1 Introduction

Fibre-reinforced polymer (FRP) composites have been used for many years in high-end industries mainly due to their excellent specific mechanical properties [1,2,3]. Glass and carbon fibres are the most used reinforcing materials in the high-end industries [4, 5]. However, their manufacturing is costly, and the life cycle of products made from carbon and glass fibres is relatively short. Their recycling and/or reusing is difficult or often not solvable [6,7,8]. Therefore, researchers developed novel fibrous composites using organic and inorganic reinforcing fibres. Hence, basalt fibres having mineral origin become the focus of attention [9]. Basalt fibres provide a suitable insulation property, high elasticity modulus, excellent heat and acoustic resistance, and are outstanding vibration isolators [10]. Czigány et al. [9] highlighted that application of basalt as reinforcing fibres in polymer composites is highly recommended and will be spread in the forthcoming years. Although material scientists revealed numerous advantageous properties of basalt fibre-reinforced polymer (BFRP) composites [11,12,13,14], their spread is inhibited due to the lack of knowledge on their machinability.

Amuthakkannan et al. [15] were among the first authors to scientifically investigate the drilling of BFRP composites. In their research, the authors focused on optimising the technological parameters of drilling BFRP composite to minimise delamination (separation between plies, which may cause the destruction of structural parts resulting in decreased resultant strength). Their results showed that optimal results could be reached by selecting the lowest feed rate tested. Several studies [16,17,18,19] have shown that the selection of the lowest feed rate for drilling fibre-reinforced polymer composites results in the lowest thrust force (cutting force component parallel to the axis of the cutting tool), which also has a positive effect on the drilling-induced delamination. However, this research did not make direct empirical connections between thrust force and delamination during drilling.

Navarro-Mas et al. [20] first studied the phenomenon of tool wear through BFRP composite edge trimming experiments. The authors performed their edge trimming experiments using an uncoated cutting insert tool. They have found that the cutting speed has the most significant effect on tool wear. The tool wear often affects the delamination, which they tested in further experiments. However, no information was published about the actual cutting force while drilling BFRP composite in this or in their upcoming studies. It has been proven that the cutting force significantly affects the occurring delamination.

Navarro-Mas et al. [21, 22] found that the drilling-induced delamination has a lower probability of formation at lower feed rates at edge trimming BFRP composites. Presumably, this is due to the fact that the smaller feed rate reduces the chip cross section, resulting in a lower cutting force, although a detailed scientific investigation has not been carried out in the past. Magyar et al. [23] conducted drilling experiments in BFRP composites and analysed the influences of process parameters on the drilling-induced burr and microstructure. They showed that the cutting speed has the most significant influence on surface roughness, and the effect of the feed is negligible. At the same time, the feed was found as the most significant factor affecting the size of burrs.

Thrust force in the drilling of carbon fibre-reinforced polymer (CFRP) composites was investigated and modelled by many scholars. Phapale et al. [24] developed a finite element (FE) model for the prediction of critical thrust force in CFRPs and validated it with experimental results. The prediction accuracy of their FE model was within 5%. Bai et al. [25] proposed a novel mechanical model for predicting thrust force with the consideration of tool wear in CFRPs. Their model provided better predictions (within 14%) than the analysed empirical methods. Zhang et al. [26] developed a theoretical model for predicting of thrust force in the drilling of CFRP stacks. The influence of tool wear was also considered in their model. They found the average absolute relative error of their model within 9%. Luo et al. [27] presented a mechanistic thrust force model in the drilling of CFRP/Ti stacks. The adequacy of their model is good, i.e. average prediction error was found to be below 10%. Wang et al. [28] developed a thrust force prediction method in the drilling of CFRPs. They considered the effect of the orientation angle, namely the angle between the cutting edge (chisel edge) and fibre direction. Although they did not provide numerical information on the prediction accuracy of their model, their results seem to be promising.

