Elaboration of analytical thermo-mechanical cutting model matched by numerical and experimental results/Analitinio termomechaninio pjovimo modelio, tenkinancio skaitinius ir eksperimentinius rezultatus, paruosimas.
Cutting processes are widely used in different industries to cut various engineering parts. Usually the optimization of these processes is made by experimental methods often expensive and not able to be extrapolated to other machining configurations. To overcome these drawbacks, numerical simulations have been carried out by many researchers but the major inconvenience of those methods are the long computing time, the high cost of numerical software, etc. For all these reasons, in manufacturing industry, a high interest in analytical methods such as that of Merchant [1, 2] and Lee and Schaffer  are usually adopted because they are practical and simple to use. Other authors such as Gilormini, Molinari, Moufki, and Oxley [4-8] have proposed pertinent thermo-mechanical cutting models but not convenient to use and not taking into account the cutting radius influence, ploughing and the spring back.
For all those reasons the present work is both complete and predictive in order to be really interesting compared to experimental and numerical methods.
It is based on a new "phenomena split approach" (PS-approach) (Fig. 1) which can be directly and efficiently exploited also from practical point of view.
[FIGURE 1 OMITTED]
This methodology stems from experimental observations in the case of turning. In fact, burrs appear during the cutting process for different manufacturing input parameters or when a threshold of tool wears is reached. Burr is the result of a plastic flow of material similar to ploughing observed in scratch tests [9-11].
Consequently, it is assumed (Fig. 1) that a cutting phenomenon can be decomposed as a sum of three elementary phenomena: pure cutting, ploughing and spring back, where R is the tool cutting radius, f is the uncut chip thickness, [f.sub.cr] is the critical feed rate characterizing the limit value between ploughing and "pure cut" (i.e. chip formation).
The proposed analytical model of the chip formation is performed using the mechanical balance formulation [1, 2] and the assumption that the shear zones thickness, where deformations are considered concentrated, are constants  but the calculation of the temperature in those zones are a function of different partition heat coefficients.
Others new hypothesis and equations are introduced by the use of FEM studies, in particular, stresses and forces distributions, triaxiality close to the tip of the tool.
Those numerical contributions substitutes the equations based on the minimization of the cutting energy like acted by Merchant and Gilormini [1, 2, 4].
The influence of tool geometry in terms of chip formation is also studied and an equivalent cutting angle is introduced to replace the cutting radius. In this way the rake face stay plane and, the forces and temperature formulations are simpler than those proposed by Albrecht  and "linked" to the thermo-mechanical approach.
First experimental tests, materials employed and the measured data such as forces and chip thickness are presented. Then a cutting numerical simulation based on ABAQUS Explicit is developed in order to identify more accurately the influences of physical phenomena. Afterwards the general PS-approach assumptions are presented but only the "pure cut" contribution is described in detail. Finally, the most important results of the sub-model "pure cut" are presented, discussed and compared to Merchant.
2. Experimental study
In this section the experimental cutting tests are presented. In order to replicate orthogonal cutting conditions, the machining operation was carried out on discs each one with a diameter of 70 mm and a thickness ([a.sub.p]) of 3 mm, which is also the cutting depth.
The cutting tool is in a carbide grade (referenced TPKN 16 03 PP R SM30) with cutting edge radius R of 30 |am. The machined material is a steel alloy AISI 4140.
The values of [a.sub.p] and R were chosen to minimize ploughing effects (lateral burrs). In fact in this paper the adopted "PS approach" is presented, but only the elementary model "pure cut" is adopted, verified and fitted with experimental and numerical methods. For that, in the next the value of[f.sub.cr] is considered equal to zero.
Cutting speed variations were made in order to measure machining forces using a dynamometer Kistler 9257 A with a sampling frequency about 2 KHz (Fig. 2).
[FIGURE 2 OMITTED]
Details of the averages of cutting forces [F.sub.x], feed forces [F.sub.y], and primary shear angle [PHI] are presented in Table 1 and 2 in the section "analytical results and discussion", compared to numerical and analytical data.
