A new model to predict the tool life in turning of titanium aluminides

Several tool wear models for machining operations have been reported in the literature. However, most of them are capable of modeling only a part of the wear curve, designed for quasi-linear curves or need to be fitted by means of differential equations. In this research, a new analytical easy-to-use wear model is proposed, taking into consideration the geometry of the cutting insert and the influence of the cutting speed and the tool-workpiece contact stress. In addition, the effect of the critical machining time, relative to the tool degradation, has also been introduced. The wear model describes completely the wear curve, from the initial rapid wear stage to the thermal-activated wear rate acceleration. The constants and parameters of the equation are determined by cutting experiments from the cutting forces and flank wear registers under a variety of technological parameters. This model has been validated for turning process on Ti48Al2Cr2Nb aluminide with uncoated carbide tool under different cutting conditions, varying the cutting speed and the geometrical position of the tool. It has been demonstrated its suitability to predict tool life, according to a defined wear criterion.


Introduction
The monitoring and prediction of cutting tool wear has always been interesting because of its influence on the efficiency and cost of the machining processes [1]. Tool wear affects the surface roughness and integrity as well as the geometrical accuracy of machined parts [2]. Several mechanisms have been identified as contributing to the progression of wear, according to their mechanical or thermally activated origins.
The most relevant are adhesion, abrasion, oxidation, and diffusion phenomena. Generally, one or more mechanisms act in combination depending on the specific existing cutting conditions, i.e., the chemical affinity and pressure in the toolworkpiece contact, the coefficient of friction, and the cutting temperature. Crater wear is mostly influenced by temperature, and diffusion phenomenon is the dominant mechanism [3]. On the other hand, adhesive and abrasive mechanisms governate the progression of the flank wear. Among the technological parameters that define the cutting processes, the cutting velocity has been reported to be the most influential variable that promotes the acceleration of the flank wear rate, which is due to the well-known existing relationship between the cutting velocity and temperature. This fact becomes more significant when difficult-to-machine materials, such as titanium alloys or titanium aluminides, are processed [4].
The wear mechanisms have been traditionally described by means of analytical models with the aim of modeling the wear rate as a function of some process parameters such as cutting speed, feed rate, depth of cut, tool-workpiece materials, and tool geometry. Some researchers have tried to find empirical equations based on the experience existing about the influence of technological cutting factors on tool wear ( Table 1). The meaning of the different letters included in the equations in Tables 1 and 2 is available in the nomenclature section. Probably, the first researcher who proposed a tool life model was Taylor in 1907 [5]. He proposed Eq. 1, correlating cutting speed, V, with tool file, T, assuming that cutting speed was the most influential variable on tool life. Although this formula gives good results, it is restricted to very narrow range of cutting process parameters. This model was subsequently extended by Colding [6] by introducing the effect of other machining variables, such as tool material and geometry, by incrementing the number of regression constants, according to Eq. 2. This expression also includes the concept of equivalent chip thickness defined in Eq. 3 by Woxén [7]. As a result, this model enabled to carry out the analysis of tool wear under multiple cutting conditions. Recently, Laakso et al. [8] proposed an empirical model that combines the Taylor´s and Colding´s expressions (Eq. 4). This model applies the logarithmic curve to model the three stages of wear, but it needs an individual calculation of the parameters for each cutting condition, not being a general model.
