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    INFLUENCE OF THE JOHNSON COOK MATERIAL MODEL PARAMETERS AND

    FRICTION MODELS ON SIMULATION OF ORTHOGONAL CUTTING PROCESS

    Amrita Priyadarshini, Surjya K. Pal and Arun K. SamantarayDepartment of Mechanical Engineering, Indian Institute of Technology Kharagpur,

    Kharagpur 721302, West Bengal, India

    ABSTRACT

    In the recent past, Finite Element (FE) modeling has emerged as one of the most effective tools

    that could substitute the conventional time consuming and expensive experimental tests to a great

    extent. This work deals with FE model-based analysis of the two critical factors during

    orthogonal cutting, namely the flow stress characterization of work material and the frictional

    boundary conditions at the tool-chip interface. We exploit the new set of Johnson-Cook constants

    reported recently along with two other sets of data that are commonly used in finite element

    modeling for orthogonal cutting of AISI 4340 steel. The new set of Johnson-Cook material data

    has not been used yet in any reported work on the orthogonal cutting process simulation. Besides,

    a comparative study has been made by considering seven most relevant friction models among

    various models proposed by researchers, including a relatively newer one. Though few works are

    already available related to friction modeling, the uniqueness of the presented results lies in the

    consideration of several friction models where friction coefficients are functions of temperature,

    stress and sliding velocity, etc. Results show that the cutting conditions are the critical parameters

    which decide the type of friction encountered in the chip formation process and, therefore, the

    friction model has to be selected based on the cutting conditions for best results.

    Keywords:Orthogonal cutting, Finite element simulation, Friction modeling, Johnson-Cook

    material model

    Corresponding author: Phone: +91-3222-282996, Fax: +91-3222-255303; E-mail: [email protected]

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    1.INTRODUCTION

    Machining process is one of the commonly used manufacturing processes. Because of its broad

    use and complexity, continuous effort is being made to solve practical problems associated with

    efficient material removal. This is achieved primarily through detailed study of the chip formation

    process. Although experimental and analytical studies have contributed significantly to the

    improvement of the machining process, researcher is now more focused in developing accurate

    models based on numerical techniques. Finite element method is one such promising tool. These

    simulations not only substitute the expensive and time consuming experimental tests for

    predicting some of the difficult to measure variables (stress, strain and machining temperature),

    but also determine results with higher accuracy as compared to the analytical models.

    One of the first FE models developed for metal cutting process was by Klamecki (1973) in the

    year 1972. Since then many works have been found to use FEM for gaining better understanding

    of the machining process. Numerous FE codes such as DEFORM, FORGE2, ABAQUS/Standard,

    ABAQUS/Explicit, Ansys/LS-DYNA have come up that are being used by the researchers; thus

    giving results closer to the experimental ones. However, it should be noted that the accuracy of

    FE modeling is determined by how adequately the characterization of selected input parameters

    reflect the deformation behaviour undergoing during the chip formation in the actual practice. In

    general, application of finite element modeling to cutting process involves consideration of

    certain key features such as type of formulation (Eulerian, Lagarangian or Arbitary Lagrangian

    Eulerian), material models, friction models and chip separation criterion. Lagrangian formulation

    is easy to implement, efficient and fast converging but unable to handle large deformation

    problems involving high mesh distortion. This necessitates the incorporation of chip separation

    criterion to simulate the cutting action at the cutting zone (Strenkowski and Moon, 1990).

    Eulerian approach, on other hand, permits simulation of machining process without the use of any

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    mesh separation criterion. But the main drawback of Eulerian formulation is that it is unable to

    model the free boundaries and may only be used when boundaries of the deformed material are

    known a priori. While Arbitrary Lagrangian Eulerian (ALE) formulation is the one which takes

    the best part of both Lagrangian and Eulerian formulations and combines them into one. ALE

    reduces to a Lagrangian form on free boundaries of the chip while maintains an Eulerian form at

    locations where significant deformation occurs, as found during the deformation of material in

    front of the tool tip (Rakotomalala, Joyot and Touratier, 1993). ALE formulation is utilized

    mostly to eliminate the termination of analysis due to excessive mesh distortion that is likely to

    occur in the vicinity of the tool tip.

    Many chip separation criteria have been used in the literature which can be broadly categorized as

    geometrical (based on distance tolerance from the tool tip) and physical types (based on stress,

    strain or strain energy density). Strenkowski and Carrol (1985) introduced the chip separation

    criterion based on the effective plastic strain and found that chip geometry and forces are

    unaffected by varying the threshold of the effective plastic strain. The simulation results of Huang

    and Black (1996) showed that the type of chip separation criterion did not greatly affect chip

    geometry and the distributions of stress and strain but it did affect the distribution of residual

    stresses on the machined surface. Some researchers tried to introduce ductile failure in chip

    separation. Rosa et al. (2007) evaluated two fracture modes, namely, specific distortional energy

    criterion and tensile cracking criterion (Cockroft-Latham ductile failure) and concluded that

    former is an appropriate criterion to evaluate ductile damage. The Johnson-Cook damage model

    is an extension of specific distortion energy criterion which has come up as an efficient model to

    simulate the chip separation and thus, is being widely used in machining problems. Though the

    energy required for the separation of work material and chip is recognized, it is usually neglected

    when compared to shearing and friction energies (Shaw, 2003). It is found that mostly the chip

    separation criterion is considered as a mere computation fix in order to permit the tool movement

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    into the work material (Atkins, 2003). While incorporating damage models, generally a thin

    sacrificial layer is implemented and chip is formed due to failure/deletion of these elements

    (Vaziri, Salimi and Mashayekhi, 2010).

    The right type of formulation and chip separation criterion are undoubtedly essential for

    simulating the cutting process but they do not affect the chip formation process, in terms of chip

    morphology and cutting forces significantly. According to Ng et al. (2002), the difficulties in

    accurate FE modeling of cutting process arise basically from two critical factors, namely,

    dependence of the chip formation process on the material models describing the flow stress

    property of the workpiece material and the friction conditions between the tool rake face and the

    chip.

    1.1.Material models

    Several classical plasticity models have been widely employed that represent, with varying

    degrees of accuracy, the material flow stresses as a function of strain, strain rate and temperature.

