the predicted function aligns reasonably well with the experimental results across the entire range of fiber orientations. The value of $β$ sharply decreases with the fiber orientation angle at $θ \leq 90 °$ , whereas it increases with the fiber orientation at $θ > 90 °$ . And the friction angle is larger than the fiber orientation in the range of [0°, 45°], which is smaller than the other fiber orientations.

Shear plane angle $ϕ$

It can be observed from Figure 1 that the chip is formed by the shear fracture of the workpiece, and the shear fracture occurs along a theoretical shear plane that makes an angle of $ϕ$ . It has been pointed out that the shear plane angle $ϕ$ is dependent on the cutting ratio, which is correlated with the depth of cut and the chip thickness. And the constant shear plane angle was modeled based on the assumption of a constant cutting ratio.¹⁸ However, the mechanism of chip formation varies with different fiber orientations in the cutting of CFRP composite, which means that the chip thickness may not always remain consistent with the depth of cut, particularly at $θ > 90 °$ .^5,25

Based on the results of the shear plane angle in the cutting simulation of UD-CFRP (see Section on “Simulation modelling”) and the experimental measurements obtained from sources,^6,28 the variation of $ϕ$ with $θ$ range from $0 °$ to $180 °$ in CFRP machining is plotted, as shown in Figure 6. It is found that the shear plane angle critically depends on the fiber orientation, which is assumed to have a nearly sinusoidal shape. The function modeling of the shear plane angle has been performed, in which the relationship between the shear plane angle and fiber orientation is expressed as

ϕ = {\begin{cases} 62.7527 - 0.00337 θ - 0.00336 θ^{2} 0 ° \leq θ < 90 ° \\ 72.6238 - 0.7714 θ + 0.0041 θ^{2} 90 ° \leq θ < 180 ° \end{cases}

(19)

Figure 6.

Variation of the shear plane angle with the fiber orientation.

The proposed function can capture the general trend of the shear plane angle in the analyzed fiber orientations, with the exception of the 30° fiber orientation. On the other hand, the assumed values of the shear plane angle are also presented in Figure 6, in which the shear plane angle $ϕ$ was assumed to be the same as the fiber orientation in $15 ° \leq θ \leq 75 °$ Ref. 17 and half of the fiber orientation angle in $θ > α + 90 °$ Ref. 24. The values used in Ref. 17 are closer to the measurements for fiber orientation ranging from 30° to 60°, while source²⁴ provides an accurate prediction for the fiber orientation range of [90°, 120°]. These indicate that the proposed function is more reasonable for predicting the shear angle across all fiber orientations.

Shear strength $τ$

The shear strength $τ$ is another fundamental coefficient that affects the shear force in the machining of UD-CFRP (see equation (10)). In previous micromechanical modeling of cutting force, the shear strength of the material was usually considered a constant value,^4,24 in which one of the shear strengths of the fiber, matrix, and interface bonding may use to represent the shear strength of the workpiece. It has been confirmed that the mechanical properties of CFRP depend on the variation of fiber orientation $θ$ . Therefore, the fiber orientation affects the shear strength of CFRP should be taken into account. Based on the experimental results of $F_{c}$ and $F_{t}$ in Ref. 9, the shear strength of UD-CFRP could be calculated using the inverse solution of equation (10). And then the variation of $τ$ with $θ$ range from $0 °$ to $180 °$ is obtained, as shown in Figure 7. The polynomial function is used to predict the variation of shear strength with fiber orientation, as follows:

τ = {\begin{cases} 143.61761 - 2.21601 θ + 0.08505 θ^{2} 0 ° \leq θ < 90 ° \\ 1932.402 - 19.22125 θ + 0.05316 θ^{2} 90 ° \leq θ < 180 ° \end{cases}

(20)

Figure 7.

Variation of the shear strength with the fiber orientation.