The opposite viewpoint consists of searching wear models that explain the physical phenomena related to tool wear ( Table 2). Archard presented Eq. 5 to establish the sliding wear rate as a function of the normal load, the tool-workpiece contact area, and a factor depending on the materials in contact [8]. This study established the basis of wear theory and stated that the rate of wear is independent of the sliding velocity between the bodies in contact. Probably, the first physical models for wear rate applied to cutting tools were proposed by Shaw and Dirke [9]. They suggested an expression based on the Archard model (Eq. 6) that explains the abrasive and adhesive phenomena by introducing the strength of the workpiece into the wear model. Almost simultaneously, Trigger and Chao [10] formulated Eq. 7 that considers the diffusion action on the tool wear rate by the introduction of the cutting temperature action. On this line, Takeyama and Murata [11] proposed a general expression (Eq. 8) to predict tool wear rate as a function of the mechanical and thermally activated processes, without considering the possibility of brittle fracture. They assumed that the abrasive and adhesive mechanisms were a function of the cutting length; meanwhile, the diffusion wear depends on the cutting temperature. Besides, this research stablished that the flank wear is proportional to the relationship between the apparent activation energy and the cutting temperature, so the tool life could be predicted only by the initial temperature. In Table 1 Summary of selected empirical models of tool wear Taylor (1907) vT n = C (1) n and C are constants Colding (1959) Shaw and Dirke (1956) Choudhury and Srinivas (2004)

, a 2 , a 3 and p are constants
Trigger and Chao (1956) Takeyama and Murata (1963) Pálmai (2013) A a and A th are constants contrast, Usui et al. [12] considered that the cutting length also influences the thermally activated wear phenomena and they rewrote the equation proposed by Shaw and Dirke introducing the effect of the cutting temperature, according to Trigger and Chao theory (Eq. 9). Usuy et al. also determined the stresses existing on the contact surface of the tool, modeling the wear on the flank and the crater in tool rake face. Taking into consideration the previous models, Zhao et al. [13] established the normal stress on the tool-workpiece contact and the tool hardness as the main parameters acting on the tool wear process. They assumed wear rate to be directly and inversely proportional to normal stress and tool hardness, respectively, and integrated Archard equation considering normal stress being constant during the machining operation. For that, wear volume was expressed as a function of the flank wear band according to the analysis performed by Shaw [14]. Finally, they enhanced the proposal by introducing thermal softening of the tool material during the machining (Eq. 10). However, the mathematical equation did not allow to describe the inflection of the wear curve in the last stage. Choudhury et al. [15] followed the expression of the Archard equation, assuming that the wear coefficient varies almost in the same manner as the diffusion coefficient of the tool material. They considered the average flank temperature as a function of the cutting speed, the feed rate, and the depth of cut, being the cutting speed the parameter that have the most significant effect Their equation also took into consideration the geometrical definition of the cutting tool, being necessary to fit five constants for its resolution, as established in Eq. 11.
On their part, Luo et al. [16] followed the theory of Childs et al. [17] to model the abrasive wear and the theory of Schmidt et al. [18] to model the diffusive phenomenon in a single equation. Their model basically jointed the concepts stated by Shaw and Dirke for abrasive wear and by Trigger and Chao for diffusive phenomenon (Eq. 12). This model required the combined use of simulations, to determine the normal force and thermal rise during cutting, together with experimentation to determine the model constants. They established that the calibrated model for a particular material could be used to find the optimum cutting speed. Attanasio et al. [19] modified the model proposed by Takeyama and Murata, introducing the variation of the diffusion coefficient with the temperature according to experimental data (Eq. 13). Subsequently, they predicted crater wear on the tool rake face by means of finite element method (FEM) by using a coupled abrasive-diffusive model [2]. They distinguished two zones in the tool, depending on whether the temperature was sufficient to activate thermal wear phenomena. Finally, Pálmai [20] proposed a wear model in the form of a non-linear autonomous differential equation that integrates the concept of Takeyama and Murata with that of Usui (Eq. 14). He also included the existing variation in cutting temperature with the progression of flank wear. He obtained very accurate results for predicting the tool life in the turning process of AISI1045 steel for some cutting conditions, being possible to describe the abrasive, adhesive and thermally activated diffusion processes in a single model. This model, along with Taylor equation, allowed to determine the tool life. However, easy-to-use models, able to adjust the flank wear curve completely, that is, the initial stage, the stable period, and the final inflection, are very lacking in literature.
