a methodology for practical cutting force evaluation

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A METHODOLOGY FOR PRACTICAL CUTTING FORCE EVALUATION

BASED ON THE ENERGY SPENT IN THE CUTTING SYSTEM

Viktor P. Astakhov1

and Xinran Xiao2

1General Motors Business Unit of PSMi, Saline MI, USA2Michigan State University, Department of Mechanical Engineering, East Lansing,Michigan, USA

& This paper presents a methodology for practical estimation of cutting force and cutting power.

Based on a previously proposed definition, the power spent in metal cutting is the summation of

four components: the power spent on the plastic deformation of the layer being removed by both

major and minor cutting edges, the power spent on the tool-chip interface, the power spent onthe tool-workpiece interface, and the power spent in the formation of new surfaces (cohesive energy).

This paper provides a complete list of mathematical expressions needed for the calculation of each

energy mode and demonstrates their utility for turning operation of two work materials: AISI bear-

ing steel E52100 and aerospace aluminum alloy 2024 T6. The calculated cutting forces were in

fairly good agreement with the experimental results. Energy partition in the cutting system and rela-tive impact of the parameters of the machining regime are discussed. For the first time, a simple and

practical method is available for the calculation of the total cutting power and the evaluation of the

relative contributions of each individual component of the cutting system.

Keywords Energy partition, force, metal cutting, power

INTRODUCTION

Machining is one of the oldest processes for precise shaping of com-

ponents in the manufacturing industry. It is estimated that 15%

of thevalue of all mechanical components manufactured worldwide is derivedfrom machining operations (Merchant, 1998). However, despite its obvi-ous economic and technical importance, machining remains to be oneof the least understood manufacturing operations due to low predictivecapability of the existing machining models (Usui and Shirakashi, 1982;Usui, 1988).

Address correspondence to Viktor P. Astakhov, 1255 Beach St., Saline, MI 48176, USA. E-mail:

[email protected]

Machining Science and Technology, 12:325347Copyright# 2008 Taylor & Francis Group, LLCISSN: 1091-0344 print/1532-2483 onlineDOI: 10.1080/10910340802306017


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One of the most important yet least understood operation parameters ofa machining operation is the cutting force. In general, this force is thoughtof as a 3D vector that is represented by three components, namely, the

power component, the radial component and the axial component in thetool coordinate system as shown in Figure 1a (Zorev, 1966). Of these threecomponents, the greatest normally is the power component, which is oftencalled the cutting force. This simplification will be used through the body ofthis paper. As this force is of high importance, one might think that theor-etical and experimental methods for its determination have been developedand are thus available in the literature. Unfortunately, this is not the case.

When it comes to a possibility of theoretical determination, the foun-dation of the force and energy calculations in metal cutting is based uponthe oversimplified orthogonal force model known as Merchants force cir-

cle diagram or a condensed force diagram (Komanduri, 1993; Merchant,2003) shown in Figure 1b. In this figure, the total cutting forceRis resolvedinto the tool face-chip friction force F and normal force N. The angle lbetween F and N is thus the friction angle. The force R is also resolvedalong the shear plane into the shear(ing) force, Fswhich, in Merchantsopinion, is responsible for the work expended in shearing the metal, andinto normal forceFn, which exerts a compressive stress on the shear plane.Force Ris also resolved along the direction of tool motion into Fc, termedby Merchant as the cutting force, and into FT, the thrust force.

The determination of the cutting force is based upon the calculation of

the shearing force, Fs. Ernst and Merchant in 1941 (Ernst and Merchant,1941) proposed the following equation to calculate this force

Fs syAc

sinu 1

FIGURE 1 The force system in cutting: (a) Turning, (b) Orthogonal force model proposed by Merchant.

326 V. P. Astakhov and X. Xiao


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where syis the shear strength of the work material, uis the shear angle, Acis the area of shearing (the uncut chip area equal to the product of theuncut chip thickness and the uncut chip width).

According to Ernst and Merchant, the work material deforms whenthe stress on the shear plane reaches the shear strength of the workmaterial. Later researchers published a great number of papers showingthat sy should be thought of as the shear flow stress, which is somehowhigher than the yield strength of the work material depending on parti-cular cutting conditions. Until today, this stress remains as the only rel-evant characteristic of the work material in terms of its resistance tocutting.