    These models include the Oxley model (Oxley, 1989), the Johnson-Cook model (Johnson and

    Cook, 1983), Zerilli-Armstrong model (Zerilli and Armstrong, 1987), Usui model (Usui and

    Shirakashi, 1982), Mechanical Threshold Stress (MTS) model (Banerjee, 2007), Litonski-Batra

    model (Litonski, 1977; Batra, 1988), Maekawa model (Maekawa, Shirakashi and Usui, 1983),

    etc. Davies et al. (2003) compared the experimental results for machining of AISI 1045 steel to

    the predictions of three different material models, namely, Johnson-Cook model, Zerilli-

    Armstrong model and Power law rate dependent model. Adibi-Sedeh et al. (2005) have presented

    a detailed comparison of various process variables using Oxley model, Johnson Cook model and

    Maekawa history dependent model for AISI 1045 steel and found that Johnson-Cook model

    predicts chip thickness better when compared with other models as well as experimental results.

    Shi and Liu (2004) found that the predicted residual stresses and the chip curl varied for the type

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    of models selected, namely, Litonski-Batra, Power law, Johnson-Cook and Bodner-Partom

    models while machining Hy-100 steel. They found that the forces predicted by Johnson-Cook

    model deviate by less than 13% and the chip thickness and shear angle do not deviate from the

    measurement by more than 9%. It is noted that Johnson-Cook model is the most popular model

    and often found to be used as the benchmark for comparison of different other models. This could

    be attributed to the fact that it not only shows less discrepancy between the predicted and the

    experimentally found chip morphology and cutting forces but also, the basic form of the model is

    readily acceptable to most computer codes, since it uses variables which are available in the codes

    (Johnson and Cook, 1983). This material model defines the flow stress as a function of strain,

    strain-rate and temperature such that, it not only considers the strain rates over a large range but

    also temperature changes due to thermal softening by large plastic deformation. Johnson-Cook

    model is not restricted to the continuous chip formation only; it is equally efficient in simulating

    the formation of segmented or saw teeth type chip which are found while machining hardened

    steels such as AISI 4340, AISI H13 as well as Titanium alloys based on adiabatic shearing

    (Mabrouki and Rigal, 2006; Ng and Apsinwall, 2002; Baker, Rosler and Seimers, 2002).

    However, every model has its own limitations and hence, it is not always true that Johnson-Cook

    model gives better results than any other models in all cases (Fang, 2003). Recently, researchers

    are working on temperature dependent flow softening based modified Johnson-Cook material

    where flow softening, strain hardening and thermal softening effects and their interactions are

    coupled in order to simulate segmented chip formation while machining Titanium alloy (Sima

    and Ozel, 2010; Calamaz, Coupord and Girot, 2008).

    Majority of these constitutive material models do not adequately represent the flow stress

    property of the workpiece that is usually heat treated to hardness levels ranging from 50-62 HRC

    as in the case of hard machining (Umbrello, Hua and Shivpuri, 2004). Gradually, researchers are

    attempting to develop innovative models that include the effect of hardness in the flow stress

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    models to reflect the heat treatment on the selected material. Umbrello, Hua and Shivpuri (2004)

    and Umbrello et al., (2008) have proposed hardness based flow stress models for AISI 52100 and

    AISI H13 tool steel by taking Johnson Cook model as the reference flow stress curve for hardness

    46 HRC as well as including an additional component that takes into account the variation of

    workpiece hardness on flow stress. It has been reported that the value of JohnsonCook constant,

    A, varies with the temper of the steel. Banerjee (2007) fitted the yield stress versus Rockwell C

    (Rc) hardness curve for AISI 4340 steel to determine the value of A for various tempers. In

    addition to the selection of suitable material model, selection of correct values of constants used

    in the material models are also critical for predicting the forces, chip morphology and cutting

    temperatures with reasonable accuracy. Umbrello, Saoubi and Outeiro (2007) studied the effects

    of five different sets of work material constants available in the literature by using them in the

    JohnsonCook constitutive equation for orthogonal cutting of AISI 316L and predicted the

    cutting forces, chip morphology, temperature distribution and residual stresses. Lesuer (200)

    defined a new set of material constants to be used in the JohnsonCook model for Ti6Al4V and

    2024-T3 aluminium based on the performed experiments and additional data from past literature.

    Similarly, Ozel and Karpat (2007) utilized the evolutionary computational methods of classical

    particle swarm optimization and cooperative swarm optimization to identify the JohnsonCook

    constants for AISI 1045, AISI 4340, AA6082 aluminium and Ti6Al4V titanium alloy. The

    present paper aims to exploit the new set of Johnson-Cook constants found by Ozel and Karpat

    (2007) along with two other sets of data that are commonly used in FE modeling of orthogonal

    cutting of AISI 4340 steel.

    1.2.

    Friction models

    Another important aspect while simulating the chip formation process is identification of the

    frictional parameters at the tool chip interface. Accuracy of the developed model depends on the

    frictional boundary conditions because of its direct effect on the forces and distribution of stresses

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    and temperatures over the rake surface. The tool-chip interfacial conditions are hard to evaluate

    experimentally due to high strain and temperature occurring in a localized region. As a

    consequence, frictional parameters are assumed based on the experimental tests conducted on

    much lower strain and temperature.

    It is known that the simplest way of characterizing tool-chip friction is to assume Coulombs

    friction law where coefficient of friction is considered constant over the entire rake surface. The

    coefficient of friction () is defined as the ratio of the cutting force parallel to the tool rake face to

    the force normal to the rake face. In past, researchers have evaluated the value of

    experimentally and found it to be ranging from 0.2 to as high as 1.8 (Zorev, 1966; Kronenberg,

    1966; Usui and Takeyama, 1960); whereas, the value of used in reported FE models is found to

    lie in a range of 0.0-0.5 (Strenkowski and Moon, 1990; Komvopoulos and Erpenbeck, 1991;

    Strenkowski and Carroll, 1985). The idea of using values below the limiting values not only suit

    the sliding conditions but also produce results that are in good agreement with the experimental

    ones. However, few researchers strictly pointed out that the concept of coefficient of friction is

    not adequate to define the tool-chip interfacial conditions. Still the concept is being widely used

    by the researchers even today because it makes the problem simpler. Besides coefficient of

    friction, constant shear friction factor mhas also been utilized as an input into the FE simulations

    to represent the friction at entire tool-chip interface in some of the works (Filice et al., 2007). This

    law again neglects altogether the low stress variation of frictional stress with normal stress n .

