1 Introduction

Due to their high resistance to corrosion and oxidation, austenitic stainless steels have been commonly used in many industrial areas including those of aircraft, nuclear, defence, food processing and particularly in that of medicine [1–4]. However, the machining of these steels is very difficult because of their high ductility, work hardening rate and low thermal conductivity [5]. The proper selection of cutting parameters in the machining of these steels plays a significant role in ensuring the quality of the product, reducing the machining costs and increasing productivity [6]. In addition, poor selection of the cutting parameters causes a decrease in the surface quality as well as rapid cutting tool wear [7–10]. Measurement of the cutting forces is of great importance in the determination of the optimum cutting parameters because the cutting forces are one of the most significant factors affecting tool life [11–15].

In scientific studies carried out to determine the optimum cutting parameters, expensive experimental setups and long test durations are needed. For this reason, different analyses, optimisation and modelling techniques have been developed, including finite element analysis, the multiple regression method, the Taguchi method, fuzzy logic, genetic algorithms and artificial neural networks (ANN) [16–20]. In recent years, many ANN-based scientific studies have been performed due to the good prediction capability of this technique [21–23]. Owing to its memorisation and analytical skills, an ANN resembles a simple imitation of the human brain. Solving systems for which the solutions are not analytically possible or for which a mathematical model cannot be completely structured has become easier through ANNs, which arose from computer programs affected by natural neural systems [24, 25]. The ANN method is used in the machining area as well. In the chip-removal process, a wide range of factors, including environmental conditions, rigidity and surface roughness, as well as cutting parameters, should be taken into account. The ANN method is mostly used for the prediction of surface roughness and cutting forces. In their studies, Hao et al. [26] used ANNs to develop a model for the prediction of the cutting forces in a self-propelled rotating tool during the turning process. In the constructed ANN architecture, while the cutting speed, feed rate, depth of cut and inclination angle were used as inputs, the main cutting force, feed force and radial force were outputs of the network. They created a new hybrid model by combining the ANN model with the genetic algorithm. Through the new hybrid model, cutting forces could be obtained that were very close to the experimental data. Suksawat [27] presented an ANN application that predicted the main cutting force and the classified chip form. The cutting speed, feed rate and depth of cut were used as inputs to a network for predicting the cutting force and classified chip form occurring during the turning of nylon material with HSS tools. As a result of the study, the convergence values for the predictions of the classified chip form and the cutting force were calculated with 86.67 and 91.13 % accuracy. Ozkan et al. [28] developed an ANN model for the prediction of the cutting forces and temperatures generated in the turning process under different cutting conditions. By means of the developed model, the cutting forces and temperature values could be predicted precisely. In another study, a back propagation neural network model has been developed for the prediction of surface roughness in turning operation by Pal and Chakraborty [18]. The convergence of the mean square error both in training and testing came out very well. The performance of the trained neural network has been tested with experimental data and found to be in good agreement. Yilmaz et al. [29] performed a study about predicting the surface roughness by means of neural network approach method on machining of a cast polyamide material. The network has two inputs called spindle speed and feed rate for this study. Gradient descent method was applied to optimise the weight parameters of neuron connections. According to the predicted results, the ANN model developed for predicting the surface roughness values in milling gave correct and acceptable results.

For the present study, an ANN model was developed for the prediction of the main cutting force and feed force occurring during orthogonal cutting. In the input layer of the ANN architectures, the coating type (C t), feed rate (f) and cutting speed (V) are included. In addition, multiple regression analysis was performed for the same input and output parameters. The cutting force values calculated using the formulas obtained from both an ANN and a multiple regression analysis were compared with the experimentally measured cutting force values.

