A subjective-objective evaluation method of hole surface quality in drilling CFRP-Ti stacks

Abstract

Hole surface quality in drilling carbon fiber-reinforced plastics and titanium alloy (CFRP-Ti) stacks is a key factor affecting high-quality assembly and reliable usage of the structure so it need be reasonably evaluated. However, there is no standardized evaluation system of hole surface quality in drilling CFRP-Ti stacks. The existing evaluation indexes cannot meet the evaluation requirements on the surface quality of the hole. This paper proposes a subjective-objective evaluation method to evaluate the surface quality of the hole in drilling CFRP-Ti stacks. Firstly, the evaluation factors are determined by experiments, including CFRP delamination, CFRP hole entrance tears, surface roughness of CFRP and Ti, and the evaluation system is established according to defect characteristics and industry standards. Secondly, the subjective evaluation model is established based on the improved analytic hierarchy process, the objective evaluation model is established based on the entropy weight method and the subjective and objective combination weight calculation method is obtained based on the principle of minimum discriminant information. Finally, the surface quality of the hole is evaluated by subjective, objective, subjective-objective evaluation methods respectively. The result shows that subjective-objective evaluation method is more reasonable and effective.

Introduction

Carbon fiber-reinforced plastics (CFRP) have many advantages, such as high specific strength, fatigue resistance, corrosion resistance and excellent designability. ¹ It is often joint with titanium alloys (Ti) in the aviation field to form CFRP-Ti stacks. A large number of holes need to be machined in aircraft assembly. In order to ensure hole coaxiality and processing efficiency, CFRP-Ti stacks are usually machined by integrated drilling. ² The properties of two materials are quite different, and the applicable cutting parameters are different. Therefore, there are some problems of hole surface quality that affect the assembly quality. Reasonable evaluation on the hole surface quality is important for the decision-making and optimization of the process plan. For Ti, the surface quality can be evaluated by using the evaluation system of the metal directly combined machining character itself. Generally, the surface roughness is used to describe the surface quality of Ti.^3,4 However, the surface characteristics of CFRP after machining are different from the metal, the evaluation system of mature metal surface quality is not fit to the CFRP.

The researches have been done on the evaluation of the surface quality in drilling CFRP. For delamination, Chen. ⁵ first proposed One-dimensional diameter delamination factor $F_{\mathrm{d}}$ as the evaluation criterion, that is, the ratio of the maximum delamination diameter to the nominal hole diameter. Faraz et al. ⁶ considered the influence of delamination area, and used two-dimensional area delamination factor $F_{\mathrm{a}}$ to evaluate CFRP delamination. Davim et al. ⁷ considered the influence of the diameter and the area synthetically, combined $F_{\mathrm{d}}$ with $F_{\mathrm{a}}$ with certain weight, and proposed the adjusted delamination factor $F_{\mathrm{da}}$ . Babu et al. ⁸ proposed to characterize the delamination with a factor called the effective equivalent delamination factor (FEED). It gave equal weightage to the crack length and damage area. Nagarajan et al. ⁹ proposed the delamination factor $F_{\mathrm{DR}}$ based on Buckingham theorem. According to the damage degree, the delamination is divided into three grades: severe, ordinary and low. The final calculation is obtained by using ANNA. Wei et al. ¹⁰ measured CFRP delamination using high frequency ultrasound scanning microscope and found that the delamination usually occurred on multi-layer materials. The three-dimensional volume delamination factor $F_{\mathrm{v}}$ was further proposed. For burrs and tears, Li et al. ¹¹ took the average value of the maximum tear length of some holes under the same process parameters as the evaluation criterion. Xu et al. ¹² discussed the cutting-induced damage in drilling T800S/250F carbon fiber–reinforced polymer. Fiber pullout, delamination, and tearing were evaluated respectively by the indicators, single maximum uncut-off fiber length, the maximum delamination diameter and the bur area. For surface roughness, Dold et al. ¹³ used $R_a$ to evaluate surface roughness of CFRP. The measurement methods, sampling conditions and evaluation parameters were studied. Çolak and Sunar ¹⁴ measured the surface quality of CFRP by three-dimensional profilometer, and found that the surface topography was uneven. Therefore, three-dimensional parameters were used to evaluate the surface roughness of CFRP.

Seldom studied the comprehensive evaluation of surface quality in drilling CFRP. Quan et al. ¹⁵ proposed a valuation method of carbon/epoxy composite hole-making damages based on the relative index in statistical analysis. In that method, the burrs need be measured when its length is more than 0.5 mm and its width is more than 0.1 mm. The length and area of the burrs are summed up, and compared with the nominal hole diameter and hole area. The burrs evaluation factor is obtained by the adding results according to a certain weight. The area of tear was compared with 3D×1.8 as a reference, and the tear evaluation factor could be obtained. Zuluaga-Ramírez et al. ¹⁶ correlated the surface irregularity factor with stiffness degradation, and proposed a new method for evaluating the damage state of CFRP. The surface irregularity factor of CFRP was measured by optical method, and the fatigue damage of material was evaluated by equal-amplitude fatigue load and real variable-amplitude load. It is found that there is good consistency between surface irregularity factor and damage evaluation indexes like stiffness degradation so surface irregularity factor can be used to evaluate the surface quality of CFRP.

