Investigation on the effects of microstructure on machining unidirectional carbon fiber reinforced polymer composites with a modified force model

Abstract

The paper aims to investigate the correlation of microstructural characteristics and machinability of unidirectional carbon fiber reinforced polymer composites by a modified analytical model. A representative volume element was selected to present the microstructure and analyze the force distribution, in which the interphase between the fiber and the matrix was considered significantly. And the microstructure can be obviously measured by using microscopic observation, especially the interphase was found to be around 0.3 μm in the used carbon fiber reinforced polymer composites. To study the representative volume element effect on the cutting behavior of unidirectional carbon fiber reinforced polymer composites, the prediction of cutting force was done by using a modified force model. Compared with the experiments, the developed model can well predict the cutting forces and subsurface damage in the carbon fiber reinforced polymer composites cutting. It was found that surface integrity as well as subsurface damage has a coincident varying trend with the fiber orientations, as confirmed by the observations of the machined surface at different fiber orientations. The good surface integrity can be obtained at the low fiber orientation of 0° and the poor surface occurs at the large fiber orientation of 135°. Moreover, the effect of interphase and the fiber volume fraction in representative volume element were further investigated. The results show that the cutting forces increase with increasing the fiber volume fraction as well as the interphase volume fraction, and the interphase affects the cutting force in the transverse direction is significantly higher than that in longitudinal direction.

Keywords

Carbon fiber reinforced polymer composites representative volume element interphase fiber orientation modified force model

Introduction

Carbon fiber reinforced polymer (CFRP) composites offer excellent mechanical properties that lead to enhanced functional performance and have been widely used in aerospace structures.^1–3 Despite the fact that composite parts are fabricated near-net shape, post machining of CFRP structures is an necessary procedure that assures the manufactured components meet their dimensional tolerances, surface quality, and other functional requirements.^4,5 However, machining of CFRP composites is considerably more difficult than machining of conventional metals and their alloys due to the obviously different material properties, such as anisotropy and nonhomogeneity nature, which are determined by the fiber type, microstructure, and matrix.⁶ And these material characteristics in turn affect the machining quality of CFRP composites, especially the variation of cutting force and damage in machined surface. Therefore, it is of great technological and practical interest to study the relationship between the material properties and cutting behavior in machining CFRP composites.

In order to understand the chip formation and material removal mechanisms of CFRP machining, the analytical approach has been used to investigate the machining behavior and model the cutting forces in the orthogonal cutting of unidirectional CFRP (UD-CFRP) composites. Everstine and Rogers⁷ presented a theoretical analysis on the cutting force based on the continuum mechanics approach, and this model was adapted to the special case of 0° ﬁber orientation. Bhatnagar et al.⁸ derived the cutting forces of unidirectional laminates by using the Merchant’s classical shear plane theory, which can predict the force responses quite accurately for fiber orientation within 0°–90°. Zhang et al.⁹ made further extension for the prediction of cutting forces by considering three distinctive cutting regions, in which the ﬁber orientation was limited to the range from 0° to 90°. These macro-mechanical models reveal the physical essence of the material failure and removal by considering the macroscopic properties of composite materials.

To better understand the cutting force of CFRP composites at the microscale, the micro-mechanical models have been developed. Pwu and Hocheng¹⁰ proposed the analytical model of cutting forces based on the beam theory for 90° ﬁber orientation. Jahromi and Bahr¹¹ also derived the force expressions by modeling the fiber as a cantilever beam. The effects of buckling force and bending force on the total cutting forces were studied for the fiber orientations exceed 90°. Qi et al.¹² developed a force prediction model based on the minimum potential energy principle (MPEP), and the supporting effect of surrounding material was considered when the fiber orientation ranging from 0° to 90°. Continuing the research of Qi et al.,¹² Chen et al.¹³ further constructed the force prediction model by using beams on elastic foundation theory and MPEP for fiber orientation in the range 0°–180°. Xu and Zhang¹⁴ established the cutting force and deformation model of UD-CFRP composites with and without vibration of tool tip for 90° fiber orientation, and the force predictive models were further proposed for both traditional cutting and EVA cutting by integrated the effect of the whole ﬁber orientation vary from 0° to 180°.¹⁵ Niu et al.¹⁶ proposed an analytical expression of cutting forces from the microscale to the macroscale for the fiber orientation in the range 45°–110°. These micromechanical models can present the material removal formation and damage at the microscale, but most of these models include the empirical parameters, which should be determined by the experiment as the empirical models of cutting force.^17–19

