Astakhov V. Tribology of Metal Cutting

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Tribology of the Tool–Chip and Tool–Workpiece Interfaces 127

It is true that, in general, the coefﬁcient of friction for sliding surfaces remains con-

stant within wide ranges of the relative velocity, apparent contact area and normal load.

In contrast, in metal cutting, the coefﬁcient of friction varies with respect to the normal

load, the relative velocity and the apparent contact area. The coefﬁcient of friction in

metal cutting was found to be so variable that Hahn [15,16] doubted whether this term

served any useful purpose. Moreover, Finnie and Shaw [6] have concluded that the con-

cept of the coefﬁcient of friction is inadequate to characterize the sliding between chip

and tool and thus recommended to discontinue using the concept of the coefﬁcient of

friction in metal cutting. An extensive analysis of the inadequacy of the concept of the

friction coefﬁcient in metal cutting was presented by Kronenberg (pp.18–25 in [4]) who

stated “I do not agree with the commonly accepted concept of coefﬁcient of friction in

metal cutting and I am using the term ‘apparent coefﬁcient of friction’ wherever feasible

until this problem has been resolved.” Unfortunately, it has never been resolved although

more than 40 years have passed since this statement of Kronenberg.

To model friction conditions in metal cutting and thus to determine the real value of

the friction coefﬁcient, a large number of theoretical analyses [17–31] have been carried

out to determine the inﬂuence of various parameters on the interfacial friction. Despite

the improvements made in the modeling of friction in machining processes over the last

30 years, there are still a number of limitations so that the results of these studies are

hardly applicable to metal cutting.

In the author’s opinion, the most severe drawbacks of the discussed theoretical

considerations are:

• At the interface, the soft material is always assumed to be rigid, perfectly plastic,

incompressible and isotropic at all stages of deformation. Moreover, no superﬁcial

and in-depth residual stresses due to previous deformation can be accounted for.

Obviously, this is not the case in metal cutting and particularly at the tool–chip

interface.

• The approaches known so far consider only two-dimensional asperity interaction

ignoring the three-dimensional nature of asperities and actual multi-asperity inter-

action. The shape of the asperity should be well deﬁned so that its initial and ﬁnal

point should lie on the same line parallel to the direction of relative motion. More-

over, this shape determines the slipline ﬁeld where the discontinuity of tangential

velocity leads to inﬁnite calculated strains. No wonder the chip formation model

predicts an increase in the coefﬁcient of friction with improved lubrication [22].

• The sliding speed was not considered to be an important factor. The temperature and

strain rate effects are poorly accounted for. The sliding speed even in the veriﬁca-

tion experiments was 0.3 mm/s and this is obviously way below the sliding speeds

found at the tool–chip interface. It was also found that the strains determined from

the experimental ﬂow ﬁelds are signiﬁcantly lower than those calculated from the

model [25].

A number of experimental techniques and tests for evaluating the interfacial friction in

metal cutting have been developed. Many of these tests actually aim to evaluate properties

of the metalworking ﬂuids and thus are known as the lubricity tests [32]. Each has its

128 Tribology of Metal Cutting

own inherent advantages and limitations. Lubricity tests can be broadly divided into three

groups [32]. One group is based upon simple rubbing or rolling action. Another group

is based upon metal removal or chipmaking processes. The ﬁnal group incorporates

forming or drawing of a metal sheet. It was soon recognized, however, that, because of

the complexity of ﬁeld conditions, no simple test machine can simulate the lubrication

requirements for different machining operations. Therefore, it is rather difﬁcult, or even

impossible, to correlate bench test data with the actual performance [32].

To understand the problem with experimental determination of the friction coefﬁcient,

one should refer to the force diagram developed by Merchant [1] and shown in Fig. 1.1(a).

Since Merchant assumed Coulomb friction is the case in metal cutting, the following

expression for the friction coefﬁcient was obtained from the force diagram shown in

Fig. 1.1(d)

µ =

tan γ

−F

tan γ

(3.4)

It directly follows from Eq. (3.4) that the friction coefﬁcient can be calculated using the

friction and normal forces obtained experimentally by resolving the measured cutting

and trust components of the cutting force. The friction coefﬁcient thus obtained does not

match with common experience. This was ﬁrst found by Kronenberg in 1927 (p.18 in

[4]). Unfortunately, Merchant, who conducted his research 20 years later, did not pay

attention to the ﬁrst edition of Kronenberg’s book.