Comparing with the constant shear strength used in sources,^4,24 the proposed function is relatively closer to describing the relationship between shear strength and fiber orientation. Meanwhile, the theoretical calculations of shear strength $τ$ at different fiber orientations, as conducted in Ref. 29, are also presented in Figure 7. It is shown that the proposed function can effectively capture the trend of shear strength in the analyzed fiber orientations.

Rebound height $δ$

As shown in Figure 1, the rebound height $δ$ is defined as the vertical distance between the foremost point of the cutting edge and the last tool-workpiece contact point on the flankface due to the bouncing back of UD-CFRP. In this modeling approach of cutting force, the rebound height as a coefficient to predict the flankface force $F_{N}$ should be determined, which is closely related to fiber orientation. It is noted that the rebound height has been obtained experimentally in the machining of UD-CFRP,^3,30 where the fiber orientation-dependent-bouncing back effect has been taken into account in the cutting force model. According to the measurements of bouncing back in Ref. 30 and combined with the rebound height calculated by the inverse solution of equation (11) based on the experimental cutting force in Ref. 9, the variation of $δ$ with $θ$ range from $0 °$ to $180 °$ is then plotted, as shown in Figure 8. The function describing the relationship between the rebound height and fiber orientation has been obtained as follows:

δ = {\begin{cases} 0.00327 + 0.00136 θ - 0.0000147889 θ^{2} 0 ° \leq θ < 90 ° \\ 0.02752 - 0.000325674 θ + 0.000000997543 θ^{2} 90 ° \leq θ < 180 ° \end{cases}

(21)

Figure 8.

Variation of the rebound height with the fiber orientation.

The fairly good agreement between predictions and measurements suggests that the proposed function can accurately assess the rebound height for fiber orientations ranging from 0° to 180°. It is obvious that extremely low rebound heights are detected when the fiber orientation exceeds 90°, which implies that the fibers still bounce back, but the effect is considered limited. Consequently, the corresponding component force can be obtained by using the function of the fiber orientation-dependent rebound height.

Experiment verification

Experimental setup

As shown in Figure 9, the machining experiments of UD-CFRP were conducted on a CNC machining center, in which the UD-CFRP laminates had fiber orientations of 0°, 45°, 90°, and 135° with a cutting thickness of 6 mm. It is noted that the UD-CFRP laminate was fabricated using the autoclave processing method. The curing profile involved heating the sample at a rate of 1.5°C/min for 100 min, maintaining the temperature at 180°C for 150 min, and then cooling it down to room temperature at a rate of 1.5°C/min. The 0.06 MPa vacuum pressure was maintained throughout the curing process of the composite samples. The machining process was performed using a diamond-coated cemented carbide cutting tool with a speed of 157 m/min, a rake angle of 20°, and a flank angle of 10°. The cutting conditions in the simulation were kept the same as in the tests. The other parameters used for the UD-CFRP laminate and tool are provided in Table 2. The Kistler 9272 dynamometer is used to measure the cutting force in each test, and the average value is calculated as the effective one after five repeated cutting trials. The Hitachi S-3400N scanning electron microscope (SEM) is utilized to analyze the machined surface area, where the depth of cut and the actual depth of cut may vary. The height measurement of the area where spring back occurs is obtained, i.e., the vertical distance between points A and B (Figure 9). The point A is the intersection of the flank face and the line of the actual depth of cut, while point B is the intersection of the cutting edge and the line of the depth of cut.

Figure 9.

Schematic of the experiment setup for CFRP composites.

Table 2.

The parameters of UD-CFRP laminate and tool used in analytical model.