Above all, some achievements have been gotten about indexes of surface quality of CFRP. Most studies use partial indexes as the evaluation criteria, which cannot fully evaluate surface quality of CFRP. The weights are mostly given subjectively. To evaluate CFRP well, standardized evaluation index system needs to be built. And there is little literature on the surface quality evaluation system of the CFRP-Ti stacks. Therefore, this paper proposes a comprehensive evaluation method combining subjective and objective evaluation of hole surface quality in drilling CFRP-Ti stacks.

Evaluation index system of hole surface quality

Evaluation factors

With the improvement of material and tool properties, some hole surface quality problems have been suppressed in drilling CFRP-Ti Stacks. Evaluation factors will be determined by experiments. The selection of the appeared defects in the experiments is important and will greatly influence the construction of the evaluation system.

Drilling experiment

The matrix material of CFRP is epoxy resin, and the laying direction is 0°. The thickness of CFRP and Ti is 3 mm respectively. Mechanical properties of CFRP and Ti are shown in Tables 1 and 2.

Table 1.

Mechanical properties of CFRP.

Material grade	Tensile strength of 0°/(MPa)	Compressive strength of 0°/(MPa)	Tensile strength of 90°/(MPa)	Compressive strength of 90°/(MPa)	Interlaminar shear strength/(MPa)
T300-12K	1205	739	50.1	183	83.9

Table 2.

Mechanical properties of Ti.

Material grade	Chemical composition	Tensile strength/(MPa)	Yield strength/(MPa)	Shear strength/(MPa)	Elastic modulus/(GPa)
TC4	Ti–6Al–4V	1205	739	50.1	183

Carbide twist drill is used without coating whose diameter is 6mm shown in Table 3.

Table 3.

Geometric parameters of twist drill.

Material grade	Rake angle/(°)	Clearance angle/(°)	Helix angle/(°)	Chisel edge length/(mm)	Drill body length/(mm)	Drill shank length/(mm)
YG8	30	12	30	1	30	30

Drilling CFRP can adopt high speed that can be more than 6000 r/min, even 20000 to 30,000 r/min. Drilling Ti is easy to break the cutting tool for the high spindle speed and need keeping very low feed which is decided by the character of Ti.^3,4 The speed of drilling Ti and the feed rate of CFRP are selected for drilling CFRP/Ti stacks. The range of process parameters in drilling CFRP-Ti Stacks should be: spindle speed is 600 to 1350 r/min, feed rate is 0.03 to 0.18 mm/r, and point angle is 100 to 135 degrees. The orthogonal experimental parameters are shown in Table 4.

Table 4.

The experimental parameters.

Group number	Spindle peed/(r/min)	Feed rate/(mm/r)	Point angle/(°)
1	600	0.03	100
2	600	0.1	135
3	600	0.18	118
4	1000	0.03	135
5	1000	0.1	118
6	1000	0.18	100
7	1350	0.03	118
8	1350	0.1	100
9	1350	0.18	135

NC machining center is adopted with dry cutting shown in Figure 1.

Figure 1.

The experimental setup.

Experimental analysis

Generally, delamination, tears and burrs often occur at the entrance and exit of drilling CFRP. The experimental result is shown in Figure 2. Burrs do not appear at hole entrance or exit. There are obvious delamination and tear at the entrance of the hole. The quality of the exit is good, only slight delamination occurs for Ti supports CFRP.

Figure 2.

The appearance of hole entrance and exit of CFRP.

In Figure 3, it can be clearly seen that surface roughness of CFRP at different fiber orientation is different.

Figure 3.

The appearance of hole wall of CFRP.

Generally, the exit burr and surface ablation often occur in drilling Ti but they have not been observed in this experiment shown in Figure 4. At the current process level, there is not almost exit burrs or surface ablation without influence of cutting heat caused by serious tool wear or continuous drilling.

Figure 4.

The appearance of hole entrance and exit of Ti.

When drilling CFRP-Ti stacks, for CFRP, the entrance delamination is larger than exit one, tears only occurs at hole entrance, burrs do not appear at hole entrance or exit, and surface roughness has a certain relationship with fiber orientation. While exit burrs and surface ablation does not appear for Ti. Therefore, the evaluation factors include CFRP delamination, CFRP hole entrance tears, surface roughness of CFRP and Ti.

Evaluation index of hole surface quality

CFRP delamination

Evaluation methods of CFRP delamination mainly include:

1. One-dimensional diameter delamination factor $F_{\mathrm{d}}$ : the ratio of the maximum delamination diameter to the nominal hole diameter is used as the evaluation criterion. It is simple to calculate and convenient to measure, but the influence of delamination area is not considered at all.

2. Two-dimensional area delamination factor $F_{\mathrm{a}}$ : the ratio of the delamination area to the nominal hole area is used as the evaluation criterion, but the influence of the delamination diameter is neglected.

3. The adjusted delamination factor $F_{\mathrm{da}}$ : by introducing two weight coefficients, the influence of delamination diameter and delamination area is considered comprehensively, but the weight distribution is not reasonable enough.

4. Three-dimensional volume delamination factor $F_{\mathrm{v}}$ :

$$F_{\mathrm{v}}=\frac{1}{N}\left(\sum _{i=1}^N\frac{A_{\mathrm{d}}^i}{A_0}\right)=\frac{1}{N}\sum _{i=1}^N{F}_{\mathrm{a}}^i$$

(1)

where

$A_0$ is the nominal hole area,

$A_{\mathrm{d}}^i$ is the delamination area of the i th material layer,

$F_{\mathrm{a}}^i$ is the two-dimensional area delamination factor of the i th material layer,

N is the total number of material layers with delamination.