It follows from Figure 1b that

Fc Fscosl ccosu l c

2

and combining Eqs. (1) and (2), one can obtain

Fc syAccosl c

sinu cosu l c 3

The cutting power Pcthen is calculated as

Pc Fcn 4

This power dictates the energy required for cutting, cutting tempera-tures, plastic deformation of the work material, machining residual stressand other parameters.

However, everyday practice of machining shows that these considera-tions do not match the reality even to the first approximation. For example,machining of medium carbon steel AISI 1045 (the ultimate tensile strengthr

R 655 MPa, the tensile yield strengthr

y0:2 375 MPa) resulted in muchlower total cutting force (Figure 1.8 in (Astakhov, 2006)), greater tool life,lower required power, cutting temperature, machining residual stressesthan those obtained in the machining of stainless steel AISI 316L(rR 517 MPa;ry0:2 218 MPa) (Outeiro, 2003). The prime reason is thatany kind of strength of the work material in terms of its characteristic stres-ses cannot be considered alone without corresponding strains, which deter-mine the energy spent in deformation of the work material (Astakhov,1998=1999, 2004). Only when the stress and corresponding strain areknown, other parameters-outcomes of the metal cutting process can be

calculated (Astakhov, 2004).

Practical Cutting Force Evaluation Methodology 327


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When it comes to experimental determination of the cutting force,there are at least two problems:

. First and foremost is that the cutting force cannot be measured withreasonable accuracy although this fact has never been honestly admittedby the specialists in this field. To appreciate the issue, one should con-sider the results of the joint program conducted by College Internationalpour la Recherche en Productique The International Academy forProduction Engineering, http://www.cirp.net (CIRP) and NationalInstitute of Standards and Technology (NIST) to measure the cuttingforce in the simplest case of orthogonal cutting (Ivester, 2004). Theexperiments were carefully prepared (the same batches of the work-piece (steel AISI 1045), tools, etc.) under the supervision of NationalInstitute of Standards and Technology (NIST) and replicated at fourdifferent most advanced metal cutting laboratories in the world. Inter-estingly, although extraordinary care was taken while performing theseexperiments, there was significant variation (up to 50%) in the measuredcutting force across these four laboratories. If less care is taken and nolaboratory conditions are available then the accuracy of cutting forcemeasurement would be much worse.

. Second, many tool and cutting inserts manufacturers (not to mentionmanufacturing companies) do not have adequate dynamometric

equipment to measure the cutting force. Many dynamometers usedin this field are not properly calibrated because the known literaturesources did not present proper experimental methodology for cuttingforce measurements using piezoelectric dynamometers (Astakhov andShvets, 2001).

Therefore, to make practical calculations of the cutting force, anotherapproach has to be sought. The objective of this paper is to present such anapproach.

PROPOSED METHODOLOGY

The proposed methodology is based on the definition of the metal cut-ting process proposed by Astakhov (1998=1999) and the model of energypartition in the metal cutting system based on this definition (Figure 2.1in Astakhov, 2006). According to this model, the power balance in thecutting system can be written as

PcFcn

Ppd PfRPfFPch Pmnce 5



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from which the cutting force is calculated

Fc

PpdPfRPfFPch Pmnce

n 6

wherePpdis the power spent on the plastic deformation of the layer beingremoved,PfRis the power spent on the tool-chip interface,PfF is the powerspent on the tool-workpiece interface, Pch is the power spent in theformation of new surfaces,Pmnceis the energy spent due to the combinedinfluence of the minor cutting edge.

As the proposed methodology is based upon the determination of thechip compression ratio, Figure 2 shows the definition of the chip com-pression ration as the ration of the chip thickness and uncut chip thickness

(Zorev, 1966). The simple methods for experimental determination of thechip compression ratio are presented elsewhere (Astakhov and Shvets,2004). Note that the chip compression ratio is reciprocal to the chip ratioused in some literature sources (Merchant, 1945; Shaw, 1984).