3. Numerical study
Numerical simulations based on Abaqus\Explicit (Lagrangian formulation) were performed in order to both understand the physics of cutting and improve the analytical method. The Johnson and Cook's law  is used for the workpiece material behaviour.
The workpiece is geometrically modelled according to Mabrouki . Parameters of the tool and the work-piece are extracted from Barge study  and the material to cut is characterized with also the Johnson and Cook damage law .
From contact point of view, the "surface to surface" interaction option and the penalty contact method were chosen. Also, it is considered that [micro] = 0.39 for [V.sub.c] = 42 m/min, [micro] = 0.30 for [V.sub.c] = 126 m/min, [micro] = 0.25 for [V.sub.c] = 378 m/min according to experimental results of Zemzemi .
In order to find the good mesh dimensions for the numerical simulations, the influence of the primary shear zone thickness [d.sub.1] versus the cutting speed [V.sub.c] for different vertical mesh sizes was made (Fig. 3).
[FIGURE 3 OMITTED]
In this specified study frictionless assumption was adopted for the chip-tool contact. The cutting angle [gamma] was considered equal to zero.
From this last figure, it is possible to underline that smaller the mesh size values, smaller the primary shear zone thickness. This is due to the strain localization phenomenon which induces high temperature (Fig. 4). The material softening induces lower forces and, by the same way, lower thickness of the primary shear zone.
[FIGURE 4 OMITTED]
The computed primary shear zone thickness for a mesh size varying from 5 to 10 [micro]m is close to the value (25 [micro]m) given by Shaw .
Consequently 10 Lim mesh size is adopted in this study and seems in agreement with Barge .
Using a 10 Lm mesh dimension and the previous friction coefficient, it is then possible to perform numerical simulation and compare results to experimental and analytical cutting forces (Fig. 5) and primary shear angle [PHI], (Table 1 and 2).
a Time [s] b Time [ms]
Fig. 5 Measured forces (a) and computed one's with FE (b) versus time in the case of a mesh 10 x 10 [micro]m for different cutting speeds, f = 0.15 mm and [gamma]= 0
[FIGURE 5 OMITTED]
In the presented FE simulations the deformation is supposed to be uniform. This assumption is justified by the manner of the calculation of the primary shear angle as a function of feed rate divided by chip thickness.
For the different outputs presented, numerical results are in concordance with the experimental ones. It is then considered that the present numerical model reproduces the real phenomena and will be used to calibrate further analytical cutting modelling.
In order to have numerical simulations without the influence of the friction coefficient, the principal outputs using the "frictionless" option in the contact between the tool and chip are now studied. The cutting radius is considered null and [f.sub.cr] too.
It is decided to chose four chip sections (Fig. 6). Primary shear zone is delimited by section 1 and 3. Section 4 is the "interface" between tool and chip.
[FIGURE 6 OMITTED]
In Fig. 7 to 9, the normal and tangential stresses distributions in the primary shear zone and the normal distribution in the secondary one are presented in the case of f= 0.15 mm, R = 0, [gamma]= 0, [micro]= 0.
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
In Fig. 7, [[sigma].sub.n1] distribution is similar in every chosen section inside the primary shear zone, close to the tool. In this zone the triaxiality is high. In the primary shear zone region close to the free surface, it is possible to capture the opposite situation; along section 1 the stress value is the bigger and along section 3 it is close to zero. In fact, in this last section, and close to the free surface, the chip is still formed and the effects of the compression in zone 2 are far. Consequently it is acted that the plastic strain can be considered concentrated in the shear zones. This hypothesis will be done in the analytical section and is in accordance with [5-8].
In the same way, the tangential stress distribution in the primary shear zone (Fig. 8) is quite similar in every chosen section close to the tool.
In this case it is decided to formalize the uniform distribution for the analytical modeling.
