Keywords: Cutting force model, Tool wear, Tool geometrical model, Additive manufactured, Inconel 718 Graphical

New mechanistic cutting force model for milling additive manufactured Inconel 718 considering effects of tool wear evolution and actual tool geometry.

Edouard Ducroux 1* , Guillaume Fromentin 1 , Fabien Viprey

Introduction

The high strength for high temperature and the high corrosion resistance of Inconel 718 make it one of the main materials used for hot parts in aero engines. Nevertheless, due to its properties, it is also considered as a hard-to-cut material. In order to limit the machining time, Additive Manufactured (AM) part are more and more produced to get a near net shape before machining operation. Yet, there is a lack of knowledge concerning the machinability of additive manufactured Inconel 718. Wang et al. [START_REF] Gong | Review on powder-bed laser additive manufacturing of Inconel 718 parts[END_REF] demonstrates that to ensure good mechanical properties of additive manufactured parts by applying heat treatment. However, AM material has still a lower yield strength and a higher elastic modulus than Wrought (WRT) one. The microstructures of AM parts are widely different and mainly depends on the applied heat treatment after manufacturing according to Chlebus et al. [START_REF] Chlebus | Effect of heat treatment on the microstructure and mechanical properties of Inconel 718 processed by selective laser melting[END_REF]. This may lead to a different machinability than wrought Inconel 718.

Cutting forces are a major concern to perform milling operation with high reliability. Altintas [START_REF] Yusuf | Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design[END_REF] shows that reducing cutting forces leads to restrain vibrations, deformations and improve tool life. Thus, to qualify machinability, the cutting forces analysis is essential. Predictive models help the connection between the global forces on the workpiece and the local interaction of the chip with the cutting edge. Armarego et al. [START_REF] Armarego | An investigation of zero helix peripheral up-milling[END_REF] are one of the first to describe both the edge and the cut effect in a unique mechanistic model to determine cutting forces in the case of a tool without helix angle. Edge effect is under influence of the cutting edge preparation and the workpiece contact whereas the cut effect models the effect of the chip formation mechanism. Then, Armarego et al. [START_REF] Armarego | Computerized Predictive Cutting Models for Forces in End-Milling Including Eccentricity Effects[END_REF] develop the previous model for end mill cutting geometry using tool discretisation. Kaymakci et al. [START_REF] Kaymakci | Unified cutting force model for turning, boring, drilling and milling operations[END_REF] later generalize this model to all cutting process. In addition to mechanical aspects, the main issues to model cutting forces in milling operation is also to consider all geometrical aspects of the end mill. These aspects are helix angle, differential pitch, radial runout and complex tool envelop. According to Fromentin et al. [START_REF] Guillaume | Geometrical analysis of thread milling-part 2: calculation of uncut chip thickness[END_REF], tools with complex definition affect the cut section over time. In this case, it is required real adapted methodology for the cut thickness determination. Therefore, Killic et al. [START_REF] Killic | Generalized modelling of cutting tool geometries for unified process simulation[END_REF] develop a generalised tool model formulation to optimise the cutting force simulation. Zheng et al. [START_REF] Zheng | Theoretical modelling and simulation of cutting forces in face milling with cutter runout[END_REF] propose a model which takes into account the runout. It substantially improves the accuracy of the simulation as it well computes the actual engagement angle of each cutting edge. Helix angle has also a large influence on the engagement angle of a cutting edge and axial cutting force (F z ). Ozturk et al. [START_REF] Erdem | Prediction of effect of helix angle on cutting force coefficients for design of new tools[END_REF] study this impact of a variable helix angle to improve machining performance. Niu et al. [START_REF] Jimbo | Mechanics and multi-regenerative stability of variable pitch and variable helix milling tools considering runout[END_REF] implement a different helix angle between pairs of cutting edges, a variable pitch and the tool runout in a mechanistic model considering edge and cut effects. Bissey-Breton et al. [START_REF] Bissey-Breton | Integration of tool geometry in prediction of cutting forces during milling of hard materials[END_REF] add the helix angle in the mechanistic model to represent the chip flow direction effect which have a great repercussion on the cutting forces for high helix angle.

Tool wear is a determinant factor for surface integrity and cutting forces. Tool life is excessively reduced in machining Inconel 718 compared to other materials used in aircraft industry such as aluminium alloys. Dudzinski et al. [START_REF] Dudzinski | A review of developments towards dry and high speed machining of Inconel 718 alloy[END_REF] summarise the work done on tool wear mechanism in machining Inconel 718 w ith carbide tools. They observe a high abrasion on the flank face and notching at the depth of cut. That is why reliable cutting model should be built including tool wear effect. Oraby et al. [START_REF] Oraby | Development of models for tool wear force relationships in metal cutting[END_REF] show the connection between tool wear and cutting forces to implement tool wear in cutting forces model for turning operation. Li et al. [START_REF] Kuan | Modeling of cutting forces in near dry machining under tool wear effect[END_REF] also develop a model considering the effect of reduced lubrication to analyse the influence of tool wear on the friction coefficient between the chip and the cutting edge. Then, Sun et al. [START_REF] Yujing | Modeling of cutting force under the tool flank wear effect in end milling Ti6Al4V with solid carbide tool[END_REF] use flank wear measurement implemented in a cutting forces model in end milling operation of Ti6Al4V. Nevertheless, these models only have considered the tool wear as a modification of the cutting process with an interaction between the chip and the cutting edge by implementing a term in their cutting forces law. The tool wear has also an impact on the tool geometry itself [START_REF] Dudzinski | A review of developments towards dry and high speed machining of Inconel 718 alloy[END_REF][START_REF] Kuan | Modeling of cutting forces in near dry machining under tool wear effect[END_REF][START_REF] Yujing | Modeling of cutting force under the tool flank wear effect in end milling Ti6Al4V with solid carbide tool[END_REF]. Indeed the flank wear induces an edge recoil which leads to a modification of the cutting edge engagement [START_REF] Jimbo | Mechanics and multi-regenerative stability of variable pitch and variable helix milling tools considering runout[END_REF]. Thus, the tool wear should be included in both tool geometry and cutting forces model.

Consequently, the machinability of AM Inconel 718 should be studied relative to microstructure changes, cutting forces and tool wear mechanism. Mechanistic model for cutting forces needs to include the most precisely possible tool wear considering the tool geometry modification and the contact interaction evolution. Based novel approaches, the present study will focus on these aspects, clearly highlight the processes of force treatment, model identification in milling and propose a way for analysis modelling precision.

Experimental materials

AM and WRT work materials

Both AM and WRT Inconel 718 are machined in this study. The AM parts are based on a laser melting of powder. It is the Direct Metal Laser Solidification (DMLS) process, which is based on melting a layer of powder and constructing a 3D piece layer by layer. The results of hardness measurements in both materials are shown in Fig. 1. For AM parts, the hardness is measured in the printed direction (different planes) and in the transversal direction (same planes a plane is too thin to be measured independently). The hardness machine has previously been adjusted on a material standard with a certified hardness equal of 50 HRC and a measurement trueness of ±1 HRC is observed.

The mean hardness measured is 44.7 HRC for AM material in the printed direction, is 44.6 HRC for AM material in the transversal direction and is 44.3 HRC for WRT material. All results show a hardness variation within the measurement trueness of the machine (±1 HRC). Hence, no visible difference between both materials in term of macroscopic hardness can be observed due to the resolution of the measuring system. Moreover, the AM parts have no difference either in macroscopic hardness regarding the print and the transversal direction.