Moreover, damages occur on the machined specimens is difficult to investigate in UD-CFRP composites machining, such as the subsurface damage induced stress concentrations and energy consumption. Thus, the mechanics modeling have been tried out. Gururaja and Ramulu²⁰ studied analytically the subsurface stress using a generalized plane strain anisotropic elasticity formulation in orthogonal cutting of UD-CFRP, and the distribution of the subsurface stress decreased along the length of fiber for 0°, 45°, and 90° fiber orientations were obtained. Li et al.²¹ developed a prediction model of cutting force by the energy-based method, and the percentage of energy consumed on subsurface debonding was identified for fiber orientations ranging from 0° to 75°. To investigate the occurrence of surface damage, Su et al.²² proposed the analytical cutting model for a single fiber based on elastic fracture mechanics theory, in which the deflection of fiber under machined surface and the reaction force vary with the fiber cutting angle were obtained. Voss et al.²³ developed an analytical force model for orthogonal machining of UD-CFRP, which considers the influence of fiber orientation, tool geometry, and increasing tool wear, the value of fiber curvature was depressed from the machined edge to the fiber root. Xu and Reifsnider²⁴ used a micro-mechanical model with a representative volume element to evaluate fiber micro-buckling, in which the potential energy of fiber and matrix dissipated over the length of the fiber were derived. Considering the cutting mechanics in relation to cutting conditions is complicated for CFRP machining, and a limited number of these proposed models with the effect of interphase, which affects the mechanical property, and consequently influence the cutting performance of CFRP composites. Thus, the theoretical analysis on the cutting force and damage is still necessary to understand the cutting characteristic of these composites.

In the present paper, the correlation of microstructure and machinability of UD-CFRP composites is investigated by using the analytical method. A representative volume element (RVE) is established to present the microstructure of composites firstly, which consists of a single fiber, matrix, and the interphase between the fiber and the matrix especially. The produced force in the RVE is derived by modifying an analytical force prediction model. Relevant experiments were also carried out to examine the reliability and accuracy of the interphase and the established model, and the effect of the microstructure on the cutting forces was further discussed. At the same time, the observations and damage measurements of the machined surfaces at different fiber orientations were also done in this paper for the validation of the proposed model.

Analysis of microstructure and force distribution in CFRP cutting

Microstructure in CFRP composites

Figure 1 shows an example of 42 × 47 μm microstructure in UD-CFRP composites sample with T800 carbon fiber, which was imaged by using scanning electron microscope (SEM). It is noted that the curing profile of the sample during autoclave processing was heated with the rate 1.5°C/min for 100 min and kept the temperature at 180°C for 150 min, and then cooling down to room temperature with the rate 1.5°C/min. The 0.06 MPa vacuum pressure was provided for the entire curing process of composites sample. A cubic sample with a size of 3 cm was then made from the cured composites sample, which was finely polished and further etched by CH₂Cl₂ before characterization using SEM. Consequently, the microscopic constituent in the microstructure can be obviously revealed in the SEM micrograph, especially the interphase between the fiber and the matrix (the white layer surrounding the fiber).

From Figure 1, it can be seen that the carbon fiber diameter is around 7 μm, and the thickness of interphase is roughly estimated with the value of 0.3 μm along the fiber cross-section and its region is denoted by the blue area. It is worth noting that the interfacial thickness of 0.3 μm was employed in the next model prediction due to the same composites sample applied to machining experiment, though the interphase greatly vary depend on the type of fiber, resin, and sizing. Moreover, it has been studied that the machinability of the CFRP composites was correlated with the material properties, which was significantly affected by the interphase.^6,25 And the model has been presented and validated to characterize the modulus distribution of the interphase between the fiber and the matrix in previous work.²⁵ As shown in Figure 1, a RVE with three different constituent phases is then established in the microstructure, i.e. single fiber, interphase, and matrix, to study the cutting performance of CFRP composites. Moreover, it is clearly shown from the figure that the fiber distribution in matrix is not a uniform distribution under the microscope scale. Therefore, this nonuniformity will be taken as a parameter of material property effect on the cutting behavior.

Analysis of force distribution

In UD-CFRP orthogonal cutting process, the machining region associate with the cutting forces could be divided into three deformation regions, which are marked as I, II, and III, as shown in Figure 2. Considering the interaction between the tool and the workpiece material, the three deformation regions are named by cutting edge region, flank contact region, and rake face region separately. In the cutting edge region, the cutting edge acts on the fiber, and fracture occurs due to the compression and bending of fiber. In the flank contact region, the material is compressed by the flank face, which will be bouncing back as the cutting tool moving forward. The matrix failure and interface debonding happen in the rake face region, the damage material slip along the rake face and then forming the cutting chip. Based on the force analysis of material deformation, the total cutting force can be obtained by adding up the forces in all the three regions.

Establishment of modified force model

It has been studied that the deformation mechanism is different with the fiber orientation in cutting the UD-CFRP composites.^15,26,27 There are two major types of ﬁber deformation and fracture according to the ﬁber orientation θ limited into the range 0° to 90° and 90° to 180°. Taking the rake angle α_t into consideration, a more accurate classification is made, in which θ divided into two section, i.e. 0°≤θ < α_t+90° and α_t+90°≤θ < 180°. When θ is less than α_t+90°, high stresses occur at the contact zone between cutting edge and workpiece, and leads to a crushing-dominated failure of workpiece. When θ is beyond α_t+90°, a larger stress happens at the rake face rather than the cutting edge as the tool moves to the workpiece, and the cutting tool pushes the ﬁbers to bend, which results in the bending-dominated fracture. This study will focus on analyzing the deformation mechanism for crushing-dominated (θ < α_t+90°) and bending-dominated (θ ≥ α_t+90°) failure in UD-CFRP cutting.