Plastic Deformation

The power spent on the plastic deformation of the layer beingremoved, Ppd, can be calculated from the chip compression ratio and

parameters of the deformation curve of the work material as follows

FIGURE 2 Definition of the chip compression ratio.



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(Astakhov, 2004, 2006)

PpdK1:15ln fn1

n1 nAw 7

whereKis the strength coefficient (N=m2) andnis the hardening exponentof the work material, f is the chip compression ratio (Astakhov, 2004,2006), Awis uncut chip cross-sectional area (m

2)

Awdwf 8

dwis the depth of cut (m), fis the cutting feed per revolution (m=rev).The practical methods of experimental determination of the chip com-

pression ratio f have been discussed for various machining operation byAstakhov (2004, 2006).

Friction at the Tool-Chip Interface

The power spent due to friction at the tool-chip interface is cal-culated as

PfRsclcb1Tn

f 9

where sc0:28rR is the average shear stress at the tool-chip contact(N=m2) (Astakhov, 2006), rR is the ultimate tensile strength of the workmaterial (N=m2),lcis the tool-chip contact length (m),b1Tis the true chip

width (m).The tool-chip contact length is calculated as (Astakhov, 2006)

lct1Tf1:5 10

where t1T

is the true uncut chip thickness (m).The true uncut chip thickness and the true chip width depend on the

configuration of the projection of the cutting edge into the main referenceplain. Formulae to calculate t1T and b1T for various configuration havebeen presented by Astakhov (1998=1999, 2006). The most common caseof machining is when the cutting insert with the tool cutting edge anglejr and the tool minor cutting edge angle jr1 is made with a noseradiusrnand set so that the depth of cutdwis greater than the nose radiusFigure 3. If the following relationships are justified

dwrn1cos jr; f 2rnsin jr1 11



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then the formulas for calculation of t1T and b1Tare as follows

t1T f

c1sin arctan

c1

1e11cos jr cotjre1sinjrg1 12

and

b1T c1dw

sin arctan c11e11cosjr cotjre1sin jrg113

where

g1 f

2rn; e1

rn

dw; c1 1e11

ffiffiffiffiffiffiffiffiffiffiffiffiffi1g1

p 14

Friction at the Tool-Workpiece Interface

The power spent due to friction at the tool-workpiece interface is calcu-lated as

PfF FfFn 15

where FfFis the friction force on the toolworkpiece interface

FfF 0:625syqcelac ffiffiffiffiffiffiffiffiffiffiffiffiBr

sin ar 16

FIGURE 3 Visualization of terms used in the considered case of machining: (a) general turning terms,(b) tool geometry terms.



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wheresyis the shear strength of the tool material (N=m2),qceis the radius

of the cutting edge (m), a is the normal flank angle (deg),lacis the lengthof the active part of the cutting edge (the length of the cutting edge

engaged in cutting) (m). In the considered case (Figure 3)

lacrn 0:018jrrncosjr

sinjr

17

Bris the Briks similarity criterion (Astakhov, 1998=1999, 2006),

Br cos c

fsin c 18

where c is the normal rake angle (deg).

Formation of New Surfaces

The power spent in the formation of new surfaces Pch is calculated asthe product of energy required for the formation of one shear plane andthe number of shear planes formed per second, i.e.,

PchEfrfcf 19

where fcf is the frequency of chip formation, i.e., the number of shearplanes formed per second, Efr is the energy of fracture per a shear oneshear plane.

The frequency of chip formation determines how many shearplanes form per second of machining time. This frequency dependsprimarily on the work material and the cutting speed as discussed by

Astakhov (1998=1999). Figure 4 provides some data for common workmaterials.

The work of fracture per a shear plane is

Efr Ech Afr 20

where Ech is the cohesive energy (J=m2) (Shet and Chandra, 2002), Afr is

the area of fracture (m2).The area of fracture is the area of the shear plane determined as

Afr Lsh b1T 21



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where the length of the shear plane Lsh calculates as

Lsh t1T

sin arctan Br22

Combined Influence of the Minor Cutting Edge

The influence of the minor cutting edge (Figure 3b) on the cuttingforce and power consumption is seldom considered in the literature onmetal cutting. At best, the influence of the tool minor cutting edge anglejr1 is mentioned in the consideration of the theoretical roughness of themachined surface or geometric component of roughness (Shaw, 1984;Stephenson, 1996). In the authors opinion, probably the term minormisled many researchers in the field causing a common perception that thiscutting edge does not affect the cutting process to any noticeable degree.