Fig. 1 Comparison of hardness for AM and WRT materials

Both materials are heat treated to obtain quite similar global mechanical properties. The heat treatment is designed to eliminate anisotropic structure, laves, lack of density and other defects of AM processed material. Their chemical compositions are presented in Table 1. According to Pan et al. [START_REF] Zhipeng | Material microstructure affected machining: a review | Manufacturing Review[END_REF], Inconel 718 strength is largely due to the γ'-Ni 3 Nb and the γ''-Ni 3 (Al,Ti) precipitates. Concerning the microstructure, Kamdar [START_REF] Kamdar Madhusudan | Embrittlement by liquid metals[END_REF] explains that a large typical grain size limits the nucleation of cracks whereas a small one limits the propagation of cracks. Indeed, grain boundaries form obstacles for the dislocation movements and so the cracks propagations. As machining is based on crack propagation along the cutting edge to create a chip, the smaller the grain size is, the harder-to-cut the material will be. The microstructures of AM and WRT studied materials are presented in Fig. 2. During microstructure observation, no anisotropy has been observed on AM parts. The grain size is evaluated according to the standardised method in ASTM E112-13 [START_REF]Standard Test Methods for Determining Average Grain Size[END_REF], the carbide size and the twin grains frequency are visually estimated on a 9 mm 2 sample. The results are given in Table 2. The grain size of AM material is about thirty times larger than the WRT one. Therefore, the crack propagation is much more limited in WRT material because of all grains boundaries. Both materials have high frequency of twin grains. This characteristic is also a barrier to the dislocation movement. High proportion of Niobium favours the formation of Niobium carbides (NbC) which are highly resistant at cutting temperatures (between 800°C and 1000°C). According to Karasev et al. [START_REF] Andrey | Characterization of nonmetallic inclusions in corrosion -resistance nickel -based EP718 and 718 alloys by using electrolytic extraction method[END_REF], other carbonitrides such as TiCN and (Nb,Ti)CN are also present in the Inconel 718 microstructure. Inclusions size and density for both materials are evaluated using SEM-EDS combined with AZtec software. Values of Equivalent Circular Diameter (ECD) are presented in Fig. 3. In the present study, these three types of carbides are analysed: NbC, TiCN and (Nb,Ti)CN. By this analysis it is shown that WRT material presents the largest carbides for all types analysed. Moreover, as seen in Table 2, WRT material presents also the highest density of carbides. Carbide size may be a sensitive indicator to evaluate machinability. Carbide is very difficult to cut during machining. Three different options would happen according to Dudzinski et al. [START_REF] Dudzinski | A review of developments towards dry and high speed machining of Inconel 718 alloy[END_REF]. Firstly, the carbide is cut and so high cutting forces are locally generated on the cutting edge. Secondly, if the carbide is small enough, the tool tears off the carbide from the material and will create a hole at the machined surface. Otherwise, the carbide is not cut and damages the tool by flank surface scratching. As the carbides of WRT material are larger than the AM material one, they are more likely to damage the tool than to be torn off. Consequently, WRT material may locally generate greater cutting forces and bigger scratch on the cutting edges compare with AM material.

Tool analysis and modelling

During this study, the used tools are solid coated carbide end milling cutters with four flutes and corner radius from Mitsubishi Materials (IMX10C4HV100R10010S EP7020). The tool parameters are shown in Table 3. In order to evaluate all parameters, the tool is scanned using an Alicona InfiniteFocusSL which generates a cloud of points. After post-processing in a CAD software to build the flank and rake face along the cutting edges, actual tool parameters can be measured and results are depicted in Fig. 4.

Table 3. Measured tool parameters

Coating

R Tool (mm) r ε (mm) Z Ψ (°) γ o (°) α o (°) r β (µm) (Al,Cr)N (2µm) 5 1 4 {0, 2, 0, 2} 6 6 3 -4
The global tool parameters (i.e. R Tool and r ε ) are used for defining the End Mill Profile (EMP) towards the point height z in the tool frame TF. This tool design presents both a variable helix angle along the cutting edge and a variable pitch Ψ. The local cutting angles (λ s , γ o , α o ) are evaluated along the cutting edge using the ISO 3002-1:1982 [START_REF]Basic quantities in cutting and grinding --Part 1: Geometry of the active part of cutting tools --General terms, reference systems, tool and working angles[END_REF] and computed as function of the height z of a cutting edge point. The flank and rake angles are measured as constant along the cutting edge. Finally, the angle λ s is described by a sum of piecewise continuous third degree polynomial functions which is 𝒞 2 (ℝ;ℝ) between two points every 0.1 mm as presented in Fig. 4.

Fig. 4. Tool scan and helix angle λ s evolution along cutting edge.

Due to a great influence of runout on the engagement angle and the cut thickness (h) along each cutting edge, the radial runout should be examined. Therefore, an in-situ scan of the tool is performed with a laser profilometer Keyence LJV7060. The radial runout R out of the i Fl th cutting edge is computed as an update of the tool radius for this edge considering the runout at the altitude z. From EMP and λ s , the Reference Cutting Edge CE ref is modelled in TF by Eq (1) considering the radial component of EMP noted EMP r , the axial wich is equal to z and the radial runout R out .

{𝐶𝐸 𝑟𝑒𝑓 [𝑖 𝐹𝑙 ,𝑧]} 𝑇𝐹 = { (𝐸𝑀𝑃 𝑟 [𝑧] + 𝑅 𝑜𝑢𝑡 [𝑖 𝐹𝑙 ,𝑧]) * cos ( 𝑧 * tan(𝜆 𝑠 [𝑧]) 𝐸𝑀𝑃 𝑟 [𝑧] + 𝑅 𝑜𝑢𝑡 [𝑖 𝐹𝑙 ,𝑧] ) (𝐸𝑀𝑃 𝑟 [𝑧] + 𝑅 𝑜𝑢𝑡 [𝑖 𝐹𝑙 ,𝑧]) * sin ( 𝑧 * tan(𝜆 𝑠 [𝑧]) 𝐸𝑀𝑃 𝑟 [𝑧] + 𝑅 𝑜𝑢𝑡 [𝑖 𝐹𝑙 ,𝑧] ) 𝑧 } 𝑇𝐹 𝑤ℎ𝑒𝑟𝑒 𝜆 𝑠 [𝑧] = ∑ 𝑎 𝑢 * 𝑧 𝑖 3 𝑢=0 (1)
Finally, all cutting edges are developed from Eq. ( 1) considering measured differential pitch Ψ following Eq [START_REF] Chlebus | Effect of heat treatment on the microstructure and mechanical properties of Inconel 718 processed by selective laser melting[END_REF].

{𝐶𝐸 [𝑖 𝐹𝑙 ,𝑧]} 𝑇𝐹 = 𝑅𝑜𝑡 𝑤 [ 2𝜋 𝑍 (𝑖 𝐹𝑙 -1) + 𝛹[𝑖 𝐹𝑙 ]] . {𝐶𝐸 𝑟𝑒𝑓 [𝑖 𝐹𝑙 ,𝑧]} 𝑇𝐹 ⏟ 𝐸𝑞 .(1) (2)
In this way, the geometrical model of the tool considers variable pitches, the progressive helix angle along corner radius and the radial runout of each cutting edge.

Experimental set-up

Experiments are performed on a Mikron HSM600U 5-axis milling machine equipped by an Heidenhain's iTNC530 CNC. The Fig. 5 represents the sample in the machine with both the tool frame TF, which is in rotation considering the spindle rotation by a θ angle, and the workpiece frame (WPF). The cutting forces are recorded using a Kistler 9119AA2 piezoelectric dynamometer with a sampling frequency of 50 kHz. The tool is attached in a BIG tool holder to limit the maximal radial runout below 20 µm. Finally, a standard 6% emulsion oil is used during machining. The studied configuration (Fig. 5) is a side milling operation in the -X WP F way. The angle θ is recorded, from spindle encoder using a Heidenhain's IBV606 derivation box, with a resolution of 0.17° (256 toothed wheel by twice interpolator by quadrature factor over 360°), and synchronized with force measurements. The trigonometric function θ is decreasing over time, from 0 to -2π in one revolution as the spindle rotates in clockwise direction. According to the derivation box resolution, a list of tool angular measured values ListθMeas is built from 0° to -360° which represents all angular position where a measure is acquired. The tool angular measured value insured no signal shift between experiments and simulations towards time and the trials, which avoid any magnitude effect on model identification. After the initial experimental identification of the tool angular origin, no angular shift is introduced in the modelling process.

Fig. 5. Machining and measurement set-up.

The optimal cutting parameters are investigated with the tool-material pair process. The cutting speed V c and the maximal depth of cut a p are set respectively at 40 m.min -1 and at 4 mm. The tool wear is measured through image analysis taken by a Dino-Lite Edge microscope. The flank wear VB is measured according to ISO 8688-2:1989 [START_REF]Tool life testing in milling -Part 2: End milling[END_REF] along the cutting edge for z from 0 to the axial depth of cut a p .