3.2.2 Contact stresses distribution – reported results

Later research has been aimed at obtaining a better understanding of the conditions at

the tool–chip interface by studying the distribution of the normal and shear stress at

this interface. A variety of experimental techniques including photoelastic tools, split-

tool dynamometer, transparent tool for the direct observation of the tool–chip interface,

metallurgical examination of “quick-stop” chip-section including experimental slipline

ﬁeld method have been developed [2,33–39].

Photoelastic method. The pioneers in the ﬁeld of photoelastic analysis of orthogonal

metal cutting, such as Coker and Filon [40] and Zeichev [41] have generally used a

photoelastic work material in their studies. This has shortcomings as the work material

undergoes plastic deformation as shown by Frocht and Thomsen [42] and Ocushiri and

Fukuii [43] who have used a photoelastic material as a cutting tool, which was pressed

against the already-made chip.

Andreev [44] and Kattwinkel [45] were probably the ﬁrst scientists who obtained some

meaningful stress distributions at the tool–chip interface using a photoelastic tool [38].

Andreev [44] pointed out that the whole contact length l

is divided into two distinc-

tive parts of approximately equal length: the plastic part that extends from the cutting

edge to the middle of the contact length (or the length of the tool–chip interface) and

the elastic part from the middle of contact to the point of chip separation from the

Tribology of the Tool–Chip and Tool–Workpiece Interfaces 129

Normal stress

Distance from the cutting edge

Shear stress

(a)

(b)

0 0.5l

Distance from the cutting edge

0 0.5l

Fig. 3.1. Stress distributions at the tool–chip interface reported in literature: (a) normal stress and

(b) shear (tangential) stress (after Astakhov and Outeiro [73]).

tool–rake face. The results obtained indicate that the normal stress, being zero at the point

of chip separation, increases exponentially towards the cutting edge (curve 1, Fig. 3.1(a)).

The distribution of the shear stress obtained by Andreev is shown in Fig. 3.1(b), curve 1.

Kettwinkel, using a similar experimental technique, obtained similar distribution of the

normal stress (curve 1, Fig. 3.1(a)). He found, however, that the shear stress, after reach-

ing the maximum value at the middle of the contact length then decreases toward the

cutting edge (curve 2, Fig. 3.1(b)).

Takeyama and Usui [46] studied the stress distribution along the tool–chip interface by

measuring the cutting forces on tools with various contact lengths. Their result for the

normal stress is shown by curve 2 in Fig. 3.1(a) and for the shear stress by curve 3 in

Fig. 3.1(b). Later on, Usui and Takeyama [7] improved the accuracy of their experiments

and thus concluded that the distribution of the normal stress is as shown by curve 3 in

Fig. 3.1(a) and that of the shear stress is as shown by curve 4 in Fig. 3.1(b). The same

experimental technique was used by Chandrasekaran and Kapoor [47]. Trying to under-

stand the inﬂuence of tool geometry, they used photoelastic tools of different rake angles.

130 Tribology of Metal Cutting

They found that the contact length l

is inversely proportional to the rake angle. For pos-

itive rake angles (0, 10 and 20

◦

), the distribution of the shear stress corresponds to that

given by curve 3 in Fig. 3.1(b) but when the rake angle was negative, the distribution

given by curve 5 in Fig. 3.1(b) was the case. For all the studied tools which had positive

rake angles, the maximum of the shear stress was found to be very nearly that of the

shear yield strength of the work material. In their opinion, a signiﬁcant difference in the

curves of normal stress distribution shows (curve 4 corresponds to a +20

◦

rake angle,

curve 5 corresponds to –10

◦

) that there is some pronounced differences in the case of a

negative rake tool.

Further studies using photoelastic tools [48,49] yielded different results. Amini [48]

concluded that both the normal and the shear stresses increase in a non-linear manner

from the point of chip separation towards the cutting edge. Ocusima et al. [49] found

that the shear stress distribution is similar to that of the given by curve 1 in Fig. 3.1(b)

and remains constant over approximately two-thirds of the contact length l

. The normal

stress, being approximately equal to the shear stress at the cutting edge, reaches its

maximum at the mid point of the contact length and then decreases gradually to zero

(curve 6 in Fig. 3.1(a)).