Everyday practice of machining and even simple observation of the chip

formed in common machining operation show that the chip side formedby the minor cutting edge is always more deformed and has a darker coloras clearly seen in Figure 5. Zorev (1966) provided a detailed analysis of thechip formation by the minor cutting edge. Zorev studied the velocity hodo-graph, associated plastic deformation and flows in this region. Using theresults of this study, one can visualize the chip cross-sectional area cut bythe minor cutting edge with the help of Figure 6. Figure 6a shows a hypo-thetical single-point cutting tool having jr1 90

, i.e., practically no minorcutting edge. Figure 6b show the cross-sectional area ABC of a tooth ofthe surface profile left after this surface was machined by this tool. Real cut-

ting tools have the minor cutting edge with jr190

so that the surface

FIGURE 4 Effect of the cutting speed on the frequency of chip formation.



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FIGURE 5 The chip side cut by the minor cutting edge is always more deformed and has darker color.

FIGURE 6 The cross-sectional area of the chip cut by the minor cutting edge: (a) hypothetic tool

having a 90 tool cutting edge angle of the minor cutting edge, (b) the cross-section of the chip cut

by the minor cutting edge when the tool minor cutting edge angle is 90, (c) geometrical model to cal-

culate the cross-sectional area of the chip cut by the minor cutting edge when the tool minor cutting

edge angle is less than 90

.



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profile left by the cutting tool is ADC as shown in Figure 6c and the heighthmof this surface profile calculates as

hm fcotjrcotjr123

Then, the part ABC shown in Figure 6c is cut by the minor cutting edge.According to Zorev (1966) the contribution of the cutting and defor-

mation process on the minor cutting edge to the overall power spent in cut-ting depends on the tool minor cutting edge angle jr1 and on the cutting

FIGURE 7 Cutting tool used in the tests and its typical wear: (a) tool used, (b) working part, (c) typical

crater wear observed at moderated feed rates, (d) nose wear observed at high feed rates.



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feed. When the feed becomes significant, the minor cutting edge takes therole of the major cutting edge so that thread cutting is the case. In real cut-ting tools, the tool nose radius is always made to connect the major and

minor cutting edges. At moderated cutting feeds, the crater tool wear, com-monly found in machining wide variety of steels occurs as shown inFigure 7c, while when the feed rate becomes greater, wear of tool nose takesplace as shown in Figure 7d. This is because the energy spend due to cut-ting by the minor cutting edge becomes great so that the prime mode oftool wear changes from crater to nose wear.

Analysis of the experimental results obtained by Zorev on the assess-ment of the cutting energy (Zorev, 1966) and the comparison of the exper-imentally obtained powers associated with the cutting tool having varioustool minor cutting edge angles suggest that when the tool minor cutting

edge angle 30 jr1 45 then the total power should be increased by14%, when 15 jr1


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Shear strength sY 125MPa,K 0.220 GPa,n 0.16. Cohesive energyEch 8000 J=m

2 (Shet and Chandra, 2002). Test pieces were preparedas rings having dimensionsD d h 180 140 50.

3. Tool standard inserts SNMG 432-MF2 TP2500 Materials Group 4(SECO) installed into a tool holder 453120141 R1-1 (Sandvik) (Figure 7aand 7b). The tool-in-machine tool geometry parameters are: the toolcutting edge angle jr 45

, tool minor cutting edge angle jr1 45,

nose radius rn 1 mm, cutting edge radius is shown in Tables 1 and 2,normal flank angle a 7, the normal rake angle c 7. Each insertused in the tests was examined using a digital vision system at a magni-fication ofx25 for visual defects such as chipping and microcracks.

4. Cutting fluid (coolant) a synthetic coolant having 12% concentration.

Tables 1 and 2 provide a summary of the input parameters and examples oftheir values for the turning process of the two work materials.

