Int J Adv Manuf Technol (2012) 62:505–515DOI 10.1007/s00170-011-3846-9

ORIGINAL ARTICLE

Behavior of austenitic stainless steels at high speed turningusing specific force coefficients

Ana Isabel Fernández-Abia · Joaquín Barreiro ·Luis Norberto López de Lacalle ·Susana Martínez-Pellitero

Received: 13 October 2011 / Accepted: 7 December 2011 / Published online: 4 January 2012© Springer-Verlag London Limited 2012

Abstract Turning operation has been widely studied,and it is a well-known process. However, still todaysome limitations exist in the processing of some mate-rials, mainly due to the poor or inexistent character-ization. Such is the case of austenitic stainless steels,which in spite of being materials of high economic andtechnological value, their behavior to machining is stillnot well understood in some aspects. There are not re-liable and updated technological data about austeniticstainless steels at industry. This fact is especially sig-nificant when considering technological developmentconducted by a continuous increment of cutting speeds.Nowadays, there is not a reliable mechanistic modelfor austenitic stainless steels turning adjusted for highcutting speeds. In this paper, a mechanistic model forcutting force prediction is presented. This model wasdeveloped for machining with nose radius tools consid-ering the effect of the edge force due to the rounded

A.I. Fernández-Abia · J. Barreiro (B) · S. MartínezDepartment of Manufacturing Engineering, Escuela deIngenierías Industrial e Informática, University of León,24071 León, Spaine-mail: joaquin.barreiro@unileon.es

A.I. Fernández-Abiae-mail: aifera@unileon.es

S. Martíneze-mail: smarp@unileon.es

L.N. López de LacalleDepartment of Mechanical Engineering, Escuela TécnicaSuperior de Ingeniería, University of the Vasque Country,Alameda de Urquijo s/n, 48013, Bilbao, Spaine-mail: norberto.lzlacalle@ehu.es

cutting edge. In addition, a set of machining tests werecarried out to obtain the specific force coefficients ex-pressions for austenitic stainless steels using the mech-anistic approach at high cutting speeds. The specificcutting coefficients were obtained applying the forcemodel as an inverse model. This paper presents expres-sions for shearing and edge cutting coefficients whichare valid for a wide range of cutting conditions. Resultswere validated by comparing the values estimated bythe model with the ones obtained by experimentation.

Keywords Austenitic stainless steel ·High speed turning · Mechanistic model ·Cutting forces · Characterization

1 Introduction

High-speed turning of austenitic stainless steels has notbeen studied in depth. The range of cutting speedsrecommended by tool manufacturers for machining ofstainless steel with coated cemented carbide insertsis very conservative (200–350 m/min). This range isunproductive at the current state of technology. Thatis, severe cutting speed conditions are demanded nowa-days, and therefore, a deep knowledge of cutting forcesimplied in the process at high cutting speeds is neces-sary. Therefore, nowadays there is not any mechanisticmodel adjusted for high cutting speeds in austeniticstainless steels, with reliable coefficients. In this re-search, a mechanistic model of cutting forces is pre-sented; also, a set of machining tests were carried outto obtain the specific force coefficients for austeniticstainless steels using the mechanistic approach at highcutting speeds (350 to 750 m/min).

506 Int J Adv Manuf Technol (2012) 62:505–515

Mechanistic models consider that cutting forces areproportional to the chip cross-sectional area. The con-stant of proportionality is called specific cutting forcecoefficient or specific cutting pressure (ks). The cuttingpressure depends on several factors, such as workpiecematerial, tool material, tool geometry, part geometry,chip section, cutting speed, lubrication, or tool wear[12]. Due to this significant number of factors that haveinfluence over the ks value and the very difficult quan-tification in some cases, the only reliable method to de-termine the cutting pressure is the direct measurementin specific machining tests at certain conditions. There-fore, in the mechanistic approach, the machining testsare conducted to establish the cutting force coefficientsover a range of cutting conditions [5]. Denkena andKöhler [3] based their research on the approach ofKienzle; the authors developed a model to estimate themachining forces for any uncut cutting section. In theresearch carried out by Weber et al. [14], the authorsanalyzed the influence of tool rake angle, tool mate-rial, and cutting edge radius on the machining forces.Redetzky et al. [10] developed a predictive model forcutting forces and chip flow in machining with noseradius tools. This predictive model is based on theintegration of two sub-models: a geometric model thatdefines the geometry of the machining operation anda force model that determines the force coefficientsfor a specific work material–cutting tool pair. Otherresearchers have used in their predictive models theconcept of equivalent cutting edge to combine the cut-ting actions that occur simultaneously at the side cuttingedge, nose radius edge, and end cutting edge [13, 15].Sambhav et al. [11] present a new nomenclature oftool geometry employing 3D rotational grinding angles,and the authors propose a mechanistic model of cuttingforces in terms of 3D angles and machining conditions.Ko et al. [6] present a mechanistic model which con-sider by the size effect. In their work, a method tocalculate the instantaneous cutting coefficient with sizeeffect is developed, which is independent of cuttingconditions.