Methods 1 to 3 only consider delamination evaluation of a single layer. Actually, delamination may occur inside of the material and have an effect on structural strength. Method 4 considers the cumulation of the delamination. However, the delamination of internal material is significantly less than that of hole entrance and exit.^17–20 The maximum of $F_{\mathrm{a}}^i$ is averaged that is, the worst delamination is ignored. The calculating result is better than the actual state. In addition, it is difficult to obtain clear delamination situation of each layer for precision calculation by the present technology.

Therefore, a more reasonable delamination evaluation method is needed that should reflect the overall situation of the delamination rather than evaluating only for a single material layer and the degree of delamination diameter and delamination area. An evaluation method, “cumulative delamination,” is proposed that is the sum of the delamination area of each layer. It need not the delamination image of each layer but the superposition image by Ultrasonic C-scan. It enlarges the delamination degree, represents the delamination limit, and is more meaningful for safe service. The evaluation result based on cumulative delamination can ensure the structure safety.

The weights cannot be directly allocate to add $F_{\mathrm{d}}$ and $F_{\mathrm{a}}$ for no study shows how much impact they will have on the structures. Therefore, two indexes are defined, the maximum delamination diameter factor ( $D_{\mathrm{d}}$ , 0< $D_{\mathrm{d}}$ <1) and the delamination area factor ( $D_{\mathrm{a}}$ , 0< $D_{\mathrm{a}}$ <1), shown in Figure 5.

$$D_{\mathrm{d}}=\frac{D_{max}-D_0}{2\times b_{max}}$$

(2)

$$D_{\mathrm{a}}=\frac{A_{\mathrm{d}}}{A_{\mathrm{b}max}-A_0}$$

(3)

where

$A_0$ is the nominal hole area,

$A_{\mathrm{b}max}$ is the area of a circle with $b_{max}$ as its diameter,

$A_{\mathrm{d}}$ is the delamination area,

$b_{max}$ is the allowable damage range of holes,

$D_{max}$ is the maximum delamination diameter,

$D_0$ is the nominal hole diameter.

The Chinese industry standard is adopted for the calculation. The value of $b_{max}$ is 2.5 mm when the diameter of the hole is 6 mm.

Figure 5.

Schematic diagram of calculation of D_d and D_a.

Tears of CFRP

Evaluation of CFRP tears mainly include:

1. Average tears length $L_{\mathrm{s}}$ : the average value of the maximum tears length on both sides of the hole is used as the evaluation criterion. The influence of tears area is not considered, and it is not suitable for holes with different diameters.

2. Tears area factor $F_{\mathrm{s}}$ : the ratio of the total area of all tears to the nominal hole area is used as the evaluation criterion. The influence of tears length is not considered. In addition, for tears area is usually much smaller than the nominal hole area, there is no obvious difference between the holes with different situations of tears.

The situation of CFRP entrance tears is shown in Figure 6. The tears must occur along the fiber direction, mostly diagonally and more than one place. Therefore, tears should be measured along the fiber direction and considered in terms of the length and the area.

Figure 6.

CFRP entrance tears.

According to the specification for riveting for CFRP, the tear damage with a length less than 2 times the hole diameter and a width less than 1.8 mm is allowed when drilling CFRP. The maximum tears length factor ( $S_{\mathrm{L}}$ ,0< $S_{\mathrm{L}}$ <1) and the tears area factor ( $S_{\mathrm{a}}$ , 0< $S_{\mathrm{a}}$ <1) are proposed to evaluate tears.

$$S_{\mathrm{L}}=\frac{L_1+L_2}{2\times D_0}$$

(4)

$$S_{\mathrm{a}}=\frac{\sum _{j=1}^M{A}_{\mathrm{s}}^j}{2D_0\times 1.8}$$

(5)

where

$A_{\mathrm{s}}^j$ is the area of the j th tears area,

$L_1$ and $L_2$ are the maximum tear length on both sides of the hole,

M is the total number of areas where tears occur.

Surface roughness of CFRP and Ti

As an anisotropic material, the geometric characteristics of hole wall in drilling CFRP are quite different from the metal. The theory of the metal cannot be used directly. At present, the evaluation methods of surface roughness in drilling CFRP mainly include:

1. Improved two-dimensional evaluation method: two-dimensional evaluation parameters are used and samples are taken from different positions of the hole wall to obtain the average surface roughness. The sampling positions are determined artificially, which do not consider the surface morphology of hole wall.

2. Three-dimensional evaluation method: Sampling the whole hole wall, and three-dimensional evaluation parameters are used. It considers the surface morphology of hole wall. However, it is difficult to measure three-dimensional surface morphology. The sampling is difficult and the evaluation process is cumbersome.

The surface morphology of hole wall is related with the angle of fiber and cutting tool. It means that the roughness is related with this angle. It is necessary to explore the regularity of surface roughness of CFRP and Ti, and accurately obtain the sampling positions. Then two-dimensional evaluation parameters can be used to achieve the same effect of three-dimensional evaluation. It is simple to measuring and evaluating. CFRP, an anisotropic material, which needs to measure the surface roughness of the hole wall at different angles and depths. While Ti is an isotropic material, which only needs to measure the surface roughness of the hole wall at different depths. The measuring equipment is MarSurf M300 with a scanning length of 1.75 mm. The arithmetic mean deviation of contour, $R_a$ , is the arithmetic mean of the distance from each point on the contour line to the center line within the sampling length. It reflects the mean level of contour fluctuation. $R_z$ is the distance between the highest peak and the lowest valley on the contour line within the sampling length. It reflects the the limit of the contour. Therefore, $R_a$ and $R_z$ are adopted as the evaluation parameters of surface roughness.