Mechanistic cutting force model

Geometrical parametrisation and cut area

The modelling approach is based on a mechanistic model and the cutting edge discretisation methodology. Thereafter, linear milling operation is parametrized, including the full definition of the milling cutter geometry, its runout and the precise computation of the cut thickness h, needed to model local cutting forces.

Considering the feed f z and the number of flutes Z, the tool centre trajectory TCT, for a straight milling operation into -X way, is evaluate in WPF as a function of spindle angle θ by the following Eq. (3):

{𝑇𝐶𝑇[𝜃]} 𝑊𝑃𝐹 = { 𝑓 𝑧 . 𝑍. 𝜃 2𝜋 0 0 } 𝑊𝑃𝐹 (3)
Then, from the end mill rotation, the i Fl th cutting edge definition from Eq. (2) and TCT, the i Fl th cutting edge trajectory S CE is computed in WPF by the Eq. ( 4) as follows. Based on this parametrisation, the cut thickness h is determined by solving numerically its real formulation [START_REF] Guillaume | Geometrical analysis of thread milling-part 2: calculation of uncut chip thickness[END_REF]. That means, in this study no approximation is made during the three steps, which are, the cutting edge definition, the surface generated by cutting edges, along their trajectories, and finally the cut thickness computation approach.

{𝑆𝐶𝐸[𝑖 𝐹𝑙 ,𝑧, 𝜃]} 𝑊𝑃𝐹 = {𝑇𝐶𝑇[𝜃]} 𝑊𝑃𝐹 + 𝑅𝑜𝑡 𝑍 [𝜃].{𝐶𝐸 [𝑖 𝐹𝑙 ,𝑧]} 𝑇𝐹 (4)
Considering the surface SCE of the i Fl th edge, the cut thickness h is the normal distance between this surface and the previous surface generated by the (i Fl -1) th edge. Thus by geometrical construction in WPF of the unit normal vector NV SCE to SCE, h is determined by finding the intersection between the line defined by NV SCE at the i Fl th edge point and SCE of the (i Fl -1) th edge. It is the solution of Eq. ( 5) at a tool angular position θ 0 near θ+(2π/Z) and a height z 0 near z.

{ 𝑆𝐶𝐸[𝑖 𝐹𝑙 -1, 𝑧 0 , 𝜃 0 ] = 𝑆𝐶𝐸[𝑖 𝐹𝑙 ,𝑧, 𝜃] + ℎ[𝑖 𝐹𝑙 ,𝑧, 𝜃]. 𝑁𝑉 𝑆𝐶𝐸 [𝑖 𝐹𝑙 ,𝑧, 𝜃] 𝜃 0 ≈ 𝜃 + 2𝜋 𝑍 𝑎𝑛𝑑 𝑧 0 ≈ 𝑧 (5)
The cut thickness computation has to be restricted to the positions for θ values where the cutting edges are engaged into work material as presented in Fig. 6; the geometrical conditions are expressed by Eq. [START_REF] Kaymakci | Unified cutting force model for turning, boring, drilling and milling operations[END_REF]. In order to reduce computation time, it is processed in the following order. Firstly, if a point M 1 of the cutting edge is at a z higher than the a p depth of cut, this point is not cutting, so h is directly set to 0. Secondly, considering the projection of a M 2 point on SCE[i Fl ], on the orthogonal axis Y WP F to the feed direction X, if its ordinate is lower than the parameter EMP r [z]-a e , this M 2 cutting edge point is actually not in work piece, then h is also set to 0. Thirdly, if a point M 3 of the SCE on backside from feed direction, the cut thickness h calculated is negative according to Eq (5), the value is replaced by 0. Finally, in any other case (i.e. in the work zone), h solution computed according to Eq. ( 5) is valid.

Cutting force simulation and measurement treatment

Cutting force simulation

During this study, in order to deal with the complex geometrical problem, the modelling approach is based on the edge discretization methodology. The tool is divided along the cutting edge every 0.025 mm in the nose radius and every 0.05 mm in the lateral edge towards z altitude. The cutting mechanics is modelled with using a mechanistic approach. Therefore, it is proposed to model local cutting forces {f t , f r , f a } in TRA frame as defined by Eq. ( 7):

This cutting law can be modelled as the combination of three effects. The first and the second are the edge effect (b iz *K ikb ) and the cut effect (b iz *K ikh *h iz ) where i k ∈{t,r,a} presented by Armarego et al. [START_REF] Armarego | An investigation of zero helix peripheral up-milling[END_REF]. The last one is the effect chip flow direction described by Bissey-Breton et al. [START_REF] Bissey-Breton | Integration of tool geometry in prediction of cutting forces during milling of hard materials[END_REF] which depends on the local helix angle, it is the cutting edge obliquity effect. This effect influences the edge effect as the helix angle modifies the apparent cut angle in P r plane. Moreover, it modifies the chip flow direction and so the friction direction between the chip and the tool. That is why it impacts also the cut effect. The model coefficients K ikb , K ikh and K ikλ , where i k ∈{t,r,a}, are later identified from cutting force measurement in WPF.

The complete process for cutting forces simulation is explained in Fig. 7. Once obtaining {f t , f r , f a } TRA , they are expressed in TF using the transfer matrix M TF←TRA 𝑖𝑧 . Then, the global cutting forces {F X , F Y , F Z } TF in TF are obtained by summing the local ones from every segments (Nb seg ) and on every teeth (Z). Finally, with the transfer matrix M WPF ←TF [𝜃]

Cutting force measurement and treatment

The complete measurement treatment is presented in Fig. 8. Thank to spindle encoder position, it includes filtering and averaging over tool angular position for characterising precisely cutting force variation towards tool revolutions.

The global cutting forces are recorded in WPF (cf. section 2.3). Force signals are only analysed during the steady state, i.e. tool fully engaged into workpiece, which stands for approximatively 40 revolutions. The time based recorded forces for multiple revolutions are evaluated towards θ measurements from ListθMeas and superimpose on a one angular revolution base. Then, in order to eliminate the noise on the signals over time or angular position, a moving average filter with 31 values is applied all along the recording data. In the following, only treated cutting forces, i.e. representative curves (last stage of treatment in Fig. 8), are used so the notation {𝐹 𝑖𝐶𝑜 ̅̅̅̅̅ } Meas_F/WP F is equal to {𝐹 𝑖𝐶𝑜 ̅̅̅̅̅ } Meas/WP F with i Co ∈{X,Y,Z}. The same notation is used for {𝐹 𝑖𝐶𝑜 𝑀𝑖𝑛 } Meas/WP F and {𝐹 𝑖𝐶𝑜 𝑀𝑎𝑥 } Meas/WP F .

Cutting force model identification and validation method

An inverse identification is carried out to determine the coefficients K ikb , K ikh and K ikλ , i k ∈{t,r,a}, of cutting laws i.e. Eq. ( 7). This process of inverse identification is described in Fig. 9. The principle is to simulate cutting forces with unknown coefficients and to establish a set of equations with the relative error towards 𝐹 𝑖𝐶𝑜 ̅̅̅̅̅ for a list ListθId of tool angular position composed of Nb θId values. This list excludes the positions where simulated cutting forces are nulls. It ensures that no equation is going to identify a noise kept from the measurements. Furthermore, no angular delay is introduced over time or the trials in order to fit the simulations with the experiments; the process is deterministic concerning this aspect thank to the measurement of spindle angular position in a reliable way. Then, the error sum ε of related to all directions and all trials, from 1 to respectively Nb Co and Nb Tr , is minimised using Levenberg-Marquardt algorithm by adjusting the coefficients of Eq. ( 7). The number Nb Co is equal to 3 reflecting i Co ∈{X,Y,Z}. The number Nb Tr is equal to the number of used trial for identification. It corresponds to the Id index in the column Usage in Table 5. In the system equations, the number of them is the product of Nb TrId , Nb θId and Nb Co . Then, residual degree of freedom (RDOF) is equal to 3* Nb TrId * Nb θId -9, because of 9 coefficients K kb , K kh and K kλ , i k ∈{t,r,a}are determined in Eq [START_REF] Guillaume | Geometrical analysis of thread milling-part 2: calculation of uncut chip thickness[END_REF]. A high RDOF insure a robust determination of the model. Nevertheless high RDOF leads to solve many equations. As consequence, the trials and the list of angular position ListθId for identification have to be carefully selected. The identified coefficients, presented in Table 4, are then implemented in the simulation process to get the global cutting forces {𝐹 𝑖𝐶𝑜 } Sim/WP F . Then a comparative analysis with {𝐹 𝑖𝐶𝑜 ̅̅̅̅̅ } Meas/WP F is performed to validate the fitness of purpose. The complete trials used for validation (Val index) trials are given in Table 5 for fresh tool.