    But, experimental investigations carried by Zorev (1963), Childs (1998) and Lee, Liu and Lam

    (1995) suggested that the normal and frictional stresses are not uniform over the tool rake face.

    The tool-chip contact consists of two distinct regions, namely, a sticking zone near the cutting

    edge and a sliding zone away from the cutting edge. In the sticking region, very high values of

    normal stress occur while the frictional stress is assumed to be equal to the equivalent shear stress

    limit of the workpiece material. No relative motion exists between the tool and the chip. On the

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    contrary, in the sliding region normal stress is small and relative sliding of chip on the rake

    surface occurs that is assumed to obey the Coulombs friction law. A modified Coulombs law as

    reported by Shi, Deng and Shet (2002) is one such model that takes care of the stick-slip zones

    along the tool-chip interface.

    Literature suggests that none of the friction models is perfect. Therefore, newer varieties of

    friction models are being developed by incorporating suitable modifications to the basic friction

    laws. Wu, Dillon and Lu (1996) extended the Zorev friction model where they assumed sticking

    and sliding regions to be of equal lengths such that the frictional stress is the function of the

    equivalent stress in the sliding region, while in the sliding region it decreases linearly to zero.

    Usui and Shirakashi (1982) described the frictional force behaviour as a function of normal force

    depending on the workpiece-tool material combination. Recently, relevant values of friction

    coefficients have been obtained under high velocity, high temperatures and high pressures by

    using a newly designed tribometer (Zemzemi et al., 2008). This system has helped to identify new

    friction models for different workpiece materials based on the average local sliding velocity. It

    has been found that the most relevant parameter affecting the adhesive friction coefficient is the

    local sliding of the workpiece material against the cutting tool (Zemzemi et al., 2009; Rech et al.,

    2009; Bonnet et al., 2008). Few researchers have suggested that the friction coefficient may vary

    with temperature (Ng et al., 2003; Guo and Liu 2002). Generally speaking, friction coefficients

    decrease with increasing relative sliding speed and as the speed increases, contact temperature

    increases due to greater frictional work being done. Hence, decrease of friction coefficient with

    increasing temperature logically follows. Besides, extreme temperature variation occurs in the

    case of machining process. Moufki, Molinari and Dudzinski (1998) have considered that the

    interface temperature has the primary role; and they used Coulomb law in which the friction

    coefficient is a monotonically decreasing function of temperature. Ozlu, Budak and Molinari

    (2009) demonstrated that true representation of the friction behaviour should involve both

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    sticking and sliding zones on the rake face for the accurate predictions. They stated that the total

    contact length on the rake face is 3-5 times the feed rate such that for high cutting speeds the

    contact is mainly sliding. While in one of the recent works, Arrazola and Ozel (2010) investigated

    the effect of limiting shear stress at the tool-chip interface on frictional conditions by coupling

    sticking and sliding friction. The investigations revealed that the stick-slip friction models should

    be used with caution since limiting shear stress value is dependent on local deformation

    conditions and temperatures. Brocail et al. (2010) proposed a friction law according to variables

    like contact pressure, interfacial temperature, sliding velocity and Coulombs friction coefficient

    and derived the constants with the help of a numerical model of upsetting sliding test. In few of

    the recent works, comparison of some selected friction models has been presented showing their

    effect on FE simulations for orthogonal cutting of AISI 1045, low carbon free cutting steel

    (LCFCS) and AISI 4140 (Filice et al., 2007; Ozel, 2006; Haglund, Kishawy and Rogers, 2008). It

    is suggested that the examined models are insufficient and further investigation is needed. Hence,

    the present work makes an attempt to give some insight in the friction modeling problem by

    considering seven most relevant friction models among various models proposed by past

    researchers, including a relatively newer one, in the FE modeling of orthogonal cutting.

    From the critical review of literature, it can be stated that the selection of appropriate material

    models and friction models is a critical issue in FE simulation of machining process. The main

    objective of the research reported here is to study the effects of three different set of material

    constants used in Johnson-Cook material model and the effects of seven different types of friction

    models on the process variables in finite element modeling of orthogonal cutting of AISI 4340 by

    using a carbide tool.

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    2.SIMULATION PROCEDURE

    2.1. Model features

    The present work focuses on the developing a 2D FE model of chip formation process based on

    orthogonal cutting conditions. A plane strain condition was assumed because the feed value is

    generally very less as compared to the depth of cut. A fully coupled thermal-stress analysis

    module of finite element software ABAQUS/Explicit version 6.7 has been employed to perform

    the study of chip formation process. This makes use of elements possessing both temperature and

    displacement degrees of freedom. It is necessary because metal cutting is considered as a coupled

    thermo-mechanical process, wherein, the mechanical and thermal solutions affect each other

    strongly. The explicit dynamics procedure used is computationally efficient for the analysis of

    metal cutting process that involve very large deformations and complicated contact conditions.

    Furthermore, Arbitrary Lagrangian Eulerian (ALE) adaptive meshing technique is more generally

    applicable in ABAQUS/Explicit. This approach provides control of mesh distortion which is

    again very much possible in the case of cutting process undergoing large deformations (Abaqus

    version 6.7 analysis users manual, 2007).

    The 2D model comprises a portion of cutting tool which participates in the cutting and a

    rectangular block representing the workpiece. The cutting tool is considered to be perfectly sharp

    based on the fact that the effect of tool edge radius hardly plays any role once a steady state is

    reached in cutting. Such assumption has been taken for the simplicity of problem by many of the

    researchers (Shi, Deng and Shet, 2002; Mabrouki et al., 2008b).The cutting tool includes the

    following geometrical angles: inclination angle 90= , rake angle 6 = and flank angle

    5= . Both the tool and the workpiece are modeled as deformable bodies and discretized with

    four-node plane strain thermally coupled quadrilateral, bilinear displacement and temperature

    (CPE4RT) type elements. As boundary conditions, the cutting tool movement is constrained in

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    both vertical and horizontal directions and the workpiece block was given the cutting velocity.