To develop the modified force model for different range of fiber orientation, the RVE in Figure 1 has the following characteristics based on the study in Xu and Zhang:^14,15 (1) the width of the workpiece is the same as that of the cutting tool, and is equal to the diameter of the RVE; (2) fiber has little elastic deformation before its breakage; (3) fiber fracture takes place when the maximum tensile stress exceeds its tensile strength; and (4) the contact of tool–fiber and tool–workpiece surface follows the Hertz contact theory.

Force analysis of cutting edge region

Crushing-dominated fiber deformation and fracture (0°≤θ < α_t+90°)

As shown in Figure 3, the deformation of a ﬁber in the RVE under crushing-dominated fracture is presented for the cutting of CFRP composites. The interphase in the RVE is present with the blue color in the figure, and $r_{m}$ is the radius of the RVE. The fiber is subjected to a pressing force F_u and is supported by a distributed reaction force behind the fiber simultaneously. The reaction force reﬂects the constraint effect of the surrounding composite. For simplicity but without losing the generality, the part of the composite that supports the ﬁber can be treated as an equivalent homogeneous material whose elastic properties can be obtained by the three-phase model developed in the previous work.²⁵

The action of the supporting composite on the fiber from both sides has been studied in Winkler.²⁸ Therefore, the reaction force from the supporting composite per unit length can be expressed as

p_{c} = k_{c} ω

(1)

where

p_{c}

is the reaction force at a fiber cross-section with deﬂection

ω

k_{c}

is called the Winkler foundation modulus, which reﬂects the normal effect from the supporting composite, and can be obtained from Biot’s formula²⁹

k_{c} = \frac{1.23 E_{c}}{C (1 - υ_{c}^{2})} {[\frac{E_{c} d_{f}^{4}}{C (1 - υ_{c}^{2}) E_{f} I_{f}}]}^{0.11}

(2)

where

C

is a coefficient varying from 1 for uniform pressure distribution to 1.13 for uniform deﬂection. Here

C

=1.13 is simply adopted.

E_{c}

and

υ_{c}

are the transverse modulus and Poisson’s ratio of composites;

E_{f}

is the transverse modulus of fiber, and

I_{f}

is the moment of inertia of the ﬁber cross-section,

I_{f} = π d_{f}^{4} / 64

d_{f}

is the diameter of fiber.

During the cutting, the interphase between the fiber and the matrix will suffer from failure, which leads to a length of debonding $L$ , as shown in Figure 3. Then, the rest of the interphase will have a bonding force to constrain the deﬂection of the ﬁber. Assuming that $p_{i}$ is the intensity of the bonding force in the un-debonding area, which is given by

p_{i} = k_{i} ω

(3)

where

k_{i}

is the equivalent modulus of the interphase, and can be obtained from the three phase model²⁵

k_{i} = \frac{1}{r_{i} - r_{f}} \int_{r_{f}}^{r_{i}} k_{i} (r) d r = \frac{E_{f} + n E_{m}}{n + 1}

(4)

where

r_{i}

and

r_{f}

are the radius of interphase and fiber in the RVE,

E_{m}

is the elastic modulus of matrix,

n

is a coefﬁcient parameter, and

n

= 1.66 is adopted by fitting the experimental modulus of the interphase in the previous work.²⁵ It is known that the intensity of the bonding force is equal to the bonding strength of interphase when the debonding occurs.

In Figure 3, there is an inﬁnitesimal element of the ﬁber with the length of ds, and the element equilibrium gives rise to

{\begin{array}{l} \sum Q = 0 Q - (Q + dQ) + (p_{c} + p_{i}) d s = 0 \\ \sum M = 0 M - (M + dM) + (Q + dQ) d s + (k_{c} + k_{i}) d s^{2} = 0 \end{array}

(5)

where Q is the shear force and M is the bending moment. In the RVE, the deﬂection of the ﬁber is modeled as a beam on an elastic foundation. Substituting equation (5) into the well-known beam bending theory³⁰

\frac{M}{E_{f} I_{f}} = \frac{d^{2} ω}{d^{2} s}

(6)

and then the second-order terms are ignored, which brings about

E_{f} I_{f} \frac{d^{4} ω}{d s^{4}} - (k_{c} + k_{i}) ω = 0

(7)

The general solution of equation (7) is

ω (s) = B_{1} e^{λ s} + B_{2} e^{- λ s} + B_{3} \cos λ s + B_{4} \sin λ s

(8)

where

ω (s)

is the deflection of the fiber,

B_{1}

B_{2}

B_{3}

, and

B_{4}

are constants of integration,

λ

is the characteristic value, which includes the flexural rigidity of the ﬁber as well as the elasticity of the supporting material. Its expression is given by

λ = \sqrt[4]{\frac{k_{c} + k_{i}}{4 E_{f} I_{f}}}

(9)

In order to solve the fiber deﬂection, the integration constants and debonding length $L$ should be determined. It is known that the stress caused by the deformation of the fiber could be calculated by³¹

σ_{f} = E_{f} η ω ″

(10)

where

η

is the distance of an arbitrary point to the neutral layer of the fiber. When

η

equals to

r_{f}

and

- r_{f}

, the absolute value of the deformation stresses reach to the maximum value, i.e., the tensile strength of the fiber. Thus, we have

E_{f} r_{f} ω ″ = σ_{fT}

(11)

where is

σ_{fT}

the fiber tensile strength, and the ﬁber fracture takes place when the maximum deformation stress exceeds its tensile strength. According to Figure 3, the boundary conditions of the fiber in this case will be