Although the above-discussed experimental studies using the photoelastic method were

great attempts to determine the stress distributions during machining, a number of points

of criticism on these results remain open. The major concerns are with the cutting speeds

used in the test which are thousand times lower (due to extremely low hot hardness of

the tools used) than those found in practice. As a result, there is no ground to believe

that the real tribological characteristics of the tool–chip contact, namely the mode of

deformation; the maximum temperature and temperature distribution; relative velocity

of the contacting bodies; abrasion, adhesion, diffusion, and chemical interactions [39],

would be even close to those in these studies. Moreover, it is not possible to determine

accurately the contact stresses immediately adjacent to the cutting edge because the

contact stresses at the tool ﬂank cause distortion of the isochromatic fringes in this

region [50].

Bagchi and Wright [37,51] overcame the problem with tool hot hardness by using a single

crystal sapphire tool although the stress birefringence effect in sapphire is relatively weak

and, of course, sapphire is inherently brittle. Despite these difﬁculties, they were able to

machine steel and brass specimens at speeds of up to 75 m/min and at a maximum feed

rate of 0.381 mm/rev to study the effect of speed and feed on stress distributions [37].

It was found that the normal and shear stress distributions qualitatively resemble those

obtained by Chandrasekaran and Kapoor [47]; the cutting speed does not have signiﬁcant

inﬂuence on the peak normal stress while the feed and the uncut chip thickness do.

Split tool method. The method was ﬁrst completely described and used by Loladze

[37,52,53]. The method is based on the use of a composite cutting tool having the

cutting wedge divided into two parts. The principles of design of the split tool is shown

schematically in Fig. 3.2. The cutting wedge and thus, the tool rake face is divided into

two parts, namely T

and T

. During the test, the forces acting only on the part T

were

measured. During the experiment, the length of T

was incrementally increased from the

smallest possible value up to l

Tribology of the Tool–Chip and Tool–Workpiece Interfaces 131

Chip

Cutting

direction

Fig. 3.2. Principle of measurement of stress distribution using the split tool technique.

The determination of the stress distribution consists of the following steps:

• A cutting test is carried out with the part T

having length l

, and the normal F

and frictional F

forces acting on this part are measured.

• The length l

is decreased to l

i+1

and the next test is carried out under the same

cutting conditions which yields in F

Ni+1

and F

fi+1

The mean normal stress σ

and shear stress τ

acting at the interface having length

(

i+1

−l

)

were calculated as

Ni+1

−F

i+1

−l

)

(3.5)

fi+1

−F

i+1

−l

)

(3.6)

Conducting his experiments using the above-described technique, Loladze [52] found

that the distributions of the normal and shear stresses correspond qualitatively to those

found by Andreev (curve 1, Fig. 3.1(a) and curve 1, Fig. 3.1(b), respectively). The values

of the maximum normal stress varied from 900 to 1600 MPa for different steels used in

the tests and were strongly related to the yield strength of the work material. In each

case, the maximum normal stress near the cutting edge was higher than the yield strength

of the work material by a factor greater than two. This maximum stress increases to a

small extent with the cutting speed and feed, and decreases as the rake angle increases.

Unfortunately, Loladze and subsequent researches did not pay attention to this important

experimental ﬁnding that resulted from well-prepared and precisely conducted cutting

tests.

132 Tribology of Metal Cutting

Kato et al. [54] conducted a wide range of cutting experiments using a split tool

dynamometer. Their results for different work materials can be qualitatively described

by the following curves in Fig. 3.1:

• For work hardened and perfectly annealed aluminum, copper and lead–tin alloy,

the normal stress distribution corresponds to curve 4 (Fig. 3.1(a)); the shear stress

distribution corresponds to curve 3 (Fig. 3.1(b)). Surprisingly, the normal and shear

stresses at the plastic part of the tool–chip contact were the same for work hardened

and for perfectly annealed aluminum although the full contact and plastic contact

lengths were twice greater for the latter. Moreover, the shear stress was found to be

25% higher than the normal stress at the plastic part of the tool–chip contact.

• For zinc, the normal and shear stresses closely (qualitatively and quantitatively)

follow each other and their distribution corresponds to curve 1 (Fig. 3.1(a)) and

curve 6 (Fig. 3.1(b)), respectively.

In the author’s opinion, the most important conclusion that can be drawn from these

results is as follows. Because the study had only one process variable, namely the work

material (all other parameters are kept the same), the state of stress in the deformation

zone and thus at the tool–chip interface should be qualitatively the same. Therefore,

if metal cutting, as it is believed now [2,34,35,37,55–60], is accomplished by pure

(or simple) shearing, the normal and shear (tangential) stress distributions should be

the same for all ductile materials. Moreover, the shear and normal stresses should be

uniquely related so that the shear stress should not exceed 0.7σ. The experimental results

obtained do not support this belief. For example, Barrow et al. [50] used a split tool to

obtain the stress distributions while machining a nickel–chromium alloy within a wide

range of cutting speeds and feeds. It was found that for some cutting conditions, the peak

shear stress was approximately equal to the peak normal stress while the magnitude of

this peak was found to depend on the cutting conditions. As argued by Astakhov [36],

the triaxial state of stress is the real phenomenon of metal cutting and, since different

materials react differently on the degree of triaxiality, they exhibit different strain–stress

behavior under the same state of stress.