1.1 Process modeling

The model developed in this paper allows the estima-tion of cutting forces in turning when the cutting actionoccurs on the side cutting edge and nose radius edge us-ing conventional turning tools. The mechanistic modelpresented here is based on the expressions deduced byAltintas [1]. This model divides the cutting forces in twomain components: a force due to the material shearingaction, which is proportional to the cross-sectional area

of the uncut chip, and a force due to the ploughing orrubbing effect, which is proportional to the cutting edgelength. This aspect is reflected in Eq. 1.

Ff = kfc · A + kfe · S

Fr = krc · A + kre · S

Ft = ktc · A + kte · S (1)

where Ft, Ff, and Fr are the force components in thedirection of cutting speed, the thrust, and the normal,respectively; ktc, kfc, and krc are the specific cuttingcoefficients related to the material shearing action andrepresent the shear force per unit of surface; kte, kfe,and kre, called edge force coefficients, represent therubbing forces per unit width; A is the area of the uncutchip thickness; and S is the cutting edge length.

The factors A and S collect the geometrical featuresof the process, whereas the specific cutting coefficientsobtained empirically collect the data related to work-piece material, tool geometry, material, and lubrica-tion conditions. Therefore, they are calculated for eachtool–part pair.

In order to apply the expressions in Eq. 1 to theturning process, it is necessary to obtain previouslythe sectional area of the uncut chip, the cutting edgelength, and the cutting force coefficients. The calculusof chip area and edge length is carried out using thetool geometry (cutting edge angle and nose radius)and the cutting conditions (feed rate and depth ofcut). With regard to the determination of cutting forcecoefficients, a series of turning tests were developedcalled characterization tests.

1.2 Geometric model

The model is developed for turning operations wherethe cutting action occurs on the side cutting edge andnose radius edge. This case takes place when the fol-lowing condition is satisfied: ap > rn (1 − cos (kr)) andfn ≤ 2rn · sin (kr1). For this combination of feeds anddepths of cut, the uncut chip section can be divided intwo different regions: one uniform thickness region (I),corresponding to the straight part of the side cuttingedge; and a variable thickness region (II), correspond-ing to the nose radius edge (Fig. 1).

In region I, the uncut chip thickness is constant,and it depends on the feed rate ( fn) and the cuttingedge angle (kr). In this region, the magnitude of forcesand their direction do not change along the straightedge. The edge length (SI) and the area (AI) for theregion I can be determined from the associated cutting

Int J Adv Manuf Technol (2012) 62:505–515 507

Fig. 1 Geometry of the uncutchip and cutting forces at thestraight and nose radius edge

geometry. They are given by the expressions in Eqs. 2and 3:

SI = ap − rn (1 − coskr)

(sinkr)(2)

AI = fn[ap − rn (1 − coskr)

] − 14

f 2n sin (2kr) (3)

In the region II, the chip thickness varies contin-uously, and consequently, the magnitude of cuttingforces and their direction varies along the nose radiusedge. In the geometrical analysis, the chip is dividedinto elements of infinitesimal width, each one withdifferent thickness and orientation. The direction ofeach infinitesimal force is considered normal to thelocal cutting edge.

The edge length (SII) for region II is expressed inEq. 4:

SII =∫ ϑ2

ϑ1

dS =∫ ϑ2

ϑ1

rn · dϑ (4)

The area of region II (AII) is calculated by means ofEq. 5:

AII =∫ ϑ2

ϑ1

h (ϑ) · dS =∫ ϑ2

ϑ1

h (ϑ) · rn · dϑ (5)

where ϑ1 and ϑ2 are the limits of integration:

ϑ1 = cos−1(

fn

2 · rn

); ϑ2 = π

2+ kr (6)

and h (ϑ) is the chip thickness and it is expressed asa function of the position of the chip element (ϑ).Applying the law of cosines to the triangle OTO’ inFig. 1, h (ϑ) can be calculated as expressed in Eq. 7:

h (ϑ)

= rn −√

f 2n +r2

n−2 fnrncos[π−ϑ−sin−1

(fn

rnsin (ϑ)

)]

(7)

1.3 Mechanistic model

Once the geometry is clearly stated, the mechanisticmodel which corresponds to the expressions in Eq. 1is applied to both geometrical region. The total cuttingforce is calculated as the sum of forces in the region Iplus forces in the region II (Eq. 8):