Table 4. Cutting law coefficients in TRA for fresh and worn tool.

Cutting law coefficient

For The identification has been done on a reduce set of position defined by ListθId, so as to reduce computation time. Nonetheless the validation may be easily analysed on much more tool angular values, ListθVal. Eventually, if there is a teeth cutting continuity, it may be validated on any measured tool angular values, i.e. on ListθMeas. If not, validation would deal with angular positions with null simulated force and pure experimental noise on the measured force. That's make no sense in term of modelling error.

Based on these considerations, the validation approach is the following. From ListθMeas a list of θ angular position with one point each degree is set. Then, every angular positions, where {𝐹 𝑖𝐶𝑜 } Sim/WPF is lower than 10% of the maximal simulated forces along i Co ∈{X,Y,Z} direction for all positions, are taken back from this list to establish ListθVal. The length of ListθVal is Nb θVal . Then, at the positions from ListθVal, the absolute gap between {𝐹 𝑖𝐶𝑜 𝑀𝑖𝑛 } Meas/WPF and {𝐹 𝑖𝐶𝑜 𝑀𝑎𝑥 } Meas/WPF for each direction in WPF is calculated with Eq. ( 8), it represents the experimental force variation over tool revolutions. Finally, the absolute error between simulated and mean measured cutting forces with Eq. ( 9) considering i Co ∈{X,Y,Z}. The results are presented in Table 5 for fresh tool.

𝛥𝐹 𝑖𝐶𝑜 = √ 1 𝑁𝑏 𝜃𝑉𝑎𝑙 .
Finally, the relative error towards all directions is given by Eq. ( 10): [START_REF] Erdem | Prediction of effect of helix angle on cutting force coefficients for design of new tools[END_REF] The result is also given for the fresh tool model in Table 5

𝐸𝑟𝑟𝑜𝑟 𝑋𝑌𝑍 =

Cutting force model analysis for fresh tool

Cutting force analysis

The model for fresh tool presented in Eq. ( 7) is identified with trials in Table 5. The cutting conditions choice covers multiple configurations such as teeth cutting continuity and discontinuity or only nose radius work (a p = r ε = 1 mm). The identified coefficients are given in Table 4. The variation proposed for coefficients correspond to different tested tools. It appears small variations in local tool geometries such as the rounded cutting edge radius r β can lead to variation of the edge effect. Concerning the cut effect variation, according to [START_REF] Devillez | Cutting forces and wear in dry machining of Inconel 718 with coated carbide tools[END_REF] it may be due to the bond between the coating and the tool material. Furthermore, the identification process admits certain uncertainty by the choice of positions in ListθId. Some of the tested conditions results are shown in Fig. 10.

Fig. 10. Cutting forces simulated and measured for fresh tool.

In Fig. 10, a difficulty to model the cutting edge exit can be seen. It is due to a formation of a build-up edge which friction and so cutting force at this location.

Model coefficients analysis

Firstly, the machined material has a limited impact on the chip flow effect. Indeed, the identified coefficients are similar in all directions in TRA. To analyse the identified coefficients, the macroscopic friction coefficient can be determined using force components projected from frame TRA to frame NGS as shown in Fig. 11. By this way the ratio of the normal force of the edge along N and the tangential one along F on the rake face is equal to a macroscopic friction coefficient Cf Rake . Hence, the coefficient Cf Rake for WRT is equal to 0.63 and is 6% higher than the AM one along the cutting edge.

Concerning the edge effect, the direction of friction velocity between the workpiece and the flute is along V c direction as seen in Fig. 11. Thus, the friction on the flank face is opposite to T direction. The macroscopic friction coefficient Cf Flank on this face is computed as the ratio of the cutting force along R and the one along V which is the plane (T,A). The coefficient for WRT is equal to 1.45 whereas it is equal to 0.71. This coefficient Cf Flank represents the friction on the flank face and also the required energy to form the chip. That is why it can be superior to 1. Finally, the edge has no effect towards the axial direction in TRA.

These two coefficients show that WRT material is more difficult to cut than AM one according to the identified coefficient K ikb , K ikh and K ikλ , i k ∈{t,r,a}.

Model Validation analysis

The absolute error of the model Error iCo (according to Eq. ( 9)) towards each direction and for each material is always lower than the absolute gap ΔF iCo (defined by Eq. ( 8)) for the same direction and the same material. Correlated with the InValue near 80%, which means that the prediction {𝐹 𝑖𝐶𝑜 } Sim is 80% of the time between {𝐹 𝑖𝐶𝑜 𝑀𝑖𝑛 } Meas/WP F and {𝐹 𝑖𝐶𝑜 𝑀𝑎𝑥 } Meas/WP F measures. Finally, the relative error Error XYZ is below the common criteria of 10 %. Hence, the model is valid on the tested cutting conditions domain.

Tool wear analysis and effects

The tool wear is investigated for WRT and AM Inconel 718 in two different configurations which are presented in Table 6. The objective is to compare a small and a large milling engagement with the same maximal cut thickness h max equal to 96 µm. This helps to see the influence of the teeth contact cutting time on tool wear. These two configurations are tested four times on each material which means that 16 tools are used during these tests. The depth of cut is equal to 2 mm for all tests. As a consequence, tool life is examined towards machined volume. The observations in Fig. 12 show the wear process in milling WRT material, but the principal manifestations of tool wear seem to be the same for both material and both configuration. One macro wear mechanism consists in a notching at the depth of cut limit, where material is going to smear in it. On the other part of the cutting edge a pure flank wear is observed. Then, the smeared material is going to be snatched from the flank face increasing by this way the notching dimension as presented in Fig. 13. Another wear manifestation is based on the previous one but the initial state is a scratch on the cutting edge due to a carbide from the machined material. However, the initial notching is not localised. So this mechanism produces random wear profile. Sometimes the material smeared on the flank face may stay for a long period creating a build-up edge and protect the cutting edge. Consequently, this protection increases tool life. All mechanisms lead to cutting edge collapse. The Fig. 14 illustrates the evolution of maximal flank wear towards all tool cutting edges VB max/max . It transposes the range of flank wear measured during 4 wear test for each condition. The tool life in WRT material is approximatively half the one in AM material, with considering VB max/max = 0.3 mm as a limit. This is due to higher cutting forces for the same cut thickness as seen in the fresh tool model which conduct quickly to a critical tool wear. Moreover as carbides are larger in WRT material, they are more likely to produce scratch on the flank face and lead to an increase of the tool wear following the second manifestation presented earlier as presented by Dudzinski et al. [START_REF] Dudzinski | A review of developments towards dry and high speed machining of Inconel 718 alloy[END_REF].

Concerning tool engagement, the higher is the engagement, the lower is the tool life in term of machined volume. This observation is due to the time contact between the material and the cutting edge. At the same cutting speed, the radial engagement sets the cutting contact time. Therefore, the higher this time is, the higher the edge temperature would be. As the temperature is a leading factor in tool wear, tool life is longer with a small engagement. Consequently, these investigations demonstrate that AM material has a better machinability, regarding cutting forces and global tool wear, than WRT one.