In this research, a wear model, derived from the theory of Archard, is proposed. As a novelty, the influence of the actual flank wear in the variation of the wear coefficient is introduced, and the degradation of the cutting tool when a critical machining time is reached is considered. It is applied and fitted for the turning process of Ti48Al2Cr2Nb titanium aluminide, which is a difficult-to-machined material for which no mathematical models have been defined previously. High thermal loads, combined with high cutting forces, lead to rapid tool deterioration, being mandatory to machine with low cutting speeds [21]. The model proposed herein has been fitted considering a wide range of cutting speeds and tool configurations, varying the tool cutting angles. As a result, a complete description of the tool weartime curve, from the initial rapid wearing to the final thermal degradation stage, is obtained.

Proposal of a new mathematical wear model
For some materials, the flank wear curve shows a nearly linear tendency, with a slight increase in the wear rate at the end of the insert life, which facilitates the applicability of different models. However, the tool wear values obtained for turning titanium aluminide Ti48Al2Cr2Nb show the typical wear curve (Fig. 1a). First, a rapid tool wear takes place, followed by a stable cutting stage. Finally, an acceleration of wear rate occurs due to thermally activated degradation processes. That is, the wear rate initially decreases and after a certain time starts to increase, as seen in Fig. 1b. Most of the reviewed models in the literature are not able to predict the tool flank wear curve completely, as they cannot define the first or the second inflections and the double curvature [13]. As a solution, some of these models start with an initial value of wear, V b0 , avoiding the modeling of the initial stage [20]. The proposed model aims to predict the behavior of the wear curve completely. It follows the principles introduced by Archard's equation, later adopted by Zhao's model. Nevertheless, the tool hardness, H, in Zhao´s differential equation before integrating [13], is replaced by a function that represents the degradation of the tool, D. This function aims to include into the tool wear equation the structural degradation phenomenon found in the cutting insert when the temperature reached is close to 1100 °C. As it was reported by the authors in a previous research [22], the cobalt phase (binder phase) combines with oxygen and loses its bonding function and wear dramatically accelerates. This sudden behavioral change can be modeled using a modified sigmoid function, defined as where ϴ is the cutting temperature and A 0 and B 0 are constants. Nevertheless, the description of the temperature evolution during the process is complex, due to the high values found. For that reason, and taking into account that the cutting temperature evolves with the flank wear, V b , that is, with the cutting time, t, it is possible to rewrite Eq. 15 as Here, A and t cr are parameters that depend on specific cutting conditions. Therefore, t cr represents the critical time when the degradation stage starts (Fig. 1a) and A is the acceleration factor. The sigmoid function is a versatile equation that permits to model abrupt changes in a curve without the need of using function parts to build up the continuous condition of it. According to that explanation, Eq. 17 defines the volume wear rate as a function of the wear coefficient, K; the normal stress in the flank wear land, σ; the cutting speed, V; and the degradation function, D.
Usually, normal stress and wear coefficient are assumed to be constant. However, considering the wear coefficient as a function of flank wear would be a more realistic hypothesis, as defined in Eq. 18, where C 1 and C 2 are constants.
Equation 18 allows the initial tool wear stage to be explained as a function of flank wear. When flank wear is zero (when the tool is new), the insert tip is sharp and geometrically weak, which promotes the initial rapid wear. As flank wear increases, the insert tip becomes blunter, reducing wear progression. Although this behavior must not be generalized for all materials (sometimes microchipping on the surface can affect in the opposite way providing tip weakness), it has been previously reported for the turning of titanium aluminides [23,24].
In Fig. 1b, the flank wear rate is represented against the cutting time. Clearly, the product of functions K(V b ) and 1/D(t) defines the complete curvature of the flank wear rate function (the purple curve). K(V b ) shows a stronger influence on the flank wear rate in an early stage, while the function 1/D(t) acts severely in the final stage, when thermal effects suddenly degrade the tool.
Besides, according to Fig. 2, the wear volume is a function of the flank wear band and the rake and clearance (16)  Rearranging V b and t in both terms of Eq. 20 and integrating, an expression of V b is obtained, as indicated in Eq. 21. The boundary condition taken into consideration for obtaining this expression was V b = 0mm at t = 0s , which corresponds to a state of "new insert".