Ff = FfI + FfII = [kfc · AI + kfe · SI

]I +

+[∫ ϑ2

ϑ1

kfc (ϑ) h (ϑ) rndϑ +∫ ϑ2

ϑ1

kfe (ϑ) rndϑ

]

II

Fr = FrI + FrII = [krc · AI + kre · SI

]I +

+[∫ ϑ2

ϑ1

krc (ϑ) h (ϑ) rndϑ +∫ ϑ2

ϑ1

kre (ϑ) rndϑ

]

II

Ft = FtI + FtII = [ktc · AI + kte · SI

]I +

+[∫ ϑ2

ϑ1

ktc (ϑ) h (ϑ) rndϑ +∫ ϑ2

ϑ1

kte (ϑ) rndϑ

]

II

(8)

508 Int J Adv Manuf Technol (2012) 62:505–515

The three force components are projected in theX-, Y-, and Z -directions, that is, the directions of theforces as measured by the dynamometer plate, whichis located underneath the tool holder. The forces soregistered correspond to the feed force (Fx) in the toolfeed direction, the cutting force (Fz) in the cuttingspeed direction, and the radial force (Fy) in the toolholder axis. Expressions for these forces are expressedin Eq. 9:

Fx = kfc AIsin (kr) + kfeSIsin (kr) +∫ ϑ2

ϑ1

kfc (ϑ) h (ϑ) rn ·

· sin (kr (ϑ)) dϑ +∫ ϑ2

ϑ1

kfe (ϑ) rnsin (kr (ϑ)) dϑ +

+ krc AIcos (kr)+kreSI cos (kr)+∫ ϑ2

ϑ1

krc (ϑ) h (ϑ) rn ·

· cos (kr (ϑ)) dϑ +∫ ϑ2

ϑ1

kre (ϑ) rn cos (kr (ϑ)) dϑ

Fy = kfc AIcos (kr)+kfeSIcos (kr)+∫ ϑ2

ϑ1

kfc (ϑ) h (ϑ) rn ·

· cos (kr (ϑ)) dϑ +∫ ϑ2

ϑ1

kfe (ϑ) rncos (kr (ϑ)) dϑ −

− krc AIsin (kr) − kreSIsin (kr) −∫ ϑ2

ϑ1

krc (ϑ) h (ϑ) rn ·

· sin (kr (ϑ)) dϑ −∫ ϑ2

ϑ1

kre (ϑ) rnsin (kr (ϑ)) dϑ

Fz = ktc AI + kteSI +∫ ϑ2

ϑ1

ktc (ϑ) h (ϑ) rndϑ

+∫ ϑ2

ϑ1

kte (ϑ) rndϑ (9)

where kr (ϑ) = ϑ − π2

This model can be used to predict cutting forcesusing as input data the cutting force coefficients, cuttingconditions, and tool geometry. Hence, to determinethe cutting coefficients for a particular workpiece–toolcombination is necessary.

2 Experimental work

Here, the mechanistic model is used in inverse wayto calculate the cutting force coefficients. For doingit, some series of experimental tests called character-ization tests were carried out. Each series consistedin cylindrical turning operations at different feed rateswhile keeping constant depth of cut and cutting speed.A new cutting edge was used for each test in order toavoid the undesirable effects originated by tool wear.The three components of cutting force were acquiredduring the operation. In order to obtain a more preciseestimation, each series was repeated three times andthe cutting force component average was calculated foreach feed rate. Figure 2 shows the specimen used inthe tests.

To expand the field of application of the model soas to be valid in a wider range of cutting conditions, tocarry out several series of experiments with differentcombinations of Vc and ap is required, so that influenceof these parameters may be analyzed and expressionsfor cutting specific coefficients can be found for makinghigher-quality predictions in a wider range of machin-ing conditions. Therefore, each series was reproducedvarying now cutting speed and depth of cut. Table 1shows the set of experiments. A total of 16 series wascarried out covering a extensive interval of cuttingspeed (300 to 750 m/min), depth of cut (1 to 2.5 mm),and feed rate (0.1 to 0.25 mm/rev). Coolant at 7% ofconcentration was used for the tests.

Fig. 2 Image of the specimenused

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Table 1 Cutting conditions for characterization tests

Vc fn1 fn2 fn3 fn4 ap Series(m/min) (mm/rev) (mm)

1 Series 1750 0.1 0.15 0.2 0.25 1.5 Series 2

2 Series 32.5 Series 41 Series 5

600 0.1 0.15 0.2 0.25 1.5 Series 62 Series 72.5 Series 81 Series 9

450 0.1 0.15 0.2 0.25 1.5 Series 102 Series 112.5 Series 121 Series 13

300 0.1 0.15 0.2 0.25 1.5 Series 142 Series 152.5 Series 16

The turning experiments were carried out in a CNClathe featured by 5,000 rpm and 15 kW. The workpiecematerial was AISI 303 austenitic stainless steel. Table 2shows the chemical composition for this steel. Cuttingforces were measured using a Kistler model 9121 three-component piezoelectric dynamometer. Signals wereprocessed and amplified using a Kistler model 5070Acharge amplifier.