Although there are some published experiences in the drilling of BFRP composites, the analysis and modelling of cutting force in the drilling of BFRPs are not conducted yet. Considering that the thrust force has a key role in drilling-induced geometrical defect (e.g. delamination) formations in fibrous composites [17, 29], the main aim of this paper is to model the thrust force during the drilling of BFRP composites. Two modelling methods were used to develop cutting force models. In order to calculate the regression coefficients and model parameters, and to validate the adequacy of the developed models, mechanical drilling experiments were performed according to the experimental setup presented in Sect. 2. Then, the results are presented and discussed in Sect. 3.

2 Experimental setup and methods

2.1 Materials, tools, and machines

A unidirectional carbon fibre-reinforced vinyl-ester-based polymer composite and a multidirectional (biaxial) basalt fibre-reinforced epoxy resin-based polymer (MD-BFRP) composite were drilled in the drilling experiments. Our industrial partner provided both composites, as they plan to replace their CFRP components with the more sustainable BFRP composite. The main material properties of the composites were measured and published in our previous study [23].

A Kondia B640 three-axis machining centre was used to perform the drilling experiments. The experiments were carried out in a dry environment, where a Nilfisk GB733 industrial vacuum cleaner provided the chip extraction. A titanium–aluminium-coated Ø10 mm Tivoly Polaris 150 Sim Dim 6537 K solid carbide drill was used to perform the drilling. The tool geometry is based on the DIN 6537 k standard. The tool has a point angle of 145°, a helix angle of 35°, and a chisel edge length of 1 mm. The composites were fixed by a special fixture providing identical supporting circumstances on the entry and exit sides.

A KISTLER 9257BA three-component dynamometer was applied to measure the cutting force in situ using a 10 000-Hz sampling frequency. A KISTLER 5070 multi-channel charge amplifier and two National Instruments USB-4431 dynamic signal acquisition modules were used for force data sampling. The tool condition was documented by capturing microscopic images on the cutting edges before, between, and after the experiments using a Dino-Lite AM413ZT digital microscope. The experimental machining setup can be seen in Fig. 1a.

Fig. 1
figure 1

Illustration of the, a experimental machining setup and b geometric representation of the CCF experimental design, where x1 and x2 denote the factors (independent variables) and the points represent the experimental setups considering that the middle point (5th) is repeated five times to gather information on the variances

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Considering that the expected interaction terms may be significant and the expected influences of the process parameters are often nonlinear, the continuous factors are recommended to be varied at least three levels. To gather helpful information on the properties of the composites in machining with relatively few drilling operations, the central-composite face-centred (CCF) design of experiments was used, which geometrical representation is illustrated in Fig. 1b. The composite type as a categorical factor with two levels (BFRP and CFRP) and the feed (mm/rev) and cutting speed (m/min) as continuous factors, each on three different levels, were varied according to the experimental table as is shown in Table 2. The levels of these continuous factors (Table 1) were defined based on previous studies and the recommendations of the tool manufacturers [7, 23, 30, 31]. The experimental design is suitable for the application of quadratic models to describe nonlinear influences of process parameters. The drilling operations were performed in a randomised order, and settings of level zero (vc = 100 m/min, f = 0.10 mm/rev) were repeated five times, which enabled us to determine the reproducibility and calculate the variances. The analysis of variances (ANOVA) was performed at a significance level of α = 0.05.

Table 1 Factor levels
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2.2 Methods

The raw data of the measured force were processed by a self-developed algorithm in Python to handle quickly, slice, filter, and evaluate the data and characterise the process. The evaluation process is automated; only the start and the end point of the useful data have to be defined manually. First, the code slices the data as defined by the user. Then, a Butterworth low-pass filter at a cut-off frequency of 300 Hz (approx. five four the frequency of the maximal spindle speed to include harmonics frequencies) was used in order to filter the high-frequency noises as shown in Fig. 2. Filtering the raw force data measured during machining experiments is necessary to remove interfering high-frequency vibrations from the experimental environment that are irrelevant to the test object. This results in a smoothed data series, better suited to the nature of the machining operation, from which the metrics describing the processes (e.g. thrust force) can be more accurately determined. The Fz denotes the filtered axial cutting force component at a constant chip cross section. Then, the thrust force (Ft) is calculated according to Eq. (1). Considering that the larger the Fz, the less beneficial the drilling process is from the point of view of tool wear and hole quality, the maximum of Fz (i.e. Ft) is selected to characterise the process. The recorded and processed cutting force is illustrated in Fig. 2.