The measuring results and analysis are as follows:

• (1) The influence of fiber orientation and transition zone on roughness of CFRP

The variation trend of surface roughness of CFRP with fiber orientation and transition zone is similar under different process parameters. The variation trend of $R_a$ and $R_z$ is respectively shown in Figure 7. The following rules can be found: with the increase of $\theta$ from 0° to 180°, both $R_a$ and $R_z$ decrease gradually at first, and reach the minimum when $\theta$ ≈ 60°. After that, they increase gradually, and the rate of increase accelerates when $\theta$ > 120°, and they reach the maximum when $\theta$ ≈ 150°. Then they decrease rapidly. When $\theta$ = 180°, they are almost the same as when $\theta$ = 0°.

Figure 7.

The variation trend of R_a and R_z of CFRP in drilling CFRP-Ti stacks.

In order to confirm that transition zone does effect on surface roughness of CFRP, the roughness of hole wall near entrance and exit is measured, the results are shown in Figure 8. There is little difference in the roughness between hole walls near entrance and exit, which proves that transition zone has effect on the surface roughness of CFRP in drilling CFRP-Ti stacks.

• (2) The influence of transition zone on surface roughness of Ti

The surface roughness of Ti is shown in Figure 9. The transition zone has little effect on surface roughness of Ti for the cutting edge never touches the hole wall of Ti when drilling in transition zone.

Figure 8.

The variation trend of R_a and R_z of hole wall near entrance and exit in drilling CFRP separately.

Figure 9.

R_a and R_z of transition zone and hole wall near exit of Ti in drilling CFRP-Ti stacks.

Thus, it can be determined that the sampling position of CFRP is transition zone when $\theta$ = 60° and $\theta$ = 150°, and there is no specific restriction on the sampling position of Ti. The evaluation indexes of surface roughness in drilling CFRP-Ti stacks are as below:

1. $R_a^{\mathrm{t}60}$ : $R_a$ of transition zone of CFRP when $\theta$ = 60°;

2. $R_a^{\mathrm{t}150}$ : $R_a$ of transition zone of CFRP when $\theta$ = 150°;

3. $R_z^{\mathrm{t}60}$ : $R_z$ of transition zone of CFRP when $\theta$ = 60°;

4. $R_z^{\mathrm{t}150}$ : $R_z$ of transition zone of CFRP when $\theta$ = 150°;

5. $R_a^{\mathrm{Ti}}$ : $R_a$ of any hole wall position of Ti;

6. $R_z^{\mathrm{Ti}}$ : $R_z$ of any hole wall position of Ti;

Evaluation index system

Before establishing evaluation index system, the indexes need the treatment of consistency or dimensionless. There are ten evaluation indexes of holes surface quality. Their expectations are as small as possible and it is not necessary for consistency treatment. At the same time, $D_{\mathrm{d}}$ , $D_{\mathrm{a}}$ and $S_{\mathrm{L}}$ , $S_{\mathrm{a}}$ have been dimensionless. The evaluation indexes of surface roughness are converted to the dimensionless value. According to the requirement of the riveting for the hole wall, $R_a$ of CFRP should be no more than 6.3 $\mu$ m and $R_z$ should be no more than 25 $\mu$ m. Therefore, $R_a$ = 6.3 $\mu$ m and $R_z$ = 25 $\mu$ m can be taken as reference standards of surface roughness. By calculating the ratio between the above evaluation indexes to the reference standards, the dimensionless evaluation indexes of surface roughness can be obtained: $R_a^{\mathrm{n}\mathrm{-}\mathrm{t}60}$ , $R_z^{\mathrm{n}\mathrm{-}\mathrm{t}60}$ , $R_a^{\mathrm{n}\mathrm{-}\mathrm{t}150}$ , $R_z^{\mathrm{n}\mathrm{-}\mathrm{t}150}$ , $R_a^{\mathrm{n}\mathrm{-}\mathrm{Ti}}$ , $R_z^{\mathrm{n}\mathrm{-}\mathrm{Ti}}$ .

Finally, evaluation index system of hole surface quality in drilling CFRP/Ti stacks can be obtained shown in Figure 10.

Figure 10.

Evaluation index system of hole surface quality in drilling CFRP/Ti stacks.

Subjective-objective evaluation method

It is still difficult to get the overall situation of hole surface quality only by some separate evaluation indexes. The comprehensive evaluation method of hole surface quality will be carried out in this paper.

The evaluation methods include subjective and objective. The former is determined by experts according to their own experience, and the latter is based on sample data. Subjective evaluation method can show knowledge of experts, but its subjectivity is too strong. Objective evaluation method is not affected by human factors, but the weighting results are independent of the importance of indexes, and very dependent on samples. This paper proposes a comprehensive evaluation method based on subjective and objective combination. First, subjective and objective evaluation models are established respectively. Then the subjective and objective combination weight is obtained by certain calculation method. The weights not only reflect experts’ subjective perception, but also make full use of sample data.