Items	Parameters	Values
UD-CFRP	Transverse elastic modulus, $E_{c}$ (GPa)	9.65
	Poisson’s ratio, $υ_{c}$	0.34
	Fiber volume fraction, $V_{f}$ (%)	65
	Fiber orientation angle, $θ$ (°)	0, 45, 90, 135
Tool	Depth of cut, $a_{c}$ (μm)	20
	Thickness of unidirectional laminate, $a_{w}$ (mm)	6
	Elastic modulus, $E_{t}$ (GPa)	1147
	Poisson’s ratio, $υ_{t}$	0.02

Model prediction and validation

The comparison between the predicted cutting forces and the experimental data, along with error bars, for various fiber orientation angles is illustrated in Figure 10. It can be observed that the theoretical values reasonably match the measurements within the fiber orientations range of [0°, 180°], in which the relative error is lower than 14.8%, except for the thrust force at 90°. Figure 10(a) shows that both the cutting force $F_{c}$ and thrust force $F_{t}$ increase significantly within the fiber orientation range of [0°, 30°]. This can be attributed to the increase in shear strength of CFRP, where the shear plane angle is larger than the fiber orientation in this range (refer to Figures 6 and 7). The cutting force starts to slightly increase within the fiber orientation range of [30°, 90°], which aligns with the change in chip formation mode from material shear to material crushing. This results in a smaller shear plane angle compared to the fiber orientation. Conversely, a significant decrease in the thrust force is observed in this range due to the noticeable decrease in shear plane angle and rebound height. This suggests that the impact of rebound height on the thrust force is greater than its effect on the cutting force. A significant decrease in the cutting force is observed within the fiber orientation range of [90°, 165°], which can be attributed to the sharp decline in the shear strength of CFRP. Additionally, a slight increase in the cutting force is observed when the fiber orientation exceeds 165°, as material bending becomes predominant. In accordance with the variation curves of the friction angle and shear plane angle, the thrust force increases as the fiber orientation range extends from 90° to 180°, indicating the significant influence of friction on the tool flank face.

Figure 10.

Comparison between the experiments and predictions of (a) cutting force and thrust force and (b) rakeface force and flankface force with fiber orientation.

The predictions of the rakeface force $F_{R}$ and flankface force $F_{N}$ also match well with the experiments in CFRP machining, as shown in Figure 10(b). It can be seen that the trend curve of rakeface force is similar to that of cutting force $F_{c}$ . This indicates that the rakeface force is greatly influenced by the chip formation mode resulting from the interaction between the fiber orientation and the rake angle of the cutting edge. The value of flankface force is small compared to that of rakeface force, which is close to zero at fiber orientations beyond 90°. This is because the flankface force is controlled by the effect of bouncing back, which may be considered limited in this range. The larger flankface force at the fiber orientation of [0°, 90°] can be found due to the large number and magnitude of bouncing back.

Furthermore, the comparison between the predicted values and the measurements of friction angle and rebound height with error bars for different fiber orientation angles is presented, as shown in Figure 11. It is found that the predictions of the proposed function for the friction angle and rebound height (see equations (18) and (21)) closely match the experimental values. It is worth noting that the experimental friction angle was obtained through the reverse tangent calculation of the friction coefficient, which was evaluated based on the variation curve of maximal $F_{c}$ and $F_{t}$ within the cutting path of 1200 mm during the machining of UD-CFRP laminate. The phenomenon of flankface force approaching zero when the fiber exceeds 90° can be further confirmed through experimental measurements of rebound height.

Figure 11.

Comparison between the experiments and predictions of (a) friction angle and (b) rebound height with fiber orientation.

Results and discussion

Figure 12 shows the impact of rake angle on the cutting force and thrust force across the entire range of fiber orientation. It is found from Figure 12 that there is a first turning point in the curve of predicted cutting force and thrust force for the fiber orientation range of [0°, 90°], which is attributed to the change in the chip formation mode. The inflection point value decreases as the rake angle increases. For instance, the turning point value changes from 60° to 45° when the rake angle range shifts from [0°, 5°] to [10°, 15°]. And the value further decreases to 30° as the rake angle reaches 20° (see Figure 10(a)), which means that a larger rake angle is beneficial for the transformation of chip formation mode to occur earlier, resulting in a decrease in cutting force. Meanwhile, the second turning point is also observed for the fiber orientation range of [90°, 180°], in which the inflection point varies from 150° to 165° as the rake angle increases within the range of [10°, 15°]. These phenomena reflect that the rake angle strongly influences the variation of the cutting force and the thrust force, both of which decrease as the rake angle increases across the fiber orientation between the two turning points. Conversely, the cutting force and the thrust force increase with the rise of the rake angle within the fiber orientation range from 0° to the first turning point and from the second turning point to 180°.