2 Materials and methods

2.1 Experimental process

Orthogonal turning tests using AISI 316L austenitic stainless steel bars (see Table 1) as the workpiece material were carried out on a Johnford T35 CNC lathe with 10-kW spindle power and a maximum spindle speed of 6,500 rpm. The experimental setup for orthogonal cutting is shown in Fig. 1. The bars, 60 mm in diameter and 240 mm in cutting length, were turned with two different coated and uncoated cemented carbide cutting inserts. The workpiece and cutting tools used in the orthogonal cutting tests are shown in Fig. 2. A Kistler piezoelectric dynamometer model 9257B with a load amplifier connected to a computer was used for the acquisition of the cutting force (F c) and feed force (F f). The orthogonal cutting tests were carried out at feed rates of 0.05, 0.1 and 0.2 mm/rev, cutting speeds of 75, 100, 150, 200 and 250 m/min and a constant depth of cut of 2 mm (Table 2). General properties of cutting inserts produced by Iscar used in the machining of AISI 316L are given in Table 3. The inserts had a −6° rake angle, 5° clearance angle and 0.8 mm nose radius. In cutting experiments, tool holder was used the tool holder code of PTGNL 2525M16 and approaching angle of 90°. The mechanical and thermal properties of the coating materials are given in Table 4 [30].

Table 1 Chemical characteristics (wt%) of AISI 316L material
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Fig. 1
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Experimental setup for orthogonal cutting

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Fig. 2
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Workpiece material used in orthogonal cutting process

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Table 2 Test parameters
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Table 3 Cutting inserts used in the experimental study
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Table 4 The mechanical and thermal properties of coating and uncoated materials [30]
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2.2 Artificial neural networks

An ANN is a data processing and modelling technique that arose in pursuit of mathematical modelling of the learning process which was inspired by the human brain. The ANN concept arose as the idea of imitating the brain’s working principles on digital computers. The studies on this subject started in 1942 with the mathematical modelling of neurons, the biological units that constitute the brain, and the application of this model to computer systems; afterwards, it was utilised in many fields in parallel with the development of computer systems. ANNs have the ability to associate the input data, defined depending on single or multiple parameters, regarding a system with that system’s outputs, defined depending on single or multiple parameters [31]. The biggest advantage of ANNs is that they are able to learn and to use different learning algorithms [32]. In ANNs, the data coming from the external world go to the input layer. These are the data that we want the network to learn. The way that data are given to the network varies with respect to the data set. The summation function (Σ) is a function that calculates the net input coming to a cell, and the net input is generally a sum of multiplications of the inputs (x) with the relevant weights (w), calculated with the formula in Eq. (1).

$${\text{NET}}_{i} = \sum\limits_{j = 1}^{n} {w_{ij} x_{j} + w_{bi} }$$
(1)

where NET i is the weighted summation of the input values, i and j are processing elements, n is the number of processing elements in the previous layer, w ij is the weight of the connections between the i and j processing elements, x j is the output of the j processing element and w bi is the weight of the biases between layers.

At the next stage, the output of the summation function is sent to the transfer (activation) function. This function transforms the received value into a real output through an algorithm. As regards the transfer function used, the output values are usually normalised between −1 and 1 [33] or between 0 and 1 [34]. The transfer functions used in ANNs are generally nonlinear functions. Using nonlinear transfer functions allows ANNs to be applied to very different complex problems. The common transfer functions in ANNs are linear, step/signum, threshold, logistic sigmoid, hyperbolic tangent sigmoid functions, etc. In the ANN model developed in this study, the logistic sigmoid transfer function was used and its formula is given in Eq. (2).

$$f\left( {{\text{NET}}_{i} } \right) = \frac{1}{{1 + e^{{ - {\text{NET}}_{i} }} }}$$
(2)

where f(NET) is the logistic sigmoid transfer function.