    The geometric details and the boundary conditions of the FE model are shown in Fig. 1. The tool

    and the workpiece are initially at the room temperature. Heat transfer from the chip surface to

    cutting tool is allowed by defining the conductive heat transfer coefficient (h) equal to 500

    2kW m K (Coelho, Ng and Elbestawi, 2007). Thermal radiation from the free surface of the

    chip to the surrounding as well as heat transfer between the boundaries of the machined surface

    and air are considered insignificant and are thus, neglected. Johnson-Cook shear failure model is

    employed as the damage model to the damage zone (Mabrouki and Rigal, 2006). This model

    along with the ELEMENT DELETION = YES module of the software allow the separation of the

    chip from the workpiece (Abaqus version 6.7 analysis users manual, 2007; Mabrouki et al.,

    2008a). The INELASTIC HEAT FRACTION and the GAP HEAT GENERATION modules of

    the software are used to incorporate heat generation rate due to plastic deformation (pq )and heat

    generation rate due to friction on the toolchip interface ( fq ) by introducing (fraction of the

    inelastic heat) andf (fraction of the frictional work), respectively (Abaqus version 6.7 analysis

    users manual, 2007; Mabrouki et al., 2008b).

    2.2.Physical properties of the cutting tool and workpiece materials

    The present work considers machining of AISI 4340 using tungsten carbide cutting tool. The

    Table 1 lists the properties of both cutting tool and the workpiece material used in simulation of

    the chip formation process (Mabrouki and Rigal, 2006).

    2.3. Material model

    Several material constitutive models have been used in FE analysis of metal cutting processes.

    Johnson-Cook model, being most widely used, is employed to describe the flow stress property of

    workpiece material AISI 4340 in this study. This material model defines the flow stress as a

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    function of strain, strain-rate and temperature such that, it not only considers the strain rates over

    a large range but also temperature changes due to thermal softening by large plastic deformation.

    According to Johnson-Cook constitutive material model, von Mises tensile flow stress of the

    workpiece is described as follows:

    ( )

    cos

    1 ln 1

    m

    n room

    o melt roomelasto plasicterm vis ity softening

    term term

    T TA B C

    T T

    = + +

    (1)

    where, is the equivalent stress, Ais the initial yield stress (MPa), Bis the hardening modulus

    (MPa), n is the work-hardening exponent, C is the strain rate dependency coefficient, m is the

    thermal softening coefficient, is the equivalent plastic strain, is the plastic strain rate, o is

    the reference strain rate (1.0 s1), roomT is room temperature, meltT is the melting temperature,

    These work material constants have been found by various researchers by applications of several

    methods, thus producing different values of data sets for a specific material. As a result, selection

    of suitable data sets along with appropriate material model becomes equally important. The

    present study selects three sets of Johnson-cook material constants, including a relatively new

    data set explored by an evolutionary computational method, namely, M1, M2 and M3 from the

    available literature. A brief description of each of the sets of data is given below, while the values

    are listed in the Table 2.

    M1: The data for the material constants of M1 were obtained by performing torsion tests over a

    wide range of strain rates (quasi-static to about 400/s); dynamic Hopkinson bar tensile tests over a

    wide range of temperatures as well as the static tensile tests. The data were then evaluated by

    comparing computation results with data from cylinder impact tests (Johnson and Cook, 1983).

    The values obtained are found to be used widely in the literature.

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    M2: Ozel and Karpat (Ozel and Karpat, 2007) developed a new methodology that utilizes an

    evolutionary computational method with the aim of minimizing error in identifying the Johnson-

    Cook constitutive model parameters. The material constants of M2 were explored and

    recalculated by cooperative particle swarm optimization (CPSO).

    M3: The M3 set of material constants were identified by conducting Split Hopkinson pressure bar

    compression tests over a wide range of temperatures (-196C to 600C) and strain rates (quasi-

    static to about 7000/s). A computer program was used to perform optimization to fit the

    experimental data (Gray et al., 1994).

    2.4. Friction models

    Friction modeling plays a significant role as it is known to considerably influence the accuracy in

    simulation of the chip formation process. Several models have already been proposed with the

    aim of achieving results closer to the experimental data. In this study, seven friction models

    developed by the researchers in the past were selected to study their effect on the process

    variables. This section briefly discusses the selected friction models.

    F1: Constant friction coefficient model

    This model simply considers the coulombs friction law on the entire contact zone (Strenkowski

    and Moon, 1990). The software defines this model by introducing a constant value of coefficient

    of friction which, in the present case, is equal to 0.2 (Komvopoulos and Erpenbeck, 1991;

    Mabrouki et al., 2008b).

    F2: Constant friction coefficient with limited shear stress model

    This model allows introduction of stick-slip conditions at the tool chip interface by defining the

    equivalent shear stress limit from the beginning for sticking zone and a constant value of

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    coefficient of friction () for the sliding zone (Shi, Deng and Shet, 2002; Mabrouki et al., 2008b).

    The constitutive mathematical model is given as:

    max maxn

    maxn n

    , for ( Sticking zone)

    , for ( Sliding zone)

    =

    (2)

    In the sticking zone, the normal stress ( n ) is very large and frictional stress ( ) is assumed to

    be equal to the equivalent shear stress limit, max . The value of max can be approximated

    as 3A , where A is the initial yield stress (Mabrouki et al., 2008). In the sliding zone, the

    normal stress is small and the frictional stress follows the simple Coulombs law with =0.2

    (Komvopoulos and Erpenbeck, 1991; Mabrouki et al., 2008b).

    F3: Variable friction coefficient as a function of temperature

    The friction law in this model accounts for the temperature effects. The model postulates the

    Coulombs law with a mean coefficient of friction ( ) in terms of mean interface temperature

    (Tint) as defined below (Moufki, Molinari and Dudzinski, 1998):

    int( )T = (3)

    It is assumed that coefficient of friction is a decreasing function of temperature i.e. decreases

    linearly to zero as the average temperature reaches the melting point of the workpiece material.

    F4: Variable friction coefficient as a function of normal stress

    This model defines the coefficient of friction as a function of the average normal stress ( n ) over

    the entire tool chip contact surface. This is expressed as follows:

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    n

    = . (4)

    The normal stress and shear stress ( ) values used for the evaluation of are taken from the

    experimentally found stress distributions by Buryta, Sowerby and Yellowley (1994).

