Elaboration of analytical thermo-mechanical cutting model matched by numerical and experimental results

F. Salvatore*, T. Mabrouki**, H. Hamdi*** *Université de Lyon, ENISE, LTDS, UMR 5513 CNRS, 58 rue Jean Parot, 42023 Saint Etienne, France, E-mail: ferdi3@free.fr **Université de Lyon, INSA Lyon, LAMCOS, UMR 5259 CNRS, 18-20 rue des Sciences, 69621 Villeurbanne, France, E-mail: Tarek.Mabrouki@insa-lyon.fr ***Université de Lyon, ENISE, LTDS, UMR 5513 CNRS, 58 rue Jean Parot, 42023 Saint Etienne, France, E-mail: hamdi@enise.fr


Introduction
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 [3] are usually adopted because they are practical and simple to use.Other authors such as Gilormini, Molinari, Moufki, and Oxley [4][5][6][7][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.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][10][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 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 [8] 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 [12] 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.

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 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 m.The machined material is a steel alloy AISI 4140.
The values of a 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 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).Fig. 2 Experimental test procedure Details of the averages of cutting forces F x , feed forces F y , and primary shear angle  are presented in Table 1 and 2 in the section "analytical results and discussion", compared to numerical and analytical data.

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 [13] is used for the workpiece material behaviour.
The workpiece is geometrically modelled according to Mabrouki [14].Parameters of the tool and the workpiece are extracted from Barge study [15] and the material to cut is characterized with also the Johnson and Cook damage law [16].
From contact point of view, the "surface to surface" interaction option and the penalty contact method were chosen.Also, it is considered that  = 0.39 for V c = 42 m/min,  = 0.30 for V c = 126 m/min,  = 0.25 for V c = 378 m/min according to experimental results of Zemzemi [17].
In order to find the good mesh dimensions for the numerical simulations, the influence of the primary shear zone thickness d 1 versus the cutting speed V c for different vertical mesh sizes was made (Fig. 3).In this specified study frictionless assumption was adopted for the chip-tool contact.The cutting angle γ 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.

Mesh 20m
Mesh 10m Mesh 5m The computed primary shear zone thickness for a mesh size varying from 5 to 10 m is close to the value (25 m) given by Shaw [18].
Consequently 10 µm mesh size is adopted in this study and seems in agreement with Barge [15].
Using a 10 µm 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 , (Table 1 and 2).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 out- Vc=42m/min Vc=126m/min Vc=378m/min 0 0.1 0.2 puts using the "frictionless" option in the contact between the tool and chip are now studied.The cutting radius is considered null and f 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.In Fig. 7,  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][6][7][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.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 l c , the distribution became triangular.This is verified for three cutting speeds and, for different Johnson and Cook parameters, in the case of V c = 378 m/min.
Afterwards, the forces distribution hypothesis, acted in Fig. 10, are validated in this section.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  n2max is a function of the tangential one  1 .The normal force N 2 in the secondary shear zone is a function of  1 .
where * 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.

Introduction
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 cr the real layer of work material removed.In fact for low values of the f/R ratio (f < f 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 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 cr .f 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 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.

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 * (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 cr " (AG).
The second one is the contribution of the angle in the "f -R" layer (AB).
It is remembered that the part "CG" of the cutting edge R is the ploughing contribution not presented in this paper.
Based on geometrical considerations the equivalent cutting angle * is given by equation It is noted that, when the ratio f/R is very big, * is close to .In the opposite situation, when f/R is close to 1 * is similar to  R .
Having now the calculation of the equivalent cutting angle it is possible to compute forces F x and F y .
Based on the ploughing F pl and friction force F 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).
where N 2 is the orthogonal force on the tool rake face, T 2 is the tangential or friction force, T 1 and N 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 ad and F pl are considered equals to zero.