Wear impact on cutting forces and tool model update

The evolution of cutting forces towards tool wear must be characterised in order to later model them precisely over a long time. In a first approach cutting forces are recorded at different wear states. Results are presented in Fig. 15 with teeth cutting discontinuity conditions. It is observed that evolution of tool wear has a modest impact on {𝐹 𝑋 ̅̅̅ } Meas/WP F . The different levels of initial cutting force peaks are a consequence of the initial runout of each teeth; the peak is higher for teeth having a high runout. Thus, a cutting edge with such high initial runout is going to generate higher cutting forces and to wear out faster. Finally, due to wear, the measured cutting forces peak increases for teeth with the smallest initial {𝐹 𝑋 ̅̅̅ } Meas/WPF whereas it decreases for teeth with the highest initial {𝐹 𝑋 ̅̅̅ } Meas/WP F . This observation shows that tool wear is first a tool geometry modification as it tends to reduce runout. That is why needs to be implemented in the tool modelling.

Concerning {𝐹 𝑌 ̅̅̅ } Meas/WPF in Fig. 15, the cutting forces gradually increase with tool wear. This observation shows that tool wear is also an alteration of the surfaces interaction between the tool and the work material. This interaction can be modelled by a modification of the local cutting model in TRA. Finally, a major increase on both {𝐹 𝑋 ̅̅̅ } Meas/WPF and {𝐹 𝑌 ̅̅̅ } Meas/WP F is located when the cutting edges exit work material whereas cutting forces are very similar at the cutting edge entrance. This is due to the non-constant wear profile along the cutting edge as show in Fig. 13. That is why the VB profile along the cutting edge has to be considered. The geometrical tool evolution due to wear and its change of runout is integrated into the tool model as following. Considering a pure flank wear, Back off Cutting Edge (BCE) is a function of the flank wear VB and the clearance angle α o by Eq [START_REF] Jimbo | Mechanics and multi-regenerative stability of variable pitch and variable helix milling tools considering runout[END_REF]:

𝐵𝐶𝐸[𝑖 𝐹𝑙 ,𝑧] = 𝑉𝐵[𝑖 𝐹𝑙 ,𝑧]. tan(𝛼 𝑜 ) (11)
Next step, the VB profile is computed based on the VB max measure on each cutting edge and takes into account the different wear zone presented in Fig. 13. An exponential function has been used to model VB along the cutting edge as shown in Fig. 16 and Eq [START_REF] Dudzinski | A review of developments towards dry and high speed machining of Inconel 718 alloy[END_REF].

Finally, the actual cutting edge is established thanks to Eq (1) based on the worn end mill profile EMP w . This profile is depicted in Eq (13) considering a BCE of EMP into -NV EMP way; the normal vector to the initial tool profile EMP.

𝐸𝑀𝑃 𝑤 [𝑖 𝐹𝑙 ,𝑧] = 𝐸𝑀𝑃[𝑧] -𝐵𝐶𝐸[𝑖 𝐹𝑙 ,𝑧]. 𝑁𝑉 𝐸𝑀𝑃 [𝑧]

For z higher than the tool nose radius r ε , the tool geometry modification generates a runout modification. Thus, it is possible to compare the model evolution with the measured runout evolution. In order to validate the propose approach, the runout is now measured with an electronic dial indicator of 1 µm resolution. The results are shown in Fig. 17. The measured runout evolution matches with its evaluation from VB measurement according to Eqs. [START_REF] Jimbo | Mechanics and multi-regenerative stability of variable pitch and variable helix milling tools considering runout[END_REF] and [START_REF] Dudzinski | A review of developments towards dry and high speed machining of Inconel 718 alloy[END_REF]. It confirms the tool model evolution towards its wear developed in this section. The VB profile considers a pure flank wear and a not random effect. Thus, only the first mechanism presented in the section 4.1 is modelled. Moreover, as the wear increasing, the edge collapse can produce a high variation in the wear profile and affect the cutting geometry and then the forces. That is why only flank wear inferior at 0.25 mm can be examined.

Cutting force model including tool wear effect

As described in section 4.2, tool wear effect must be included into the local mechanistic model to integrate the modification of the interaction between the tool and the machined material. It is proposed in Eq. ( 14) a mechanistic model for local cutting forces in TRA considering tool wear.

{𝑓 𝑡 [𝑖 𝐹𝑙 ,𝜃]} 𝑇𝑅𝐴 𝑖𝑧 = 𝑏 𝑖𝑧 (𝐾 𝑡𝑏 (

This model is identified with the process shown in section 3.3. Nevertheless, the coefficients representing the edge, the cut and the obliquity effect are limited around the ones identified previously for a fresh tool. Identified coefficients for Eq. ( 14) are presented in Table 4 and the trials used for the identification are shown in Table 7. As for fresh tool, the coefficients variations are due to identification deviation or local tool geometry variation. The cut, edge and obliquity effects coefficients are the same as for fresh tool because for VB equal to 0 the model for used tool is equal to the model for fresh tool. The wear coefficient K aw for axial direction in TRA is negligible as the edge one K ab for both material. It means that the chip flow direction does not change towards wear evolution. Nevertheless, they are not set to 0 to limit identification instability.

The same analysis illustrated in section 3.4 concerning the macroscopic friction coefficient is done with the wear coefficient added to the computation. The wear has no impact on the coefficient Cf Rake for both material as wear coefficient is applied to the edge effect. The coefficient Cf Flank,w is equal to 1.22 for WRT and 0.61 for AM. It confirms the conclusion seen in section 3.4.

The Fig. 18 presents the evolution of cutting forces over tool wear, modelled vs experiments, for one milling test in AM Inconel 718. For these conditions, there is no teeth cutting continuity and flank wear levels of each tooth are given. This results show that tool wear has not only a repercussion on the cutting forces peak but also on the curves forms. It also shows that greater is the tool wear, lower is the prediction precision. The causes of this effect are the model hypotheses, indeed in high wear state the tool has not been only damaged on the flank face and not only according to the profile presented in section 4.2. Furthermore, the fresh tool is coated and this coating (2 µm) may be not present anymore at high wear state which highly changes the friction coefficient between the tool and the chip creating hazard effect such as build up edge. The Table 7 presents the absolute error and the absolute difference between minimal and maximal measured cutting forces. It shows that only Error Y in the Y direction is higher than experimental force variation over revolution ΔF Y , but for AM material the difference is only of 2 N. Moreover, the InValue for both materials is around 75 %, but on Fig. 18 the out pipe simulation are concentrated on trials with high tool

Conclusion

The purpose of this work is to analyse the machinability of additive manufactured Inconel 718 in side milling with rounded solid carbide cutters. To this end, the work material microstructure, wear mechanisms, the cutting forces affected by tool wear are studied for both the additive manufactured Inconel 718 and a wrought Inconel 718. Then, it is developed a mechanistic approach to model cutting forces based on a fully geometrically defined parametrisation including real runout along cutting edge with a precise cut thickness determination, with using the cutting edge discretisation methodology. It includes at first step a fresh tool and then considering tool wear. The main conclusions are:

-In machining additive manufactured Inconel 718, cutting forces are lower in all direction compare to machining wrought Inconel 718 which could be explained by the difference of microstructure -The main tool wear mechanism is abrasion on the flank face which leads to edge collapsing through attrition of the cutting edge by material stocked and snatched from the flank face. -A second tool wear process is notching at the depth of cut and where carbides damage the cutting edge. -A final process is the cutting edge protection by smeared material on the flank face and built-up edge formation. -Tool life in wrought Inconel 718 is half the tool life in additive manufactured one.

-The tool wear has an impact on the tool geometry and its model has to follow this geometrical evolution (runout). -The tool wear also affects significantly and directly cutting forces and so has to be considered into local model formulation as proposed. -It is difficult to consider critical tool wear effect (VB max greater than 0.25 mm) in a predictive model as it creates too many random wear profile.

In addition, this article deeply formulated several methods to have a robust model identification and to characterize its precision, these aspects are often eclipsed from comparable study. The first method consists in considering the cutting forces variation, over tool revolutions, in the measurement treatment to create the measured pipe. The second one is to validate the model identification and simulation data through several criteria. The last one is to examine identified coefficients and their relation to physical properties observed.

