1. Introduction
In recent years, carbon fiber-reinforced polymer composites (CFRP) have been widely used in aviation, aerospace, subway tracks, and other fields. However, due to their anisotropy, multiphase, and heterogeneous nature, they are prone to drilling delamination damage in the drilling process, which seriously affects the reliability of CFRP parts [
1]. To reduce the delamination damage in CFRP drilling, scholars have conducted a lot of research on the delamination damage in the drilling process around the drilling force. It is generally believed that there is a critical axial force (
PC) [
2]. When the drilling axial force is lower than
PC, delamination damage will not occur during drilling; on the contrary, delamination damage will occur when the force is higher than
Pc [
3]. Based on these studies, a series of analytical models for predicting the
PC has been established [
4]. For example, Hocheng et al. [
5] proposed the concept of
PC in 1990 and established the mechanical analytical model of
PC based on linear elastic fracture mechanics and classical beam–plate bending theory. They believe that delamination defects do not occur when the axial force is less than a certain critical value in the process of composite drilling. However, their model assumes that the composite material is isotropic, which is inconsistent with the actual situation of the material, resulting in large errors when predicting the
PC.
Since then, many scholars have improved the model of
PC based on the Hocheng model. For example, Lachuad et al. [
6] established an analytical model of
PC with anisotropic mechanical properties of composites based on the classical laminated plate theory and Hocheng model, and they proved that the analytical model established by distributed load has higher accuracy than the concentrated load model for the first time. Jain et al. [
7] first introduced the idea that the shape of the drilling delamination area is elliptic and characterized the delamination area by a coefficient, which improved the prediction accuracy of the
PC analytical model. However, neither the model of Lachuad [
6] nor Jain [
7] considers the bending torsion coupling effect of laminates. Therefore, their models can only predict the
PC of unidirectional laminated plates. Zhang et al. [
8] considered the bending torsion coupling effect of laminates based on Jain [
7] and established an analytical model of
PC that can analyze any laminate stack sequence. To further improve the accuracy of
PC, Ojo et al. [
4] subdivided the chisel edge and the main cutting edge of a drill. They assume that the force caused by the chisel edge is a concentrated force and that the force caused by the main cutting edge is a uniformly distributed force. In their model, they put forward the hypothesis that the concentrated force and the uniformly distributed force both exist, and they confirmed that the ratio of concentrated force and distributed force has an impact on the
PC. In addition, the model also puts forward the assumption that the drilling delamination crack propagation form is I/II mixed type and analyzes it. Saoudi et al. [
9] first studied the analytical model of the
PC under mechanical–thermal coupling. The results showed that the drilling
PC would change under the influence of drilling temperature, but the mechanical properties of the CFRP materials were not considered in the model.
Table 1 is a summary of the
PC model in the above references, where
PC is the critical axial force,
GIC is a mode I crack, and
C3, K,
D11,
D22, and
D are the stiffness coefficients of undrilled materials.
Through the above analysis of the
PC model, it can be found that the “fracture toughness” is an important factor that affects the
PC. However, due to the complexity of the CFRP drilling process, it is difficult to directly observe the failure mode of drilling delamination from the experimental results. Most of the above models assume that the failure mode of drilling delamination crack is the mode I crack failure form. In fact, due to the complexity of force in the drilling process of composite materials, the failure mode is often a combination of different failure modes [
1], and thus, using the mode I crack alone cannot reflect the actual cracking situation. In addition, for CFRP, because the temperature has a great impact on the mechanical properties of the resin matrix, some mechanical properties related to the resin, especially the fracture toughness
GIC or
GIIC, will change greatly with an increase in temperature, but the above models do not consider the influence of the temperature-dependent characteristics of CFRP mechanical properties on
PC. This research found that
GIC,
C3,
K,
D11,
D22, and
D in the classical calculation equation of
PC are affected by temperature changes and that the influence trends are different. The drilling process of composite materials produces a lot of heat, which increases the drilling temperature. Therefore, the influence of drilling temperature on the
PC should also be considered when analyzing the
PC of CFRP drilling delamination.