Comparison

Tables 3 and 4 show the comparison of the calculated and experimentalresults for the steel and aluminum used in the test. At least three tests wereperformed for each combination of cutting conditions indicated in thetable using the guidelines for test preparation and evaluation of the

obtained results discussed by Astakhov (1997a). Fairly good agreementbetween the calculated and the experimental results confirms the adequacyof the proposed methodology. The major advantage of the proposed

TABLE 1 Values of Input Parameters for AISI Steel E52100

Variable Symbol Unit Value

Depth of cut dw m 3.00E03Cutting feed per revolution f m=rev 4.00E04

Strength coefficient K N=m2

1.34E09Hardening exponent n 2.50E01Chip compression ratio f 2.41E00Cutting speed n m=s 1.50E00Ultimate tensile strength of the work material rR N=m

2 8.50E08

Tool nose radius rn mm 1.00E03Tool cutting edge angle Kr rad 1.57E00Normal rake angle c rad 2.44E 01Radius of the cutting edge qce m 5.00E05Shear strength of the work material sy N=m

2 5.20E08Normal flank angle a rad 2.44E01Cohesive energy per unit fracture area Ech J=m

2 4.20E04

Frequency of chip formation fcf Hz 1.60E03



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TABLE 2 Values of Input Parameters for Al 2024 T6

Variable Symbol Unit Value

Depth of cut dw m 4.00E 03

Cutting feed per revolution f m=rev 4.50E 04Strength coefficient K N=m2 2.20E 08Hardening exponent n 1.60E 01Chip compression ratio f 4.96E 00Cutting speed n m=s 1.00E 00Ultimate tensile strength of the work material rR N=m

2 1.85E 08

Tool nose radius rn mm 1.00E 03Tool cutting edge angle Kr rad 0.00E 00Normal rake angle c rad 2.44E 01Radius of the cutting edge qce m 1.00E 04Shear strength of the work material sy N=m

2 1.25E 08

Normal flank angle a rad 2.44E 01

Cohesive energy per unit fracture area Ech J=m

2

8.00E 03Frequency of chip formation fcf Hz 1.00E 03

TABLE 3 Comparison of the Experimental and Calculated Results for AISI steel E52100

Cutting Speed(m=s)

Feed(mm=rev)

Depth ofcut (mm) CCR

Frequency(kHz)

Cutting forcecalculated through

the measured power (N)

Cutting forcecalculated using

the proposedmethodology (N)

1 1 0.20 3 3.12 1.0 1580 1608

2 1.5 0.20 3 2.54 1.6 1348 13893 3 0.20 3 2.03 3.2 1076 1104

4 4 0.20 3 1.67 4.7 873 9455 1.5 0.30 3 2.08 1.6 1562 16066 1.5 0.40 3 1.76 1.6 1640 16787 1.5 0.20 2 2.64 1.6 940 9988 1.5 0.20 5 2.52 1.6 2202 2256

TABLE 4 Comparison of the Experimental and Calculated Results for Aluminum 2024 T6

CuttingSpeed (m=s)

Feed(mm=rev)

Depth ofcut (mm) CCR

Frequency(kHz)

Cutting forcecalculated through

the measuredpower (N)

Cutting forcecalculated using

the proposedmethodology (N)

1 1 0.45 4 4.96 1.0 1223 12562 3 0.45 4 3.84 2.6 1038 10763 5 0.45 4 2.65 4.2 794 8544 7 0.45 4 1.92 5.8 601 6255 3 0.75 4 2.82 2.6 1393 14766 3 0.50 3 3.75 2.6 906 932

7 3 0.50 2 3.82 2.6 632 6588 3 0.30 4 3.94 2.6 787 834



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methodology is that it allows not only calculating the total power and thusthe cutting force, but also provides a valuable possibility to analyze theenergy partition in the cutting system.

It should be noted that the results presented in Tables 3 and 4 are validfor new tools (a fresh cutting edge of a cutting insert). Tool wear would sig-nificantly increase the cutting force. For steel E52100, VBB 0.45 mmcaused 2.02.5 times increase in the cutting force when no plastic loweringof the cutting edge (Astakhov, 2004) occurred (for cutting speeds 1 and1.5m=s) and 3.03.5 times increase when plastic lowering was the case(for cutting speeds 3 and 4 m=s).