Subjective evaluation method based on improved AHP

Analytic hierarchy process (AHP) is an evaluation method that divides the problem into multiple levels and objectives, and then quantifies it through mathematical operations, finally obtains the weighted results. It can make people’s thinking level and mathematical, and introduce quantitative analysis on the basis of qualitative decision-making. It is simple and practical, and has profound theoretical basis. ²¹ Experts evaluation methods are too subjective and lack of the theorical support in the execution process. Fuzzy evaluation can solve the evaluating problem of the fuzzy objects. The objects in this paper are not fuzzy and has concrete value. Therefore, AHP is adopted in the subjective evaluation.

The specific evaluation process is as follows:

• (1) Establish hierarchical model

The hierarchical model for evaluating hole surface quality in drilling CFRP-Ti stacks is shown in Figure 11.

• (2) Constructing judgment matrix

The quantitative values of relative importance between elements are recorded in judgment matrix, that is, scales. The judgment matrix constructed by AHP often fails to pass the consistency check. It means that the evaluator’s judgment on importance of elements is self-contradictory. Therefore, it is necessary to improve AHP by re-establishing judgment matrix. In addition, AHP compares all elements under the same node. When the number of elements is large, the workload is enormous. Improved AHP first ranks the elements as $X_1\ge X_2\ge ···\ge X_n$ in a decreasing order of importance. Then, n-1 scales can be obtained by comparing each two elements from left to right, which $x_{n-1}$ represents the scale of $X_{n-1}$ compared with $X_n$ . Finally, the judgment matrix shown in equation 6 can be obtained. The judgment matrix is completely consistent, and there is no need for consistency checking.

$$\mathit{R}=\left[\begin{array}{ccccc}1\hfill & x_1\hfill & x_1{x}_2\hfill & ···\hfill & x_1{x}_2···x_{n-1}\hfill \\ \multicolumn{1}{c}{\frac{1}{x_1}}& 1\hfill & x_2\hfill & ···\hfill & x_2{x}_3···x_{n-1}\hfill \\ \multicolumn{1}{c}{\frac{1}{x_1{x}_2}}& \frac{1}{x_2}\hfill & 1\hfill & ···\hfill & x_3{x}_4···x_{n-1}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{\frac{1}{x_1{x}_2···x_{n-1}}}& \frac{1}{x_2{x}_3···x_{n-1}}\hfill & \frac{1}{x_3{x}_4···x_{n-1}}\hfill & ···\hfill & 1\hfill \end{array}\right]$$

(6)

In improved AHP, the value of scales is given. Comparing two elements $X_i,X_{i+1}$ ,

•  1)  $X_i$ is important as same as $X_{i+1}$ , $x_i$ equals 1;

•  2)  $X_i$ is slightly more important than $X_{i+1}$ , $x_i$ equals 1.2;

•  3)  $X_i$ is obviously more important than $X_{i+1}$ , $x_i$ equals 1.4;

•  4)  $X_i$ is strongly more important than $X_{i+1}$ , $x_i$ equals 1.6;

•  5)  $X_i$ is absolutely more important than $X_{i+1}$ , $x_i$ equals 1.8.

Four judgment matrixes are constructed:

•  1)  $\mathit{A}$ for evaluation factors $B_1$ , $B_2$ , and $B_3$ ;

•  2)  ${\mathit{B}}_1$ for evaluation indexes $C_1$ , $C_2$ of delamination;

•  3)  ${\mathit{B}}_2$ for evaluation indexes $C_3$ , $C_4$ of tears;

•  4)  ${\mathit{B}}_3$ for evaluation factors $C_5$ ∼ $C_{10}$ of surface roughness.

• (3) Single-layer weight calculation

According to judgment matrix $\mathit{A}$ , ${\mathit{B}}_1$ , ${\mathit{B}}_2$ , ${\mathit{B}}_3$ , the weight $v_i$ of the ith element can be calculated by equation 7. It is necessary to calculate four weight vectors ${\mathit{v}}_0$ , ${\mathit{v}}_1$ , ${\mathit{v}}_2$ , ${\mathit{v}}_3$ .

$$v_i=\frac{{\left(\underset{j=1}{\overset{n}{\Pi }}{r}_{\mathrm{ij}}\right)}^{1/n}}{\sum _{i=1}^n{\left(\underset{j=1}{\overset{n}{\Pi }}{r}_{\mathrm{ij}}\right)}^{1/n}}$$

(7)

where n is the order of judgment matrix (judgment matrix must be square matrix), $r_{\mathrm{ij}}$ is the element of row i and column j in judgment matrix.

• (4) Total weight calculation

According to the hierarchical model, the weights of the lower elements relative to the higher elements can be calculated. By multiplying each element in ${\mathit{v}}_1$ , ${\mathit{v}}_2$ , ${\mathit{v}}_3$ and ${\mathit{v}}_0$ , the subjective weight vector $\mathit{v}$ of evaluation indexes can be obtained.

$$\mathit{v}=\left[\begin{array}{c}{\mathit{v}}_1\times {{\mathit{v}}_0}^{(1)}\hfill \\ \multicolumn{1}{c}{{\mathit{v}}_2\times {{\mathit{v}}_0}^{(2)}}\\ \multicolumn{1}{c}{{\mathit{v}}_3\times {{\mathit{v}}_0}^{(3)}}\end{array}\right]$$

(8)

where ${{\mathit{v}}_0}^{(1)}$ , ${{\mathit{v}}_0}^{(2)}$ , ${{\mathit{v}}_0}^{(3)}$ represent the 1st, 2nd, and 3rd element of ${\mathit{v}}_0$ respectively.