Frames

  WPF = {O, {X, Y, Z}} Workpiece Frame TF = {O T , {u, v, w}} Tool Frame TRA = {O S , {N Ps ,N Pr ,P s ∩P r }} Local discretisation Frame where O s is a point of the cutting edge NGS = {O S , {N Rf ,N P n ,R f ∩P n }} Local rake face Frame with R f = Rake face of the tool Index I Fl i Fl th flute/cutting edge of the tool i z i z th segment of the cutting edge i k i k th component of the cutting forces in TRA

Fig. 2 .

 2 Fig. 2. Inconel 718 (a) AM and (b) WRT microstructure before machining.

Fig. 3

 3 Fig. 3 Carbides size (ECD) distribution measure using SEM-EDS and Aztec software.

  top.rev -1 256 encoder top.rev -1 512 encoder top.rev -1 {FX, FY, FZ}WPF (N) θ (°)

Fig. 6 .

 6 Fig. 6. Cut thickness limitation zones.

Fig. 7 .

 7 Fig. 7. Computation algorithm of cutting forces in workpiece frame.

  Finally, based on 40 revolutions signals, i.e. 40 values for any tool angular position, three representative curves are computed from the measurement for each of the three force components in WPF. The 𝐹 𝑖𝐶𝑜 ̅̅̅̅̅ , with i Co ∈{X,Y,Z}, which describes the mean of the measured values at the i θ th position from ListθMeas, the 𝐹 𝑖𝐶𝑜 𝑀𝑖𝑛 and 𝐹 𝑖𝐶𝑜 𝑀𝑎𝑥 which respectively describes the minimum and the maximum measured value at the i θ th position fromListθMeas. As a consequence, these representative curves characterise the natural fluctuations of force components over multiple tool revolution, and a model precision has to be analysed in the light of them.

Fig. 8 .

 8 Fig. 8. Cutting forces measurement treatment.

Fig. 9 .

 9 Fig. 9. Cutting forces simulation and inverse identification process.

Fig. 11 .

 11 Fig. 11. Rake face projection from TRA to NGS.

4 .

 4 Modelling of cutting forces including tool wear

Fig. 12 .

 12 Fig. 12. Tool wear characterisation in end milling of WRT Inconel 718.

Fig. 13 .

 13 Fig. 13. Tool wear observation from flank wear to cutting edge collapse in end milling of AM Inconel 718.

Fig. 14 .

 14 Fig. 14. Flank wear evolution range (4 wear tests) as a function of machined volume respectively forAM and WRT material and for both engagements.

Fig. 15 .

 15 Fig. 15. Cutting forces measurement and their evolutions in X (a) and Y (b) directions towards tool wear.

Fig. 16 .

 16 Fig. 16. Modelling wear profile with VB max measurement.

Fig. 17 .

 17 Fig. 17. Tool radial runout computed considering VB measurement and actual measured runout for AM material and a e = 1 mm.

Fig. 18 .

 18 Fig. 18. Cutting forces simulated and measured during milling of AM Inconel 718 with worn tool.

  1 , David Prat 1 , Alain D'Acunto 21 : Laboratoire Bourguignon des Matériaux et Procédés [LABOMAP], Arts et Métiers Institut de Technologie 2 : Laboratoire d'Etude des Microstructures et de Mécanique des Matériaux [LEM3], Arts et Métiers Institut de Technologie * Corresponding author at :edouard.ducroux@ensam.eu

	i θ	i θ th angular position of the rotation angle θ in TF
	i Tr	i Tr	th trial
	i Co	i Co	th component of the cutting forces in WPF
	Tool parameters
	Z	Number of flutes
	R Tool	Tool radius (mm)
	r ε	Corner radius of the tool (mm)
	z	Altitude of a cutting edge O s point in TF (mm)
	θ	Rotation angle of the tool in TF (°)
	P r	Normal plane to V c direction and containing O S and Z TF
	P s	Normal plane to P r and tangent to the cutting edge
	P n	Normal Plane to P r and P s and containing O s
	Ψ[i Fl ]	Differential pitch of the i Fl	th flute (°)
	λ s [z]	Helix angle function of z and defined in P s (°)
	a u	u th Coefficient of the function λ s [z] (mm -u )
	γ o	Rake angle of the cutting edge measured in P o (°)
	α o	Flank angle of the cutting edge measured in P o (°)
	Cutting parameters
	V c	Cutting speed (m.min -1 )
	f z	Feed (mm.rev -1 .tooth -1 )
	a e	Tool radial engagement (mm)
	a p	Axial depth of cut (mm)
	h	Cut thickness (mm)
	b	Discretised cutting edge length engaged (mm)
	Nb seg	Number of segments for each cutting edge after discretisation
	Geometrical objects
	EMP[z]	End Mill Profile
	TCT[θ]	Tool Centre Trajectory
	CE ref [z]	Reference Cutting Edge
	CE[i Fl , z]	i Fl	th Cutting Edge
	R

out [i Fl , z] i Fl th cutting edge Runout at the altitude z SCE[i Fl , z, θ] Surface generated by the i Fl th flute/cutting edge

Table 1 . Chemical composition for AM and WRT Inconel 718 (wt%).

 1

	Material Ni	Cr Fe Nb Mo Ti Al Co
	AM	52.4 18.6 17 5	3	1 0.5 1
	WRT	52	19 17 5.1 3.1 1 0.5 1

Table 2 . Microstructure analysis of Inconel 718 before machining.

 2

	Material	Grain size (G)	Carbide AZtec (mm -2 ) density with	Twin grains
	AM	6.63	43	Frequent
	WRT	9.21	68	Frequent

  the global forces are determined in WPF.{𝑓 𝑡 [𝑖 𝐹𝑙 ,𝜃]} 𝑇𝑅𝐴 𝑖𝑧 = 𝑏 𝑖𝑧 ( 𝐾 𝑡𝑏 ⏟

		+ 𝐾 𝑡ℎ . ℎ 𝑖𝑧 [𝑖 𝐹𝑙 ,𝜃] ⏟	) ( 1 + 𝐾 𝑡𝜆 . 𝜆 𝑠 𝑖𝑧 ⏟	)
	𝐸𝑑𝑔𝑒 𝑒𝑓𝑓𝑒𝑐𝑡	𝐶𝑢𝑡 𝑒𝑓𝑓𝑒𝑐𝑡			𝐶ℎ𝑖𝑝 𝑓𝑙𝑜𝑤 𝑒𝑓𝑓𝑒𝑐𝑡
	{𝑓 𝑟 [𝑖 𝐹𝑙 ,𝜃]} 𝑇𝑅𝐴 𝑖𝑧 = 𝑏 𝑖𝑧 ( 𝐾 𝑟𝑏 ⏟	+ 𝐾 𝑟ℎ . ℎ 𝑖𝑧 [𝑖 𝐹𝑙 ,𝜃] ⏟	) ( 1 + 𝐾 𝑟𝜆 .𝜆 𝑠 𝑖𝑧 ⏟	)
	𝐸𝑑𝑔𝑒 𝑒𝑓𝑓𝑒𝑐𝑡	𝐶𝑢𝑡 𝑒𝑓𝑓𝑒𝑐𝑡			𝐶ℎ𝑖𝑝 𝑓𝑙𝑜𝑤 𝑒𝑓𝑓𝑒𝑐𝑡
	{𝑓 𝑎 [𝑖 𝐹𝑙 ,𝜃]} 𝑇𝑅𝐴 𝑖𝑧 = 𝑏 𝑖𝑧 ( 𝐾 𝑎𝑏 ⏟	+ 𝐾 𝑎ℎ . ℎ 𝑖𝑧 [𝑖 𝐹𝑙 ,𝜃] ⏟	) .	𝜆 𝑠 𝑖𝑧 ⏟
	𝐸𝑑𝑔𝑒 𝑒𝑓𝑓𝑒𝑐𝑡	𝐶𝑢𝑡 𝑒𝑓𝑓𝑒𝑐𝑡			𝐶ℎ𝑖𝑝 𝑓𝑙𝑜𝑤 𝑒𝑓𝑓𝑒𝑐𝑡