[FIGURE 9 OMITTED]
In Fig. 9, the normal stress distribution in the region of the chip close to the tool is represented (section 4). For every condition there is a uniform part from the tip of the tool to the "f" value and, from f to the contact length lc, the distribution became triangular. This is verified for three cutting speeds and, for different Johnson and Cook parameters, in the case of [V.sub.c] = 378 m/min.
Afterwards, the forces distribution hypothesis, acted in Fig. 10, are validated in this section.
[FIGURE 10 OMITTED]
Afterwards the triaxiality is big in the region of the piece close to the tip of the tool. The normal stress value in the primary shear zone is close to the normal stress value in the interface tool-chip (Figs. 7-9).
From those considerations it is possible to extract new simple Eqs. (1) and (2) where the maximum value of the normal stress [[sigma].sub.n2max] is a function of the tangential one [[tau].sub.1]. The normal force [N.sub.2] in the secondary shear zone is a function of [[tau].sub.1].
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where [gamma]* is the equivalent cutting angle explained in the next section. Eqs. (1) and (2) are verified for different Johnson and Cook parameters (Fig. 9). For those reasons it is considered that it can be applied for different types of materials.
4. Cutting analytical model
Orthogonal cutting represented by a 2D model is considered in the present study. The cutting tool removes a specific layer of work material (Fig. 11). f is the theoretical uncut thicknesses to remove with the tool, R is the cutting edge radius, f - [f.sub.cr] the real layer of work material removed. In fact for low values of the f/R ratio f < [f.sub.cr]) the chip formation does not occur; only spring back and ploughing appear (in case of 3D approach). The chip formation is only possible if f is greater than [f.sub.cr]: in this case it is supposed that the cutting process is made with a virtual tool, with a cutting edge radius R = 0 (Fig. 11), and the effective layer work material removed is f--[f.sub.cr].
[FIGURE 11 OMITTED]
[f.sub.cr] gives the supposed position of the separation line between the material that will became chip and the part of metal that will slide under the tool (spring back) or will be displaced laterally (ploughing).
The ploughing and spring back phenomena are concentrated in the layer defined from the point C to the point G.
It is also supposed that the ploughing and spring back variables only depend on [f.sub.cr] value. It will thus be possible to apply the effect superposition principle and study separately all those phenomena.
In the proposed approach the ploughing and spring back phenomena are caused by the cutting radius and, if it is zero, only the pure cutting condition will exist.
Moreover the effects of the cutting radius exists in "pure cutting case" too; it will be explained in the next section.
4.2. Modeling of cutting radius contribution
The cutting radius contribution must be taken into account when sliding and normal forces on the tool rake face are considered. Both cutting radius and rake face are modeled by a single equivalent rake face without cutting radius (R = 0) and consequently with an equivalent cutting angle [gamma]* (Fig. 11).
The latter angle is the summation of 2 contributions. The first one is the local cutting angle caused by the cutting radius; its action is localized in the layer "R--[f.sub.cr]" (AG).
The second one is the contribution of the angle [gamma]in the "f--R" layer (AB).
It is remembered that the part "CG" of the cutting edge [gamma]* is the ploughing contribution not presented in this paper.
Based on geometrical considerations the equivalent cutting angle r* is given by equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
It is noted that, when the ratio f/R is very big, [gamma]* is close to r. In the opposite situation, when f/R is close to 1 [gamma]* is similar to [[gamma].sub.R].
Having now the calculation of the equivalent cutting angle it is possible to compute forces [F.sub.x] and [F.sub.y].
Based on the ploughing [F.sub.pl] and friction force [F.sub.ad] in the layer defined by "CG", and, according to Fig. 11 the cutting and feed force can be presented respectively by Eqs. (4) and (5).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
where [N.sub.2] is the orthogonal force on the tool rake face, [T.sub.2] is the tangential or friction force, [T.sub.1] and [N.sub.1] are the tangential and normal forces in the primary shear zone, respectively.
All those last variables are to be computed, as what it is suggested in section 4.3.