Therefore, we established a three-dimensional finite element drilling model of CFRP, including an interface phase, to study the interlaminate damage mechanism of CFRP drilling and obtain the interlaminate damage mode of CFRP caused by drilling; then, based on the
PC model of Zhang [
8], we deduced a new
PC model that considers the temperature-dependent characteristics of CFRP mechanical properties and the damage mode of CFRP interlaminate; finally, the
PC model was verified by static compression experiments at different temperatures.
3. Analysis of Simulation Results
Figure 2 shows the simulation results of CFRP drilling. The white part is the CFRP element, the blue part is the interlamination interface element, and the red part represents the damage degree of the interface layer. To more clearly show the position and failure mode of the interlamination element in the laminated plate during the drilling process, some composite elements are hidden in
Figure 2.
In order to more clearly show the damage and failure process of interlaminate interface elements during the drilling process, part of the CFRP element layer and the interlaminate element layer of the middle part are hidden in
Figure 3 and
Figure 4, and only the first layer of CFRP, the first layer of interlaminate interface elements (short for first interface), and the last layer of interlaminate interface elements (short for last interface) are reserved for location reference.
Figure 3 is a comprehensive diagram of
Figure 4.
Figure 4 is a representative figure of the drill bit passing through every layer of CFRP and every layer of the interface at different stages.
Figure 4(a1–d1) is the front view of
Figure 4 (a2–a4), (b2–b4), (c2–c4), and (d3,d4), respectively.
It can be seen from
Figure 4(a1–a4) that, when the drill bit is drilled to stage I, the chisel edge makes contact with the first layer of the CFRP. Although the removal amount of CFRP material in the first layer is very small
Figure 4(a2), the interlaminar element in the first layer also begins to be damaged due to the downward bending load. When the drill bit is drilled to stage II
Figure 4(b1–b4), the main cutting edge starts to work, the contact delamination area between the first layer of the CFRP and the drill increases
Figure 4(b2), and the damage delamination area of the first interface increases
Figure 4(b3). At this time, the last interface is still not affected. When the drill bit is drilled to stage III
Figure 4(c1–c4), the main cutting edge starts working, the drilling axial force increases, the contact delamination area between the first layer of the CFRP and the drill increases
Figure 4(c2), and the damage delamination area of the first interface increases
Figure 4(c3). At this time, the last interface begins to be damaged
Figure 4(c4). When the drill bit is drilled to stage IV
Figure 4(d1–d4), the main cutting edge is fully working, and the drilling axial force increases to its maximum. Accordingly, the damage delamination area of both the first layer of the CFRP and the first interface increase to their maxima
Figure 4(d2,d3). As the drill bit is not in contact with the last interface, the damage evolution area of the last interface at this stage does not expand significantly. The above results show that the [0/45/90/−45]s layers can cover all the drilling processes of this bit, and all CFRP layers and interface layers need to go through stages I to IV.
Due to the laminated structure characteristics of CFRP, the crack form often occurs between layers during drilling, accompanied by a coupling effect. To judge the cracking form of the laminated plate in the drilling process, it is necessary to analyze the interlamination failure form in FEM results combined with different drilling stages.
Figure 5 shows the section view of the finite element simulation results when the chisel edge just contacts the material (drilling stage I). It can be seen from
Figure 5 that due to the width of the chisel edge, it will exert a pressing force on the vertical feeding direction of the first layer of the CFRP material of the laminate, which will cause the material of this layer to produce a shear force (κ) in the vertical extrusion direction and then cause a type II crack. At the same time, the downward feeding movement will produce a downward extrusion force (
σ) on the uncut layer and then cause a type I crack. As the shear force and extrusion force exist simultaneously, the CFRP at the entrance is affected by the chisel edge, resulting in the coupling of I/II cracks (
Figure 5).