Figure 11.

The hierarchical model for evaluating hole surface quality in drilling CFRP-Ti stacks.

Objective evaluation method based on entropy weight method

Entropy weight method determines the weight of each index according to the amount of information carried by sample data. The process of the evaluation is objective without the human subjective judgment. The value of entropy reflects the degree of difference between the observed values of indexes, so as to determine the weight of the index should be given. The specific evaluation process is as follows:

1. Forming decision matrix

According to the results of hole surface quality measurement in drilling CFRP-Ti stacks, m evaluation objects and n index values of each evaluation object can be determined, and the decision matrix shown in equation 9 can be formed.

$$\mathit{X}=\left[\begin{array}{cccc}{x}_{11}\hfill & x_{12}\hfill & ···\hfill & x_{1n}\hfill \\ \multicolumn{1}{c}{x_{21}}& x_{22}\hfill & \cdots \hfill & x_{2n}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{x_{m1}}& x_{m2}\hfill & \cdots \hfill & x_{\mathrm{mn}}\hfill \end{array}\right]$$

(9)

where $x_{\mathrm{ij}}$ is the j th index value of the i th evaluation object.

• (2) Normalized decision matrix

Each evaluation index value in this paper represents the defect degree of hole surface quality. The smaller the value, the better. Their normalization method is as follows:

$$w_{\mathrm{ij}}=\frac{max(x_j)-x_{\mathrm{ij}}}{max(x_j)-min(x_j)}$$

(10)

where $w_{\mathrm{ij}}$ is the normalized value of $x_{\mathrm{ij}}$ , $max(x_j)$ and $min(x_j)$ represent the maximum and minimum values of the jth evaluation index in the sample respectively. After that, the normalized decision matrix $\mathit{W}$ can be obtained.

• (3) Calculating the entropy value of each evaluation index

According to the normalized decision matrix $\mathit{W}$ , the specific gravity can be calculated by equation 11.

$$p_{\mathrm{ij}}=\frac{w_{\mathrm{ij}}}{\sum _{i=1}^m{w}_{\mathrm{ij}}}$$

(11)

where $p_{\mathrm{ij}}$ is the specific gravity of the j th evaluation index of the i th evaluation object.

Then the entropy value of each evaluation index can be calculated by equation 12. The smaller the entropy value, the more information the index carries and the higher weight it should be given.

$$e_j=-\frac{1}{\ln (m)}\sum _{i=1}^m{p}_{\mathrm{ij}}\ln (p_{\mathrm{ij}})$$

(12)

where $e_j$ is the the entropy value of the jth evaluation index. When $p_{\mathrm{ij}}$ = 0 or 1, $p_{\mathrm{ij}}\ln (p_{\mathrm{ij}})=0$ .

• (4) Determining the entropy weight of each index

The difference coefficient $d_j$ of the jth index can be calculated by equation 13, then the objective weight of that can be calculated by equation 14.

$$d_j=1-e_j$$

(13)

$${\mathit{w}}^{(j)}=\frac{d_j}{\sum _{k=1}^n{d}_k}$$

(14)

where ${\mathit{w}}^{(j)}$ is the jth element in the objective weight vector $\mathit{w}$ , that is, the entropy weight of the jth index.

The subjective and objective combination weight calculation method

The subjective and objective combination weight is calculated based on the principle of minimum discriminant information to ensure that it is as close as possible to the subjective and objective weights. The objective function and the constraints are shown in equations 15 and 16.

$$f({\mathit{Q}}^{(j)})=\sum _{j=1}^n{\mathit{Q}}^{(j)}\ln (\frac{{\mathit{Q}}^{(j)}}{{\mathit{v}}^{(j)}})+\sum _{j=1}^n{\mathit{Q}}^{(j)}\ln (\frac{{\mathit{Q}}^{(j)}}{{\mathit{w}}^{(j)}})$$

(15)

$$\sum _{j=1}^n{\mathit{Q}}^{(j)}=1$$

(16)

where

${\mathit{Q}}^{(j)}$ is the subjective and objective combination weight of the jth index,

${\mathit{v}}^{(j)}$ is the subjective weight of the jth index,

${\mathit{w}}^{(j)}$ is the objective weight of the jth index.

The minimum of $f\left(Q^{\left(j\right)}\right)$ can be obtained by Lagrange multiplier. Then, ${\mathit{Q}}^{(j)}$ can be obtained as shown in equation 17.

$${\mathit{Q}}^{(j)}=\frac{\sqrt{{\mathit{v}}^{(j)}\times {\mathit{w}}^{(j)}}}{\sum _{j=1}^n\sqrt{{\mathit{v}}^{(j)}\times {\mathit{w}}^{(j)}}}$$

(17)

Validation of the comprehensive evaluation method of hole surface quality

Drilling experiment of CFRP-Ti stacks

The materials, drills, equipment and processing conditions are consistent with section 2.1.1. In order to improve the accuracy of the objective weight calculation, more experimental groups are added, as shown in Table 5.

Table 5.

The experimental parameters in drilling CFRP-Ti stacks.