The proposed equation includes the influence of the cutting speed, the normal stress, the depth of cut, the rake and clearance angles of the cutting insert and permits the flank wear-time curve to be completely defined. This wear model takes into consideration the contact stress state, the effect of the cutting parameters, the tool geometry, and its thermal degradation, all in all in an easy-to-use equation. The model must be fixed by determining the constants C 1 and C 2 , which depend on the tool-workpiece pair materials, and the parameters A and t cr , which depend on the specific cutting conditions applied.

Analytical determination of flank normal stress
As mentioned, the proposed model (Eq. 21) involves the knowledge of the normal stress on the tool flank, σ, which can be estimated according to the evolution of the cutting forces with the flank wear. The cutting forces consist of the forces on the rake face, F rk , and the wear effect forces on the flank face, F w (Eq. 22).
The forces on the rake face of the tool are composed of the tool-workpiece contact press and the tool-chip friction, which are a function of the selected cutting parameters.
On the other hand, the forces on the flank depend on contact stress and evolve with the increment of the flank wear band. In this research, Waldorf's theory [25] is considered to determine the normal stress in the flank face of the tool from the experimental radial and axial forces measured. The contact area in the flank wear band can be divided into the plastic flow area, near the cutting edge, and the elastic contact area, far from the cutting edge (Fig. 3).
Waldorf [25] defines the normal stress distribution along the plastic flow area and the elastic contact zone according to Eq. 23. Thus, the normal stress in the plastic flow area is constant; meanwhile, it is assumed to follow a quadratic function in the elastic contact area. V bp is the width of the plastic flow area, and σ is the normal stress in this zone.
Waldorf stated that for small values of V b , only an elastic contact between the workpiece and the flank of the tool takes place. When the flank wear reaches a critical value, V b *, the plastic flow is initiated at the front of the wear band [26]. From here on, the width of the elastic contact area remains invariable, defined by V b -V b *, while the plastic flow area increases with the flank wear. According to the previous explanation, V bp y-axis position is defined by Eq. 24.
The normal force per flank length-unit (Eq. 25) can be obtained by integrating the normal stress expressed in Eq. 23, along the flank wear band, taking into consideration Eq. 24. Fig. 3 Normal stress distribution in the flank wear band It is commonly assumed that the flank wear band evolves along the cutting speed direction, so that the normal stress in the flank is perpendicular to the cutting speed. Consequently, the normal stress action for finishing operations can be divided into the radial and axial directions, considering the tool-workpiece geometrical relationship (Fig. 4a).
Dividing the cutting edge into infinitesimal units (Fig. 4b), the axial, F wa , and radial, F wr , forces due to the flank wear are obtained by integrating the normal force per flank length-unit (Eqs. 26 and 27).
Finally, introducing Eq. 25 into Eqs. 26 and 27, the expressions of F wa and F wr are established in Eqs. 28 and 29, respectively.
Equations 28 and 29 describe the evolution of the axial and radial forces because of the flank wear as a function of V b , σ, r and V b *, and show a marked tendency change when the critical value V b * is reached. These expressions allow the normal stress, necessary to use the wear model (Eq. 21), to be obtained for each experimental case by fitting the measured cutting forces.

Experimental tests
Long-term turning experiments were conducted on Ti48Al-2Cr2Nb gamma titanium aluminide, whose chemical composition is indicated in Table 3. The initial workpieces were cylindrical rods of 57-mm diameter, melted via vacuum arc furnace from consumable electrodes, followed by hot isostatic pressed (HIP).
The cylinders were turned on a semi-automatic Microcut H-2160 lathe equipped with a variable frequency drive in order to be able to select the exact cutting speed for each test, so that the experimental conditions are maintained. TNMG 160,408-SM H13A uncoated carbide tool on MTJNR 2525 M 16M1 toolholder, from Sandvik Coromant, was used (Fig. 5a). Rough turning preliminary passes were made, with a depth of cut of 0.25 mm, feed rate of 0.1 mm/rev and cutting speed of 15 m/ min, to remove the cast skin and ensuring the uniformity of the cylinder surface. During the turning tests, the involved cutting forces were continuously registered by means of a dynamometer built up by the research team [27]. The flank wear band, V b , was measured after each cutting trial (approximately 200 m of cutting length) by using an Olympus SZX7 stereomicroscope until the end of the tool life (Fig. 5b-c). This threshold was  established for a flank wear value of 0.100 mm, for which a significant deterioration of the surface quality of the machined parts was visually detected and the process began to be unstable. This threshold has already been applied by other authors on the machining of titanium aluminides [28].