{\begin{cases} s = 0 : ω (0) = 0, \frac{d ω}{ds} |_{s = 0} = 0 \begin{matrix}  \end{matrix} \\ s = L : \frac{d ω}{ds} |_{s = L} = 0, E_{f} r_{f} \frac{d^{2} ω}{d s^{2}} |_{s = L} = σ_{fT} \end{cases}

(12)

At the bottom of the ﬁber (s→+∞)

ω (s) |_{s \to + \infty} = 0

(13)

Furthermore, at point A, i.e. the onset point of the interphase failure, the intensity of the bonding force which calculated from equation (14) is equal to the bonding strength

k_{i} ω_{i} = σ_{iT}

(14)

where

ω_{i}

is the deﬂection of the ﬁber at the debonding point A,

σ_{iT}

is the bonding strength of interphase. Equation (14) provides a condition to calculate the critical debonding length

L

. At the same time, the force effects on the fiber will be determined from the fracture conditions of the ﬁber, which is the minimum force to cut off the ﬁber. Thus, the damage length

L

can be determined by the conditions of interphase failure and ﬁber fracture, and satisfying

\begin{array}{l} \min {{[\max (E_{f} r_{f} \frac{d^{2} ω}{d s^{2}}) - σ_{fT}]}^{2}} \\ k_{i} ω_{i} \leq σ_{iT} \end{array}

(15)

As shown in Figure 3, there are three different supporting conditions along the ﬁber as it interacts with the cutting tool, which including: (1) the part of the fiber above the contact point A, i.e. $s < (a_{c} - r_{t}) / \sin θ$ ; (2) that of the fiber between points A and B, i.e. $(a_{c} - r_{t}) / \sin θ \leq s \leq (a_{c} + L) / \sin θ$ ; and 3) that of the fiber below point B, i.e. $s > (a_{c} + L) / \sin θ$ . Therefore, the pressing force $F_{u}$ vertical to the ﬁber can be obtained by summing the reaction forces on the three parts, i.e.

F_{u} = \int_{0}^{(a_{c} - r_{t}) / \sin θ} k_{c} ω d s + \int_{(a_{c} - r_{t}) / \sin θ}^{(a_{c} + L) / \sin θ} k_{c} ω d s + \int_{(a_{c} + L) / \sin θ}^{+ \infty} (k_{c} + k_{i}) ω d s

(16)

where

a_{c}

is the depth of cut,

r_{t}

is the radius of cutting edge. Letting

F_{fu}

represents the friction force between the ﬁber and the cutting edge, which can be determined by the Coulomb friction law as

F_{fu} = μ F_{u}

(17)

where μ is the friction coefﬁcient between the tool and ﬁber. Finally, the cutting forces along the

x

and

y

direction in the cutting edge region under

θ < 90 ° + α_{t}

could be written as

{\begin{matrix} F_{ux} = F_{u} \sin θ + μ F_{u} \cos θ \\ F_{uy} = F_{u} \cos θ + μ F_{u} \sin θ \end{matrix}

(18)

Bending-dominated fiber deformation and fracture (α_t+90°≤θ < 180°)

As shown in Figure 4, the RVE undergo bending due to an exerted pressing force $F_{u}$ when $θ \geq 90 ° + α_{t}$ . The pressing force perpendicular to the fiber axis is toward out of the cutting plane, and hence the fiber gets a weaker support from the surrounding composite materials. With no surrounding materials supporting the RVE near the surface, the fiber tends to glide over the tool edge instead of being shorn off. In this process, the fiber in front of tool is bent until the resulted stress exceeds fiber tensile strength and fracture takes place often below the cutting plane.

As explained before, the composites may also fail due to fiber bending and fracture. If the normal stress in the composites exceeds its flexural strength at a certain point, the composite material will break and a chip will form. In Figure 4, the deﬂection of the ﬁber is also modeled as a slender beam on the elastic foundation. By using the beam-deflection equation, the tensile stress caused by the bending effect of the fiber is given by

σ_{f} = \frac{η M}{I_{f}}

(19)

The tensile stress reaches to the maximum value when $η$ equals to $r_{f}$ . As the tensile stress exceeds the fiber tensile strength $σ_{fT}$ , the bending force break the fiber, which can be obtained by inverse solution of equation (19)

M_{\max} = \frac{I_{f} σ_{fT}}{r_{f}}

(20)

where

M_{\max}

is the maximum bending moment. Based on the beam bending theory, the bending failure occurs when the bending stress of the fiber reaches the maximum bending moment, and then we have

E_{f} I_{f} ω ″ = \frac{σ_{fT} π r_{f}^{3}}{4}

(21)

Using the analysis method of force equilibrium, the expression of fiber deflection $ω (s)$ keeps the same as equation (8) at $θ \geq 90 ° + α_{t}$ . In order to solve the ﬁber deﬂection, the integration constants $B_{1}$ , $B_{2}$ , $B_{3}$ , $B_{4}$ , and debonding length $L$ need to be determined. According to Figure 4, the deﬂection of the ﬁber is continuous, and the boundary conditions of the ﬁber with different supporting constraints are as follows