The significant advantages of ANNs are their learning ability and their use of different learning algorithms. In order to obtain the output values closest to the experimental values, the best learning algorithm and the optimal number of neurons in the hidden layer should be determined. Different types of learning algorithms used in the network training process include gradient descent back propagation (GD), quasi-Newton back propagation (BFGS), Levenberg–Marquardt back propagation (LM), scaled conjugate gradient back propagation (SCG), resilient back propagation (RP), conjugate gradient back propagation with Polak–Ribiére updates (CGP) and Bayesian regulation back propagation (BR). The most important factor that determines its success in practice, after the construction of ANN architecture, is the learning algorithm. In this study, the LM, SCG, BFGS, RP and CGP learning algorithms were used for network training. In addition, the number of hidden layers and neurons in each hidden layer were determined. Of the 60 experimental data obtained as a result of an experimental study, 48 data were randomly selected as training data, while 12 data were selected as testing data. An investigation of previous studies in the literature found the data numbers used in this study to be adequate for ANN modelling (Kohli and Dixit [35]—31 data; Pal and Chakraborty [36]—27 data; Cus and Zuperl [37]—30 data; Al-Ahmari [38]—28 data; Davim et al. [39]—30 data; Karnik et al. [40]—36 data; Chavoshi and Tajdari [41]—18 data; Korkut et al. [17]—50 data; Asiltürk et al. [42]—27 data). The input and output values were normalised between 0.1 and 0.9. The normalisation values of the network are given in Table 5. Finally, the iteration number was entered and the training process was started. The data obtained from the ANN training were compared with the cutting force values obtained experimentally from the cutting tests. For comparison, the root mean square error (RMSE), R 2 (coefficient of determination) and mean error percentage (MEP) values were used. These values are calculated with the formulas given in Eqs. (3–5).

$${\text{RMSE}} = \left( {\left( \frac{1}{p} \right)\sum\limits_{j} {\left| {t_{j} - o_{j} } \right|}^{2} } \right)^{1/2}$$
(3)
$$R^{2} = 1 - \left( {\frac{{\sum\nolimits_{j} {\left( {t_{j} - o_{j} } \right)^{2} } }}{{\sum\nolimits_{j} {\left( {o_{j} } \right)^{2} } }}} \right)$$
(4)
$${\text{MEP}}\,(\% ) = \frac{{\sum_{j} \left( {{{\left( {t_{j} - o_{j} } \right)} \mathord{\left/ {\vphantom {{\left( {t_{j} - o_{j} } \right)} {t_{j} }}} \right. \kern-0pt} {t_{j} }}} \right) \times 100}}{p}$$
(5)

where t is the target value, p is the number of patterns and o is the output value. The cutting parameters and coating types used for turning AISI 316L austenitic stainless steel under orthogonal conditions and the experimental cutting force values obtained as a result of the cutting tests are given in Table 6. As shown in Table 6, the digits for the coating type to be entered into the multiple regression analysis and the ANNs were determined as: uncoated = 1, TiCN + Al2O3 + TiN-coated = 2 and Al2O3-coated = 3.

Table 5 Normalization values of inputs and outputs
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Table 6 Experimental cutting force values
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The ANN architectures for the prediction of the cutting forces are shown in Fig. 3. In the ANN architecture constructed for the main cutting force, three neurons (coating type, cutting speed and feed rate) were included in the input layer, seven neurons were included in its hidden layer and one neuron was included in the output layer. In addition, in the ANN architecture constructed for the feed force, three neurons were included in the input layer, six neurons were included in its hidden layer and one neuron was included in the output layer.

Fig. 3
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Single hidden layer ANN architectures created for cutting forces

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3 Results and discussion

3.1 Experimental evaluation of the main cutting force

As a result of the experimental studies, the change in the main cutting force obtained under the same conditions for each tool used is given in Fig. 4. As can be seen clearly from the diagrams, increases in the main cutting force depend on an increased feed rate. Since the increase in the feed rate leads to an increase in the chip cross section, the pressure affecting the cutting tool will also increase. Consequently, the force required for cutting will increase. In addition, another common point in the main cutting force results obtained for each of the three cutting tools is that the cutting force rates obtained for low cutting speeds are high. In a sense, as the cutting force decreases, it becomes harder to break chips off the workpiece. As the cutting speed increases, the cutting temperature in the first deformation zone increases and thermal softening occurs in the cutting zone. This in turn leads to a decrease in the cutting force. Similar results were found in the literature [43, 44].