With the aim of evaluating the applicability of the wear mathematical model developed, a wide range of cutting conditions, based on the variation of the cutting speed and the geometrical positioning of the tool, was analyzed. The selected cutting speed levels were 40, 50, and 60 m/min according to a previous tool optimization research [24]. Three tool configurations, consisting of using different shims on the tool-holder, were taken into consideration. Therefore, the configurations led to obtaining a different combination of effective rake and clearance angles [24]. These configurations are collected in Table 4. As the rake and clearance angles were linked to each other by the design of the insert, the clearance angle was taken as the variable that defined the insert position. The feed rate and the depth of cut were fixed, at 0.1 mm/rev and 0.25 mm, respectively, since cutting speed is the most influential variable in the evolution of tool wear [4]. The tests were developed under dry machining conditions. Therefore, a total number of 9 experimental cases, that is 3 2 , were carried out, measuring the flank wear progression after each trial (Fig. 6). Although each test was conducted once, to ensure the repeatability of the wear measurements, three random cases, one for each tool configuration, were replicated  ig. 6 Flank wear evolution: a = 6.3 • , b = 11.6 • , and c = 15 • from the beginning to the end. The results verified the stability of the machining process and the experimental methodology applied.
It is clearly observed in Fig. 6 the influence of increasing the cutting speed, leading to a rapid acceleration of the flank wear due to temperature increment. Besides, the effect of the clearance angle increment is remarkable, enhancing the tool life. According to a previous study, abrasive wear on the tool flank is the predominant phenomenon during the cutting process, due to the high friction forces and temperatures generated in a very small area of the tool tip. This results in an accelerated increase of the wear band, leading to an increment of the temperature up to the thermal degradation of the tool. The increment of the clearance angle helps limit the flank face-workpiece interface contact area, leading to an effective reduction of the temperature and friction effects [24]. Besides, thanks to the increment of the tool insert inclination, the evolution of the built-up-layer (BUL) is shared between the flank and rake faces (Fig. 7), which improves the geometrical strength of the tool tip [23].
The evolution of the radial force and the axial force is shown in Figs. 8 and 9, respectively. Comparatively, Figs. 5, 6, and 7 show the same tendency with the machining time, which is consistent with the analytical linear relationship between the axial and radial forces and the flank wear.

Determination of model constants and parameters
In model fitting processes, the number of wear curves evaluated together influences the determination of the constants and the correlation of the fit. If the number of experimental wear curves is low, the correlation of the model with the experimental data will be higher, but the values of the constants obtained may not be useful for describing curves at different cutting conditions to those considered for the fitting process. When many conditions are evaluated together, the constants are determined to minimize the error for all cases. This increases the applicability of the model to different cutting conditions, although it could lead to a lower correlation in the fit. Anyway, it is necessary to design a strategy to obtain the best possible correlation between the experimental data and the fitted curves. Thus, the strategy considered herein consists of evaluating the constants C 1 , C 2 , and the parameters A and t cr that made the regression error the lowest for all experimental conditions once the normal stress is determined for each cutting condition. The normal stress for each cutting condition was estimated according to Eq. 22. The axial and radial forces, F wa , F wr , due to the flank wear were calculated by subtracting the value of the forces at the beginning of the machining, F a0 , F r0 , from the total forces registered by the dynamometer. Effectively, it can be assumed that F a0 and F r0 are the initial rake face force components (t = 0 s, V b = 0 mm) and remain invariable during the turning operation. Then, Eqs. 28 and 29 were fitted to the calculated F wa and F wr forces. Both components must be simultaneously fitted to obtain the normal stress, σ, and the critical wear band, V b * , so that the least square method minimizes the error, e, between the experimental values and the worked-out ones, F wa c and F wr c (Eq. 30).