$${F}_{t}=max\left({F}_{zi}\right), i=0, 1\dots n$$
(1)

First, response surface methodology (RSM) was used to statistically model the thrust force of the drilling process. According to previous investigations [32,33,34,35,36], the effects of the applied process parameters on the thrust force are expected to be nonlinear; therefore, a second-degree polynomial model was used, as expressed by Eq. (2).

$${F}_{t}\left({x}_{1},{x}_{2}\dots {x}_{n}\right)={b}_{0}+\sum_{i=1}^{n}{b}_{i}{x}_{i}+\sum_{i=1}^{n}{b}_{ii}{x}_{ii}^{2}+\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}{b}_{ij}{x}_{i}{x}_{j}+\delta$$
(2)

where Ft is the corresponding response value (i.e. thrust force), xi are the factors (i.e. feed and cutting speed), b0, bi, bij, and bii are the regression coefficients of the parameters, and δ is a random experimental error. Then, the significance of the factors and their interactions were determined through the analysis of variance (ANOVA) technique at a significance level of α = 0.05.

Fig. 2
figure 2

Illustration of the workflow of the thrust force (Ft) generation: a raw dataset (blue), and the trimmed and filtered (red) force signal of BFRP (Fz), b enlarged part of the raw dataset and Fz of BFRP, c raw dataset (green), and the trimmed and filtered (orange) force signal of CFRP (Fz), d enlarged part of the raw dataset and Fz of CFRP

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Second, the Fz was modelled through advanced statistics considering the actual cutting tool edge position and feed, as expressed by Eq. (3). The applied advanced statistical model takes account of the symmetry observed in the structure of the material and therefore uses a periodic function. Furthermore, it was important that the function should be differentiable and computable easily, providing the possibility of real-time process monitoring for future industrial applications.

$${F}_{z}\left(\varphi \right)=A\bullet \mathrm{sin}\left({\omega }^{-1}\bullet \varphi +\Delta \right)+k$$
(3)

where φ (°) denotes the angular position of the spindle, ω = 2πn/60 (rad/s) is the angular frequency of the spindle; moreover, A, Δ and k are regression coefficients denoting amplitude, phase shift and vertical shift, respectively. Considering that the advanced model includes the effect of φ, which is closely correlated with the fibre cutting angle, this model is able to predict not only the thrust force but also the axial cutting force tool. In addition, as the feed has a significant statistical and physical influence on the cutting force, we implemented it in our model, as expressed by Eq. (4).

$${F}_{z}\left(\varphi ,f\right)={C\bullet F}_{z}\left(\varphi \right)\bullet {f}^{x}$$
(4)

where f (mm/rev) denotes the feed; moreover, C and x are regression coefficients used to model the effect of the feed. The thrust force (Ft) can be predicted through the maximum of Eq. (4), as expressed by Eq. (5).

$${F}_{t}\left(\varphi ,f\right)=max\left({F}_{z}\left(\varphi ,f\right)\right)$$
(5)

The coefficient of determination (R2) was used to determine the adequacy of the model fitting as it is a widely used measure to describe how well a statistical model can predict the outcome of a phenomenon, which is the thrust force in this study. The absolute percentage error (APE) was used to characterise the adequacy of the Ft predictions, as expressed by Eq. (6).

$$APE=abs\left(\frac{{F}_{t,p}-{F}_{t,m}}{{F}_{t,m}}\right)\bullet 100$$
(6)

3 Results and discussion

This section presents the developed models, the predictions, and their comparison and discussion. The measured and predicted thrust forces and the adequacy of models are summarised in Table 2 and discussed separately in the following subsections.