    F5: Variable friction coefficient as a function of sliding velocity

    This model considers a variable friction coefficient as a function of the average local sliding

    velocity at the tool-chip contact. Zemzemi et al. (2009) identified a new empirical friction model

    based on the local sliding velocity lsV as follows:

    057.0)ln(07.0 += lsV (5)

    This friction relates the friction coefficient and the local sliding velocity in the range of 2-170

    m/min corresponding to the cutting velocity up to 200 m/min.

    F6: Rate dependent friction coefficient model

    This recent friction model developed by Tawfiq (2007) is expressed by the following relation:

    += eksk )( (6)

    where, k is the coefficient of kinetic friction, s is coefficient of static friction, is the

    exponential decay coefficient and v is the relative sliding velocity of the slave and master

    surfaces. The optimum values of coefficients, s = 0.4 and k = 0.2 were determined through

    trial and error approach (Tawfiq, 2007).

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    F7: Two friction coefficients model

    In this model, basic Coulombs law is implemented with two different coefficients for specified

    portions of the contact length ( cL ), namely sticking region ( stickingL ) and sliding region. On the

    basis of past research, the length of the sticking region was assumed equal to the uncut chip

    thickness (Ozel, 2006). In order to apply this at the tool-chip interface, the tool rake surface was

    split into two sections such that

    23.0= when stickingLx0 and 15.0= when csticking LxL ,

    wherexis the distance measured from the tool tip along the tool rake face. The constant values of

    were taken based on the existing literatures (Komvopoulos and Erpenbeck, 1991; Ozel, 2006).

    3. RESULTS AND DISCUSSION

    In this section, numerical results obtained from FE simulations with different types of Johnson-

    Cook material constants and the friction models are presented.

    3.1. Johnson-Cook material constants

    Since chip morphology affects the stress, strain and temperature distributions, prediction of the

    right kind of chip morphology through finite element simulations is an important aspect in the

    study of metal cutting. This motivates us to carefully select the material model as well as the

    corresponding material constants to be used in the flow stress equation of the chosen material

    model. The results dealing with three different sets of Johnson-Cook material constants, namely,

    M1, M2 and M3 (see Table 1) are presented in this section. The friction model for each of the

    three cases are kept same i.e. constant friction model with limited shear stress, so that the results

    can be compared on the same conditions. Fig. 2 shows the predicted chip morphology and the

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    cutting force ( cF ) variation over a simulation time of 2 ms at cutting velocity of 100 m/min and

    uncut chip thickness of 0.2mm for different values of material constants (M1, M2 and M3). The

    chip morphology may be described in terms of the average values of segmented chip geometry

    such as distance between two saw-teeth (D), peak (a) and valley (b). In addition, the

    corresponding cutting force profiles are presented which not only show the predicted values of

    cutting forces but also aid in deriving the segmentation frequency. The segmentation frequency

    (fSN in Hz) can be calculated as:

    ( )1000

    N

    =

    initialfinalSN

    TTf (7)

    where, N = number of saw teeth produced between a given time interval of initialT to finalT .

    The results in Figure 1 show too much variation in the predicted chip morphology, cutting force

    values and the segmentation frequency with the change in the values of material constants. When

    the sets M2 and M3 are compared with M1, cutting forces showed a deviation of as high as 68%.

    While 63% of deviation in chip geometry and almost 50% of deviation in segmentation frequency

    were observed. As discussed earlier, the constantA used in the Johnson-Cook model is known to

    vary with the temper of steel and hence can be correlated to the hardness of the work material

    (Banerjee, 2007). It is also evident from literature that as the hardness changes, there is a change

    in the flow stress of the work material (Umbrello et al., 2007). That the value ofAaffects the chip

    morphology, segmentation frequency as well as cutting force values is therefore, a logical

    conclusion. Generally, harder the workpiece material, higher is the tendency for the deformation

    to localize and thus produce more prominent segmented chips even at low cutting speeds

    (Umbrello, Hua and Shivpuri, 2004). This can be figured out in the case of M3 where the value of

    Ais highest. Not only the saw teeth are well defined for M3 but also the segmentation frequency

    is higher. Furthermore, as the material hardness increases, more deformation energy is required

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    for the chip formation (Qian and Hossan, 2007). This explains the increasing trend of cutting

    forces from M1 to M3. In addition, mand n are also considered as the critical parameters that

    influence the segmentation of chip (Baker, 2003). Baker (2003) suggested increase in mleads to

    increase in chip segmentation and decrease in cutting forces, while increase in n causes lesser

    tendency to undergo chip segmentation keeping cutting force nearly constant. This could be the

    reason possibly for higher value of segmentation frequency ( SNf = 3663 Hz) for M1 as compared

    to the other two values. In the former case, greater value of mand lower value of n(see Eq. (1))

    increases the number of saw teeth or segmentation frequency, but comparatively lesser distinct

    and smaller chips are produced due to lower value ofA.

    To give a clear insight, the temperature distributions, being the most important factor responsible

    for the varied chip morphology, are presented for all the three cases in Fig. 3 at a cutting velocity

    ( cV ) of 120 m/min and uncut chip thickness ( f ) of 0.2 mm for a simulation time of 2 ms. As

    expected, the distribution patterns of temperatures are different from each other, reflecting

    difference in the chip morphology. It is known that the chip segmentation basically results from

    the thermal softening due to very high temperatures in highly localized regions of the shear plane

    during the cutting of hardened steels. The temperature distribution of M1 is not as localized as the

    other two, in the shear zone. While in case of M2 and M3, higher values of temperatures are

    attained in a highly localized region in the form of a band extending from tool tip towards the

    back of the chip. Consequently, highly pronounced saw teeth are produced due to the thermal

    softening caused by very high temperatures at the back of the chip. The maximum temperature at

    the rake face ( rakeT ) of the tool and shear zone ( shearT ) of the workpiece can also be determined

    from the simulated results presented in Fig. 3. In Fig. 3 (c), the chip appears to penetrate the

    workpiece. But in actual sense, the chip flows out at the side of the workpiece and it is simply the

    overlapping of the images of chip and workpiece (hidden boundary shown by dashed lines). The

    predicted results conform well to the existing ones which have concluded that greater hardness,

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    which in the present context is higher value of A, yields higher cutting temperatures (Matsumoto

    and Hsu, 1987). This, as stated earlier, can be attributed to the higher energy involved during the

    cutting of the harder steels (Qian and Hossan, 2007).

