1 Introduction

In machining processes, as in many other fields of science and technology, there is now a strong emphasis on the implementation of the goals of sustainable development. Its assumptions, including a reduction of negative impact on natural environment, an increase in energy efficiency, and a decrease in the amount of production waste, require the development of reliable predictive models that will enable better understanding and optimization of manufacturing processes [1, 2]. The possibility of predicting state of a process by means of various models allows avoiding many difficulties already at the stage of their design [3, 4]. Due to the complexity of machining problems and despite long-term development work, progress in modeling these processes is slow and the existing models still have many imperfections [1].

Important issues in the field of machining are related to the temperature in the cutting zone. It strongly affects both the quality of the manufactured items and the durability of the cutting tools [5,6,7,8]. Shalaby et al. showed its dependence on density, specific heat, and thermal conductivity of a workpiece [9]. A lot of attention in the literature is devoted to various methods of determining the temperature in the cutting zone, but no common method has been established to date. Experimental research requires a lot of attention, time, and investment. Difficult access to the cutting zone is also a significant problem [5]. Finite element method (FEM), although very effective, often requires a long calculation time which negatively affects its efficiency. The disadvantage of many analytical models is their high level of complexity [3, 10].

As a result of experimental tests, average or maximum values of the temperature measured at the chip-tool interface or over the entire area of heat flow are usually determined. Contact and non-contact methods are used for this purpose. They are based on phenomena of conduction and radiation, respectively [9, 11]. Examples of the use of these two methods, i.e., the use of a thermocouple and a thermal imaging camera, are presented, e.g., in Grochalski et al. [12].

Numerical methods based on FEM or BEM (boundary element method) allow obtaining detailed information in the form of color maps presenting the distribution of temperature or stress within the workpiece and the tool [6, 13,14,15]. An important aspect in this case is proper definition of geometric and material models of the workpiece and the tool. Preparation of an adequate simulation model of the turning process is a difficult task, as it requires detailed knowledge about the method itself, the properties of the workpiece and tool, and the phenomena occurring in the process [6, 14, 16]. Expensive software as well as relatively long simulation time are significant disadvantages of these methods.

Analytical models are the third group of methods used to determine the temperature in the cutting process. Various models based on different assumptions and inputs, e.g., the modified chip formation model, the Komanduri-Hou model, or the Ning-Liang model, can be found in the literature [17]. By means of them, attempts are made to describe the behavior of a given material during its decohesion [11]. In case of orthogonal cutting, two main zones are distinguished: the primary shear zone (PSZ) which lies between the workpiece and the chip and the secondary shear zone (SSZ) related to the flow of the chip on the rake face of a tool [18]. Many models described in the literature were built [3, 5, 8, 11, 19,20,21] based on such scheme. Models of this type, most often based on the Johnson–Cook (J-C) constitutive equation and Oxley’s shear theory [22, 23], are based on several assumptions. Firstly, the tool is not worn; secondly, the resulting chip is continuous and its thickness over the entire length of chip-tool interface is constant; thirdly, loss of heat caused by convection and radiation from the surface of the workpiece, the tool, and the chip is negligibly small; and fourthly, the directions of the shear forces on the shear plane and the force tangential to the rake face of the tool are collinear to the respective directions of the vectors of shear velocity and chip velocity [11]. Despite the simplifications used, analytical methods generally provide results similar to FEM simulations [10].

Ongoing technological progress means that various industries are looking for modern solutions in the field of engineering materials and methods of their processing. There is an increasing use of materials with excellent chemical and physical properties which, on the other hand, often cause difficulties in their processing [4, 7]. Titanium and its alloys are good examples of such materials. They are characterized by an exceptionally good strength-to-weight ratio; they are also non-magnetic, biocompatible, and corrosion resistant [24]. However, high strength of these materials causes high forces and large amounts of heat generated in the cutting process, which, with their poor thermal conductivity, leads to the occurrence of high temperatures in the cutting zone. This, together with strong chemical reactivity of titanium, contributes to very rapid failures of the tools [4, 7]. Relatively high elasticity and ductility of titanium also cause problems with maintaining proper shape of made products, chip control, and the formation of burrs [2, 25]. In order to overcome these difficulties, machining at low cutting parameters is used together with various types of coolants/lubricants. This, however, increases the cost and the process duration. Grade 2 titanium is used, e.g., in power generation, chemical, medical, automotive, and aerospace industries [2].

The ability to determine the thermal state of the cutting process is one of the most important aspects of its modeling and optimization. The authors of this paper attempted to expand and verify the mathematical model of orthogonal turning of grade 2 titanium. An uncoated tool with a flat rake surface made of cemented carbide was used in the research. As a result, average temperature values in the PSZ and the SSZ were determined. The main difference between the presented method and FEM-based applications lies in obtaining single temperature values in the PSZ and the SSZ instead of comprehensive temperature distribution maps. This method does not require detailed information about the workpiece or tool material, and moreover, the algorithm used can be easily implemented in any programming language.

