Astakhov V. Tribology of Metal Cutting

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Energy Partition in the Cutting System 117

where dε

,dε

and dε

are the plastic strain increments in the principal directions.

Analyzing stress–strain relations proposed by Prandtl for plane strain deformation, Reuss

[63] assumed that the plastic strain increment is, at any instant of loading, proportional

to the instantaneous stress deviation and the shear stress such that

dε



dε



dε



dγ

= dλ (2.99)

or, more compactly, in tensor notation

dε

= σ



dλ, (2.100)

where σ



is the deviator stress tensor, dλ is a scalar non-negative constant of propor-

tionality which is not a material constant and may vary throughout the stress history. As

before, the superscript “p” denotes the plastic strain increment.

Considering the principal directions, Eq. (2.100) can be stated as

dε



dε



dε



= dλ (2.101)

If Eq. (2.101) is valid, µ

should be numerically equal to υ

. It directly follows from

Eqs. (2.101) and (2.95) that

2dε

−dε

dε

−dε

2σ



−σ



−σ



−σ



2σ

−σ

(2.102)

Thus υ

= µ

To prevent confusion, it should be appreciated that the Lode strain parameter υ

is not

related to Poisson’s ratio [21].

A generalization of the available experimental material on deformation of metals results

in the following conclusions [2]:

• Failure strain depends signiﬁcantly on the characteristic of the state of stress, as

shown in Fig. 2.21. Therefore, by changing this state, the fracture strain and thus

the energy consumption per unit volume of the layer being removed in cutting can

be minimized. In the author’s opinion, the easiest way to do this is to change the

geometry of the cutting tool used to achieve the necessary µ

. This approach appears

to be a new ground for selecting both the cutting geometry and the cutting regime.

• The dependence of the fracture strain ε

on the hydrostatic stress p

may be char-

acterized by three distinct zones (Fig. 2.22): (I) zone of insigniﬁcant dependence,

wherein ε

is independent or depends insigniﬁcantly on p

until a certain limit p

This behavior is observed in tensile tests of brittle materials (cast iron, chromium);

118 Tribology of Metal Cutting

hII

I II III

Fig. 2.22. Generalized dependence of the fracture strain (ε

) on the hydrostatic stress (p

) (after

Astakhov [2]).

(II) zone of signiﬁcance, approximately linear, dependence. This behavior was

observed in the testing of all the metals; (III) zone of a parabolic increase in plas-

ticity. The boundary between the zones II and III (the limit p

hII

) is not one line as

shown in Fig. 2.22. Rather, it is a small zone. The exact behavior of metals in the

zone III is not yet known.

−m

0.5 0

−0.5

0.4

0.3

0.2

0.1

q = 700°C

q = 250°C

Fig. 2.23. Effect of temperature on the inﬂuence of the Lode parameter (µ

) on the fracture strain

(ε

) (after Astakhov [2]).

Energy Partition in the Cutting System 119

UTS

tan

∆s

+∆s

∆l

−∆s

tan q

∆s

( )

Fig. 2.24. Determining the stress and elongation at fracture for different stressed states.

• The temperature under which the test is carried out has a signiﬁcant inﬂuence on

the fracture strain. However, over the range of temperatures in the deformation

zone (high tool–chip contact temperatures do not affect the resistance of the work

material to cutting), temperature may not signiﬁcantly affect the dependence of the

fracture strain on the Lode stress parameter, as illustrated in Fig. 2.23.

• Although the inﬂuence of superimposed hydrostatic pressure on the deformation and

fracture behavior of work materials is very signiﬁcant, a concern which remains is

the practical determination of this inﬂuence. In other words, the question can be

rephrased as: “Should one carry out a series of materials tests over a wide range of

hydrostatic pressures?” In the author’s opinion, it is not feasible because a bulk of

data obtained using special testing equipment (for example Material Testing System

(MTS)) has to be generated. Obviously, it may take years to collect these data.

Therefore, a much more practical way should be suggested.