  𝑁𝑏 𝑇𝑟𝑉𝑎𝑙 ∑ ∑ ({𝐹 𝑖 𝑇𝑟 ,𝑖 𝜃 ,𝑖 𝐶𝑜

		𝑁𝑏 𝑇𝑟𝑉𝑎𝑙	𝑁𝑏 𝜃𝑉𝑎𝑙					2
		𝑖𝑇𝑟=1	𝑖𝜃=1	𝑀𝑎𝑥	} 𝑀𝑒𝑎𝑠/𝑊𝑃𝐹	-{𝐹 𝑖 𝑇𝑟 ,𝑖 𝜃 ,𝑖 𝐶𝑜 𝑀𝑖𝑛	} 𝑀𝑒𝑎𝑠 /𝑊𝑃𝐹	)	(8)
	𝐸𝑟𝑟𝑜𝑟 𝑖𝐶𝑜 = √	1 𝑁𝑏 𝜃𝑉𝑎𝑙 .𝑁𝑏 𝑇𝑟𝑉𝑎𝑙	𝑁𝑏 𝜃𝑉𝑎𝑙 ∑ ∑ ({𝐹 𝑖 𝑇𝑟 ,𝑖 𝜃 ,𝑖 𝐶𝑜 } 𝑆𝑖𝑚/𝑊𝑃𝐹 -{𝐹 𝑖 𝑇𝑟 ,𝑖 𝜃 ,𝑖 𝐶𝑜 𝑁𝑏 𝑇𝑟𝑉𝑎𝑙 ̅̅̅̅̅̅̅̅̅̅̅ } 𝑀𝑒𝑎𝑠/𝑊𝑃𝐹 ) 𝑖𝜃=1 𝑖𝑇𝑟=1	2

  . A final validation is done by computing the InValue which represents the percentage of positions from ListθVal included in the experimental pipe, that means where {𝐹 𝑖𝐶𝑜 } Sim/WP F is greater than {𝐹 𝑖𝐶𝑜 𝑀𝑖𝑛 } Meas/WP F and lower than {𝐹 𝑖𝐶𝑜 𝑀𝑎𝑥 } Meas/WP F for all directions i Co ∈{X,Y,Z} (i.e. ∀i Co ∈{X,Y,Z} and ∀θ∈ListθVal, {𝐹 𝑖𝐶𝑜 𝑀𝑖𝑛 [θ]} Meas/WP F < {𝐹 𝑖𝐶𝑜 [θ]} Sim < {𝐹 𝑖𝐶𝑜 𝑀𝑎𝑥 [θ]} Meas/WP F ).

Table 5 . Identification and validation trials for mechanistic model with fresh tool.

 5

	N° T rial	Usage	V c (m/min)	f z (mm/rev/tooth)	a e (mm)	a p (mm)
	1	Id/Val	40	0.07	2	2
	2	Id/Val	40	0.10	2	2
	3	Id/Val	40	0.15	2	2
	4	Id/Val	40	0.13	3	2
	5	Id/Val	40	0.11	3	2.5
	6	Id/Val	40	0.15	1	3
	7	Id/Val	40	0.30	2	2
	8	Id/Val	40	0.13	2	1
	9	Val	40	0.11	2	4
	10	Val	40	0.11	4	2
	11	Val	40	0.08	1	4

Table 6 . Milling conditions for small and large engagement configurations (h max = 96 µm).

 6

	Configuration	V c (m/min)	f z (mm/rev/tooth)	a e (mm)	a p (mm)
	Small	40	0.16	1	2
	Large	40	0.098	4	2

  1 + 𝐾 𝑡𝑤 . 𝑉𝐵 𝑖𝑧 [𝑖 𝐹𝑙 ]) ⏟ 𝑇𝑜𝑜𝑙 𝑤𝑒𝑎𝑟 𝑒𝑓𝑓𝑒𝑐𝑡 + 𝐾 𝑡ℎ . ℎ 𝑖𝑧 [𝑖 𝐹𝑙 ,𝜃]) (1 + 𝐾 𝑡𝜆 . 𝜆 𝑠 𝑖𝑧 ) {𝑓 𝑟 [𝑖 𝐹𝑙 ,𝜃]} 𝑇𝑅𝐴 𝑖𝑧 = 𝑏 𝑖𝑧 (𝐾 𝑟𝑏 (1 + 𝐾 𝑟𝑤 . 𝑉𝐵 𝑖𝑧 [𝑖 𝐹𝑙 ]) ⏟ 𝑇𝑜𝑜𝑙 𝑤𝑒𝑎𝑟 𝑒𝑓𝑓𝑒𝑐𝑡 + 𝐾 𝑟ℎ . ℎ 𝑖𝑧 [𝑖 𝐹𝑙 ,𝜃]) (1 + 𝐾 𝑟𝜆 .𝜆 𝑠 𝑖𝑧 ) {𝑓 𝑎 [𝑖 𝐹𝑙 ,𝜃]} 𝑇𝑅𝐴 𝑖𝑧 = 𝑏 𝑖𝑧 (𝐾 𝑎𝑏 (1 + 𝐾 𝑎𝑤 . 𝑉𝐵 𝑖𝑧 [𝑖 𝐹𝑙 ]) ⏟

	+ 𝐾 𝑎ℎ . ℎ 𝑖𝑧 [𝑖 𝐹𝑙 ,𝜃]) . 𝜆 𝑠 𝑖𝑧
	𝑇𝑜𝑜𝑙 𝑤𝑒𝑎𝑟 𝑒𝑓𝑓𝑒𝑐𝑡


Rachel Waschbüsch

John Carran

Philippe Savignac

A High Yielding Synthesis of Diethyl 1-Fluoromethylphosphonate in pure form

We describe a high yielding synthesis of the title compound (2) in pure form. The synthesis consists of : a) a Michaelis-Arbuzov reaction between triethylphosphite and fluorotribromomethane in hexane at 50°C giving diethyl dibromofluoromethylphosphonate (1) in 79-85% yield and b) a halogen-metal exchange reaction between (1) and n-BuLi-ClSiMe 3 at -90°C yielding (2) in 94% yield.

Une synthèse efficace du diéthyl 1-fluorométhylphosphonate de grande pureté Résumé Nous avons mis au point une préparation efficace du diéthyl 1fluorométhylphosphonate (2) pur. Elle comprend: a) une réaction de Michaelis-Arbuzov entre le triéthylphosphite et le fluorotribromométhane dans l'hexane à 50°C conduisant à (1) (79-85%) puis b) une réaction de double échange halogéne-métal en présence de n-BuLi-ClSiMe 3 à -90°C convertissant (1) en (2) avec un rendement de 94%

Version française abrégée

Le diéthyl 1-fluorométhylphosphonate 2 est un réactif trés utile dans la préparation d'analogues stables de phosphates d'intérét biologique (Schéma 1). Les voies d'accés à 2, essentiellement par réaction électrophile, sont peu efficaces et conduisent réguliérement à des mélanges de produits mono et difluorés. Nous avons mis au point une préparation efficace de 2 pur que nous décrivons en détail. Le procédé se décompose en deux étapes. La première conduit au diethyl dibromofluorométhylphosphonate 1 (79-85%) par réaction de Michaelis-Arbuzov entre (EtO) 3 P et CFBr 3 dans l'hexane à 50°C (Schéma 2) .

La seconde est un double échange halogéne-métal réalisé à -90°C entre 1 et le n-BuLi en présence de ClSiMe 3 . Aprés éthanolyse et distillation 2 est isolé non contaminé de composés phosphorés indésirables et avec un excellent rendement (94%). A notre connaissance cette préparation de 2 est la seule qui, actuellement, conduise efficacement à un produit pur.

Difluoro-and monofluoromethylenephosphonates, (CF 2 -phosphonate) and (CHF-phosphonate), are currently being explored as phosphatase stable phosphate mimics for biological systems (1). The pK a of the second deprotonation (ca. 6.4 for a phosphate group) is generally considered to be an important electronic factor in the binding of such compounds to enzymes. The CH 2 -phosphonate has a pKa of ca. 7.6, less acidic than the phosphate, when the CHF-phosphonate has a pKa of ca. 6.5, almost identical to that of the natural phosphate (2) (Scheme 1).