Group number	Spindle speed/(r/min)	Feed rate/(mm/r)	Point angle/(°)
1	600	0.03	100
2	600	0.06	118
3	600	0.09	135
4	600	0.12	100
5	600	0.15	118
6	600	0.18	135
7	750	0.03	135
8	750	0.06	100
9	750	0.09	118
10	750	0.12	135
11	750	0.15	100
12	750	0.18	118
13	900	0.03	118
14	900	0.06	135
15	900	0.09	100
16	900	0.12	118
17	900	0.15	135
18	900	0.18	100
19	1050	0.03	100
20	1050	0.06	118
21	1050	0.09	135
22	1050	0.12	100
23	1050	0.15	118
24	1050	0.18	135
25	1200	0.03	135
26	1200	0.06	100
27	1200	0.09	118
28	1200	0.12	135
29	1200	0.15	100
30	1200	0.18	118
31	1350	0.03	118
32	1350	0.06	135
33	1350	0.09	100
34	1350	0.12	118
35	1350	0.15	135
36	1350	0.18	100

Delamination is detected by ultrasonic scanning microscope SAM 302 HD2. Tears are detected by ordinary industrial microscope. Surface roughness is measured by MarSurf M300.

Evaluation example of hole surface quality

Calculate the subjective weight

The judgment matrices are constructed as follows.

• (1) Evaluation factors layer

The order of importance of elements is $B_1>B_2>B_3$ , and the scales are 1.6 and 1.2 respectively. The judgment matrix $\mathit{A}$ is as follows:

$$\mathit{A}=\left[\begin{array}{ccc}1\hfill & 1.6\hfill & 1.92\hfill \\ \multicolumn{1}{c}{0.625}& 1\hfill & 1.2\hfill \\ \multicolumn{1}{c}{0.5208}& 0.8333\hfill & 1\hfill \end{array}\right]$$

• (2) Delamination evaluation indexes layer

The order of importance of elements is $C_1>C_2$ , and the scale is 1.4. The judgment matrix ${\mathit{B}}_1$ is as follows:

$${\mathit{B}}_1=\left[\begin{array}{cc}1\hfill & 1.4\hfill \\ \multicolumn{1}{c}{0.7143}& 1\hfill \end{array}\right]$$

• (3) Tears evaluation indexes layer

The order of importance of elements is $C_3>C_4$ , and the scale is 1.4. The judgment matrix ${\mathit{B}}_2$ is as follows:

$${\mathit{B}}_2=\left[\begin{array}{cc}1\hfill & 1.4\hfill \\ \multicolumn{1}{c}{0.7143}& 1\hfill \end{array}\right]$$

• (4) Surface roughness evaluation indexes layer

The order of importance of elements is $C_5>C_6>C_7>C_8>C_9>C_{10}$ , and the scales are 1.2, 1.4, 1.2, 1.4, 1.2 respectively. The judgment matrix ${\mathit{B}}_3$ is as follows:

$${\mathit{B}}_3=\left[\begin{array}{cccccc}1\hfill & 1.2\hfill & 1.68\hfill & 2.016\hfill & 2.8224\hfill & 3.3869\hfill \\ \multicolumn{1}{c}{0.8333}& 1\hfill & 1.4\hfill & 1.68\hfill & 2.352\hfill & 2.8224\hfill \\ \multicolumn{1}{c}{0.5952}& 0.7143\hfill & 1\hfill & 1.2\hfill & 1.68\hfill & 2.016\hfill \\ \multicolumn{1}{c}{0.496}& 0.5952\hfill & 0.8333\hfill & 1\hfill & 1.4\hfill & 1.68\hfill \\ \multicolumn{1}{c}{0.3543}& 0.4252\hfill & 0.5952\hfill & 0.7143\hfill & 1\hfill & 1.2\hfill \\ \multicolumn{1}{c}{0.2953}& 0.3543\hfill & 0.496\hfill & 0.5952\hfill & 0.8333\hfill & 1\hfill \end{array}\right]$$

Then, calculating weight vectors ${\mathit{v}}_0$ , ${\mathit{v}}_1$ , ${\mathit{v}}_2$ , ${\mathit{v}}_3$ by equation 7.

$$\begin{array}{c}{\mathit{v}}_0=(0.466,0.2913,0.2427{)}^{\mathrm{T}}\hfill \\ {\mathit{v}}_1=(0.5833,0.4167{)}^{\mathrm{T}}\hfill \\ {\mathit{v}}_2=(0.5833,0.4167{)}^{\mathrm{T}}\hfill \\ {\mathit{v}}_3=(0.2798,0.2332,0.1665,0.1388,0.0991,0.0826{)}^{\mathrm{T}}\hfill \end{array}$$

Finally, calculating the subjective weight vector $\mathit{v}$ by equation 8.

$$\begin{array}{c}\mathit{v}=(0.2718,0.1942,0.1699,0.1214,0.0679,\hfill \\ 0.0566,0.0404,0.0337,0.0241,0.02{)}^{\mathrm{T}}\hfill \end{array}$$

Calculating the objective weight

According to the results of hole surface quality measurement, the normalized decision matrix $\mathit{W}$ can be obtained by equation 10. The entropy vector $\mathit{e}$ can be obtained by equations 11 and 12.

$$\begin{array}{c}\mathit{e}=(0.9594,0.9828,0.9694,0.9753,0.9643,0.9695,\hfill \\ 0.9728,0.9765,0.984,0.9799{)}^{\mathrm{T}}\hfill \end{array}$$

Finally, calculating the objective weight vector $\mathit{w}$ by equations 13 and 14.