When the drill bit continues to feed along the axial direction, the main cutting edge begins to work (drilling stage II). Any point on the main cutting edge will generate a shear force (κ) in the direction of the cutting speed, which will form a type III crack. At the same time, the chisel edge feeds downwards, and the downward feeding movement will produce a downward extrusion force (
σ) on the uncut layer, causing a type I crack. Because the shear force in the direction of the cutting speed and the downward extrusion force exist simultaneously, this results in the coupling of I/III cracks (
Figure 6).
When the main cutting edge is fully working (drilling stage IV), the material removal form and the delamination form caused by the chisel edge and the main cutting edge are the same, type I/II cracks are caused by the chisel edge, and type I/III cracks are caused by the main cutting edge. However, when drilling to the exit, as the number of uncut layers decreases, the residual stiffness also decreases, and the I/II cracks caused by the chisel edge are removed by the movement of the main cutting edge. Because the shear force (κ) in the direction of the cutting speed and the downward extrusion force (
σ) exist simultaneously, I/III cracks couple at the drilling exit. Therefore, the main delamination form of the drilling exit is the I/III mixed crack (
Figure 7).
4. Modeling of CFRP Drilling Delamination PC
In this section, based on the drilling axial force model of Zhang [
8], the temperature-dependent characteristics of CFRP mechanical properties and the mixed I/III delamination failure mode at the drilling exit, a prediction model of the drilling
PC is established. The model derivation process is as follows:
First, assuming that the shape of the CFRP drilling exit layer is elliptical [
5], the force state of the delamination damage at the drilling exit is shown in
Figure 8. In
Figure 8a,
h is the thickness of the undrilled layers,
a is the longitudinal radius of the ideal damaged ellipse, and b is the transverse radius of the ideal damaged ellipse (
Figure 8b).
Assuming that all the work accomplished via the drilling axial force at the drilling exit (
Figure 8) is converted into energy released by delamination and energy required by material strain, then, based on the linear elastic fracture mechanics and the law of energy conservation, the balance relation equation of energy required for delamination can be established:
where
Pc is the critical axial force (N);
dω0 is the differential cross-section deflection of uncut laminates;
GC is the critical energy release rate (N/mm);
dA is the differential layered area (mm); and
dU is the differential of the strain energy required for the elastic strain of the material (N).
The process of solving the Pc is also the process of solving dω0, GC, dA, and dU in the equation. In previous studies, the influence of cutting temperature has not been considered when solving the above parameters. This study analyzes the equation based on the temperature-dependent characteristics of CFRPs’ mechanical properties. The detailed process is as follows.
4.1. The Solution of Section Deflection dw0 of Undrilled Layer Material
According to the classical theory of laminated plates, the constitutive relationship of CFRPs considering the mechanical and thermal coupling properties is shown in Equation (3).
where [
Ni] is the total internal force under mechanical–thermal coupling; [
Mi] is the total bending moment under mechanical–thermal coupling; [
Nim] is the internal forces caused by the mechanical load; [
Mim] is the bending moments caused by mechanical forces; [
NiT] is the internal forces caused by thermal loads; [
MiT] is the bending moment induced by the thermal load; [
Ai] is the components of the extensional stiffness matrix; [
Bi] is the components of the extension–bending coupling matrix; and [
Di] is the components of the bending coupling matrix. The calculation methods of [
Ai], [
Bi], and [
Di] are from [
8].
Assuming that the displacement is very small during drilling, the relationship between strain, curvature, and displacement is:
where
ε is the in-plane strain; u and v are the displacements in the x and y directions, respectively;
τ is the curvature of the laminate midplane; and
ω is the displacement perpendicular to the laminate.
The drilling force model of undrilled CFRP laminate can be simplified as a plate model with a concentrated force in the middle. According to the basic bending equilibrium differential equation of plate shell theory:
Assuming that the load is uniformly distributed on the surface of the failure zone: , where .