Table 2 Experimental design table containing the measured values of the thrust force (Ft) with the predictions obtained by the RSM model (Ft,RSM) and the advanced statistical method (Ft,adv), along with the absolute percentage error of both models, and the R2 belonging to the Fz predictions achieved by the advanced statistical models
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Representative filtered force diagrams of BFRP and CFRP composites are shown in Fig. 3. Force diagrams clearly show that the BFRP composite has much smoother characteristics (Fig. 3a), while the CFRP composite shows a higher-amplitude characteristic (Fig. 3b). This difference is resulted by the different reinforcement structures of the composites. The BFRP composite with a biaxial reinforcement results in a relatively more anisotropy structure compared to CFRP with a UD reinforcement, which explains why the thrust force diagrams of the BFRP composite show a much more flattened shape. Furthermore, it can also be observed that the thrust force (Ft) was much larger for the CFRP composite (Table 2), as was expected based on the strength tests [23]. The diagrams also indicate that the smaller the feed, the longer the operation time, as it was expected.

Fig. 3
figure 3

Representative filtered force diagrams of a BFRP and b CFRP composites at a cutting speed of vc = 100 m/min

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3.1 RSM models

The developed thrust force prediction models based on the RSM for BFRP and CFRP composites are expressed by Eq. (7) and Eq. (8), respectively. The response surface diagrams (graphical illustrations of the RSM models) are illustrated in Fig. 4. These diagrams represent the effect of the technological parameters (i.e. feed and cutting speed) on the thrust force. The averages of the absolute percentage errors are 3.26% and 3.03% for the BFRP and CFRP composites, respectively. These errors are within the scope of mechanical engineering and manufacturing science expectations, thus proving their applicability.

Fig. 4
figure 4

The effect of technological parameters (f and vc) on the thrust force (Ft) at drilling a BFRP and b CFRP composites using RSM models

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$${F}_{t,RSM}^{BFRP}=20.6+1536f-0.054{v}_{c}-4980f\bullet f+0.001{v}_{c}\bullet {v}_{c}-0.71 f\bullet {v}_{c}$$
(7)
$${F}_{t,RSM}^{CFRP}=158.3+2284f-0.066{v}_{c}-4980f\bullet f+0.001{v}_{c}\bullet {v}_{c}-0.71 f\bullet {v}_{c}$$
(8)

The combination of F-values and P-values is examined to determine whether the factors and their interactions have a significant effect on the thrust force. The applied null hypothesis is that the influence of the x1 factor on the Y response parameter is not significant considering the given significance level. In the case where the P-value is greater than α = 0.05, the null hypothesis is failed to be rejected, meaning that the particular effect will not be considered to be significant. The response graphs suggest that the influence of the cutting speed is negligible. This is in a good correlation to the ANOVA results, as the P-value of the cutting speed is large (P-value = 0.416), as Table 3 shows. Furthermore, it can be seen that the influence of the feed on the thrust force is significant, as significantly larger force values are belonging to the larger feed levels. The interaction effect of the feed and cutting speed is not significant (F-value = 0.12; P-value = 0.734), as the influence of feed is not affected by the level of cutting speed in the analysed factor space. This is proved by the ANOVA table (Table 3). Based on the results of the ANOVA, the type of the composite has by far the most significant (F-value = 1626.12 P-value = 0.000) effect on the thrust force, followed by the feed (F-value = 104.49; P-value = 0.000).

Table 3 The result of ANOVA in the case of thrust force
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The main effect plots of the significant factors can be seen in Fig. 5. The diagrams show that the composite type has the most significant influence on the cutting force, i.e. the thrust force in the BFRP composites is significantly lower than that of CFRPs (Fig. 5a). This may be due to the stronger (i.e. tensile, interlaminar, impact) carbon fibrous polymer composites than basalt fibrous polymer composites [23]. Figure 5b indicates that the larger the feed, the larger the thrust force in each composite. This was expected because the larger the feed, the larger the chip cross section and the larger the cutting force is [37, 38].