{\begin{array}{l} s = 0 : ω (0) = 0, \frac{d ω}{ds} |_{s = 0} = 0 \\ s = L : \frac{d ω}{ds} |_{s = L} = θ - \frac{π}{2} - α_{t} \\ E_{f} I_{f} \frac{d^{2} ω}{d s^{2}} |_{s = L} = M_{\max} \end{array}

(22)

Based on the method to determine the debonding length at $θ < 90 ° + α_{t}$ , the debonding length $L$ at $θ \geq 90 ° + α_{t}$ is solved from the following optimization problem by an optimality iterative algorithm.

\begin{array}{l} \min {{[\max (E_{f} I_{f} \frac{d^{2} ω}{d x^{2}}) - \frac{σ_{fT} π r_{f}^{3}}{4}]}^{2}} \\ k_{i} ω_{i} \leq σ_{iT} \end{array}

(23)

Using the principle of superposition to calculate the force in the parts of the ﬁber sections, the pressing force $F_{u}$ can therefore be described as

\begin{array}{l} F_{u} = \int_{0}^{(a_{c} - r_{t}) / \sin θ} k_{c} ω d s + \int_{(a_{c} - r_{t}) / \sin θ}^{(a_{c} + L) / \sin θ} k_{c} ω d s \\ + \int_{(a_{c} + L) / \sin θ}^{+ \infty} (k_{c} + k_{i}) ω d s \end{array}

(24)

And then the cutting forces along the $x$ and $y$ directions in the cutting edge region under the case of $θ \geq 90 ° + α_{t}$ can be determined by

{\begin{array}{l} F_{ux} = F_{u} \sin θ + μ F_{u} \cos θ \\ F_{uy} = F_{u} \cos θ + μ F_{u} \sin θ \end{array}

(25)

Force analysis of flank contact region

In the flank contact region, the nose and the flank face of the cutting tool make the workpiece below the machined surface deform. This deformation can be viewed as that a two-dimensional contact between a blunt wedge (tool nose and clearance face) and a flat plane (workpiece material), as shown in Figure 5. By using the contact mechanics of the blunt wedge in contact with and the plane surface,³² the distribution of the normal pressure in the contact zone can be described by the following equation

\begin{array}{l} p (x) = \frac{E^{*} \cot α}{2 π} {\frac{2 l}{{(l^{2} - x^{2})}^{1 / 2}} - \ln [\frac{l + {(l^{2} - x^{2})}^{1 / 2}}{l - {(l^{2} - x^{2})}^{1 / 2}}]} \\ + \frac{F_{P}}{π {(l^{2} - x^{2})}^{1 / 2}} \end{array}

(26)

It is known from Figure 5 that when the smooth faces of the wedge extend beyond the edges of the contact, the pressure must fall to zero at the edges to avoid tension or interference outside the contact,³² and then the pressure force can be obtained by

F_{P} = \frac{1}{2} l d_{m} E^{*} \cot α = \frac{1}{2} l d_{m} E^{*} \tan γ_{t}

(27)

where

E^{*}

is the effective elastic modulus of the workpiece material in the contact zone,

α

is the semi-angle of the blunt wedge,

F_{P}

is the flank contact force perpendicular to the machined surface,

d_{m}

is the diameter of the RVE,

γ_{t}

is the clearance angle of tool, and l is the length of contact region, which related to the position of fiber rupture and induced the bouncing back of fiber in actual cutting.

In order to solve the contact length $l$ , the minimum bending radius of fiber $r_{f \min}$ is introduced, i.e. the smallest allowed radius when the fiber bending fracture, as shown in Figure 6(a). $r_{f \min}$ depends on the fiber diameter and its proportional limit of tensile, and the following formula is used

r_{f \min} = \frac{d_{f}}{2} (\frac{1}{ε_{fT}} - 1)

(28)

where

ε_{fT}

is the ultimate tensile strain of fiber. For simplicity, assume that all the fibers experience the minimum bending radius as the cutting tool moving forward, and then rupture occurs when the tensile stress beyond the fiber tensile strength. Thus, the contact length

l

in the case of

θ < 90 ° + α_{t}

(Figure 6(a)) can be obtained as

l = r_{f \min} (\frac{π}{2} - θ + γ_{t}) \sin γ_{t}

(29)

On the other hand, the fiber is often bending fracture below the cutting surface at $θ \geq 90 ° + α_{t}$ , and a small bouncing back of fiber occurs under the flank faces, as shown in Figure 6(b). This induces the contact between the fiber and tool is limit, and the contact length $l$ for $θ \geq 90 ° + α_{t}$ can be simply defined by

l = d_{f}

(30)

So the relationship between the contact length and bouncing back value $δ$ is given by

δ = l \sin γ_{t}

(31)

In addition, the equivalent elastic modulus $E^{*}$ can be obtained through the following equation

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

(32)

where

E_{t}

and

υ_{t}

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

E_{c}

and

υ_{c}

are the transverse elastic modulus and Poisson’s ratio of composite materials. The flank contact force

F_{P}

works not only to press the workpiece located below the flank face, but also as a friction force parallel to the flank surface. Using the Coulomb friction law with the friction coefficient

μ

, the friction force between the flank face and the workpiece is expressed by

F_{f} = μ F_{P} \cos γ_{t}

(33)

When adding the friction force $F_{f}$ between the flank face and the workpiece material, the cutting forces along the $x$ and $y$ directions in the flank contact region are given by

{\begin{array}{l} F_{px} = μ F_{P} \cos^{2} γ_{t} \\ F_{py} = F_{P} (1 + μ \cos γ_{t} \sin γ_{t}) \end{array}

(34)

Force analysis of rake face region

In the orthogonal cutting of CFRP composites, the chip formation mechanism was basically determined by the fiber orientation.³³ As shown in Figure 7, fiber fracture occurs in front of the rake face and slip along the fiber–matrix interfaces, which results in a chip. An interacting force is induced by the tool to the chip on the rake face in the rake face region, which is named the rakeface force and denoted by $F_{R}$ .