COHESIVE ENERGY

The proposed methodology implies the total power supplied into thecutting system may be thought of as consisting of four components,namely: the power spent on the plastic deformation of the layer beingremoved, Ppd, the power spent on the tool-chip interface, PfR, the powerspent on the tool-workpiece interface, PfF, and the power spent in the for-mation of new surfaces Pch.

Although it is conclusively proven that metal cutting is the purposefulfracture of the layer being removed (Atkins and Mai, 1985; Astakhov,1998=1999; Atkins, 2003), the notions and theory of traditional fracturemechanics are not applicable in the metal cutting studies as this analysis

presupposes the existence of infinitely sharp crack leading to the singularcrack tip fields. In real materials, however, neither the sharpness of thecrack nor the stress levels near the crack tip region can be infinite. Further,for cracks along material interfaces, the crack tip will no longer be embed-ded in a square-root singular stress field leading to a condition that stressintensity may either be zero or infinity (Atkinson, 1979), As an alternativeapproach to this singularity driven fracture approach, Barrenblatt (1962)and Dugdale (1960) proposed the concept of the cohesive zone model.This model has evolved as a preferred method to analyze fracture problems

in monolithic and composite materials, as discussed by Shet and Chandra(2002). This is due to the fact that this method not only avoids the singu-larity but also can be easily implemented in analytical and numericalmethods of analysis.

Although a particular cohesive zone model for metal cutting is yet to beselected and justified among many available models (Shet and Chandra,2002), the simplest, practical way to account for the fracture (and thusfor the energy associated with the formation of new surfaces) in metal cut-ting is the use of the so-called cohesive energy, J (J=m2), which can bedetermined experimentally for any work material using a relatively simple

test (Shet and Chandra, 2002). Then, this energy multiplied by the area



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of fracture in metal cutting, which is the area of the shear plane, definesthe mechanical work involved in the fracture and formation of newsurfaces. The problem then arises what to do with the result obtained,

i.e., how to incorporate this result in the metal cutting model to calculatethe cutting force, power and other characteristics of a practical machiningoperation.

For many years, Atkins (1985, 2003) has been arguing that fractureis the case in metal cutting even of ductile materials and the energy asso-ciated with this fracture is significant so it has to be accounted for inmetal cutting models and calculations. Atkins (2003) and Rosa et al.(2007) proposed a method of experimental determination of the cohesiveenergy and incorporation this energy in the metal cutting model to calcu-late the cutting force. In the authors opinion, however, this attempt to

combine the improper chip formation and thus force model (Astakhov,2005) with the concept of cohesive energy does not account for the real dis-crete metal cutting process, i.e., for the number of shear planes formed perunit time.

Although it is well known and depictured in any book on metalcutting that the chip formation is discrete, i.e., at some point, a transitionfrom one shear plane to the next has to happen, this simple fact hasnever been accounted for in the known models of chip formation, as dis-cussed by Astakhov (2005). As the cohesive energy is associated with a sin-gle surface of fracture, the number of surface of fracture that occur per

unit time is essential to the determination of the power needed for suchfracture process.

To clarify the issue with the number of shear planes occurred per unittime (discussed previously as the frequency of chip formation), a specialexperiment was carried out (Astakhov et al., 1997b). Two specimens the first made of AISI steel 1045 (yield strength rY0:2 525 MPa, ultimatestrengthrR 585 MPa, elongation at breakd 10%), the second made ofmuch more ductile steel AISI 302 (rY0:2 250 MPa, rR 610 MPa,d 67%) were machined using the same cutting regime (cutting speed

90 m=min; feed

0.12 mm=rev; depth of cut

1.5 mm; no cutting fluid, aP10 carbide cutter with rake angle8). In the experiment, the chipcompression ratio, fwas measured and the deformed structure of the pro-duces chip was analyzed. For the first specimen, it was found thatf1 1:87,

while for the second f2 5:22. As seen, the degree of plastic deformationin machining of steel AISI 302 is much greater that that in machining of

AISI steel 1045. Moreover, it was also found in this test that the temperatureof the chip is almost 200% higher in machining of steel AISI 302.