Scheme 1

The introduction of fluorine on the carbon in the a-position to phosphorus has been shown to be a useful operation, reducing the disparity in important physical properties between alkylphosphonates and phosphates leading to isopolar, isosteric analogues of biological phosphates (3).This advantage of CHF-phosphonate over CH 2 -phosphonate suggests that it is of interest to master a good synthetic approach to CHF-phosphonate on preparative scale and in pure form. We report here a high yielding synthesis of the parent compound, the 1-fluoromethyl-phosphonate 2 which is a valuable synthetic intermediate.

Previous methods for incorporating fluorine into a phosphonate moiety are based on two main procedures (4). The first involves P-CF bond formation using CH 2 FCl or CH 2 FBr and dialkylphosphite anions (5). The yields of 2 are generally modest (45%) and moreover this route has been abandoned since the two starting freons are no longer commercially available. The second involves PC-F bond formation. This can be accomplished by fluorination of phosphonate carbanions by reagent sources of positive halogen which are frequently hazardous fluorinating reagents. The preferred reagents for these reactions were found to be perchloryl fluoride (FClO 3 ) (3b), and N-fluorobenzenesulfonimide ((C 6 H 5 SO 2 ) 2 NF) (6). With the former the yields of monofluorinated phosphonates are modest (43%), with the latter the yields are significantly lower (11%). Full utilization of this chemistry is handicapped by the competitive formation of the CF 2 -phosphonates resulting from an acid-base equilibrium during the reaction. Another method of access to CHFphosphonates is based on the reaction of diethylaminosulphur trifluoride (Et 2 NSF 3 ) (DAST) (3a) with 1-hydroxymethylphosphonates, however these intermediates are not easily obtained in good yields.

Our synthetic approach to 2, based on the P-CF approach, exploited a halogenmetal exchange reaction with the diethyl dibromofluoromethyl-phosphonate 1 obtained by a Michaelis-Arbuzov reaction. Each of the two steps has been tested under a wide range of conditions with the aim of improving the reaction conditions and isolating compounds on preparative scale with the highest purity and in high yield (7).

Several variations on the preparation of 1 from (EtO) 3 P and CFBr 3 including, time, solvent, thermally or photolytically conditions have been reported (8). CFBr 3 is an expensive reagent and its consumption has thus to be complete. Best yields are obtained when the reaction is realized in a solvent in order to minimize the competitive halophilic attack of 1 by (EtO) 3 P. Use of polar solvents, such as diethyl ether or THF, leads to notable proportion of byproducts. These disadvantages are overcome in nonpolar solvent where the formation of byproducts is negligible, but reaction time is far too long (15-20 days) at room temperature. Hexane is particularly suited to the reaction and heating the reaction mixture at 50°C for 48h offers a shorter and more convenient route to 1 in 79-85%. With these conditions there is no isomerization of triethylphosphite and the only byproducts are the triethylphosphate and a phosphorus compound containing fluorine at low field (Scheme 2).

Scheme2

The second step is the key step (Scheme 3). The halogen-metal exchange reaction is realized in THF at low temperature in the presence of chlorotrimethylsilane as trapping agent. The intermediate carbenoid species has to be generated at low temperature where the carbon-lithium bond has mainly a covalent character and behaves as an ordinary nucleophile. It has been found more efficient to accomplish the reaction at -90°C rather than -78°C. The reaction is fast and clean and after ethanolysis with an excess of anhydrous ethanol and cleavage of the C-Si bond, 2 is isolated in pure form as a colorless liquid and distilled in almost quantitative yield.

Scheme 3

The procedure described herein for the preparation of 2 is representative of a general method for the synthesis of a-fluoromethyl and a-fluoroalkylphosphonates previously reported on a small scale (9). It is characterized by good chemical yield, simple operation and ready availability of the starting materials. To the best of our knowlegde it is presently the sole available preparative access to 2 in pure form and high yield.

Experimental Section

Diethyl 1,1-dibromo-1-fluoromethylphosphonate 1. An oven dried, 500 mL pressure bottle purged with nitrogen is charged with 100 mL of hexane triethylphosphite (29 g, 0.175 mol) and fluorotribromomethane (50 g, 0.184 mol). The pressure bottle is carefuly closed with a rubber stopper and immersed in a water bath at 50°C. On heating the hexane solution a yellow colour appears. After heating during 48 h the pressure bottle is cooled to 0°C and opened. Hexane is evaporated on a rotary evaporator and the yellow crude liquid (56-57g) is transferred to a pearshaped flask fitted with a short column and distilled under reduced pressure to give 45-48.5 g (79-85%) of diethyl 1,1-dibromo-1-fluoromethylphosphonate 1 as a colorless liquid, b.p. 62-64°C / 0.05 mmHg.

Product 1 displays the following spectroscopic data: 31 P NMR (CDCl 3 ) d: +1.6 (d, 2 J PF 76.5); 19 F NMR (CDCl3) d: -51.2 (d, 2 J FP 76.4); 1 H NMR (CDCl 3 ) d: 1.38 (t, 6H, 3 J HH 7.1, CH 3 CH 2 O), 4.37 (qd, 4H, 3 J HP 8.2, 3 J HH 7.1, CH 3 CH 2 O); 13 C NMR (CDCl 3 ) 16.5 (d, 3 J CP 5.8, CH 3 CH 2 O), 67.0 (d, 2 J CP 6.5, CH 3 CH 2 O), 88.4 (dd, 1 J 333.6, 1 J 202.3, PCFBr 2 ).

Diethyl 1-fluoromethylphosphonate 2. An oven-dried, 1-L, four-necked, roundbottomed flask is fitted with an efficient mechanical stirrer, a thermometer, a reflux condenser surmounted with a bubbler, and a 200 ml pressure equalizing dropping funnel with a nitrogen inlet. Under a gentle flow of nitrogen the flask is charged with 150 mL of THF. The dropping funnel is charged with 135 mL of a 1.56 M solution of n-butyllithium in hexane (0.210 mol) . The flask is cooled to -70°C with a Dewar partially filled with liquid nitrogen, stirring is started, and n-butyllithium is added to THF over a few minutes. At -90°C a solution of diethyl 1,1-dibromo-1fluoromethylphosphonate 1 (32.8 g, 0.100 mol) and chlorotrimethylsilane (12 g, 0.110 mol) in THF (50 mL) is added dropwise from the dropping funnel over 15 min. The resulting clear slightly yellow solution is stirred for an additional 10 min at -90°C. The reaction mixture is then treated at this temperature by a dropwise addition of a solution of anhydrous ethanol (16 g, 0.35 mol) in THF (10 mL). The resulting mixture is then allowed to warm up gently to 0°C in 45 min. At this temperature the reaction mixture is poured into a beaker containing a stirred mixture of 3N hydrochloric acid (50 mL), an equal volume of crushed ice and CH 2 Cl 2 (60 mL). The solution became completely colourless. The organic layer is then separated, and the aqueous layer is extracted with CH 2 Cl 2 (2 x 60 mL). The extracts are combined with the original organic layer and dried over anhydrous magnesium sulfate. The magnesium sulfate is removed by filtration and the solvents are evaporated on a rotary evaporator. The crude liquid is transferred to a pear-shaped flask fitted with a short column and distilled under reduced pressure to give 15.98 g (94%) of diethyl 1fluoromethylphosphonate 2 as a colorless liquid, b.p. 40-45°C / 0.1 mm Hg .

Product 2 displays the following spectroscopic data: 31 P NMR (CDCl 3 ) d: + 14.2, (d, 2 J PF 63.5); 19 F NMR (CDCl 3 ) d: -223.5, (d, 2 J FP 62); 1 H NMR (CDCl 3 ) d: 1.20 (t, 6H, 3 J HH 7.1, CH 3 CH 2 O), 4.0 (qd, 4H, 3 J HP 8.1, 3 J HH 7.1, CH 3 CH 2 O), 6.08 (dd, 2H, 2 J PH 4.7, 2 J FH 46.8, PCH 2 F); 13 C NMR (CDCl 3 ) 16.0 (d, 3 J CP 4.8, CH 3 CH 2 O), 62.1 (d, 2 J CP 6.1, CH 3 CH 2 O), 76.2 (dd, 1 J CF 179.9, 1 J CP 169.2, PCH 2 F).