In Figure 7(a), the chips are formed by the shear fracture of the matrix and interphase along an overall shear plane when $α_{t} < θ < 90 ° + α_{t}$ . The force in the RVE associated with the shear failure could be estimated by the shear plane theory

F_{s} = 2 r_{m} L' S_{m} + 2 r_{i} L' S_{i} = \frac{2 a_{c}}{\sin ϕ_{s}} (r_{m} S_{m} + r_{i} S_{i})

(35)

where

F_{s}

is the shear force of RVE,

r_{m}

(= r_{f} / \sqrt{V_{f}})

is the radius of the RVE,

r_{i}

is the radius of the interphase, which is equal to

r_{f} + t_{i}

(

t_{i}

is the interphase’s thickness),

L'

is the length of shear failure,

S_{m}

is the shear strength of the matrix,

S_{i}

is the shear strength of the interphase, and

ϕ_{s}

is the shear plane angle. Considering the chips produced by shearing the matrix slip along the fiber direction under the case of

α_{t} < θ < 90 ° + α_{t}

. Thus,

ϕ_{s}

can be replaced by fiber orientation angle

θ

. Based on the force equilibrium method, the rakeface force normal to the tool-chip interface can be expressed by

F_{R} = F_{s} / \cos (θ - α_{t} + β_{t})

(36)

where

β_{t}

is friction angle on the rake face, and

β_{t} = \arctan μ

. Unlike the chip slip along the fiber orientation, the chips will split off due to the force produced by the rake face when

0 \leq θ \leq α_{t}

.⁷ Then, the rakeface force for the case of

θ \leq α_{t}

could be given by the consideration of the interphase

F_{R} = \frac{2 a_{c} \cot α_{t}}{\cos θ \cos β_{t}} (r_{m} S_{m} + r_{i} S_{i})

(37)

Thus, the rakeface force in the rake face region in the case of $θ < 90 ° + α_{t}$ could be written as

F_{R} = {\begin{array}{l} \frac{2 a_{c} \cot α_{t}}{\cos θ \cos β_{t}} (r_{m} S_{m} + r_{i} S_{i}) & 0 \leq θ \leq α_{t} \\ \frac{2 a_{c}}{\sin θ \cos (θ + β_{t} - α_{t})} (r_{m} S_{m} + r_{i} S_{i}) & α_{t} < θ < 90 ° + α_{t} \end{array}

(38)

On the other hand, the fiber bending fracture and pelt off from the rake face at $θ \geq 90 ° + α_{t}$ , as shown in Figure 7(b). The rakeface force act on the RVE could be estimated by

F_{R} = \frac{F_{s}}{\sin β_{t}} = \frac{2 a_{c}}{\sin θ \sin β_{t}} (r_{m} S_{m} + r_{i} S_{i})

(39)

And the cutting forces along the $x$ and $y$ directions in the rake face region are given by

{\begin{matrix} F_{Rx} = F_{R} \cos (β_{t} - α_{t}) \\ F_{Ry} = F_{R} \sin (β_{t} - α_{t}) \end{matrix}

(40)

The mechanical analysis above is the force generated by a single fiber in the RVE, i.e. the cutting width is $2 r_{m}$ . And the total cutting force act on the single fiber is a superimposition of the forces produced in the three deformation regions. Considering an arbitrary cutting width $a_{w}$ in the CFRP composites machining, the macroscopic cutting forces along the $x$ and $y$ directions could be expressed as

{\begin{matrix} F_{x} = \frac{a_{w} \sqrt{V_{f}}}{2 r_{f}} (F_{ux} + F_{px} + F_{Rx}) \\ F_{y} = \frac{a_{w} \sqrt{V_{f}}}{2 r_{f}} (F_{uy} + F_{py} + F_{Ry}) \end{matrix}

(41)

Experiment and prediction condition

As a view to validate the modified force model derived in the above sections, trimming experiments were carried out on a high-speed machining center. The experimental setup and the geometry feature of the cutting tool are shown in Figure 8. The workpiece was fixed by the fixture mounted on a Kistler 9272 dynamometer, which was used to measure the cutting force together with a charge ampliﬁer and a data collection device. The forces were acquired with a sampling frequency of 5 kHz and at sample intervals of 0.2 ms, and the cutting force signals were filtered with a low-pass filter. The cutting parameters employed for each test were as follows: the cutting speed is 157 m/min, the feed rate is 1 m/min, the axial depth of cut is 6 mm, and the radial depth of cut is 1 mm, as listed in Table 1. For the edge trimming experiments, each test was repeated five times and their average values were taken as the effective ones.