$$\begin{array}{c}\mathit{w}=(0.1525,0.0646,0.1151,0.0927,0.1341,0.1146,\hfill \\ 0.1023,0.0884,0.0602,0.0755{)}^{\mathrm{T}}\hfill \end{array}$$

Calculating the subjective and objective combination weight

According to $\mathit{v}$ and $\mathit{w}$ , calculating the subjective and objective combination weight vector $\mathit{Q}$ by equation 17.

$$\begin{array}{c}\mathit{Q}=(0.2182,0.1199,0.1498,0.1136,0.1023,0.0863,\hfill \\ 0.0689,0.0585,0.0408,0.0417{)}^{\mathrm{T}}\hfill \end{array}$$

Analysis of evaluation results

From the calculation results of all weights, the subjective and objective combination weight not only makes use of sample data, but also conforms to professional knowledge. Taking $D_{\mathrm{a}}$ and as examples, the objective weights of the two are 0.0646 and 0.1151 respectively. It is not consistent with the initial conditions of calculation, $B_1>B_2$ that is, “delamination is more serious than tears.” The subjective weight can modify them and make the results more consistent with cognition. On the other hand, the subjective weights of the two are 0.1942 and 0.1699 respectively. The results are in line with the practical condition but the variation range of $D_{\mathrm{a}}$ is much lower than that of $S_{\mathrm{L}}$ .The objective weight can make the calculation results more suitable to actual situation.

The scoring results of hole surface quality obtained by subjective, objective, subjective and objective comprehensive evaluation methods respectively are shown in Table 6.

Table 6.

The scoring results of hole surface quality.

Hole number	Subjective score	Objective score	Combination score
1	0.3029	0.4276	0.3682
2	0.2818	0.3927	0.3397
3	0.4571	0.4930	0.4803
4	0.2653	0.3613	0.3156
5	0.5231	0.4982	0.5186
6	0.5877	0.4779	0.5422
7	0.2280	0.2995	0.2638
8	0.2234	0.3001	0.2618
9	0.4357	0.4600	0.4533
10	0.5813	0.6225	0.6100
11	0.2465	0.3180	0.2839
12	0.5007	0.4602	0.4891
13	0.2822	0.3611	0.3232
14	0.3695	0.5249	0.4513
15	0.3098	0.4464	0.3798
16	0.2821	0.3842	0.3343
17	0.3674	0.4101	0.3936
18	0.3072	0.4015	0.3573
19	0.3776	0.5197	0.4491
20	0.3282	0.5046	0.4163
21	0.6101	0.7093	0.6667
22	0.2580	0.3389	0.2999
23	0.2860	0.3232	0.3083
24	0.5554	0.5159	0.5444
25	0.3092	0.4065	0.3605
26	0.4551	0.5673	0.5120
27	0.6260	0.7059	0.6725
28	0.6078	0.6929	0.6604
29	0.2818	0.3837	0.3350
30	0.3606	0.3507	0.3617
31	0.3325	0.4638	0.4002
32	0.3412	0.4625	0.4059
33	0.3535	0.5798	0.4665
34	0.4099	0.4429	0.4312
35	0.3856	0.4445	0.4208
36	0.2647	0.3326	0.3006

It can be found from Table 6 that the subjective and objective combination weight leads to a compromise in the final score, thus ensuring that the evaluation results are more reasonable. Taking hole 25 and hole 30 as examples, the subjective and objective scores of the former are 0.3092 and 0.4065 respectively, while the subjective and objective scores of the latter are 0.3606 and 0.3507 respectively. From the perspective of subjective or objective evaluation result alone, it is difficult to distinguish which hole has better surface quality. After comprehensive consideration of professional knowledge and sample statistical information, the two holes scored 0.3605 and 0.3617 respectively, indicating that their surface quality has little difference.

In general, the subjective and objective comprehensive evaluation method can not only eliminate randomness of the subjective evaluation method, but also avoid the phenomenon that the objective evaluation method may appear contrary to professional knowledge. It is undoubtedly a reasonable and effective evaluation method.

Conclusion

Aiming at hole surface quality evaluation in drilling CFRP-Ti stacks, this paper proposes a subjective-objective comprehensive evaluation method. The main conclusions are as follows:

1. The evaluation factors are determined by drilling experiments, including CFRP delamination, CFRP hole entrance tears, surface roughness of CFRP and Ti.

2. The evaluation index system of hole surface quality is established. Considering the effect of delamination on diameter and area and its three-dimensional properties, the maximum delamination diameter factor $D_{\mathrm{d}}$ and the delamination area factor $D_{\mathrm{a}}$ are proposed. Considering the effect of tears on length and area, the maximum tears length factor $S_{\mathrm{L}}$ and the tears area factor $S_{\mathrm{a}}$ are proposed. Considering the regularity of surface roughness of CFRP and Ti, Surface roughness indexes including $R_a^{\mathrm{t}60}$ , $R_a^{\mathrm{t}150}$ , $R_z^{\mathrm{t}60}$ , $R_z^{\mathrm{t}150}$ , $R_a^{\mathrm{Ti}}$ , $R_z^{\mathrm{Ti}}$ are proposed.

3. In subjective-objective comprehensive evaluation method, the subjective evaluation method based on improved AHP, the objective evaluation method based on entropy weight method, the subjective, and objective combination weight calculation method are proposed and illustrated by the example. The rationality of the method is proved by experiments, which is not only in line with subjective cognition, but also in line with objective reality.