The relationship between the internal force, bending moment, and displacement can be obtained by substituting the strain component and stiffness matrix of [
Ai], [
Bi], and [
Di] into Equation (3). Then, the internal force and bending moment are substituted into Equation (12) to obtain the relationship between the displacement and
PC (Equations (13)–(17)). C
j is the material performance coefficient of the undrilled material.
The deflection of the undrilled layer material is obtained by deriving the diameter of the elliptical long axis in Equation (15):
4.2. Solution of Elastic Strain Energy dU of Undrilled Material
According to classical plate theory, the strain energy of the plate model can be obtained via the following equation:
Here, the relationship between stress and strain under thermal–mechanical coupling is:
where
is the total strain
is the strain caused by mechanical force; and
is the coefficient of thermal expansion.
Equation (21) is obtained by expanding Equation (20) and substituting the expanded Equation (20) into Equation (19):
Assuming that the temperature difference between the drilling temperature and the drilling ambient temperature is 0,
, and the strain energy generated by pure mechanical strain is:
Pure mechanical strain can be obtained by substituting Equations (13)–(17) into Equations (4) and (9):
By substituting Equations (23)–(28) into Equation (22), the strain energy of undrilled laminates caused by mechanical force can be obtained when the drilling temperature difference is 0:
where:
Referring to the solution of thermal strain in Saoudi [
9], when the temperature difference between the drilling temperature and the drilling ambient temperature is not zero, the total strain energy including thermal strain caused by the undrilled laminate is as follows:
The basic assumptions of classical laminated plate theory are:
Then, in Equation (31):
where:
where Δ
T is the temperature difference between the drilling temperature and the ambient temperature.
Then, the calculation equation of undrilled materials’ strain energy under mechanical-thermal coupling can be deduced:
From the derivation of Equation (39), we can obtain:
4.3. Solution of Fracture Randomness
Through the analysis of the results of a three-dimensional drilling finite element analysis, it can be seen that, when the drill bit is drilled at the exit, the delamination failure mode is type I/III mixed mode, so the coupling of I and III cracks should be considered in the analytical model of
PC. According to the BK damage criterion:
Then,
where
r is the mixing coefficient of the fracture toughness of type I/III cracks and
. This can be solved via the conjugate gradient method according to inverse problem theory.
When substituting the Gc, dA, dw, and
dU obtained above into the critical layered energy balance relationship, the analytical equation of the
PC can be obtained as follows:
The temperature-dependent parameters in the equation are G
IC, G
IIIC, K,
Cj, and
D*. The mechanical parameters of
GIC that vary with temperature are obtained via the
ASTM D5528-01 standard [
17], the experimental results of
GIC are shown in
Figure 9, and a detailed experimental process is shown in [
18]; the mechanical parameters of
GIIIC that vary with temperature are obtained from the measurement results of
GIIC [
19], the mechanical parameters of
GIIC that vary with temperature are obtained via the
ASTM D7905M-14 standard [
20], the experimental results of
GIIC are shown in
Figure 10, and the detailed experimental process is shown in
Appendix A; the values of
K are calculated via Equation (30); the calculation methods of
Cj are from [
8].
Both
K and
Cj are calculated by the combination of the [
Ai], [
Bi], and [
Di] stiffness matrices. As the mechanical properties of carbon fibers hardly change with a change in drilling temperature in this temperature range, the parameters affecting the values of the [
Ai], [
Bi], and [
Di] stiffness matrices are calculated using the modulus of the resin matrix with temperature, which is obtained via the tensile test of the resin at different ambient temperatures (
Figure 11). The detailed experimental process is shown in [
18]; in addition to the stiffness matrix of [
Ai], [
Bi], and [
Di], the value of
D* at different temperatures is also affected by the thermal expansion coefficient. The thermal expansion coefficients of the unidirectional CFRP were measured using a dilatometer (NETZSCH DIL 402C). The test sample laying method was [0]
40, the sample size was 10 mm × 10 mm × 5 mm, and the measuring temperature range was −50–200 °C. The thermal expansion coefficients of the unidirectional CFRP in the longitudinal and transverse directions are shown in
Figure 12.