Figure 8a and 8b show two models and two chip micrographs forthe obtained experimental results. Figure 8a shows the model for machin-

ing of a medium carbon steel 1045 and Figure 8b shows the model for



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machining of steel AISI 302. In both models, the right-hand coordinatesystem is set as following:

. The x axis coincides with the direction of the sliding plane (approxi-mation of the surface of the maximum combined stress) in the currentchip formation cycle.

. Theyaxis is perpendicular to thexaxis, as shown in Figure 8(a) and (b).

. The axis (not shown) is perpendicular to the xand yaxes.

Because it was found experimentally that in metal cutting the change inthe volume in the chip plastic deformation is negligibly small (Astakhov1998=1999), the following expression for strains is valid

ex eyez0 24

whereex; eyand ezare the true strains along the corresponding coordinates.

FIGURE 8 Model showing that the number of shear planes increases with plastic deformation of

the layer being removed: (a) Chip compression ratio f 1:87, (b) Chip compression ratio f 5:22,(c) Typical chip structure when f 1:87 (x200), (d) Typical chip structure when f 5:22 (x200).



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Referring to Figure 8a and 8b, consider a volume of the work materiallocated between two successive sliding planes (shown as hatched areas inFigure 8a and 8b). The plastic deformation of this volume can be repre-

sented by corresponding strains as

exlnx1

x0; eyln

y1

y0; ezln

z1

z025

where x0;y0; z0 and x1;y1; z1 are the dimensions of the discussed volumealong the corresponding coordinates before and after deformation,respectively.

It is known (Astakhov, 1998=1999) that, when properly measured, thechip width is practically equal to the width of cut, which yields z0z1 thus

ez0. Accounting for this result, substituting Eq. (24) into Eq. (25), onecan obtain

lnx1

x0ln

y1

y00 or

x1

x0

y0

y126

Because x0 and y0 do not change,

x1y1 Const 27

A very important conclusion immediately follows from Eq. (27):increasing chip thickness x1 (that is, the chip compression ratio f) leadsto the directly proportional reduction of distance y1between two successivesliding planes, increasing the number of sliding planes per unit length ofthe chip. To support this conclusion, Figure 8(c) and (d) show micro-graphs of the chip structure formed according to the discussed models.

As seen in Figure 8d, a great number of traces of the sliding planes, oneclosely followed by another, can be observed with a very small distance

between two successive fragments. The very similar picture can be observedin the micrograph of the quick-stop section through cupper chip presentedby Wright and Trent (2000, Fig. 4.27).

As the number of sliding planes per unit time increases, the power (thework done per unit time) associated with fracture and formation of newsurfaces should increase. To account for this power properly, the cohesiveenergy J (J=m2) should be multiplied by the area of fracture (m2) to obtainthe work done in fracture (J) on a single shear plane. Multiplying this work(J) by the frequency of chip formation (1=s), one obtained the powerneeded for the formation of new surfaces (W). This sequence is used in

equations (19) through (22).



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INFLUENCE OF THE CUTTING SPEED, DEPTH OF CUT

AND CUTTING FEED ON POWER PARTITION

The proposed methodology allows accessing the absolute and relativeimpacts of various variables of a metal cutting operation on the powerrequired and thus on the cutting force. Figures 9 through 11 present someresults for steel E52100.

The relative impact of the cutting speed on the energy partition is

shown in Figure 9. As seen, the power required for the plastic deformationof the layer being removed in its transformation into the chip is the great-est. However, the greater is the cutting speed, the greater powers on therake and flank faces of the cutting tool. When the cutting speed is 1 m=s,the power of the plastic deformation, Ppd is 67% while the power spent

FIGURE 9 Relative impact of the cutting speed on the energy partition.

FIGURE 10 Relative impact of the depth of cut on the energy partition.



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on the tool-chip interface, PfR is 18% and the power spent on the tool-workpiece interface, PfF is 9%. When the cutting speed is 4 m=s then Ppdis 45%, PfR is 25%, and PfF is 22%, i.e., the sum of powers spent on thetool-chip and tool-workpiece interfaces (PfR and PfF) is greater than thepower spent for the plastic deformation Ppd. This result signifies the roleof tribology in high-speed machining (Astakhov, 2006). The power spentin the formation of new surfaces Pch is 6% in both considered cases,although the frequency of chip formation is much greater when v 4 m=s.