Fig. 4
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The variations in main cutting force depending on the coating type and cutting speed

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When Fig. 4 is examined, it can be observed that the increase in the feed rate leads to a linear increase in the main cutting force. Additionally, it can be seen that as the cutting speed increases, the main cutting force decreases proportionally. It is possible to say that the cutting force obtained for the uncoated cutting tool is higher when compared with that of the TiCN + Al2O3 + TiN-coated and the Al2O3-coated cutting tools. This condition can be seen clearly, especially when the maximum cutting forces obtained for 75 m/min cutting speed and 0.3 mm/rev feed rate are compared with each other. The reason for this is the low thermal conductivity coefficient of the coating materials [30]. In particular, the thermal conductivity of the Al2O3 coating decreases with increased temperature [45]. Thus, the spread of the heat occurring within the tool is delayed [46]. Most of the occurring heat is focused on the cutting zone. This in turn leads to a decrease in the cutting force because the increased cutting temperature decreases the friction coefficient in the tool–chip interface [47] and causes thermal softening in the workpiece.

3.2 Experimental evaluation of the feed force

The variations in feed forces obtained from the experimental studies conducted are given in Fig. 5. Depending on the increased feed rate, the feed forces in the three tools also increase. In addition, depending on the increased cutting speed, the feed forces decrease [44, 48]. The fact stands out that the feed forces obtained with an uncoated cutting tool at cutting speeds of V = 75 and 100 m/min are higher than the results obtained with the other two cutting tools. However, it is possible to say that the feed forces obtained at other cutting speeds are closer to each other. According to Fig. 5, the feed forces obtained with uncoated tools are higher than the feed forces obtained with the Al2O3-coated cutting tool. In addition, the F f obtained with the TiCN + Al2O3 + TiN-coated cutting tool for V ≥ 150 m/min and f ≥ 0.2 mm/rev is higher than that obtained with the uncoated tool. However, it can be said that there are no significant differences between TiCN + Al2O3 + TiN-coated and uncoated tools for V ≥ 150 m/min and f ≤ 0.1 mm/rev. This was attributed to low thermal conductivity coefficient of the coating material. Low thermal conductivity of the coated cutting tools exhibits better performance compared with uncoated cutting tools [30, 49, 50].

Fig. 5
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The variations in feed force depending on the coating type and cutting speed

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When both main cutting and feed forces were evaluated together, the cutting of AISI 316L stainless steel material at 150 m/min and higher cutting speeds may be more advantageous in terms of tool life because it is clear that at cutting speeds lower than 150 m/min in all three types of tools, the main cutting and feed forces are very high. On the other hand, when the cutting tools are evaluated within themselves, it is possible to say that the forces obtained for the Al2O3-coated cutting tool are lower. The thermal conductivity of the Al2O3 coating decreases with increased temperature [45]. Therefore, the Al2O3-coated cutting tool exhibits better performance compared with TiCN + Al2O3 + TiN-coated and uncoated tools [30, 49].

3.3 Prediction of cutting forces using multiple regression analysis

The variance analyses performed for the main cutting force and feed force are given in Table 7. As is apparent from the table, the effects of all the cutting parameters (Pr = 0.0067, Pr < .0001, Pr < .0001) on the main cutting force are statistically significant. Here, Pr < 0.05 value indicates statistically significant of parameters. This situation shows that the coating type, cutting speed and feed rate have a significant effect on the main cutting force in the orthogonal cutting process. Nevertheless, when the F statistical values are considered, it is possible to say that the feed rate has a greater effect on the main cutting force when compared with the coating type and cutting speed.