The cutting speed values chosen are adequate forhigh productivity environments. These cutting speedsare out of the range recommended by tool manufac-turers that recommend values lower than 350 m/min.However, as indicated in the research developed byFernández-Abia [4], AISI 303 austenitic stainless steelundergoes a significant change in its behavior abovecutting speed of 450 m/min which favors the machiningoperation. Values chosen for depth of cut and feed ratesatisfy the condition of ap > rn (1 − cos kr) and fn ≤2rn · sin kr1, which assures that cutting action takes placeon the side cutting edge and nose radius edge, whereasthe end cutting edge is not engaged in the cutting.

All tests were performed using multi-layer coated(TiCN − Al2O3 − TiN) cemented carbide cutting tools.The insert geometry was TNMG 160408-MM mountedon a DTGNL 2020K16 tool holder. This tool is recom-mended for general turning of austenitic stainless steels.The main geometric features of the tool are indicated inTable 3.

Table 3 Tool geometry used for the characterization tests

Property Value

Rake angle −6◦Inclination angle −6◦Side cutting edge angle 91◦End cutting edge angle 29◦Nose radius 0.8 mm

3 Determination of cutting force coefficients

Each series of the tests was used to calculate the cuttingforce coefficients (kic and kie) by applying the algo-rithm of least squares fitting. Later on, the coefficientsobtained for the different series at different values ofap and Vc were analyzed using statistical techniquesin order to state dependency of these coefficients withdepth of cut and cutting speed.

With the purpose of calculating the coefficients, theexpressions in Eq. 9 were rewritten to individualize theforce due to material shearing effect and the force dueto ploughing or rubbing effect. Besides, if cutting coeffi-cients are considered independents of ϑ , they can bepulled out of the integral, given place to the expressionsin Eq. 10:

Fx = kfe · Gfx + kre · Grx + kfc · Z fx + krc · Zrx

Fy = kfe · Gfy + kre · Gry + kfc · Z fy + krc · Zry

Fz = kte · Gtz + ktc · Z tz (10)

where

Gfx = sin kr SI +∫ ϑ1

ϑ2

sin(ϑ − π

2

)rn dϑ

Gfy = cos kr SI +∫ ϑ1

ϑ2

cos(ϑ − π

2

)rn dϑ

Grx = cos kr SI +∫ ϑ1

ϑ2

cos(ϑ − π

2

)rn dϑ

Gry = − sin kr SI −∫ ϑ1

ϑ2

cos(ϑ − π

2

)rn dϑ

Z fx = sin kr AI +∫ ϑ1

ϑ2

sin(ϑ − π

2

)h (ϑ) rn dϑ

Z fy = cos kr AI +∫ ϑ1

ϑ2

cos(ϑ − π

2

)h (ϑ) rn dϑ

Zrx = cos kr AI +∫ ϑ1

ϑ2

cos(ϑ − π

2

)h (ϑ) rn dϑ

Table 2 Chemical composition of AISI 303 workpiece material

AISI C P S Si Mn Cr Ni Mo Ti N Cu Fe

303 0.050 0.033 0.273 0.365 1.776 17.773 8.783 0.271 0.003 0.041 0.273 70.392

510 Int J Adv Manuf Technol (2012) 62:505–515

Zry = − sin kr AI −∫ ϑ1

ϑ2

sin(ϑ − π

2

)h (ϑ) rn dϑ

Gtz = SI +∫ ϑ1

ϑ2

rn dϑ

Gtz = AI +∫ ϑ1

ϑ2

h (ϑ) rn dϑ (11)

where ϑ1 and ϑ2 are the limits of integration (see Eq. 6).Using the expressions in Eqs. 10 and 11 and once

the Gij and Zij terms have been calculated, which only

a

b

c

Fig. 3 Evolution of cutting forces with feed rate (Vc =750 m/min): a Fx, b Fy, c Fz

Table 4 Analysis of variance for the cutting force coefficients

Source P value (95%) R2 R2adj

Vc ap

kfc 0.005 0.766 0.75 0.58krc 0.000 0.725 0.91 0.85ktc 0.000 0.804 0.86 0.76kfe 0.064 0.299 0.62 0.37kre 0.009 0.006 0.84 0.73kte 0.603 0.418 0.36 0.00

depends on tool geometry and cutting conditions, thesystem of Eq. 12 is solved in order to determine thecutting force coefficients for each series of experiments.⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

Fx1...