The relative impacts of the depth of cut and the cutting feed are shown

in Figures 10 and 11. As seen in Figure 10, a 2.5-fold increase in the depthof cut does not affect the energy partition. A 2-fold increase in the cuttingfeed reducesPpdfrom 62% to 54% whilePfRincreases from 20% to 27%.

Practically the same results were obtained for aluminum. When the cut-ting speed is 1 m=s, the power of the plastic deformation, Ppd is 67% whilethe power spent on the tool-chip interface,PfRis 20% and the power spenton the tool-workpiece interface,PfF is 6% andPch is 7%. When the cuttingspeed is 7 m=s then Ppd is 50%, PfR is 25, and PfF is 25%, and Pch is 6%.

CONCLUSIONS

A methodology to evaluate the cutting force and the required cuttingpower is proposed. The proposed methodology uses the major parametersof the cutting process and the chip compression ratio as the one of themost important process output (in terms of process evaluation and optimi-zation). The apparent simplicity of the proposed methodology is basedupon a great body of the theoretical and experimental studies on theestablishing the correlations among the parameters in metal cutting.This simplicity allows the use of this methodology even on the shop floor

for practical evaluations and optimization of machining operations.

FIGURE 11 Relative impact of the cutting feed on the energy partition.



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The results of calculations indicate that the power required for thedeformation of the layer being removed is the greatest in the metal cuttingsystem within the practical cutting speed limits. When cutting speed

increases, the relative impact of this power decreases while the powersspent at the tool-chip and tool-workpiece interfaces increase. At highcutting speeds, the sum of the later powers may exceed that requiredfor the plastic deformation of the layer being removed. This result signifiesthe role of metal cutting tribology at high cutting speed. The effects ofcutting feed and the depth of cut on the energy partition seem to beinsignificant.

Although an increasing attention is played to the role of the so-calledcohesive energy in metal cutting, the obtained results show that, whenaccounted for properly, the relative impact of this factor is insignificant.

This can be readily explained by very small area of fracture in metal cutting.

NOMENCLATURE

Ac area of shearing (the uncut chip area equal to the product of theuncut chip thickness and the uncut chip width) (m2)

Ach area of fracture (m2)

Aw uncut chip cross-sectional area (m2)

b1T true chip width (m)Br Briks similarity criteriondw depth of cut (m)

Ech cohesive energy (J=m2)

Efr energy of fracture per a shear one shear plane (J=m2)

f cutting feed per revolution (m=rev)fcf frequency of chip formation (Hz)FfF friction force on the toolworkpiece interface (N)Fc cutting force in Eq. (2) (N)Fs shearing force in Eq. (1) (N)hm height of this surface profile (m)

K strength coefficient (N=m2

)lc tool-chip contact length (m)lac length of the active part of the cutting edge (the length of the

cutting edge engaged in cutting) (m).Lsh length of the shear plane (m)n hardening exponent of the work materialPc cutting power (W)Pch power spent in the formation of new surfaces (W)PfF power spent on the tool-workpiece interface (W)PfR power spent on the tool-chip interface (W)



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Pmnce energy spent due to the combined influence of the minor cuttingedge (W)

Ppd power spent on the plastic deformation of the layer being

removed (W)rn nose radius (m)t1T true uncut chip thickness (m)

v cutting speed (m=min)VBB width of the flank wear land (m)a flank angle (deg)a normal flank angle (deg)c normal rake angle (deg)ex; ey; ez the true strains along the corresponding coordinatesf chip compression ratio

jr tool cutting edge angle (deg)jr1 tool minor cutting edge angle (deg)l friction angle is Eqs. (1) and (2) (deg)qce radius of the cutting edge (m)rR ultimate tensile strength of the work material (MPa)ry0:2 tensile yield strength of the work material (MPa)sy shear strength of the work material (MPa)u shear angle (deg)

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a methodology for practical cutting force evaluation

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