One important issue in machining of titanium alloys is its strong affinity with oxygen. Ti and its alloys react with oxygen even at room temperature, and as a result of this reaction, an oxide layer is formed on the titanium surface. An oxide layer formation on the Ti surface causes strong corrosion passivity at temperatures up to 500 °C. The oxidation resistance decreases drastically as the temperature exceeds 500 °C and continues to decrease at a rate parallel with the increasing temperature; the material loses protection against oxygen, nitrogen, and hydrogen embrittlement [26, 27]. In the experimental studies, the average values of the forces in the stabilized range were determined. Exemplary results of cutting force measurements are shown in the Appendix.

Based on the analysis of the literature, a relatively small number of publications on the treatment of grade 2 titanium alloy can be noticed in relation to often described Ti-6Al-4 V alloy. In [2, 28], the authors describe the properties of grade 2 titanium and point out the good ratio of density to mechanical properties. Its tensile strength is at the level of alloy steels—in the 210–1380 MPa range—with density equal only to 40% of the steel density. Other advantages of grade 2 titanium include a low modulus of elasticity and high corrosion resistance.

Titanium, as a low-density, biocompatible, and corrosion-resistant material, has become the most commonly used material for bone implants. Titanium is also used as material for condensers and heat exchangers in power and heating plants. Furthermore, it is used for elements of infrastructure used in the chemical industry, desalination, and desulphurization plants, for elements in wastewater treatment plants, and finally as a construction material in the aerospace and automotive industries.

2 Methodology

The presented calculation method is based mainly on the approach described by Ning and Liang [5], as well as Ślusarczyk [11]. It enables determination of average temperature values in the PSZ and in the SSZ during orthogonal turning using a tool with a flat rake face. In the first part of the study, experimental tests consisting of orthogonal turning of a grade 2 titanium tube were carried out. Cutting force measurement setup was used in order to register stabilized values of the tangent component Fc and the feed component Ff. A novelty presented in this paper is the use of a video tracking process that utilizes a high-speed camera which made it possible to record the process of chip formation and its flow on the rake face of the tool. The area recorded by the high-speed camera was specially prepared—marks were applied by means of knurling (Fig. 1). Such treatment allowed for the identification of characteristic points present on the side of the chip. The chip flow velocity vch on the rake face of the cutting tool was determined by means of specialized, single-point tracking software.

Fig. 1
figure 1

Workpiece surface with visible marks

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In the second part of the study, the values of shear stress in the PSZ and the SSZ (Fig. 2) were calculated using the J-C constitutive equation and Oxley’s model of cutting mechanics. In the PSZ, mean temperature on the shear line (TAB) was determined by finding the minimum difference between shear stress values obtained from the J–C model (k’AB) and Oxley’s model (kAB) according to Eqs. (1) and (2), respectively:

Fig. 2
figure 2

Orthogonal cutting model

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$${\overset'k}_{AB}=\frac{\sigma_{AB}}{\sqrt3}=\frac1{\sqrt3}\left(A+B{\varepsilon^n}_{AB}\right)\left(1+C\ln\frac{\varepsilon_{\dot AB}}{{\dot\varepsilon}_0}\right)\left(1-\left(\frac{T_{AB}-T_r}{T_m-T_r}\right)^m\right)$$
(1)
$${k}_{AB}=\frac{{F}_{s}}{{l}_{AB}w}$$
(2)

The values of material constants \(A, B, C, m\), and \(n\) used in Eq. (1) are presented in Table 5. The values of strain \({\varepsilon }_{AB}\) and strain rate \({\dot\varepsilon}_{AB}\) were determined from Eqs. (3) and (4):

$${\varepsilon }_{AB}=\frac{{\gamma }_{AB}}{\sqrt{3}}=\frac{\mathrm{cos}\gamma }{2\sqrt{3}\mathrm{sin}\phi \mathrm{cos}(\phi -\gamma )}$$
(3)
$${\dot\varepsilon}_{AB}=\frac{\gamma\dot AB}{\sqrt3}=C_0\frac{V_s}{\sqrt3l_{AB}}$$
(4)

The value of shear angle ϕ, occurring in Eq. (2), was calculated based on the distribution of velocities in the orthogonal cutting model (Fig. 3), according to Eq. (5) and experimentally determined value of chip flow velocity vch.