4.4. The Solution of the Layered Area
Assuming that the shape of the exit layer is an ellipse, the area of the layer is the area of the ellipse minus the area of the ideal hole. The assumptions are:
where
is an ellipse ratio. Then, the differential of the stratified area is:
5. Critical Axial Force Verification Test
5.1. Experimental Setup
To verify the accuracy of the drilling PC model derived above, a temperature-controlled PC experiment must be carried out. However, due to the influence of the process parameters on the drilling temperature and drilling force, it is difficult to obtain a PC that only produces delamination at different drilling temperatures through drilling tests. In addition, the experimental data from past studies on drilling PC cannot be referenced because the influence of drilling temperature is not considered, but the experimental method can be referenced. In previous studies, a static compression experiment was used to measure the PC.
To verify the theoretical model introduced in the previous section, this section draws on this test method to design a PC equivalent test at different ambient temperatures. However, because the PC experiment can only approximately simulate the axial force generated by the downward feed of the drill bit and cannot simulate the torque caused by the rotation of the drill bit, the failure mode of the exit stratification of the PC experiment is the only mode I open failure.
The CFRP composite used in this study was a carbon T300/epoxy unidirectional prepreg with a ply thickness of 0.125 mm. The paving sequence of materials used in the experiment is [0/45/90/−45]4S. The reason why we use [0/45/90/−45]4S to verify the analytical mode is that a thick layer can form better-quality blind holes. After laying the plate preform manually, we put it into the autoclave for heating and curing. The curing conditions are heating to 80 °C, holding for 30 min, then pressurizing to 0.5 MPa, heating to 120 °C, holding for 90 min, and finally cooling in a furnace. During the experiment, non-drilled conical blind holes with a thickness of 1–6 layers are prefabricated on the tested sample. Due to the taper of the drill, when the number of layers is small, the horizontal edge of the drill drills the blind holes. To ensure that drilling stratification does not occur during the prefabrication of blind holes, a back plate is placed under the laminated plate to be drilled during drilling to increase the exit stiffness. The spindle speed is 4000 rpm, and the feed rate is 0.03 mm/r.
Figure 13 shows the
PC testing bench at different ambient temperatures. During the experiment, the sample was fixed on the platform of the universal testing machine. The exit temperature of the blind hole was locally heated by a silica gel heater. The temperature of the heater was set to 23 °C, 60 °C, 90 °C, and 120 °C, respectively. The experimental pressure head was a carbide drill bit with a diameter of 8 mm. The compression speed was set to 2 mm/min during the test.
5.2. Result Discussion
Figure 14 shows a comparison between the
PC results predicted by Equation (45) and the experimental results under different drilling ambient temperatures. It can be seen from
Figure 14 that the theoretically predicted
PC value is in good agreement with the experimental value when the drilling temperature is 120 °C, and there are some errors when the drilling temperature is 23 °C, 60 °C, or 90 °C. There are two reasons for this error. Firstly, the influence of the geometry of the drill bit on the drilling axial force distribution is not considered in this
PC model. Secondly, when measuring the fracture toughness, the influence of different ply angles is not considered.
However, both the prediction results and the experimental results show that the drilling temperature has a great impact on the critical axial force, and the critical axial force increases with an increase in the drilling temperature. Additionally, the change trend of the curve is consistent when the drilling temperature is 23 °C, 60 °C, and 90 °C. From the general trend, it can be seen that the PC of CFRP drilling delamination is greatly influenced by the mechanical properties of the CFRP. When the drilling temperature is not greater than the glass transition temperature range of the material itself (about 120 °C), due to the increase in fracture toughness, the PC increases with an increase in drilling temperature.