In the "pure cutting case" [F.sub.ad] and [F.sub.pl] are considered equals to zero.
4.3. Pure cutting case
Pure cutting involves f > [f.sub.cr]. Chip formation is made using an "equivalent" tool with a cutting radius R = 0, a cutting equivalent angle r* (Eq. (3)), and the layer of material removed is f - [f.sub.cr].
The chip is assumed rigid and uniform except in zone 1 and 2 where all the deformations are localized [5-8]. This last assumption is in concordance with the results given by FE cutting model. The thicknesses of these zones are [d.sub.1] and [d.sub.2]. In both zones, the assumption of thermo-viscous-plastic-hardening material behaviour is made .
According to Fig. 10 and when applying the force equilibrium conditions on the chip (forces and momentum), it is possible to write Eqs. (6)-(8).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
[xi] is the distance between the tip of the tool and the point where the force N2 is applied. This variable can be computed using Eqs. (1) and (2) in Eq. (8).
From Eq. (6) it will be possible to extract N1, from Eq. (7) the shear angle [PHI] and from Eq. (8) the contact length lc.
[[sigma].sub.n2max] and [N.sub.2] can be computed with Eqs. (1) and (2). Forces and stresses distribution are in accordance with Fig. 10.
By applying the Johnson and Cook law (Eq. (9)) in the zone 1, the tangential force T1 can be calculated by Eq. (11). In this case the tangential stress r1 calculation is proposed in shear formulation.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
where l1 is the length of the primary shear zone, A, B, n, C, m and are Johnson and Cook parameters.
The unknown parameters in Eq. (9) are now computed; in particular the strain is given by Eq. (11) and the strain rate by Eq. (12).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
In Eqs. (11) and (12), d1 and [V.sub.1] are, respectively, the thickness and the shearing speed along [l.sub.1] in the primary shear zone. The first of those equations is similar to Moufki  and it can be considered a generalised version because it takes into account the cutting radius R with the equivalent cutting angle [gamma]*.
Now, in order to compute the temperature [T.sub.emp1] in the primary shear zone, it is considered that the force T1 produce the work [Q.sub.1] to shear the primary zone with a strain done by equation [l.sub.1]. It is also considered that 90% of this work became heat and this last variable is multiplied by a partition heat coefficient to take into account the part of heat that is conduced into the workpiece. Afterwards the volume where the heat is generated is defined by [d.sub.1], [l.sub.1] and [a.sub.p] (Fig. 11).
Consequently it is possible to compute equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
In order to have a low sensibility of [d.sub.1] in the case of very negatives angles ([gamma]*) computed values, it is decided to use Eq. (14). It is still close to Oxley estimation  where d1 is equal to 10% of the primary shear zone length.
[d.sub.1] = 0.2 (f-[f.sub.cr]) (14)
In Eq. (13), 31 is the heat partition coefficient between the chip and the workpiece and Temp10 is the initial temperature in the primary shear zone, equals to the initial temperature of the workpiece. In this paper the method of Gilormini  to compute 3 is adopted
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
where [k.sub.2] and [C.sub.p] are, respectively, the conductivity and the specific heat of the workpiece.
The last variable to be computed to solve the system is the friction force at the interface tool-chip [T.sub.2].
It is supposed a sticky contact from the point B to G and a sliding contact from B to C. This is based on numerical simulations (Fig. 9) where from the point B to G the normal stress is constant and from B to C it degrease to zero.
In the present work the partition of the secondary shear zone is similar to the Bahi works  where the sticking and sliding contact is defined and based on the measurement of the friction coefficient. In our case the approach is more easy and predictive.
It is also supposed that the thickness of the secondary shear zone [d.sub.2] is equal to d1.
Consequently, the strain and the strain rate are given by the following equations, respectively
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
In the same way as it was computed [T.sub.emp1], it is now suggested the temperature [DELTA][T'.sub.emp2s] generated in the sticky zone in the secondary shear zone (Eq. (18)).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
[[delta].sub.2s] is heat partition coefficient between the tool and the workpiece computed with Jaeger method  in the case of sticky contact, as shown in equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)
where [k.sub.1] and [k.sub.2] are the conductivities of the tool and the workpiece, respectively. [[sigma].sub.2] is the diffusivity of the steel.