Fig. 5
figure 5

Main effect plots of the a type of the composite and b the feed

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Although the developed RSM models predict accurately (APE is below 11%) the value of thrust force at different levels of feed and cutting speed and may be used for real-time process control—as the polynomial form of the model makes it easy and fast to calculate with—these are not describing the nature of the axial cutting force (Fz) component. Considering that these RSM models fail to represent the influences of fibres on the cutting force, an advanced statistical method was required to be developed, as presented in Sect. 2.2 and Sect. 3.2.

3.2 Advanced statistical models

Considering that the fibre cutting angle (i.e. the angle between the fibre and cutting speed direction) continuously changes, the filtered thrust force data are sinusoidal (shown in Fig. 2b and Fig. 2d). Thus, we fitted a sinusoidal function to the filtered datasets to model the characteristics of each force diagram. Furthermore, the influence of the feed is also added to the advanced model, as it was found to have a significant influence on the thrust force. The model coefficients were calculated through the least squares method. The developed advanced statistical models to predict Fz are expressed by Eq. (9) and Eq. (10) for the BFRP and CFRP composites, respectively.

$${F}_{z,adv}^{BFRP}\left(\varphi ,f\right)=2.4546\bullet \left(50.5881\bullet \mathrm{sin}\left(0.0347\bullet \varphi +0.0000\right)+85.1331\right)\bullet {f}^{0.3804}$$
(9)
$${F}_{z,adv}^{CFRP}\left(\varphi ,f\right)=2.4178\bullet \left(102.6220\bullet \mathrm{sin}\left(0.0347\bullet \varphi +0.5629\right)+180.6572\right)\bullet {f}^{0.3739}$$
(10)

In the case of these advanced statistical models, the averages of the absolute percentage error of the calculated thrust forces (i.e. the maximums of Eq. (9) and Eq. (10)) are 4.99% and 4.35% for the BFRP and CFRP composites, respectively. Although these average APEs are slightly larger than the APEs found in the RSM models, these errors are still within the scope of mechanical engineering and manufacturing science expectations. In addition, these advanced models are capable of predicting not only the maximums of the Fz, but also its characteristics, as it is illustrated in Fig. 6, where the best fitting sections of the whole force dataset are shown. While the diagrams qualitatively indicate that the advanced models fit well the measured and filtered force dataset, the averages of the coefficient of determination (R-squared) values of 0.6751 and 0.8481 indicate the adequacy of the fitting of the models in the case of BFRP and CFRP composites, respectively.

Fig. 6
figure 6

Illustration of the fitting of the prediction models to the measured and filtered cutting forces in a–c BFRP and d–f CFRP composites

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3.3 Comparison and discussion

The absolute percentage errors of the developed RSM models indicate that the prediction of the thrust force (Ft) is accurate; thus, these models can be applied for thrust force modelling and drilling process optimisation. Considering that the RSM models are polynomial, these models can be efficiently used in real-time (i.e. in situ) process diagnostics and monitoring of drilling CFRP and BFRP composites because polynomial calculations do not require a significant amount of computational capacity. In addition, these models can be implemented in Industry 4.0-driven digital-twin models too. Although the thrust force is accurately predicted by these RSM models, they provide no in-depth understanding of the machining mechanisms, i.e. does not include the mechanistic effect of fibre cutting angle, chip cross section. Consequently, another drawback of these RSM models is that only the maximum axial cutting force (i.e. thrust force) can be predicted and not the characteristics of the axial cutting force component. In order to further improve these models, the effect of the feed at more levels is recommended to be analysed to increase the performance of the models.