One of the most promising directions in this respect is use of available tensile data to

calculate the strain and stress at fracture at different states of stress. Brownrigg et al.

[64] proposed the following relationship

(

)

(

)

= 1 +

1.68p

2¯σ

, (2.103)

where ε

(

)

is the fracture strain at superimposed hydrostatic pressure p

, ε

(

)

is the

fracture strain in uniaxial tension with no additional superimposed pressure and ¯σ is the

effective stress at fracture [60].

120 Tribology of Metal Cutting

Oh [65] proposed a simple method that, in the author’s opinion, can be used to the

ﬁrst approximation to determine the stress and strain at fracture under different state

of stress. The essence of this method is shown in Fig. 2.24 (bilinear representation

of elongation–stress diagram) is self evident. As such the elongation at fracture is

calculated as



∆l



f , ∆σ



∆l





−1

tan θ



∆σ

, (2.104)

where



∆l



f , ∆σ

is the elongation at fracture at mean stress variation ∆σ



∆l



is the elongation at fracture in uniaxial tension with no additional superimposed

pressure.

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Energy Partition in the Cutting System 123

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CHAPTER 3

Tribology of the Tool–Chip and

Tool–Workpiece Interfaces

3.1 Introduction

This chapter considers the tribology of the tool–chip and tool–workpiece interfaces,

namely, the distribution of contact stresses, nature and interdependence of contact pro-

cesses, correlations between characteristics of the contact process with important input

and output parameters and characteristics of the cutting process. Particular attention is

paid to the properties of the tool and work materials.

The tool–chip and tool–workpiece interfaces are those contact areas that directly par-

ticipate in the cutting process. Considerably different functions of the tool–chip and

tool–workpiece interfaces in this process deﬁne the differences in the tribological pro-

cesses at these interfaces. Although there are a number of differences, there are also a

number of similarities in the tribology of these interfaces as high contact pressures and

temperatures, contact with freshly formed ( juvenile) counter-surfaces and high sliding

velocities. There is also a correlation between the stresses at these interfaces as they both

are related to the same state of stress in the deformation zone. As such, however, the

tool–chip interface plays a leading role that deﬁnes to a large extent this state. There-

fore, a greater part of the considerations presented in this chapter is dedicated to the

tribological conditions at the tool–chip interface.

The major results of Chapter 2 show that only 30–50% of the energy required by the cut-

ting system is spent for the useful work, i.e. for the separation of the layer being removed

from the workpiece. The rest is spent at the tool–chip and tool–workpiece interfaces. This

fact can be easily appreciated if one realizes that nearly all the energy spent in the cutting

process is converted into thermal energy. Therefore, the temperature of a certain zone in

the cutting system is a very good and relevant indicator of the energy spent in this zone as

this energy generates heat, so the higher the temperature of a particular zone, the greater

is the energy spent in this zone. If one compares the temperature in the deformation zone

and that at the discussed interfaces, one can come to a surprising yet well-known conclu-

sion. The essence of it is that the temperature in the deformation zone, where the major

124

Tribology of the Tool–Chip and Tool–Workpiece Interfaces 125

work of plastic deformation and separation of the layer being removed from the rest of

the workpiece takes place, is relatively low (normally 80–250

◦

C) as discussed later in

this chapter. On the contrary, the maximum temperatures at the tool–chip interface can

reach more than 1000

◦

C. Therefore, more energy required by the cutting system is spent

at the tool–chip and tool–workpiece interfaces although the deformation zone prevented

much more attention of the researchers in the ﬁeld.

The main objective of this chapter is to discuss the tribological conditions at the tool–

chip and tool–workpiece interfaces. The understanding of these conditions and the proper

utilization of this understanding in the design of the cutting process should result in the

increased efﬁciency of the cutting system and in the reduction of tool wear. These results

can be used in the meaningful selection of the machining regime, tool geometry and tool

materials (including coatings).