Fig. 3
figure 3

Distribution of velocities in the orthogonal cutting model

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$$\frac{{v}_{c}}{\mathrm{cos}\left(\phi -\gamma \right)}=\frac{{v}_{\mathrm{ch}}}{\mathrm{sin}\left(\phi \right)}=\frac{{v}_{s}}{\mathrm{sin}\left(90-\gamma \right)}$$
(5)

In Eq. (3), the value of shear force Fs was determined from Eq. (6):

$${F}_{s}=R\mathrm{cos}(\phi +\uplambda -\upgamma )$$
(6)

where \({\varvec{\lambda}}\) is the friction angle determined from (7)

$$\lambda =a\mathrm{tan}\left(\frac{{F}_{t}}{{F}_{c}}\right)+\gamma$$
(7)

The length of the shear line lAB was determined from Eq. (8):

$${l}_{AB}=\frac{{t}_{1}}{\mathrm{sin}\phi }$$
(8)

The value of parameter w is constant and corresponds to the thickness of the tube wall. The determination of constant C0, appearing in Eq. (4), is performed iteratively by comparing the values of normal stress in the SSZ, determined according to the J–C model (9) and Oxley’s model (10). The value of C0 is established when the difference between \({\overset'\sigma}_N\) and \({{{\sigma}}}_{{{N}}}\) reaches its minimum.

$${\overset'\sigma}_N=k_{AB}\left(1+\frac\pi2-2\gamma-2C_0n_{\mathrm{eq}}\right)$$
(9)
$${\sigma }_{N}=\frac{N}{hw}$$
(10)

where h is the length of the chip-tool interface determined from Eq. (11) and \({n}_{\mathrm{eq}}\) is the strain hardening constant determined from Eq. (12):

$$h=\frac{{t}_{1}\mathrm{sin}\theta }{\mathrm{cos\lambda sin\varphi }}\left(1+\frac{{C}_{0}{n}_{\mathrm{eq}}}{3\left(1+2\left(\frac{\pi }{4}-\phi \right)-{C}_{0}{n}_{\mathrm{eq}}\right)}\right)$$
(11)
$${n}_{\mathrm{eq}}\approx \frac{{{nB\varepsilon }^{n}}_{AB}}{(A+{{B\varepsilon }^{n}}_{AB})}$$
(12)

The mean temperature in the SSZ (Tint) was determined in a way analogous to the PSZ, i.e., by finding the minimum difference between the shear-flow stress \({{\varvec{\tau}}}_{\mathbf{i}\mathbf{n}\mathbf{t}}\) and the shear-flow stress kint, calculated according to Eqs. (13) and (14):

$${\tau }_{\mathrm{int}}=\frac{F}{hw}$$
(13)
$$k_{\mathrm{int}}=\frac1{\sqrt3}\left(A+B{\varepsilon^n}_{\mathrm{int}}\right)\left(1+Cln\frac{\varepsilon_{int}}{{\dot\varepsilon}_0}\right)\left(1-\left(\frac{T_{\mathrm{int}}-T_r}{T_m-T_r}\right)^m\right)$$
(14)

A block diagram of the developed algorithm is shown in Fig. 4.

Fig. 4
figure 4

Block diagram of the algorithm for determining temperature values in the PSZ and SSZ

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3 Materials and methods

Laboratory tests included a series of trials of orthogonal turning of a tube with an outer diameter of D = 60 mm and wall thickness of ap = 2.77 mm, made of grade 2 titanium. Its chemical composition in accordance with the EN 10,204–3.1 standard is presented in Table 1, while its selected properties are presented in Table 2.

Table 1 Chemical composition of grade 2 titanium
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Table 2 Selected properties of grade 2 titanium alloy
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An uncoated P10 sintered carbide parting tool with a flat rake face was used in the research. The rake angle in the tool coordinate system was γ = 13° while the clearance angle was α = 10°. The laboratory stand was equipped with a KNUTH Masterturn 400 precision lathe and a measuring setup suited for recording the cutting forces. It consisted of a Kistler 9257B piezoelectric dynamometer mounted on the lathe carriage and a Kistler 5070B charge amplifier. Signals representing cutting force components were recorded at the frequency of 1000 Hz. An analysis of the obtained waveforms was performed by means of the DynoWare computer software (version 2825A). The method of mounting the dynamometer on the lathe carriage is shown in Fig. 5.

Fig. 5
figure 5

Dynamometer mounted on the lathe carriage. 1—workpiece; 2—piezoelectric dynamometer; 3—parting tool mounted in a tool holder

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The setup for recording fast-changing images consisted of a PHANTOM V5.2 high-speed camera (Vision Research) with a fixed-focus lens NIKON NIKKOR f = 200 mm, a PC with CineViewer dedicated software, and a Dedolight Dedocool Colt 3 lighting system. Such setup made it possible to record the chip flowing down the rake surface of the tool. The recording process was carried out at the frequency of 3000 fps, in the resolution of 512 × 512 px. The Tracker computer software was used to determine the chip flow velocity. The position of the characteristic point of the chip was analyzed frame by frame. Examples of frames from the analyzed video sequence are shown in Fig. 6.