Table 7 Variance analysis for main cutting force and feed force
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When the significance values in the variance analysis conducted for the feed force are considered, it can be seen that the cutting parameters (coating type Pr = 0.0221, cutting speed Pr < .0001, feed rate Pr < .0001) are effective on the feed force. The F statistical values of the feed rate, coating type and cutting speed parameters are 230.1489, 5.5416 and 78.8310, respectively. These values show that the feed rate that has the highest F statistical value has a greater effect on the feed force when compared with the coating type and cutting speed. The statistical results obtained through multiple regression analysis for the main cutting force and feed force are given in Figs. 6 and 7. The mathematical statements established with multiple regression analysis for the main cutting force and feed force are given in Eqs. (6) and (7).

$$F_{\text{c}} = \left( {(400.851){-}((23.425) \cdot C_{\text{t}} ){-}((1.080) \cdot V) + ((4,074.169) \cdot f)} \right)$$
(6)
$$F_{\text{f}} = \left( { (664.757 ){-}((36.650) \cdot C_{\text{t}} ){-}((1.762) \cdot V) + ((2,008.542) \cdot f)} \right)$$
(7)
Fig. 6
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Comparison of cutting force values predicted by using regression analysis with residual cutting force values

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Fig. 7
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Comparison of experimental results with cutting force values predicted by using regression analysis

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The variations in the predicted cutting forces depending on the experimental results are given in Fig. 6. While the maximum deviation values vary between 150 and −100 for the main cutting force, the maximum deviation values for the feed force are between 300 and −250. This is also an indication that the coefficient of determination of the feed force is lower. The variations in the cutting force values predicted with multiple regression analysis, as per the experimental cutting force values, are given in Fig. 7. The predicted main cutting force values are very close to the experimental cutting force values. It was determined that the best convergence value was 99.96 % and the worst convergence value was 77.65 %, whereas for the feed force, it was found that the best convergence value was 99.69 % and the worst convergence value was 65.42 %. Also, the coefficients of determination for the main cutting force and feed force resulting from the regression analysis were 0.98 and 0.85, respectively.

3.4 Prediction of cutting forces using artificial neural networks

In the training of the ANN model developed for obtaining the cutting forces, five different learning algorithms, namely LM, SCG, BFGS, RP and CGP, were used to determine the optimum learning algorithm. Also, by using a range of network architectures, from three neurons to ten neurons, the optimum network architecture was determined. As a result of trials, while the optimum results for the main cutting force were obtained by the network architecture with seven neurons in its hidden layer and by the SCG learning algorithm, the optimum results for the feed force were obtained by the LM learning algorithm and by the network architecture with six neurons in its hidden layer. The network architectures developed for the main cutting force and the statistical results obtained for the learning algorithms are given in Table 8.

Table 8 Statistical data for the main cutting forces using five different algorithms
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The matching of the experimental and ANN values for testing and training sets of the main cutting forces is shown in Figs. 8 and 9, respectively. Additionally, the matching of the experimental and ANN values for testing and training sets of feed forces is shown in Figs. 10 and 11, respectively. They consist of the results of 60 tests that were performed using uncoated, TiCN + Al2O3 + TiN- and Al2O3-coated cutting tools. The most significant point here is that the prediction values are very close to the experimental values. As shown in Figs. 8, 9, 10 and 11, the predictive ability of the network for F c and F f was found to be satisfactory. This means that the selection of three input parameters as influencing factors for predictions of main cutting forces and feed forces provided satisfactory results. Among the algorithms developed for the prediction of cutting forces, the statistical values of the learning algorithm and network architecture that provided the best results are given in Table 9.

Fig. 8
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Matching of the experimental and ANN values for testing sets of main cutting forces

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Fig. 9
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Matching of the experimental and ANN values for training sets of main cutting forces

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Fig. 10
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Matching of the experimental and ANN values for testing sets of feed forces

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Fig. 11
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Matching of the experimental and ANN values for training sets of feed forces

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Table 9 Statistical values for the cutting forces
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Having selected the most suitable network architectures and learning algorithms, the mathematical statements established using the ANN model developed for the prediction of cutting force values are given in Eqs. (8) and (9).