Further researchers [58,61,62] conducted a great number of experimental studies with

different work and tool materials in order to determine particular values of maximum

normal and shear stress and their distribution along the contact length. Since more sen-

sitive techniques for force measurement became available, the ﬂuctuation of the cutting

forces became evident. However, instead of understanding the nature of such ﬂuctua-

tions (as argued above, this ﬂuctuation is a result of cyclical nature of the chip formation

process), they just used smoothing or averaging of the experimental forces to obtain

steady-state stress distributions [63].

Experimental slipline ﬁeld method. In order to determine the distribution of the nor-

mal and shear stresses at the tool–chip interface, Roth and Oxley [64] analysed an

experimental ﬂow ﬁeld obtained under plain strain conditions using the slipline ﬁeld

technique. According to their perception, a single slipline ﬁeld which would describe

the ﬂow in the two deformation zones in metal cutting (the primary deformation zone

which stands for the shear plane and the secondary deformation zone which stands for the

Tribology of the Tool–Chip and Tool–Workpiece Interfaces 133

plastic part of the tool–chip contact) could be constructed using experimental ﬂow ﬁeld

and the measured cutting forces. The following drawbacks of this method are listed as

follows:

• The cutting speeds used to obtain a specimen to study the experimental slipline ﬁeld

are more than 1000 times lower than those used in the practice of cutting of the same

work material. For example, the cutting speed was 0.0125 m/min for free machin-

ing steel SAE 1112 in Roth and Oxley experiments [64] while the recommended

cutting speed used for the same work material exceeds 300 m/min (Table 21.2 in

Ref. [65]). As well-discussed by Zorev [2], there is no similarity between the cutting

processes at low cutting speeds (see Chapter 2 “Experimental studies of chip for-

mation and contact processes on the tool face at low cutting speeds” and Chapter 4

“Experimental studies of chip formation and contact processes on the tool face at

high cutting speeds” in Zorev’s book [2]).

• In order to apply the slipline method, simple shearing was assumed to be the prime

mode of deformation in the primary and secondary deformation zones which is

obviously not the case in metal cutting [36]. The state of stress in the mentioned

deformation zones has not been analysed, hence the z component of the normal

stress was ignored.

• The method is extremely subjective. For example, the crucially important parameters

namely the radius of chip curvature and the tool–chip contact length were a subject

of assumptions rather than results of measurements.

The normal and shear stress distributions obtained by Roth and Oxley [64] are qual-

itatively shown in Fig. 3.3(a) by curve 1 and in Fig. 3.3(b) by curve 1. As seen, a

stress singularity exists at the point of chip separation from the rake face that cannot be

physically justiﬁed.

Later, Oxley in Chapter 7 “Predictive machining theory based on a chip formation model

derived from analyses of experimental ﬂow ﬁelds” of his book [35], concluded that

“a plastic state of stress exists in the chip over the full contact length and that the

deformation in the chip can be represented by a rectangular plastic zone with no sliding

at the interface.” (p. 100 in Ref. [35]). Further followers of this method tried to take

into account the so-called ploughing force components due to the radius of the cutting

edge [38]. After a number of assumptions, the distributions of the normal and shear

(tangential) stress were obtained to be uniform over the tool–chip contact length, as

shown in Figs. 3.3(a) and (b), curves 2. These results are in direct contradiction with

those of previous experimental studies where the division of the contact length into two

distinctive parts (plastic and elastic) was clearly observed. Moreover, the singularity of

stresses at the point of chip separation from the rake face has never been explained. These

distributions of the normal and shear stresses suggest that a constant friction coefﬁcient

is the case at the tool–chip interface according to Eq. (3.3).

A critical analysis of the above-mentioned and other studies related to stress distribution

beyond those analysed above (for example, Refs. [2,34,57,66–69]) shows that it is quite

possible that actual stresses and stress distributions may have not yet come to light due

to a signiﬁcant scatter in the results obtained.