Fig. 6
figure 6

An exemplary sequence of the chip flowing on the rake surface of the tool

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Additionally, the temperature of chip side surface was measured by means of a FLIR SC 620 thermal imaging camera with a fixed-focus lens f = 38 mm. Sampling frequency was 30 fps, while the image resolution was 640 × 480 px. ThermaCam Researcher 2.9 computer software was used to analyze gathered data and to determine the average temperature in its maximum range. The assumed value of the emissivity factor was \(\varepsilon =0.6\). In Fig. 7, an example of the sequence recorded by the thermal imaging camera is presented and the area where maximum temperature was determined is marked.

Fig. 7
figure 7

Example of the sequence recorded by the thermal imaging camera with marked analyzed area

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The research plan was developed according to the Taguchi method. The control variables were vc and f. Their ranges, presented in Table 3, were selected on the basis of a catalog data (Tables 4 and 5).

Table 3 Ranges of values of control parameters used in experimental tests
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Table 4 Plan of experimental tests
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Table 5 Johnson–Cook model constants for grade 2 titanium (\(\dot{{{\varvec{\varepsilon}}}_{0}}=1\); Tr = 25 °C) [29]
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4 Results

Table 6 includes mean values of cutting force components Fc and Ff as well as values of chip flow velocity vch, shear angle ϕ, and temperature obtained experimentally for ap = 2.77 mm. Values of the maximum average temperature presented in the table were determined in the stabilized phases of the process.

Table 6 Summarized results of experimental tests
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Table 7 includes values determined in the analytical part of the study.

Table 7 Summary of results obtained in the analytical part of the study
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Figure 8 presents the impact of cutting parameters vc and f on cutting forces Fc and Ff.

Fig. 8
figure 8

Impact of cutting parameters vc and f on mean cutting forces Fc and Ff

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Temperature values determined by means of the presented method in the PSZ (TAB) and the SSZ (Tint) are presented in Figs. 9 and 10, respectively.

Fig. 9
figure 9

Temperature values in the PSZ determined by means of the presented method

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Fig. 10
figure 10

Temperature values in the SSZ determined by means of the presented method

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Figure 11 presents the values of shear angle and chip flow velocity obtained from specific experimental trials. Figure 12 shows the value of chip compression ratio calculated according to Eq. (15).

Fig. 11
figure 11

Values of shear angle and chip flow velocity for specific tests

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Fig. 12
figure 12

Chip compression ratio as a function of cutting speed

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$${\Lambda }_{h}=\frac{{v}_{c}}{{v}_{\mathrm{ch}}}$$
(15)

Figure 13 presents a comparative summary of the results obtained by means of experiments and analytical studies. The relative error calculated for each trial according to Eq. (16) is presented in Fig. 14.

Fig. 13
figure 13

Summarized results obtained from thermal imaging measurements and analytical calculations in the PSZ

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Fig. 14
figure 14

Relative error between the results obtained from thermal imaging measurements and analytical calculations in the PSZ

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$$D=\frac{abs({T}_{AB}-{T}_{\mathrm{max}\_\mathrm{AVG}})}{{T}_{\mathrm{max}\_\mathrm{AVG}}}$$
(16)

5 Conclusions

The article presents an original, experimental-analytical method for determination of mean temperature values in the PSZ and the SZZ during the orthogonal machining process. The input parameters include experimentally determined values of cutting force components and chip flow velocity. The values of shear and normal stresses in the PSZ and the SSZ are determined based on two different models, i.e., Oxley’s model of cutting mechanics and the J–C model. Temperature values in the PSZ and the SSZ are estimated by an iterative determination of the minimum difference between stress values obtained by means of those two different models. A novelty in this paper is the analysis of fast-changing images obtained by means of a high-speed camera, which enables to determine the chip flow velocity.

According to experimental measurements of the chip flow velocity, it was found that the feedrate has a significant impact on the chip compression ratio—as the former increases, the latter decreases. It was also found that the cutting speed does not affect values of the said coefficient. In turn, analytical research indicated that the temperature in the PSZ increases with an increase of the cutting speed and that temperatures in both the PSZ and the SSZ strongly depend on the adopted material constants used in the J–C model. A section of the algorithm related to estimation of the PSZ temperature tends to be more sensitive to the values of input parameters (cutting forces, tool geometry) than the part related to the temperature in the SSZ.

The comparison of the results obtained by means of the analytical method and the measurements carried out with a use of the thermal imaging camera shows their good conformity. The mean relative error is 18.1%, with maximum at 27.4%.

The presented method can be used as a simple alternative to complicated numerical simulations. The generic form of the algorithm shown in this paper enables its implementation in any programming language. Moreover, due to its low complexity, it is characterized by a short calculation time, and hence, it can be used as a tool for quick, indicative verification of the temperature in the cutting zone.