condition. This fitting process was iteratively repeated until all wear values meet the condition applied for its calculation, solving the final σ and V b * values for each experimental case. Thereafter, the experimental wear values, V b , were fitted by using the wear model (Eq. 21), and considering all the experimental curves together (m = 9 = 3 cutting speed levels × 3 clearance angle levels). As explained before, the acceleration factor, A, and the critical time, t cr , are parameters that vary with every specific cutting condition. However, it was checked that the acceleration factor does not vary significantly with the clearance angle and, for that reason, A was constrained to be only function of the cutting speed, v. Note that this assumption implies a significant restriction of the variability of this parameter, decreasing from 9 to 3 possible values. Nevertheless, the parameter t cr is highly influenced by the cutting speed and the clearance angle, what it is clearly observed in the flank wear evolution established in Fig. 6. As a result, t cr (v,α) takes a different value for each experimental case evaluated.
The model was fitted to the experimental data by regression analysis. The constants C 1 and C 2 , and the parameters A and t cr were determined by minimizing the normalized error, e, between the fitted and the experimental values of V b according to where m is the total number of experimental curves evaluated together, V b c are the calculated flank wear values and V b the experimental ones. The total number of flank wear values taking into consideration, n, is different in each case depending on the tool life for every experimental condition. Therefore, n has been included in the error expression to normalize the weight of each experimental curve in the fitting process.
With the objective of optimizing the correlation between the experimental and the fitted values with the proposed model, a regression methodology was developed based on the individual influence of the functions K(V b ) and D(t) on the model that define the inflections of the wear curve. Firstly, the initial and middle parts of the wear curve were fitted assuming that the function D(t) = 1 , since this function only acts in the final inflection of the wear curves, that is, just on the onset of the degradation of the tool. The equation, avoiding the last part of the wear curve, can be expressed as Equation 32 permits starting values for C 1 and C 2 to be obtained, agreeing to the requirements of Eq. 31. This planning is graphically explained in Fig. 10 and simplifies the calculus process for the final procedure. Then, beginning with the calculated values of C 1 and C 2 , the complete wear model (Eq. 21) was fitted for the entire time range, solving the definitive values for C 1 , C 2 , A, and t cr .
Finally, to evaluate the correlation of the wear model with the experimental curves, the coefficient of determination, ρ 2 , was calculated according to Eq. 33, where V b means the average flank wear value for the particularly experimental case evaluated.
To make easier the understanding of the fitting methodology developed, Fig. 11 establishes the different stages carried out to complete the numerical process.

Results and discussion
With the aim of providing an example of the capacity of the wear model, one of the experimental curves is fitted individually (v = 40 m/min, = 11.6 • ). Figure 12a shows the axial and radial wear forces worked out according to Eqs. 28 and 29, respectively, obtaining a normal stress of2375MPa and a critical flank wear value V * b of 0.056mm . The normal stress is of a similar magnitude order as the values obtained by Cheng et al. that experimented with a different Ti-aluminide and comparable technological conditions [29]. It is remarkable the good correlation found, mostly once the critical wear value, V * b , is reached and the plastic flow area begins to increase. As expected, according to Eqs. 29 and 30, the radial force increment due to wear is higher than the axial component. In Fig. 12b, the fitted wear curve, evaluated for this individual case, is shown.
A good agreement between the experimental values and the fitted wear model is remarkable, considering a coefficient of determination 2 of 0.978 . It is remarkable that the proposed model is able to describe the wear curve  In Fig. 13, the values of worked out from the experimented conditions (Eqs. 28 and 29) are collected. As observed, strongly increases with cutting speed. This result can explain the acceleration of the wear phenomena when the cutting speed rises, along with the higher cutting temperature generated. On the other hand, the increment of the clearance angle allows the normal stress to be lower, which, according to the mentioned, produces a beneficial effect that extends the life of the tool considerably. The normal stress values have been fitted to the expression Eq. 31 with an excellent correlation, 2 = 0.991.