[[tau].sub.2s] in Eq. (18) is the tangential sticky shear stress in zone 2 computed with Johnson and Cook law (Eq. (20)).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)
[T.sub.emp2s] is the temperature in this sub zone (sticky). It is the summation of the temperature of the zone 1, displaced to the secondary shear zone during the chip evolution, and [DELTA][T.sub.emp2s].
Consequently it is possible to write
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)
In the zone BC, a sticky-sliding contact is assumed. Here the strain and the strain rate are not constants like considered in the sticky zone. In fact it is supposed that in the point B the contact is still sticky ([V.sub.g] = 0) and in the point C totally sliding ([V.sub.g] = [V.sub.t]). In order to easy compute variables of this zone, an only sticky contact is supposed in half distance BC. In the other half distance a totally sliding contact is considered ([micro] = 0). Consequently
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)
The temperature generated by heat [DELTA][T'.sub.emp2g] is presented in equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)
where, [[delta].sub.2g] and [[gamma].sub.2g] are computed with Jaeger and Johnson and Cook formulations, respectively (Eqs. (25) and (26)).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)
[T.sub.emp2g] is done by Eq. (27), where [T.sub.emp2s] is the "initial" temperature of this sub-zone, because of the displacement of the chip. [T.sub.2] is resolved using Eq. (28).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)
Now it can be allowable to compute the friction coefficient in the interface between the tool and the chip as the ratio between [T.sub.2] and [N.sub.2].
4.4. Analytical results and discussions
According to Fig. 11, the most important variables of the study are resumed in Table 1 and 2, compared to numerical and experimental results.
The values of analytical forces and temperatures presented in Tables 1 and 2 are similar to experimental  and numerical ones. Concerning the primary shear zone angle [PHI], the analytical calculation is accurate for cutting speeds major than 126 m/min if f is bigger than 0.15 mm.
In the case of feed rate equal to 0.07 mm, it appears that, for low cutting speed (42 m/min) the analytical calculation of [PHI] is accurate; this is due to the equivalent cutting angle [gamma]* equal to -0.33 rad. Feed force [F.sub.y] is also close to experimental data.
The presented "pure cut" sub-model is based on a thermo-mechanical approach using Johnson and Cook's law in the shear zones 1 and 2. Stresses value in those zones are a function of the shear strain, shear strain rate and temperature. The firsts one are function of the equivalent cutting angle [gamma]* and, consequently the cutting radius R. The temperatures are function of the strains but also of the heat partition coefficients. In particular those last variables depend on the diffusivity of the material and the cutting parameters f and [V.sub.c].
For low values of f and [V.sub.c] the heat partition coefficients [[delta].sub.1] are close to 1 and for high values of those "machining parameters" it is close to 0. This leads to low temperatures and high forces because the "softening" contribution in Johnson and Cook equations became small.
Finally, concerning the comparative with Merchant calculation, it is possible to act (Table 1) that the precision in terms of [F.sub.x] and [F.sub.y] is lower than the proposed "pure cut" model. Only the angle [PHI] is more accurate here because it is the "set parameter". [F.sub.y] values are smalls because the cutting radius effect is not taken into account.
In this paper an analytical model of material removal is presented. This model is based on the new assumption that cutting operation can be defined as a sum of three contributions: ploughing, spring back and pure cut.
It was decided to present and study in details the "pure cut" sub-model where the chip formation is investigated. In this case the analytical model was fitted with FEM simulations and verified with experimental tests.
In particular the stresses distribution is numerically studied and the outputs of this investigation were acted in new formulations based on triaxiality and forces distributions applied to the chip.