3.2 Tool–Chip Interface

When a metal is cut, the cutting force acts mainly through a small area of the rake

face, which is in contact with the chip and thus, is known as the tool–chip interface.

Therefore, it is of interest in determining the cutting force, developing the theory of tool

wear and understanding the mechanics of chip formation to establish the tribological

characteristics of the tool–chip interface.

The basic tribological characteristics of the tool–chip interface are:

• The contact length – the length of the tool–chip contact, l

• The sliding velocity, ν

= ν/ζ,

• The friction force at the tool–chip interface, F ,

• The speciﬁc frictional force which is the mean shear stress, τ

• The normal force at the tool–chip interface, N,

• Mean normal stress at the tool–chip interface, σ

• Mean contact temperature at the tool–chip interface, θ

r−av

3.2.1 Friction coefﬁcient

In Merchant’s analysis, it is implied that the contact between the tool and the chip is a

sliding contact, where the coefﬁcient of friction is constant [1].

In most engineering and physical situations, friction effects at a tribological interface are

described by a constant coefﬁcient of Coulomb friction µ

, (3.1)

126 Tribology of Metal Cutting

where N is the normal force acting at the considered interface (Fig. 1.7) and F is the

frictional force at this interface (Fig. 1.7).

Although it is well established that contact between the two bodies is limited to only a few

microscopic high points (asperities), it is customary to calculate the stresses by assuming

that the forces are distributed over the total (apparent) contact area. Such an approxima-

tion, however, is not far from reality in machining, where the actual and apparent contact

areas are practically the same due to high contact pressures [2]. If it is so, the numerator

and denumerator of Eq. (3.1) can be divided over the tool–chip contact A

and then

recalling that the mean normal stress at the interface is σ

= N/A

and the mean shear

(frictional) stress at the interface is τ

= F/A

, one can obtain

(3.2)

Equation (3.2) reveals that if the friction coefﬁcient at the tool–chip interface is constant,

the ratio of the shear and normal stresses should be the same along the entire contact

length.

As discussed by Dieter [3], the above analysis was for sliding friction at the interface,

as in our ﬁrst encounter with friction in elementary physics. At the other extreme, we

can envision a situation where the interface has a constant ﬁlm shear strength. The most

usual case is sticking friction, where there is no relative motion between the chip and the

tool at the interface. For sticking friction, this shear strength should be equal to the ﬂow

stress in shear, k

, and the normal stress to the yield stress of the work material σ

. Using

the von Mises yield criterion, the coefﬁcient of friction under sticking conditions is



√

= 0.577 (3.3)

Therefore, the value of the friction coefﬁcient µ

deﬁned by Eq. (3.3) should be con-

sidered as the limiting value so that if µ

≥ 0.577, no relative motion can occur at the

interface.

In the practice of metal cutting, however, this is not the case. In experimental stud-

ies, Zorev [2] obtained µ

= 0.6–1.8, Kronenberg [4] – 0.77–1.46, Armarego and

Brown [5] – 0.8–2.0, Finnie and Shaw [6] – 0.88–1.85, Usui and Takeyama [7] –

0.4–2.0, etc. In the simulations of metal cutting, Stenkowsky and Moon [8] used

= 0.2, Komvopoulos and Erpenbeck [9] – 0.0–0.5, Lin, Pan and Lo [10] – 0.074,

Lin and Lin [11] – 0.001, Stenkowsky and Carroll [12] – 0.3, Endres, DeVor and

Kappor [13] – 0.05, 0.10, 0.25, and 0.5, Olovsson, Nilson and Simonsson [14] – 0.1,

etc. As seen, the reported values of µ

obtained in metal cutting tests are well above

0.577. On the other hand, the values of µ

used in modeling (more often in FEM mod-

eling), are always below the limiting value to suit the sliding condition at the interface.

Interestingly, the results of FEM modeling were always found to be in good agreement

with the experimental results regardless of whether the particular value of the friction

coefﬁcient is selected for such a modeling.