Fxm

Fy1...

Fym

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

exp1..16

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

Gfx1 Grx1 Z fx1 Zrx1...

......

...

Gfxm Grxm Z fxm Zrxm

Gfy1 Gry1 Z fy1 Zry1...

......

...

Gfym Grym Z fym Zrym

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

1..16

×

⎡

⎢⎢⎣

kfe

kre

kfc

krc

⎤

⎥⎥⎦

1..16

⎡

⎢⎣

Fz1...

Fz4

⎤

⎥⎦

exp1..16

=⎡

⎢⎣

Gtz1 Z tz2...

...

Gtz4 Z tz4

⎤

⎥⎦

1..16

×[

kte

ktc

]

1..16(12)

a

b

Fig. 4 Evolution of shearing coefficients with a ap and b Vc

Int J Adv Manuf Technol (2012) 62:505–515 511

Table 5 Relationships andtheir analysis of variance forthe shearing coefficients andedge coefficients

Relationships for shearing R2 (%) R2adj (%) P value Edge

coefficients coefficients

kfc = 158 + 0.132 · Vc + 5.6 · ap 77.9 73.6 0.014 kfe = 115.280krc = 169 − 0.150 · Vc + 5.78 · ap 87.0 83.5 0.000 kre = 8.674ktc = 1,537 − 0.533 · Vc − 21.6 · ap 87.9 84.5 0.000 kte = 78.835

where Fx1, . . . , Fxm; Fy1, . . . , Fym; Fz1, . . . , Fzm are theaverage values for cutting force components measuredduring the test series and m is the number of levels forfeed rate ( fn1, . . . , fnm) in each series. As aforemen-tioned, variability of cutting coefficients with regardto depth of cut and cutting speed was studied usingstatistical techniques.

4 Results and discussion

Figure 3 shows the relation between cutting forcesand feed rate for different depths of cut at a cuttingspeed of 750 m/min. Force components show a linear

relationship with feed rate. This observation agrees wellwith the proposed mechanistic model.

If the cutting force component expressions in themechanistic model are examined (Eqs. 10 and 11) andthey are compared with the curves in Fig. 3, it can be de-duced that friction force at the straight zone of cuttingedge (zone I) is equivalent to the force extrapolatedat a feed rate equal to zero. The slope of the curverepresents the friction force associated to the curvedzone of cutting edge (zone II) and the shearing forcesin zones I and II. If the effect of tool nose radius isnot taken into account, the slope of the curve wouldrepresent the full effect of the shearing force and thefriction force at feed rate equal to zero [2].

Table 6 Results of the first stage of validation (replication with the same conditions)

Test Cutting parameters Fx(N) Fy(N) Fz(N)

Vc ap fn Exp Estim %Err Exp Estim %Err Exp Estim %Err(m/min) (mm) (mm/rev)

1.1 750 1 0.1 126.60 129.07 1.95 118.53 121.16 2.22 225.44 234.94 4.211.2 750 1 0.2 143.16 148.05 3.41 143.83 145.88 1.42 362.79 365.10 0.641.3 750 1.5 0.15 210.94 216.27 2.53 139.82 141.94 1.51 418.94 419.26 0.081.4 750 2 0.2 302.42 317.38 4.95 176.81 167.59 −5.22 662.49 658.12 −0.661.5 750 2.5 0.25 440.38 432.8 −1.72 208.34 198.62 −4.66 978.31 950.07 −2.891.6 600 1.5 0.15 210.19 210.73 0.25 142.95 145.13 1.52 445.49 437.96 −1.691.7 600 2 0.1 273.60 266.77 −2.50 144.96 138.47 −4.48 445.08 437.26 −1.761.8 600 2.5 0.25 399.58 418.32 4.69 219.73 209.42 −4.69 990.40 1002.14 1.191.9 450 1 0.15 128.49 130.45 1.53 130.32 136.25 4.55 314.65 324.12 3.011.10 450 1.5 0.2 208.28 217.41 4.38 160.81 164.87 2.52 557.23 563.31 1.091.11 450 2 0.2 287.12 298.40 3.93 179.59 180.48 0.50 717.28 724.70 1.031.12 450 2.5 0.2 377.61 379.96 0.62 198.37 196.67 −0.86 887.75 883.93 −0.431.13 300 1.5 0.1 191.02 189.19 −0.96 129.84 134.47 3.57 349.05 365.39 4.681.14 300 1.5 0.2 212.85 209.90 −1.38 167.87 169.08 0.72 563.15 588.62 4.521.15 300 2 0.15 273.22 273.32 0.04 161.04 165.68 2.88 597.98 612.30 2.391.16 300 2.5 0.15 353.57 347.41 −1.74 175.65 180.30 2.64 780.60 747.60 −4.23

512 Int J Adv Manuf Technol (2012) 62:505–515

The influence of depth of cut and cutting speed onthe cutting forces coefficients was analyzed by meansof analysis of variance. Table 4 shows the P valueobtained for the six coefficients at a 95% confidenceinterval.