Table 1.

Cutting parameters used in the experiment.

Parameter items	Symbol	Value
Cutting speed (m/min)	$V_{c}$	157
Feed rate (m/min)	a_f	1
Nominal depth of cut (μm)	$a_{c}$	30, 60
Width of cut (mm)	$a_{w}$	6
Spindle speed (r/min)	$n_{t}$	5000

The workpiece used were fabricated from the unidirectional IMS/X850 prepregs with T800 carbon fiber. The composite coupons used in the experiment were the same as that in the previous work.^3,5 The mechanical properties of the UD-CFRP composites as well as its constituents are given in Tables 2 and 3, in which properties of composites, fiber, and matrix were obtained from the supplier’s specifications sheet, and the interfacial properties was calculated from the model developed in the previous work.²⁵ The cutting process was performed by using a diamond-coated and cemented carbide multi-edge cutter with a 10 mm diameter, 36 mm cutting length, 12 teeth, and a 15° helix angle, as shown in Figure 8(b). The edge radius of the coated tooth $r_{n}$ was then measured under microscope and was found to be around 10 μm. It must be noted that the coating thickness of 5 μm was given by the producer (Chengdu Tool Research Institute Co., Ltd), and the edge radius of 5 μm (without coating $r_{t}$ ) was then used in model calculation for ignoring the effect of coating. The detailed geometry and property parameters of tool used in the model prediction and experiment are given in Table 4. The value of tool/workpiece friction in the table was obtained from Li et al.²¹

Table 2.

Material parameters of CFRP composites.

Property items	Symbol	Value
Transverse elastic modulus of CFRP composites (GPa)	E _c	9.65
Poisson’s ratio of CFRP composites	υ _c	0.34
Fiber volume fraction (%)	V_f	65
Fiber orientation angle (°)	$θ$	0, 45, 90, 135

CFRP: carbon fiber reinforced polymer.

Table 3.

Material properties of fiber, matrix, and interphase.

Property items	Symbol	Value
Transverse elastic modulus of fiber (GPa)	E_f	19.6
Poisson’s ratio of fiber	υ_f	0.3
Tensile strength of fiber (GPa)	$σ_{fT}$	5.49
Ultimate tensile strain of fiber	$ε_{fT}$	0.019
Fiber radius (μm)	r_f	3.5
Elastic modulus of matrix (GPa)	E_m	4
Poisson’s ratio of matrix	υ_m	0.4
Shear strength of matrix (MPa)	S_m	70
Elastic modulus of interphase (GPa/μm)	k_i	14.4
Shear strength of interphase (MPa)	S_i	50
Bonding strength of interphase (MPa)	$σ_{iT}$	50
Thickness of interphase (μm)	t_i	0.3

Table 4.

Geometry and property parameters of cutting tool.

Parameter items	Symbol	Value
Rake angle of tool (°)	$α_{t}$	25
Clearance angle of tool (°)	$γ_{t}$	10
Radius of cutting edge (μm)	$r_{t}$	5
Tool diameter (mm)	$d_{t}$	10
Elastic modulus of tool (GPa)	$E_{t}$	1147
Poisson’s ratio of tool	$υ_{t}$	0.07
Friction coefficient between tool and workpiece	$μ$	0.2

Results and discussion

Analysis of machined surface

Figure 9 shows the observation position of the machined surface after trimming. The Hitachi S-3400N SEM was used to analyze the machined specimen area (Figure 9(a)), and the measurement of damage area where delamination occurs is obtained (Figure 9(b)). It is found that the type I delamination is mainly observed along the machined edge, which describes the surface broken fibers removed some distance inward from the trimmed edge.³⁴ Then the depth of subsurface damage can be determined by the intensity of the microcrack and the fiber breakage in the type I delamination, as shown in Figure 9(c). Figure 10 shows the machined surface morphologies of workpiece for different fiber orientations, as obtained in our previous work.³ It is clear that the machined surface was mainly composed of bare fibers and resin ridges at 0° (Figure 10(a)), which led to a rather smooth surface with few bouncing back of fiber. While the large pits of carbon fiber bundles and resin ridges were obviously observed at 135° and a rough surface was generated. Although the bouncing back of fiber still occurs in the region surrounding the pitting damage, the number and magnitude of the bouncing back may be considered limited. It can be found that the resin is plough on the machined surface of 45°, which is induced by the significant bouncing back of material. Moreover, it is important to note that the bouncing back of 5 μm was kept the same as the radius of cutting edge to predict the cutting forces when $θ \leq α_{t}$ , which is due to the change of chip formation mechanism.

Figure 11 shows the subsurface damage of the machined workpiece with different fiber orientations. The subsurface damage of debonding and fiber fracture is found at 0°, and the depth of damage is limited due to the failure mainly extends along the fiber’s axis. When the ﬁber orientation is 90°, as illustrated in Figure 11(b), it is obvious that the cut surface was generated by crushing-induced fiber fracture. However, since the fiber cracking point was a little above the cut surface, which resulted in a small interphase failure and reduced the fiber–matrix debonding. At the fiber orientation of 135° (Figure 11(d)), the bending-dominated fiber fracture ahead of the cutting tool and led to a deep fiber fracture beneath the surface and increased the subsurface debonding damage. The depth of debonding was 46 μm and 61 μm at the fiber orientation of 90° and 135°, respectively.