In the sub-model "pure cut", the chip is considerate rigid and the static equilibrium of the chip leads to the definition of three simple equations like Merchant [1, 2]. The thermo-mechanical assumption was made in the shear zones 1 and 2 where the strain is supposed to be concentrated [5-8].
The temperature calculation is made using the definition of specific heat in the shear zones.
The cutting radius of the tool is modelled using an equivalent cutting angle what is new compared to the pre-cited works. The ratio f/R is important and define 2 different conditions. If it is big the cutting radius R can be considered null, but if it is low, it is necessary to model the radius R. It is the industrial case of damaged tools.
The calculations of the variables of the model are only functions of input parameters (process parameters and thermo-mechanical data) and no experimental setting like chip thickness measurements are required.
The analytical equations can be directly used for industrial application or for scientific purpose, in order to have easier data to understand the removal material problem before a more intense research.
The proposed analytical model can be employed for several steels if parameters of Johnson and Cook law exist.
"PS approach" can be exploited to predict residual stresses after machining, where the cutting radius and the ploughing are important for the accuracy of the study.
Received August 29, 2011 Accepted November 15, 2012
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F. Salvatore*, T. Mabrouki**, H. Hamdi***
* Universite de Lyon, ENISE, LTDS, UMR 5513 CNRS, 58 rue Jean Parot, 42023 Saint Etienne, France, E-mail: firstname.lastname@example.org
** Universite de Lyon, INSA Lyon, LAMCOS, UMR 5259 CNRS, 18-20 rue des Sciences, 69621 Villeurbanne, France, E-mail: Tarek.Mabrouki@insa-lyon.fr
*** Universite de Lyon, ENISE, LTDS, UMR 5513 CNRS, 58 rue Jean Parot, 42023 Saint Etienne, France, E-mail: email@example.com
Table 1 Analytical, experimental and numerical results in case of AISI 4140 steel, carbide tool, f = 0.15 mm, [gamma]* = -0.15 rad. In brackets Merchant [1, 2] calculation [V.sub.c], Computing Cutting force Feed force m/min method [F.sub.x], N F[gamma], N 42 Analytical 1056 (1447) 611 (434) Experimental 1118 563 Numerical 1150 556 126 Analytical 981(826) 512 (207) Experimental 1052 495 Numerical 1102 457 378 Analytical 949 (699) 474 (140) Experimental 998 412 Numerical 982 393 Contact [T.sub.emp1], [V.sub.c], Primary shear length/feed, [degrees]C m/min angle [PHI], Rad [l.sub.c]/f (zone 1) 42 0.58 (0.45) 3.02 (2.12) 432 0.45 0.48 2.41 451 126 0.61 (0.58) 2.81 (1.7) 485 0.58 0.57 2.31 495 378 0.62 (0.60) 2.74 (1.57) 509 0.6 0.58 2.19 521 [T.sub.em2s], [T.sub.em2g,] [V.sub.c], [degrees]C [degrees]C m/min (zone 2 sticky) (zone 2 sliding) 42 885 1048 931 1024 126 960 1130 991 1051 378 1006 1171 1021 1146 Table 2 Analytical and experimental results in case of AISI 4140 steel, carbide tool, f = 0.15 mm, [gamma]* = -0.15 rad Cutting Feed f, Computing force Feed force mm/rev method [F.sub.x], N [F.sub.y], N 0.07 Analytical 554 311 Experimental 512 304 0.225 Analytical 1459 741 Experimental 1552 798 Temperature Contact zone 1 Feed f, Primary shear length/feed [T.sub.emp1], mm/rev angle [PHI], Rad [l.sub.c]/f [degrees]C 0.07 0.51 3.81 375 0.44 0.225 0.62 2.75 439 0.49 Temperature Temperature zone 2 zone 2 Feed f, [T.sub.emp2s], [T.sub.emp2g], mm/rev [degrees]C [degrees]C 0.07 830 1001 0.225 960 1129
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|Author:||Salvatore, F.; Mabrouki, T.; Hamdi, H.|
|Date:||Nov 1, 2012|
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