The results show that only cutting speed is significantfor the three shearing coefficients (kfc, krc, and ktc);that is, the P value is lower than 0.05. Nevertheless,although the statistical analysis indicates independenceof the shearing coefficients with regard to the ap para-meter, a model that includes both Vc and ap variableswas considered more appropriate. The explicit depen-dence of the coefficients with cutting speed and depthof cut assures the sensibility of the model to changesin these two cutting parameters. In many cases, therange of variation for a variable is not wider enoughto detect variations in the response, and therefore, the

significance analysis results negative. However, this factdoes not mean that the response will be not modifiedby the variable when the variation range increases;the model that includes the variable will have morepredictive capacity outside of the tested range of condi-tions [9].

For the kfe and kte edge coefficients, all terms areinsignificant. Therefore, these coefficients were consid-ered constant. For the kre edge coefficient, ap and Vc

parameters are both significant. However, to outlinesimilar expressions for the three friction coefficientswas considered more correct. If nature of the cut-ting specific coefficients is analyzed, the shearing andfriction coefficients pick up information of the tool–part binomial, and therefore, they should be simi-lar at the three radial, axial, and tangential direc-tions. Several authors use variable shearing coefficients

Table 7 Results of the second stage of validation (in the range of conditions and the same tool geometry but different cuttingparameters)

Test Cutting parameters Fx(N) Fy(N) Fz(N)

Vc ap fn Exp Estim %Err Exp Estim %Err Exp Estim %Err(m/min) (mm) (mm/rev)

2.1 375 2.5 0.13 325.75 340.67 4.58 171.44 173.15 1.00 659.64 664.22 0.702.2 375 2.5 0.17 344.44 358.74 4.15 189.79 192.10 1.22 794.69 800.80 0.772.3 375 2.5 0.23 364.12 385.62 5.91 212.67 221.24 4.03 973.14 1,009.55 3.742.4 375 2.5 0.3 396.03 416.58 5.19 241.04 256.40 6.37 1,146.94 1,259.62 9.822.5 675 1 0.13 143.37 131.10 −8.56 141.03 131.21 −6.97 286.52 278.34 −2.862.6 675 1 0.17 152.48 138.07 −9.45 151.34 141.28 −6.65 339.41 332.18 −2.132.7 675 1 0.23 158.72 148.47 −6.46 166.53 157.12 −5.65 415.28 416.44 0.282.8 675 1 0.3 172.53 160.49 −6.98 184.67 176.83 −4.25 499.06 520.62 4.322.9 600 2.25 0.13 294.99 314.40 6.58 168.91 158.94 −5.90 569.66 568.58 −0.192.10 600 2.25 0.17 310.22 333.38 7.47 184.41 174.22 −5.53 680.70 681.44 0.112.11 600 2.25 0.23 338.25 361.75 6.95 203.03 197.88 −2.53 822.82 854.28 3.822.12 600 2.25 0.3 375.30 394.65 5.16 233.02 226.74 −2.69 981.26 1,061.86 8.212.13 750 1.75 0.13 235.68 245.52 4.18 158.28 144.33 −8.81 447.20 437.00 −2.282.14 750 1.75 0.17 247.10 261.13 5.68 170.28 156.46 −8.12 528.57 521.30 −1.372.15 750 1.75 0.23 270.66 284.53 5.12 191.71 175.43 −8.49 658.99 651.08 −1.202.16 750 1.75 0.3 302.66 311.76 3.01 217.72 198.85 −8.67 793.13 808.08 1.892.17 500 2.2 0.13 298.85 302.57 1.25 168.25 161.02 −4.30 570.23 572.94 0.472.18 500 2.2 0.17 315.11 319.64 1.44 184.88 177.05 −4.24 686.91 688.53 0.242.19 500 2.2 0.23 337.91 345.07 2.12 202.88 201.82 −0.52 834.03 865.61 3.792.20 500 2.2 0.3 366.53 374.46 2.16 227.65 231.94 1.88 1,002.82 1,078.42 7.54

Int J Adv Manuf Technol (2012) 62:505–515 513

whereas friction coefficients are considered con-stant [7, 8].

Therefore, in accordance to the performed analy-sis, the shearing coefficients were considered variablewith regard to ap and Vc parameters; however, frictioncoefficients were considered as constant, and their val-ues are the average of the ones obtained in the testseries.