Figure 12 compares the predicted depth of subsurface damage by the model with that from the experimental measurement when fiber orientation varies. It shows that the fiber orientation affects the subsurface damage signiﬁcantly, and the varying trend of the model predictions coincides with those of experimental results in the whole range of ﬁber orientations. Thus, it can be further concluded that the model can be used to evaluate the damage under the machined surface to some extent.

Effect of microstructure on the cutting force

Figure 13 compares the theoretical cutting forces with the average measurements of the experimental cutting force for different fiber orientation angles and different depths of cut. It is found that the theoretical predictions have a reasonably well with the experimental results at the entire range of fiber orientations. This means that the mechanics model established above has captured the major fiber fracture and deformation mechanisms in machining CFRP composites. It can be seen from Figure 13 that the fiber orientation affects the cutting force and the thrust force markedly. When $0 \leq θ < 90 °$ , the cutting force $F_{x}$ increases with the increase of the fiber orientation, and $F_{x}$ decreases with the increase of the fiber orientation as $90 ° \leq θ < 180 °$ . On the contrary, the thrust force $F_{y}$ decreases and then increase with the fiber orientation increases from 0° to 180°. This discontinuity in the cutting force is indicating the deformation mechanism changes with the fiber orientation. Moreover, the depth of cut also significantly influences the cutting forces, which are due to forces applied on the region of cutting edge and rake face occupy an important proportion of the total cutting forces.

Figure 14 shows the effects of interphase on the cutting force in the entire range of fiber orientation. It is obvious that the forces (interfacial force $F_{ix}$ and $F_{iy}$ ) produced by the interphase depend on the fiber orientation, and the trend of the interfacial force is identical to that of the total cutting force $F_{x}$ . This is because the chips are formed by the shear fracture of interphase on the rake face region. As shown in Figure 14(a), the interfacial force in the transverse direction ( $F_{ix}$ ) was significantly higher than that in the longitudinal direction ( $F_{iy}$ ), which indicates that the effect of interphase on cutting force is mainly reflected in the transverse direction. In addition, the proportions of $F_{ix}$ and $F_{iy}$ are close to each other at 90°, which are different at 0° (see Figure 14(b)), this further indicates the different effect of interphase on the machinability of CFRP composites in the longitudinal and transverse direction.

Another important parameter affecting the inconsistency of the cutting forces is the random-fiber distribution, and causing variable fiber volume fraction V_f among the material. As shown in Figure 1, V_f varies from 0.58 to 0.7 in a small area of CFRP composites. Therefore, the cutting forces may vary from point to point. Figure 15 shows the fiber volume fraction influence on the predicted cutting force and thrust force in CFRP composites trimming. As can be seen, both of the machining forces increase with increasing the fiber volume fraction across the fiber orientation range from 0° to 180°. The fiber volume fraction correlates with the fraction of interphase has been studied by Gao et al.²⁵ and Gohil and Shaikh,³⁵ in which the interphase volume fraction vary linearly with fiber volume fraction. Consequently, the cutting forces increase with the increase of the interphase volume fraction in the CFRP composites machining.

Conclusions

This paper has successfully investigated the correlation of microstructure and machinability of UD-CFRP composites by the modified force prediction model. As conﬁrmed by the corresponding experimental analysis, the following conclusions can be drawn:

With consideration of the effect of interphase, the modified force prediction model can predict the cutting forces and the depth of subsurface damage reliably and precisely. The surface integrity and the subsurface damage have a coincident varying trend with the fiber orientations based on the observations of the machined surfaces at different fiber orientations. And the good surface integrity is obtained at the low fiber orientation of 0° and the poor surface occurs at the large fiber orientation of 135°.

The effect of the interphase on the cutting forces has been expressed accurately, and the interphase affects the cutting force in the transverse direction is significantly higher than that in longitudinal direction. The cutting forces increase with increasing the fiber volume fraction as well as the fraction of interphase.

The proposed model can be applied to study the relationship between microstructure and machinability for other unidirectional long fiber reinforced polymer composites.

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 Science Fund for Creative Research Groups of National Natural Science Foundation of China (No. 51821093) and the Basic Public Welfare Research Project of Zhejiang Province (LGG19E050010).

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Investigation on the effects of microstructure on machining unidirectional carbon fiber reinforced polymer composites with a modified force model

Abstract

Keywords

Introduction

Analysis of microstructure and force distribution in CFRP cutting

Microstructure in CFRP composites

Analysis of force distribution

Establishment of modified force model

Force analysis of cutting edge region

Crushing-dominated fiber deformation and fracture (0°≤θ < αt+90°)

Bending-dominated fiber deformation and fracture (αt+90°≤θ < 180°)

Force analysis of flank contact region

Force analysis of rake face region

Experiment and prediction condition

Results and discussion

Analysis of machined surface

Effect of microstructure on the cutting force

Conclusions

Footnotes

Declaration of conflicting interests

Funding

References

Crushing-dominated fiber deformation and fracture (0°≤θ < α_t+90°)

Bending-dominated fiber deformation and fracture (α_t+90°≤θ < 180°)