    The predicted results show that material constants significantly affect various aspects of the

    simulation of the chip formation process. As far as the chip morphology is concerned, the

    distance between the two consecutive saw-teeth in case of M2 (D=0.326 mm) is fairly closer to

    the experimental one (D=0.3 mm) found by Belhadi et al. (2005). But at the same time, M2

    underestimates the segmentation frequency to some extent. When cutting forces were considered,

    experimental values were taken from the work of Lima, Avila and Abrao (2005). Tests were

    conducted for machining AISI 4340 steel of hardness varying from 23 to 42 HRC with coated

    carbide tool and the corresponding cutting forces measured were around 475 N and 575 N,

    respectively. It is found that the M2 gives a cutting force value that is closer to the one obtained

    by Lima, Avila and Abrao (2005) for 42 HRC steel, while M1 gives a cutting force of 408.19 N

    comparable with the value obtained in the case of 23 HRC, (though an underestimated value) and

    M3 yields cutting force of 687.7 N which of course is a much higher value. Hence, this work

    demonstrates the importance of choosing optimum values of material constants that may prove as

    a satisfactory compromise between the hardness of the workpiece material and the cutting

    variables such as chip morphology, cutting forces and cutting temperatures.

    3.2. Influence of friction models

    This section aims to investigate the importance of implementing the right kind of frictional

    boundary conditions in the FE simulation. Seven different friction models (F1 to F7) are selected

    from the review of past research and are incorporated into a finite element model by keeping all

    the other factors constant, say, the workpiece material model, tool geometry and the cutting

    conditions. The Johnson-Cook material model, with M1 set of parameters, was considered for the

    workpiece material AISI 4340 of hardness 48 HRC (Mabrouki and Rigal, 2006). Table 3 shows

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    the predicted values of various process variables such as the cutting force ( cF ), thrust force ( tF ),

    tool chip interface temperature ( intT ), contact length ( cL ) and chip thickness ( 2a ) for the seven

    cases by varying the cutting velocity from 60 to 180 m/min for uncut chip thickness of 0.2 mm.

    The predicted values show the trend that accord well with the basic theory. The forces, contact

    length and chip thickness tend to decrease while the interface temperature increases with the

    increase in the cutting velocity in all the seven cases. However, percentage change in the

    variables varies with the type of friction model used. The predicted cutting force and thrust force

    are compared with the experimental values obtained by Lima et al. (2005)and Lima, Avila and

    Abrao (2007) for machining AISI 4340 of hardness 42 HRC and 50 HRC using negative rake

    ( 6 = ) carbide tool under equivalent cutting conditions. Although the workpiece hardness

    considered in this work is marginally different from those considered in Lima et al. (2005) and

    Lima, Avila and Abrao (2007), the forces can be assumed to lie in the range identified in these

    cited sources and thus, a qualitative assessment of the correlation between the published

    experimental results and our numerical results can be made. Experimental findings of Lima et al.

    (2005) show a percentage decrease of 30.78% and 21.88% for cutting forces when machining

    AISI 4340 of hardness 42 HRC and 50 HRC, respectively as the cV is increased from 60 m/min

    to 180 m/min. The numerical results show a maximum decrease of 15% in the cutting force and

    27.8 % decrease in the thrust force (both observed in the case of F3) for the same variation in

    cutting speed. The friction model F4 showed 11.8 % decrease in the cutting force while the rest of

    the models, except F5, show a percentage decrease of 8-10% in cutting force. In case of F5,

    variation in force is found to be lowest, i.e. 5 %. Lima, Avila and Abrao (2007) observed an

    increase in the cutting and thrust forces with the increase in hardness (23 to 42 HRC) while

    cutting AISI 4340. In their research (Lima, Avila and Abrao, 2007), cutting force and thrust force

    came out to be around 550 and 240 N, respectively, while machining AISI 4340 of hardness 42

    HRC at cV = 120 m/min and f = 0.2 mm/rev. However, lower forces were observed when

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    machining work material AISI 4340 hardened to 50 HRC in comparison with 42 HRC, i.e. cF =

    425 N and tF= 100 N in the work of Lima et al. (2005). Since the present work deals with a

    work material having hardness 48 HRC, the values of cutting and thrust forces ought to lie in

    between the mentioned values (450550 N for cF and 150250 N for tF at cV = 120 m/min and f

    = 0.2 mm/rev). It can be seen from Table 3 that the predicted values of the cutting and thrust

    forces appear to fall in the concerned range; thrust force being slightly underestimated. The

    model F5 seems to give one of the best values of cutting force ( cF = 500.5 N) with tF =187.9 N,

    while F4 gives a better value for thrust force ( tF = 192.6 N) with a corresponding value of cutting

    force ( cF = 486.8 N) slightly lower than the value obtained from F5.

    The percentage increase in the interface temperature is found to be as high as 40-45 % in case of

    F1, F2, F6 and F7, while the friction models F5, F3 and F4 showed a percentage increase of 30,

    29 and 23%, respectively. However, the predicted values of average interface temperatures for all

    the cases are found to be underestimated when compared with the experimental results by Dhar,

    Kamruzzaman and Ahmed (2006). The average interface temperatures for cutting velocities of 63

    and 128 m/min were found to be 737.5 C and 788 C, respectively, when measured

    experimentally while machining AISI 4340. Though the frictions models F1, F2, F6 and F7

    exhibit large temperature increase with the increase in the cutting velocity, the intT values are

    much lower especially at cV = 60 m/min. Hence, F5 and F4 can be considered as models showing

    fairly better results for intT because not only they show higher values as compared to others but

    also they exhibit temperature increase fairly well. Since temperature at interface is directly

    influenced by the relative velocity, velocity dependent friction model F5 could possibly predict

    the value of intT better. Fig. 4 shows the variation in the temperature distribution over the rake

    face for friction models F4, F5 (that showed better results in terms of intT ) and F2 (one of the

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    commonly used friction model) at cV = 120 m/min and f = 0.2 mm/rev. As expected, all three

    models show that the temperature is non-uniformly distributed along the rake face with peak

    interface temperature occurring at some distance away from the cutting edge. The difference is

    mainly observed in the position and the value of the peak interface temperature in models F2, F4

    and F5. This may be attributed to the distribution of frictional stress along the tool chip contact

    length. Fig. 5 presents the normal and frictional stress distribution along the tool chip interface. It

    can be seen that F5 has the highest value of frictional stress followed by F4 and F2. This reflects

    the trend observed in the case of temperature distribution in Figure 4. It is also noted that there is

    almost no change in the stress distribution for model F1 (constant friction coefficient) and F2

    (constant friction coefficient with limited shear stress model). Hence, it may be inferred that the

    ABAQUS-Explicit software is unable to simulate the stick-slip zones based upon the equivalent

    shear stress value effectively in case of F2.