Mathematical models for shearing coefficients weredeveloped using regression analysis, and significance ofthese models was verified using analysis of variance.Firstly, evolution of the three shearing coefficients isrepresented graphically with respect to Vc and ap (seeFig. 4). In this figure, it can be observed that variationof three shearing forces coefficients with cutting speedand depth of cut fits well to a linear model. Least squareestimation method was used to develop the linearmodels.

The models for the shearing coefficients and the in-dicators of analysis of variance are included in Table 5.The P value indicates that linear models are statisticallysignificant for the three shearing cutting coefficients.Correlation coefficients (R2 and R2

adj) are higher than70%, which means the relationship between shearing

coefficients and cutting speed depth of cut has beenadequately represented in the proposed models.

The values for the edge coefficients are also includedin Table 5.

4.1 Validation of the model

A series of experimental tests were carried out to studythe consistency of the proposed models. The expres-sions in Table 5 for the specific cutting coefficients wereintroduced in the corresponding models together withcutting conditions and tool geometric features. So, thevalues for Fx, Fy, and Fz forces are estimated. Theseforce values are then compared with the average valueof cutting forces as measured in the experimental testswith the dynamometer plate.

The validation tests were carried out in four stages:

1. First stage: The same experiments used to calculatethe specific cutting coefficients were replicated toestimate the fitting level.

2. Second stage: A series of tests different to the onesused to determine the specific cutting coefficients

Table 8 Results of the third stage of validation (out of range of conditions and the same tool geometry)

Test Cutting parameters Fx(N) Fy(N) Fz(N)

Vc ap fn Exp Estim %Err Exp Estim %Err Exp Estim %Err(m/min) (mm) (mm/rev)

3.1 250 2.5 0.13 358.65 334.55 −6.72 185.40 177.89 −4.05 680.71 686.30 0.823.2 250 2.5 0.17 382.27 350.67 −8.27 208.72 198.27 −5.01 835.59 829.87 −0.683.3 250 2.5 0.23 403.84 374.56 −7.25 241.53 229.53 −4.97 1,043.63 1,049.32 0.553.4 250 2.5 0.3 418.95 401.92 −4.07 263.38 267.13 1.43 1,255.74 1,312.20 4.503.5 850 1 0.13 145.78 135.16 −7.28 135.48 129.67 −4.29 280.44 265.58 −5.303.6 850 1 0.17 153.61 143.48 −6.60 146.04 139.29 −4.62 332.39 315.21 −5.173.7 850 1 0.23 169.82 155.99 −8.14 164.44 154.50 −6.04 408.12 392.88 −3.733.8 850 1 0.3 183.63 170.63 −7.08 190.26 173.55 −8.78 497.24 488.89 −1.683.9 775 0.8 0.13 106.00 103.42 −2.43 123.33 126.89 2.89 221.38 227.09 2.583.10 775 0.8 0.17 109.65 109.03 −0.56 131.03 136.18 3.93 260.81 269.86 3.473.11 775 0.8 0.23 117.41 117.44 0.02 145.05 150.86 4.01 321.94 337.38 4.803.12 775 0.8 0.3 128.75 127.23 −1.19 162.80 169.26 3.97 392.16 421.78 7.55

514 Int J Adv Manuf Technol (2012) 62:505–515

were carried out. These additional tests wereperformed in the same range of cutting conditionsusing the same tool geometry.

3. Third stage: A series of tests out of the range ofcutting conditions used in the characterization testswere carried out using the same tool geometry.

4. Fourth stage: A series of tests both in and out of therange of cutting conditions were performed, but inthis case, the tool geometry was different (positionangle and tip radius were changed).

Tables 6, 7, 8, and 9 show the comparison ofthe forces obtained experimentally with the estimatedforces using the proposed model. The “%Error” col-

umn represents the relative error calculated using thefunction in Eq. 13:

%Error =(

Fpred − Fexp

Fexp

)· 100 (13)

In the first validation stage, relative errors lower than5% were obtained for the three components of thecutting force with regard to the values estimated forthe proposed model. This fact indicates that the pro-posed model provides a good fitting. Errors obtainedin the second and third stage were around 8%, whichindicates that the model provides a good capacity forextrapolation. In the fourth validation stage, the model

Table 9 Results of the forth stage of validation (in and out of range of conditions and different tool geometry)

Test Tool Cutting parameters Fx(N) Fy(N) Fz(N)

kr rn Vc ap fn Exp Estim %Err Exp Estim %Err Exp Estim %Err(deg) (mm) (m/min) (mm) (mm/rev)