The global model is obtained taking into consideration the total experimented in this research, m = 9 . Thus, the constants and parameters involved are calculated to minimize the fitting error for all the cutting conditions. According to this, the values obtained were C 1 = 3.59 • 10 −16 and C 2 = 4.60 for the pair Ti48Al2Cr2Nb-WC/Co insert. On the other hand, each combination of the input variables gives a particular value for the parameters, A and t cr , as indicated in Fig. 14. Similarly to the consideration taken into account for normal stress, A and t cr were established as functions of v and α (Eqs. 35 and 36).
As observed in Fig. 14, A evolves with the cutting speed following an exponential function, which means that the tool degradation behaves in this way with the cutting speed. Thus, this result is consistent with classical cutting theory that states the rise of the cutting temperature and tool wear be predicted as a function of the cutting speed (between 40 and 60 m/min) and the clearance angle (between 6.3° and 15°) for a depth of cut of 0.25 mm and a feed rate of 0.1 mm/rev. The fitted curves, considering the nine cutting curves together, are shown in Fig. 15. According to that, the average coefficient of determination for the global model is 2 = 0.921 . Nevertheless, the lowest 2 evaluated if the application of the function is carried curve by curve is 0.880, which means a significative correlation of the model under all experimental conditions.
The initial wear stage is very fast. It actually occurs in the early stages of machining, where the cutting edge is lost, and the tool starts becoming blunt rapidly. As can be seen in the curves, the length of this stage is very short compared to the stable stage, so it is only possible to obtain 1 or 2 points experimentally. Actually, they are enough to describe this stage, which is almost linear up to a limit value where the stable working zone starts. The last stage of the curve (accelerated) depends on the cutting speed, so it has been possible to obtain more values for 40 and 50 m/min than for 60 m/min, existing at least a 26% of the data, with respect to the total ones, what it is a sufficient condition to evaluate the whole fitting. The high coefficients of determination obtained show a good fit, which takes into consideration all three stages of the curve (especially the stable and final stages). The fitting procedure minimizes the sum of errors in all stages, and the weight of the last zone behavior is relevant, which pointed to the authors that a good methodology was employed.
Using the wear model and considering the criterium, V bmax = 0.1mm, the tool life may be predicted and compared with the experimental one (Fig. 16), what demonstrates that the proposed wear model is suitable for the description of the tool life in the range of the machining conditions experimented herein.

Conclusions
In this research, a review of the classical tool wear models for machining operations has been carried out. Some deficiencies have been found as either they are limited to the modeling of the uniform tool wear period or they are not capable of taking into consideration a very different wear tendency together. For these reasons, a new tool wear model is proposed introducing the influence of the cutting temperature by means of a degradation function. This function aims to consider the machining time when the critical temperature, which caused the acceleration of the flank wear progression, is reached. The model has been evaluated for the turning process of Ti48Al2Cr2Nb titanium aluminide with an uncoated carbide tool, considering the variation of the cutting speed and the clearance angle as input variables. It demonstrated the suitability of the model to describe the full tool wear behavior that, along with the application to a high potential material for high quality requirements, is the main novelty of the work. The global model is suitable to faithfully describe wear curves and to predict tool life in the range of cutting speeds and tool configurations taken into consideration in this research, and that guarantee its industrial viability. Moreover, the normal stress in the flank face of the tool has been evaluated and modeled by means of an analytical procedure demonstrating that it is a good indicator of tool wear. Finally, the model is highly manageable as corresponds to an explicit equation from which a direct solution can be obtained.
The wear model proposed should be enlarged in future research by considering a wider number of variables, such as the depth of cut and the feed rate, in order to extend the application of the function to any applicable working conditions, that constitutes one of the limitations in its current shape. In addition, it should be introduced the cutting temperature directly in the model, which requires it was accurately registered during continuously during the cutting operation. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This research has been carried out thanks to a contract co-financed by FEDER funds through Axis 1: "Strengthening research, development and innovation" at a rate of 80% of the total costs of the total costs of the project.

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