    Chip reduction coefficient (), one of the most important process output (in terms of process

    evaluation and optimization) is defined as the ratio of chip thickness (a 2) to uncut chip thickness

    (f). This has been calculated and plotted forc

    V = 60180 m/min, as shown in Fig. 6 for each of

    the seven models. The reason to determine chip reduction coefficient is that it is an important

    machinability index giving much idea about the nature of the tool-chip interaction, chip contact

    length, and chip form. As expected, decreases with the increasing cutting velocity for all the

    cases. But it is observed that the decrease in is more prominent in F5 followed by F4 and F3.

    Note that all the three models are variable friction models, F5 as a function of sliding velocity, F4

    based on experimentally measured normal and frictional stresses and F3 being temperature

    dependent model. Interestingly, similar type of behaviour was observed by Ozel (2006), who

    suggested that the predicted process variables are clearly found to be most accurate when utilizing

    variable friction models based on the experimentally measured normal and frictional stresses on

    the tool rake surface.

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    In general, it is observed that the predicted values tend to vary with the change in the friction

    model. The constant Coulomb friction coefficient model (F1) is one of the most widely used

    models as it is considered to be a fair compromise from simplicity and accuracy point of view of

    the problem. When rest of the models are compared with the model F1, the variable friction

    models F4 (stress dependent) followed by F5 (sliding velocity dependent) and F3 (temperature

    dependent) showed significant deviations in cutting force, thrust force and interface temperature,

    while F2, F6 and F7 produced nearly same values for all the three variables. F4 showed a

    deviation of 6 % in cF , 37 % in tF and 20 % in intT . Such observation is in agreement with the

    results predicted by Filice et al. (2007) and Ozel (2006). Filice et al. (2007) found tF to differ by

    29 %, intT by 24 % and cF by 12 % among different friction models for machining AISI 1045.

    Similarly, Ozel observed the thrust force varying by 8 % and cutting force by not more than 3 %

    with the friction models while machining Low Carbon Free Cutting steel (LCFCS). It can be said

    that influence of friction models is larger over the thrust force followed by temperature as

    compared to cutting force (Arrazola and Ozel, 2010). However, variation in contact length and chip

    thickness are found to be less among all the models as compared to the variation in the contact

    length with higher predicted values for models F4 and F5. Likewise, Filice et al. (2007) and Ozel

    (2006) have also found out that the main mechanical results such as cutting forces, contact length

    etc. are not as sensitive to the friction models as compared to the cutting temperatures.

    From the predicted results, it is reasonable to state that the friction modeling is not very critical at

    higher cutting speeds but it is fairly prominent in case of low cutting speed ( cV = 60 m/min)

    showing a difference of 37% in thrust force and 20% in interface temperature by model F4. The

    model F4, being the function of average normal stress over the rake face, reflects the stick-slip

    zone in a much realistic way. This could be a probable reason for which F4 comes out to be one

    of best models when compared with the rest of the models as well as with the existing

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    experimental results. The friction models tend to be crucial at low cutting speeds can be attributed

    to the fact that sliding contact strongly depends on the cutting speed. At low cutting speeds

    sticking zone can be up to 30% of the total contact while for high cutting speeds the contact is

    mainly sliding (Ozlu, Budak and Molinari, 2009). Consequently, at higher cutting speeds all the

    friction models behave like constant Coulomb friction coefficient model, thus showing similar

    kind of results.

    4. CONCLUSIONS

    Incorporation of different friction models were used in an FE model to predict cutting force and

    thrust force within a satisfactory range during orthogonal machining. It is observed that cutting

    temperatures followed by thrust force seem to be affected more by the type of frictional

    conditions implemented as compared to the cutting forces. In the present work, the overall values

    of cutting temperatures for all the seven considered friction models are found to be lower than the

    experimental ones. The one possible reason could be the inability to include the effect of hardness

    in the flow stress models which properly include the effect of the heat treatment on the selected

    work material in the simulation. This necessitates the incorporation of the right kind of material

    models and their corresponding material constant values depending upon the hardness along with

    the appropriate friction model in the FE simulation. Therefore, the first part of this work dealt

    with the variation of chip morphology, forces and temperatures with three different sets of

    material constants used in the Johnson-Cook model. Significant differences were observed in the

    values of cutting forces and chip geometry. The deviation in cutting force value was found to be

    as high as 68%. The chip geometry and segmentation frequency were found to differ by 63% and

    50%, respectively. The new set of material parameters developed by Ozel and Karpat (2007)

    through evolutionary computational method, M2, gives values of cutting force and chip geometry

    that are closer to the existing experimental results (Lima, Avila and Abrao, 2005). The results

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    indicate that suitable selection of the material constants incorporates the effect of hardness of the

    workpeice material on the flow stress behaviour of the work material.

    The models F4 followed by F5 gave the best results in terms of cutting force, thrust force and

    interface temperature when compared with the existing experimental findings. The decrease in

    with the increase in cutting speeds is also more prominent in F4 and F5. Interestingly, both the

    models are variable friction models, F4 based on normal and frictional stresses and F5 as a

    function of sliding velocity. F4 showed the maximum deviation in the values of thrust force

    (37%) and interface temperature (20%) when compared with the commonly used model F1.

    However, the cutting forces, contact length and chip thickness are not as sensitive to the friction

    models as compared to the thrust force and cutting temperatures. It is also noted that there is

    almost no change in the stress distribution for model F1 (constant friction coefficient) and F2

    (constant friction coefficient with limited shear stress model) reflecting the inadequacy of the

    software to model stick-slip zones based upon the equivalent shear stress value effectively in case

    of F2.