Figure 12.

Effect of rake angle on the (a) cutting force and (b) thrust force for different fiber orientation in machining CFRP.

Figure 13 shows the impact of flank angle on the cutting force and thrust force within the fiber orientation range of [0°, 180°] with a rake angle of 20°. It is evident that the cutting force and the thrust force influenced by the flank angle are limited. The trends of the cutting force and the thrust force show that both forces decrease as the fiber orientation range increases from 0° to 180°. The flank angle has a significant effect on the thrust force compared to the cutting force, which indicates the interaction between the workpiece and the tool’s flank face is more likely to act in the normal direction of the machined surface. Moreover, the inflection point value in the curve of predicted cutting force and thrust force will not change with the increase of the flank angle. This implies that the change in chip formation mode may be independent of the flank angle throughout the fiber orientation. This implies that the elastic bouncing back effect of CFRP on the flank surface is the main factor affecting the variation of cutting force and thrust force.

Figure 13.

Effect of flank angle on the (a) cutting force and (b) thrust force for different fiber orientation in machining CFRP.

Conclusions

The numerical-analytical method was used to derive the mechanical model for cutting force in machining UD-CFRP laminates. Experimental measurements were conducted to validate the model, and the results indicate good agreement between the predicted and experimental values, with relative errors below 14.8%, except for the thrust force at 90°. The force coefficients, such as friction angle, shear plane angle, shear strength, and rebound height, were strongly influenced by the fiber orientation. The results of the functional modeling align well with the experimental findings.

The impact of rake angle and flank angle on the cutting force and thrust force was also investigated. Two turning points were observed in the curve of predicted cutting force and thrust force across the entire range of fiber orientation, and the position of inflection points changed with the increase in the rake angle. It can be further observed that a larger rake angle is beneficial for the transformation of chip formation mode to occur earlier, resulting in a decrease in cutting force. Additionally, the impact of flank angle on cutting force and thrust force is limited, and these forces decrease as the flank angle increases within the fiber orientation range of [0°, 180°]. The inflection points in the curve of predicted cutting force and thrust force remain constant with the increase in flank angle. This indicates that the change in chip formation mode is not influenced by the flank angle. Furthermore, the results obtained from UD-CFRP laminates can be utilized to investigate the changes in cutting force and shear plane angle during the machining of multi-directional CFRP (MD-CFRP) laminates.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research work was supported by the Zhejiang Provincial Natural Science Foundation of China (LQ21E050006), the Special Project of Center for Scientific Research and Development in Higher Education Institutes of Ministry of Education (No. ZJXF2022174), the Basic Public Welfare Research Project of Zhejiang Province (LGG19E050010), the Scientific Research and Cultivation Fund Project of Hangzhou City University (204000-58162208).

ORCID iD

Jianzhang Xiao

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Force coefficient characterization in machining of UD-CFRP using numerical-analytical approach

Abstract

Keywords

Introduction

Analytical modelling of UD-CFRP

Simulation modeling of UD-CFRP

Description of 3D micromechanical cutting model

Failure model of fiber and matrix

Failure model of interphase

Force coefficients determination

Friction angle β

Shear plane angle ϕ

Shear strength τ

Rebound height δ

Experiment verification

Experimental setup

Model prediction and validation

Results and discussion

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Friction angle $β$

Shear plane angle $ϕ$

Shear strength $τ$

Rebound height $δ$