Advanced statistical models—compared with the RSMs—predict not only the thrust force but the Fz(ϕ) function also. Although the absolute percentage errors of these advanced models are slightly larger than those of the RSM models, these can be applied to analyse the characteristics of the axial cutting force in the drilling of CFRP and BFRP composites. The regression coefficients (R2) of the advanced models are 0.6751 and 0.8481 for the BFRP and CFRP, respectively. The R2 of the CFRP composite model is in the range achieved by similar research works [37, 39,40,41,42]; thus, it is accurate enough to use. Although the method for the model building of the BFRP and CFRP composites is identical, the R2 of the BFRP becomes significantly smaller than that of CFRPs. However, in the case of advanced statistical models, there are opportunities to improve the fit to the measured datasets. This may require the development of more complex models, which might complicate the application in real industry scenarios due to the high computational demand. Nevertheless, there are no existing similar results published to prove the adequacy of this prediction, according to the best knowledge of the authors.

The experimental results show that the thrust force is significantly lower in the BFRP than in the CFRP composite. Considering that the lower the thrust force, the lower the tool wear rate is [25], the tool wear speed is expectedly lower in the BFRPs than expected in the CFRP composites. However, this has to be proved by further investigations. In addition to the beneficial ability of BFRPs to have low thrust forces, the sustainability of natural basalt fibres-reinforced polymer composites is larger than that of CFRPs. Therefore, if the mechanical properties of a BFRP composite part are suitable for replacing existing CFRP parts, it is recommended to use the BFRP composites from the point of view of drilling.

Considering that the dominancy of the ploughing phenomena is significantly more decisive when microholes (diameter less than 1 mm) are drilled, the findings of the current study are valid only conventional-sized dry drilling of BFRP and CFRP composites. Therefore, the analysis of the influences of the cutting tool diameter and machining conditions (e.g. cooling techniques, mechanical supporting circumstances of composite plies, application of support plates, application of tool vibration and laser assistance) is required in the future.

Although the authors experienced similar chip removal, axial force generation, and similar effects of drilling parameters when drilling BFRP composites compared to the CFRPs, the in-depth analysis of chip removal mechanisms and tool wear mechanisms in the machining of BFRPs is recommended to support the spread of BFRP applications in the future. As the number and depth of similar research works are moderate, this study directly supports the current trends in sustainable material usage in high-end industries.

4 Conclusions

In the present study, mechanical drilling experiments were conducted in basalt and carbon fibre-reinforced polymer (BFRP and CFRP) composites and the thrust force was modelled through response surface methodology and advanced statistics. According to the present study, the following conclusions can be drawn:

  • Response surface methodology (RSM) models were developed to describe the influences of process parameters on the thrust force in drilling BFRP and CFRP composites. The averages of the absolute percentage errors of the RSM models are 3.26% and 3.03% for the BFRP and CFRP composites, respectively. These models include the effect of feed (mm/rev) and cutting speed (m/min).

  • The analysis of variances (ANOVA) results prove that the type of the composite has by far the most significant effect on the thrust force (whether it is BFRP or CFRP), followed by the feed. The influence of the cutting speed is negligible on the thrust force. The larger the feed, the larger the thrust force in each composite. Furthermore, the thrust force in the BFRP composites was significantly lower than that of CFRPs.

  • We fitted a sinusoidal function to the filtered datasets to model the characteristics of each force diagram. This model includes the influence of the angular position of the cutting tool and the feed. The averages of the absolute percentage error of the calculated thrust forces are 4.99% and 4.35% for the BFRP and CFRP composites, respectively. Thus, this sinusoidal model is suitable for modelling the thrust force in the drilling of BFRP composites.

  • This experimental study suggests that if the mechanical properties of a BFRP composite part are suitable for replacing an existing CFRP part, it is recommended to use the BFRP composites from the point of view of drilling.

Although the machining behaviour of BFRPs seems to not differ significantly from that of CFRPs, further research works are recommended to support the spread of BFRP applications, i.e. an in-depth analysis of chip removal mechanisms and tool wear mechanisms in the machining of BFRPs.