    The effect of friction model is not very significant at higher cutting speeds but it is reasonably

    significant in case of low cutting speed showing a difference of 37% in thrust force and 20% in

    interface temperature by model F4. The present investigation points out that the selected friction

    models does not affect the chip formation process at high cutting speeds which may be attributed

    to the fact that chip-tool contact is mostly sliding type, i.e, governed by Coulomb friction law, at

    high cutting speed. But at low cutting speeds, it is observed that the friction modeling plays a

    crucial role in predicting the correct thrust force and the temperature. Though the chip thickness

    prediction accuracy is inadequate, as far as other results are concerned, F4 friction model may be

    considered as most suitable friction model among the selected models.

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    Most of the works so far, including the present study consider the effect of material modeling and

    the friction modeling individually on the simulation of chip formation process. It is, however felt

    that emphasis should be given to the combining of the friction models with suitable material

    models or at least the material constant values depending upon the hardness of the work material,

    instead of considering the friction models or material models alone to overcome the inadequacies.

    This will definitely characterize the metal cutting process in a more realistic way that may

    possibly lead to results closer to the experimental ones.

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    FIGURES

    Fig. 1 Geometric details and boundary conditions of the model

    X

    Y

    Z

    Chip surface

    Work piece

    Damage zone

    Vc

    Cutting tool

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    Model Chip morphology Cutting force

    M1

    D =0.199 mm

    a = 0.309 mm

    b = 0.272 mm

    0.0 0.3 0.5 0.8 1.0 1.3 1.5 1.8 2.0

    0

    100

    200

    300

    400

    500

    Tfinal

    Tinitial

    Cuttingforce(N)

    Time (ms)

    cF = 408.19 N

    SNf = 3663.3 Hz

    M2

    D =0.326 mm

    a = 0.368 mm

    b = 0.217 mm

    0.0 0.3 0.5 0.8 1.0 1.3 1.5 1.8 2.0

    0

    100

    200

    300

    400

    500

    600

    700

    800

    Tfinal

    Tinitial

    Cuttingforce(N)

    Time (ms)

    cF = 597.25 N

    SNf = 1855Hz

    M3

    D=0.217 mm

    a= 0.313 mm

    b= 0.222 mm

    0.0 0.3 0.5 0.8 1.0 1.3 1.5 1.8 2.0

    100

    200

    300

    400

    500

    600

    700

    800

    900

    Tfinal

    Tinitial

    Cuttingforce(N)

    Time (ms)

    cF = 687.73 N

    SNf = 2798.5 Hz

    Fig.2Predicted chip morphology and cutting force for M1, M2 and M3 at cV = 120 m/min and

    f = 0.2 mm for t = 2 ms

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    (a)

    (b)

    (c)

    Fig. 3Temperature distributions (C) for (a) M1, (b) M2 and (c) M3 at cV = 120 m/min and f =

    0.2 mm

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    0.2 0.4 0.6 0.8 1.0

    0

    100

    200

    300

    400

    500

    600

    700

    Temperature(C)

    Distance from the cutting edge (mm)

    F5

    F4

    F2

    Fig. 4Temperature distributions along the rake surface of the tool predicted by using friction

    models F2, F4 and F5 at t = 2 ms

    0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

    0.0

    0.2

    0.4

    0.6

    0.8

    1.0

    1.2

    1.4

    Stress(GPa)

    Distance from the cutting edge (mm)

    Normal Stress (F1)

    Normal Stress (F2)

    Normal Stress (F4)

    Normal Stress (F5)

    Frictional Stress (F1)Frictional Stress (F2)

    Frictional Stress (F4)

    Frictional Stress (F5)

    Fig. 5Stress distributions along the rake surface of the tool predicted by using different friction

    models at t = 2 ms

    60 90 120 150 180

    1.80

    1.95

    2.10

    2.25

    2.40

    2.55

    2.70

    2.85

    3.00

    Chipreductioncoefficient

    Cutting velocity (m/min)

    F1 F5

    F2 F6

    F3 F7

    F4

    Fig. 6Variation of chip reduction coefficient with cutting velocity with different friction models

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    Table 1 Physical properties of cutting tool and workpiece

    Parameter Work piece

    (AISI 4340)

    Tool

    (Tungsten carbide)

    Thermal Conductivity, k 44.5W/m/oC 50 W/m/ oC

    Density, 7850 kg/m3 11900 kg/m3

    Youngs modulus, E 205 GPa 534 GPa

    Poissons ratio, 0.3 0.22

    Specific heat, Cp 475 J/kg/oC 400 J/kg/ oC

    Expansion coefficient, (10-5) 1.37

    Table 2Johnson-Cook Constants for AISI 4340

    Model A(MPa) B(MPa) C n m

    M1(Johnson and Cook, 1983) 792 510 0.014 0.26 1.03

    M2 (Ozel and Karpat, 2007) 1523.3 1022.6 0.001512 0.5358 0.89438

    M3 (Gray et al., 1994) 2100 1750 0.0028 0.65 0.75

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    Table 3 Predicted values from FE simulations at cV = 120m/min and f = 0.2 mm

    ModelcV (m/min) cF (N) tF(N) intT (C) cL (mm) 2a (mm)

    60 503.29 163.19 558.84 0.59 0.526

    120 471.28 152.7 651.01 0.487 0.484

    F1

    180 448.9 146.34 778.4 0.428 0.423

    60 503.07 163.34 558.8 0.570 0.517

    120 471.28 152.7 651.01 0.448 0.47

    F2

    180 448.9 146.34 778.403 0.408 0.414

    60 520.40 176.38 561.01 0.448 0.548

    120 463.0 139.76 617.12 0.415 0.461

    F3

    180 441.15 127.22 721.16 0.407 0.409

    60 532.41 223.66 667.07 0.530 0.590

    120 486.88 192.687 683.16 0.489 0.501

    F4

    180 469.22 187.31 820.68 0.448 0.416

    60 514.39 191.54 653.66 0.529 0.518

    120 500.5 187.99 709.968 0.448 0.486

    F5

    180 486.34 179.97 847.19 0.407 0.461

    60 502.88 162.64 526.85 0.448 0.507

    120 470.15 152.25 646.15 0.428 0.464

    F6

    180 450.35 147.02 778.18 0.407 0.464

    60 500.13 169.92 550.0 0.467 0.511

    120 477.24 168.19 683.82 0.457 0.464

    F7

    180 457.52 163.15 800.71 0.416 0.424