4.1 60 0.8 450 1.5 0.1 174.10 178.10 2.30 195.87 204.75 4.53 341.00 361.99 6.164.2 60 0.8 450 1.5 0.15 178.14 186.03 4.43 216.73 224.48 3.58 447.78 461.79 3.134.3 60 0.8 450 1.5 0.2 181.19 193.65 6.88 235.09 244.52 4.01 544.50 563.13 3.424.4 60 0.8 450 1.5 0.25 184.14 200.94 9.13 248.54 264.92 6.59 618.45 666.28 7.734.5 60 0.8 600 2 0.1 244.23 246.12 0.77 254.80 252.17 −1.03 436.34 452.47 3.704.6 60 0.8 600 2 0.15 248.81 260.52 4.70 276.08 276.85 0.28 563.84 574.62 1.914.7 60 0.8 600 2 0.2 256.48 274.63 7.08 299.09 301.88 0.93 683.04 698.20 2.224.8 60 0.8 600 2 0.25 265.58 288.45 8.61 319.64 327.31 2.40 780.99 823.46 5.444.9 91 1.2 450 1.5 0.1 174.41 183.38 5.14 163.24 185.33 13.54 329.75 371.01 12.514.10 91 1.2 450 1.5 0.15 186.35 193.70 3.95 183.84 203.07 10.46 440.07 474.39 7.804.11 91 1.2 450 1.5 0.2 199.18 203.94 2.39 200.82 221.34 10.22 550.08 580.61 5.554.12 91 1.2 450 1.5 0.25 207.46 214.09 3.19 214.79 240.15 11.80 643.61 689.80 7.184.13 91 1.2 600 2 0.1 250.07 257.52 2.98 176.15 193.21 9.68 435.44 455.19 4.544.14 91 1.2 600 2 0.15 266.69 276.18 3.56 195.57 212.75 8.79 571.56 580.66 1.594.15 91 1.2 600 2 0.2 281.05 294.80 4.89 213.56 232.84 9.03 700.61 708.76 1.164.16 91 1.2 600 2 0.25 300.67 313.39 4.23 230.97 253.52 9.76 821.66 839.63 2.194.17 60 1.2 525 2 0.13 221.90 246.94 11.28 271.78 295.32 8.66 518.54 550.74 6.214.18 60 1.2 525 2 0.17 228.99 256.59 12.05 294.05 315.69 7.36 622.33 652.07 4.784.19 60 1.2 525 2 0.23 236.71 270.75 14.38 320.98 346.62 7.99 757.92 805.66 6.304.20 60 1.2 525 2 0.3 248.69 286.79 15.32 350.44 383.32 9.38 888.85 987.59 11.114.21 60 1.2 800 1.5 0.13 181.34 191.98 5.87 235.46 241.74 2.66 395.29 399.50 1.074.22 60 1.2 800 1.5 0.17 182.34 201.74 10.64 247.88 257.26 3.78 466.76 468.12 0.294.23 60 1.2 800 1.5 0.23 192.84 216.15 12.09 272.91 280.99 2.96 568.28 572.48 0.744.24 60 1.2 800 1.5 0.3 212.78 232.59 9.31 307.30 309.41 0.69 685.53 696.67 1.63

Int J Adv Manuf Technol (2012) 62:505–515 515

provides good predictions for those tests where onlyone of tool position angle or tool tip radius (tests 4.1 to4.16) were changed. Tests where both tool parameterswere changed at the same time (4.17 to 4.24) providerelative errors significantly higher, where the maximumerrors corresponds to the Fx component (15%) andlower for the other two Fy and Fz components.

5 Conclusions

A mechanistic model for cutting forces in turning withnose radius tools has been proposed. In our approach,the effect of the edge force due to the rounded cuttingedge is considered. This model of forces requires areduced number of machining tests for determiningthe cutting specific coefficients for a particular workmaterial–cutting tool pair. The expressions for the cut-ting specific coefficients have been deduced for theturning of AISI 303 austenitic stainless steel at highcutting speeds (350 to 750 m/min). Therefore, the val-ues of the coefficients so obtained are adequate forhigh productivity machining. The expressions derivedfor shearing coefficients are variable with depth of cutand cutting speed; this consideration allows the exten-sion of the proposed model to other cutting conditionsrange. The edge force coefficients have been consid-ered constant since they are largely unaffected by depthof cut and cutting speed, which enable to simplify themodel.

From the validation tests, it may be deduced thatthe model provides a very good capacity of prediction.The set of validation tests demonstrates the validityof the model when machining both in and out of thecutting conditions range used for developing the pro-posed model, with errors lower than 8% in the worsecase. When changing the tool geometry, the errors forthe predicted values are worse but always acceptable,which assures a good estimation prediction even inchanging conditions.

Acknowledgements We gratefully acknowledge the financialsupport provided by the Spanish R2-TAF initiative through aresearcher mobility grant.

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