134 Tribology of Metal Cutting

Normal stress

0.5I

(b)

(a)

Shear stress

Distance from the cutting edge

0 0.5I

Distance from the cutting edge

Fig. 3.3. Qualitative representation of the normal (a) and shear (tangential) (b) stress distributions

at the tool–chip interface obtained by Roth and Oxley for a free-machining steel (after Astakhov

and Outeiro [73]).

3.2.3 Contact stress distribution – modeling

Experimental evidences. Multiple experimental studies conducted by Zorev [2],

Loladze [52], Poletica [69] and many others conclusively proved that the tool–chip

interface consists of plastic and elastic parts. Figure 3.4 shows that the plastic part of

the tool–chip interface can be clearly observed on the tool rake face. Therefore, the

uniform stress distribution shown in Fig. 3.3 is in direct contradiction with the known

experimental observations.

Zorev [2] studied the length of the plastic part using a quick-stop device and conclu-

sively proved that the whole contact length l

is divided into two distinctive parts: the

plastic part, l

c−p

which extends from the cutting edge and the elastic part l

c−e

, from

the plastic part to the point of tool–chip separation. Zorev showed that the contact

length, l

is a function of the cutting speed, as shown in Fig. 3.5. Similar experimental

results were obtained by Poletica [69]. Summarizing the results of multiple experiments,

Tribology of the Tool–Chip and Tool–Workpiece Interfaces 135

Fig. 3.4. The tool–chip interface consists of two parts, namely plastic and elastic.

1.5

1.0

0.5

n (m/min)

, l

c−p

(mm)

2.0

60 90 120 150

c−p

Fig. 3.5. Inﬂuence of cutting speed (ν) on the tool–chip contact length (l

) and on the plastic part

c−p

of the tool–chip contact length. Orthogonal cutting, work material – steel 4130, rake angle

γ = 5

◦

and uncut chip thickness t

= 0.15 mm.

136 Tribology of Metal Cutting

Abuladze [70] proposed the following expression to calculate the length of the plastic

part of the tool–chip interface

c−p

= t

[

(

1 − tan γ

)

+sec γ

]

(3.7)

Model and FEM analysis. The contact problem at the tool–chip interface can be mod-

eled as the contact of two bodies made of dissimilar materials. As such, one body

deforms elastically within the contact interface (the tool), and the other body (the chip)

contains elastic and plastic regions within this interface. Moreover, there is relative slid-

ing between these two contacting bodies and thus the traction develops over the contact

length. In the considered case of the tool–chip contact, it is not simple to determine the

exact contours of the deformed surfaces and other boundary conditions so that the exact

solution of the stated contact problem is rather difﬁcult. Therefore, simpliﬁcations have

to be introduced to develop a suitable model.

All real contacts, like all real engineering components, are three-dimensional, and there-

fore theoretically demand a solution in the three-dimensional theory of elasticity of

elastoplasticity [71]. However, there are some simple cases where it is feasible to approx-

imate the geometry to one or two dimensions. As discussed in Chapter 1, orthogonal

cutting is always considered as a two-dimensional problem. Thus, in order to investi-

gate the tool–chip contact problem, plane contact problems will be considered, i.e. those

where it is assumed that displacements are restricted to a single plane, the xy plane.

Among the two-dimensional contact problems known, those where the contacting bodies

are considered to be half-planes, i.e. semi-inﬁnite bodies with remote boundaries are

looked at, provided that the width of the resulting contact is negligible in comparison with

the bodies’ radii of curvature. Contacts where this is so are said to be non-conformal [71].

A wide range of non-conformal contact problems may be readily solved in two stages.

First, the mixed boundary value problem (where displacements are speciﬁed within the

contact, and the remainder of the free surface is free of tractions) is reduced to an elasticity

problem of the ﬁrst kind, i.e. where the tractions are speciﬁed everywhere. Second,

the interior stress ﬁeld due to these tractions is found by applying the Muskhelishvili

potential [72].

Figure 3.6 shows the general class of contact problems. Subscripts 1 and 2 will be

added to the elastic constant relating to the upper and lower bodies respectively and a

mathematical sign conversion is adopted for the tractions, i.e. direct tractions (contact

stresses) p

(

)

are positive tensile and shear tractions q

(

)

are positive when acting

to the right on the lower body. According to the known results [71], the equations for

displacements υ

and υ

are

∂υ

∂x



(

)

dς

x − ς

−

(

)

(3.8)

∂υ

∂x



(

)

dς

x − ς

−

(

)

(3.9)