$$F_{\text{c}} = \text{ }\left( {\frac{\text{1}}{{\text{1} + e^{{ - \text{( 5}\text{.1916 } \times \text{ }\varvec{F1 } + \text{ 0}\text{.5208} \times \text{ }\varvec{F2 } - \text{0}\text{.4575} \times \text{ }\varvec{F3} - \text{7}\text{.0998} \times \varvec{F4}\text{ } + \text{ 2}\text{.3128 } \times \varvec{F5 } + \text{1}\text{.0023 } \times \text{ }\varvec{F6} - \text{1}\text{.9508 } \times \text{ }\varvec{F7 } + \text{1}\text{.9509)}}} }}} \right) \times \text{ 1,850}$$
(8)
$$F_{\text{f}} = \text{ }\left( {\text{ }\frac{\text{1}}{{\text{1} + e^{{ - \text{( 0}\text{.5889} \times \varvec{F1 } + \text{ 0}\text{.4594} \times \varvec{F2} + \text{0}\text{.7798} \times \varvec{F3} - \text{5}\text{.9822} \times \varvec{F4} - \text{132}\text{.2598} \times \varvec{F5} - \text{0}\text{.6642} \times \varvec{F6} + \text{132}\text{.7910)}}} }}} \right) \times \text{1,550}$$
(9)

where F i (i = 1, 2, 3,…, 6 or 7) can be calculated according to Eq. (10).

$$F_{i} = \text{ }\frac{1}{{1 + e^{{ - E_{i} }} }}$$
(10)

where E i is the weighted sum of the input and is calculated by the equation given in Tables 10 and 11.

Table 10 Weight values between the input and hidden layers for main cutting force (F c)
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Table 11 Weight values between input and hidden layers for feed force (F f)
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The weight values show the effects of the parameters existing in the input layer on the cutting force values. The weight values obtained for the main cutting force and feed force are given in Tables 10 and 11. As seen in Table 10, the most effective parameter, which affects the main cutting force positively and negatively, is the feed rate.

It can be seen clearly from the weight values in Table 11 that, while the most effective input affecting the feed force positively is the feed rate, the most negatively effective input is the cutting speed. The maximum error values for the cutting force values obtained as a result of the ANN predictions are given in Table 12. The deviation values in Table 10 provide the evidence that the ANN model developed is a valid model.

Table 12 Maximum errors for cutting forces
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4 Conclusions

In this study, the multiple regression method and an ANN were used for the prediction of the cutting forces that occur during the orthogonal turning of AISI 316L austenitic stainless steel. Several cutting experiments were conducted during the experimental stage of the study by taking different cutting parameters and coating types into consideration. The findings of the study are as follows:

  1. 1.

    When the experimental and ANOVA results were examined, the dominant factor affecting both the main cutting force and the feed force was the feed rate. The feed rate was followed by the cutting speed and coating type, respectively.

  2. 2.

    The best results in the prediction of the main cutting force were obtained by the network architecture with seven neurons in its hidden layer and the SCG learning algorithm, whereas the best results for the feed force prediction were obtained with the network architecture having six neurons in its hidden layer and the LM learning algorithm. Because coefficients of determination were (R 2) higher, the RMSE lower and the MEP values within acceptable error limits (±5 %) for these learning algorithms.

  3. 3.

    While the coefficients of determination for the main cutting force and feed force resulting from the regression analysis were 98 and 85 %, respectively, in the ANN model, the coefficients of determination of the cutting forces for both the training and the testing set were more than 99 %. In addition, when the results of the predictions made with both modelling techniques were considered, it was concluded that the error values obtained with ANNs were rather lower than the error values obtained with the regression analysis.

  4. 4.

    It was determined that the ANN results were within acceptable error limits (±5 %). Therefore, it is recommended that ANNs be used for the prediction of cutting forces instead of conducting experimental studies, which are complicated, expensive and time consuming.

  5. 5.

    ANNs have significant advantages when compared with other conventional modelling techniques in terms of speed, simplicity and learning capacity from samples. This study has shown that ANNs constitute a good alternative to conventional modelling techniques for the prediction of cutting forces.