Pure cutting case
Pure cutting involves f > f cr .Chip formation is made using an "equivalent" tool with a cutting radius R = 0, a cutting equivalent angle * (Eq.( 3)), and the layer of material removed is f -f cr .
The chip is assumed rigid and uniform except in zone 1 and 2 where all the deformations are localized [5][6][7][8].This last assumption is in concordance with the results given by FE cutting model.The thicknesses of these zones are d 1 and d 2 .In both zones, the assumption of thermo-viscous-plastic-hardening material behaviour is made [13].
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).
x is the distance between the tip of the tool and the point where the force N 2 is applied.This variable can be computed using Eqs.( 1) and (2) in Eq. ( 8).From Eq. ( 6) it will be possible to extract N 1 , from Eq. ( 7) the shear angle and from Eq. ( 8) the contact length l c .
 n2max and N 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 T 1 can be calculated by Eq. (11).In this case the tangential stress  1 calculation is proposed in shear formulation.
where l 1 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).
In Eqs. ( 11) and ( 12), d 1 and V 1 are, respectively, the thickness and the shearing speed along l 1 in the primary shear zone.The first of those equations is similar to Moufki [7] and it can be considered a generalised version because it takes into account the cutting radius R with the equivalent cutting angle *.Now, in order to compute the temperature T emp1 in the primary shear zone, it is considered that the force T 1 produce the work Q 1 to shear the primary zone with a strain done by equation l 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 1 , l 1 and a p (Fig. 11).
Consequently it is possible to compute equation In order to have a low sensibility of d 1 in the case of very negatives angles (*) computed values, it is decided to use Eq.(14).It is still close to Oxley estimation [8] where d 1 is equal to 10% of the primary shear zone length.
In Eq. ( 13),  1 is the heat partition coefficient between the chip and the workpiece and T emp10 is the initial temperature in the primary shear zone, equals to the initial temperature of the workpiece.In this paper the method of Gilormini [4] to compute  1 is adopted where k 2 and C 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 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 [19] 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 2 is equal to d 1 .
Consequently, the strain and the strain rate are given by the following equations, respectively In the same way as it was computed T emp1 , it is now suggested the temperature T' emp2s generated in the sticky zone in the secondary shear zone (Eq.( 18)).
 2s is heat partition coefficient between the tool and the workpiece computed with Jaeger method [20] in the case of sticky contact, as shown in equation where k 1 and k 2 are the conductivities of the tool and the workpiece, respectively. 2 is the diffusivity of the steel.
 2s in Eq. ( 18) is the tangential sticky shear stress in zone 2 computed with Johnson and Cook law (Eq.( 20)).
T 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 Consequently it is possible to write 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 g = 0) and in the point C totally sliding (V g = V 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 ( = 0).Consequently The temperature generated by heat T' emp2g is pre- where,  2g and  2g are computed with Jaeger and Johnson and Cook formulations, respectively (Eqs.( 25) and ( 26)).
T emp2g is done by Eq. ( 27), where T emp2s is the "initial" temperature of this sub-zone, because of the displacement of the chip.T 2 is resolved using Eq.(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 2 and N 2 .

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 [21] and numerical ones.Concerning the primary shear zone angle , 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  is accurate; this is due to the equivalent cutting angle * equal to -0.33 rad.Feed force F 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 equiva- 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 c .For low values of f and V c the heat partition coefficients  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 x and F y is lower than the proposed "pure cut" model.Only the angle  is more accurate here because it is the "set parameter".F y values are smalls because the cutting radius effect is not taken into account.

Conclusions
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][6][7][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 precited 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.

S u m m a r y
The global aim of this paper is to propose a complete analytical model concerning the material removal.In order to take into account all the aspects of the process, a "phenomena split approach", based on the assumption that the material removal is the summation of three major contributions, ploughing, spring back and "pure cut" was adopted.This new methodology is developed on the base of experimental tests and industrial experience.In fact the chip is not systematically present.When the chip exists, a part of theoretical layer of the material to be removed is transformed in lateral burrs and elastic compression under the tool tip.In this paper this new approach is presented and the "pure cut" contribution is developed in details.This analytical sub-model of chip formation is calibrated and fitted with a finite element modeling, in order to presents new hypothesis and new formula based on the physics close to the tip of the tool.The chip is considered rigid and uniform, and the regime is supposed stationary.Thermo-mechanical law is applied in shear zones where plastic strain, temperature and strain rate are concentrated.The model takes into account the cutting edge radius too, using an equivalent cutting angle.The friction coefficient at the interface tool-chip is also analytically computed and the present model is considered predictive.

Fig. 1
Fig.1Schematization of the cutting process with the "phenomena split approach"

Fig. 5
Fig. 5 Measured forces (a) and computed one's with FE (b) versus time in the case of a mesh 1010 m for different cutting speeds, f = 0.15 mm and  = 0

Fig. 6 Fig. 7 Fig. 8
Fig. 6 Different sections employed in numerical simulations 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,  = 0,  = 0.

Fig. 9
Fig. 9 Normal stress  n2 in zone 2 (section 4). a -different cutting speed using the adopted behavior law; b -the "pure plastic" case is presented and 2 different values of the m exponent in Johnson and Cook law in the case 378 m/min

Fig. 10
Fig. 10 Stresses and forces distributions applied to the chip

Fig. 11
Fig. 11 Schematization of the "PS-approach".Forces, geometrical variables and the equivalent cutting angle * are presented