Astakhov V. Tribology of Metal Cutting

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Energy Partition in the Cutting System 107

190

170

12345

370

470

= 1.0 mm

Cutting speed, n (m/s)

Cutting force, P

(N)

= 0.5 mm

= 0.1 mm

Fig. 2.17. Scatter for cutting inserts No. 1 and No. 2.

determined. Figure 2.18 shows that the experimental points ﬁt this curve fairly good,

accounting for the accuracy with which the cutting force can be measured.

The experimental results show that the cutting force changes sinusoidally with the cutting

speed. If the tool geometry and location of the cutting tool with respect to the workpiece

surface are kept constant, the energy of failure (fracture) of the layer to be removed

remains constant too. Therefore, according to Eq. (2.66), the mechanical energy required

should be a function of the cutting speed. If the interaction of deformation and thermal

waves causes their reinforcement then the required mechanical energy decreases. On the

contrary, if this interaction results in destructive interference, the required mechanical

energy increases.

When the cutting feed is constant, the path difference of the coherent waves does not

change. According to Eq. (2.66), a change in the cutting speed results in the change

in the time interval between the two successive tool positions (between the neighboring

trajectories of the cutting tool), which is, according to Eq. (2.66), equivalent to a change in

the intensity of the heat source. Therefore, it becomes clear that the revealed dependence

of the interaction of the energy waves on the cutting speed is possible only if the frequency

of deformation and heat waves depends on this speed.

The phenomenon of interaction of the deformation and thermal longitudinal waves was

additionally examined using the following logic: the interaction of the waves takes place

108 Tribology of Metal Cutting

160

190

12345

220

250

Cutting speed, n (m/s)

Cutting force, P

(N)

Fig. 2.18. Experimental results for cutting inserts No. 3 and No. 4.

due to the fact that these waves are coherent. Therefore, if the coherence of the waves

is disturbed, the interaction should not be observed. To verify this statement, the second

series of experiment was carried out using face cutting (Fig. 2.19a). Table 2.5 lists the

ranges of the cutting parameters used in the test. Other parameters of the cutting system

and the measuring rig were kept the same.

The experimental results obtained in this test show that no force variation has been

observed over a considerably wide range of cutting conditions. This result can be

explained as follows: in face cutting, the cutting speed is not constant but differs for

(a)

Workpiece

Tool

(b)

Fig. 2.19. (a) Face cutting and (b) the trajectory of the cutting tool (after Astakhov [8]).

Energy Partition in the Cutting System 109

Table 2.5. Ranges of values of cutting parameters for the end turning

tests.

Parameter Low value High value

Workpiece diameter (mm) 60.00 120.00

Cutting speed of the workpiece (o.d.)* (m/s) 0.15 4.00

Feed (mm/rev) 0.07 0.25

Depth of cut (mm) 0.10 3.00

*Outer diameter.

each successive point of the tool trajectory (Fig. 2.19b). Therefore, the thermal wave

generated on the neighboring trajectory of the cutting tool (Point 3) does not affect the

cutting force (energy consumption) on the current trajectory (Point 4), because this ther-

mal wave has been formed at different cutting speeds and, therefore, the deformation

and the thermal waves are not coherent. Hence, the results prove that, on the one hand,

the cutting speed affects the wavelength, while on the other hand, the interaction of the

energy waves is a real phenomenon of the cutting process.

The correlation of the wavelengths under different cutting speeds that result in the max-

imum reinforcement of the coherent waves can also be established using the following

considerations. It follows from Eqs. (2.69) and (2.70) that, if the cutting feed is kept

constant, the reinforcement (constructive interference) and destructive interference of

the total energy ﬂux are possible only when the wavelength is a function of the cutting

speed, i.e. when l

= f(ν) is the case. If a certain ith crest of this function takes place

when f = kl

, then the next crest appears when the number of waves increases by 1, i.e.

when f = (k +1)l

i+1

. Consequently, if the cutting force varies according to a sinusoidal

function with the cutting speed,

i+1

1 + l

(2.80)

Inﬂuence of cutting feed When the path difference (the cutting feed) remains constant,

a change in the cutting speed causes a periodic increase and decrease of the cutting force

due to the variation in the frequencies of the energy waves generated at different cutting

speeds (the velocity of deformation). If this happens due to the interference of the energy

waves discussed, a change in the cutting feed should also affect the cutting force because

the cutting feed deﬁnes the path difference.

However, force F

is affected not only by the wave-interaction force component F

(

)

but also by the force component F

(

)

, which depends on the uncut chip thickness

because the uncut chip thickness changes with the cutting feed [2]. Mathematically,

it can be represented by analogy with Eq. (2.75) as

= F

(

)

(

)

= F

(

)

sin



2π



f +f





(2.81)

If the wavelength l

ν1

of the sine wave which approximates the variation in the cut-

ting force F

with the cutting speed ν and the frequency of the coherent waves l

are

110 Tribology of Metal Cutting

known for the cutting speed ν

accounting Eq. (2.80), then the wavelength l

for any

other cutting speed which differs from ν

by a number divisible by l

can be found as

follows:

f +l

(

k − 1

)

, where k = 2, 3 ... n (2.82)

Points

(

,ν

)

calculated using Eq. (2.82) can be approximated by a line so that the

wavelength of the deformation wave corresponding to any cutting speed can be calculated

knowing ν

, l

ν1

and l

(providing that cutting feed f remains invariable) as

l =

1/l

(

ν −ν

)

(

ν1

)

(2.83)

The third series of experiments was carried out using the same experimental conditions.

The workpiece diameter was 88 mm, the depth of cut d

= 0.1 mm. The cutting force

was measured as a function of the cutting feed f for different speeds of rotation of

the workpiece n. The experimental results for ν = 2.9, 2.3, 1.8, 1.4, 1.2 and 0.9 m/s are

shown in Table 2.6.

The graphical analysis of the experimental results shown in Table 2.6 showed that the

maximum and minimum of F

locate in the “f −F

” coordinate system on two inclined

parallel lines. Therefore, the mean of the sine wave (for the experimental conditions

Table 2.6. Experimental results (Series No. 3 of experiments).

f (mm/rev) ν = 2.9 m/s ν = 2.3 m/s ν = 1.8 m/s

(N) P

(N) F

(N) P

(N) F

(N) P

(N)

0.070 75 −2.84 56.2 −3.05 56.2 −0.01

0.074 84 3.91 60.0 −2.06 60.0 1.67

0.084 81 −4.70 71.2 2.29 65.6 2.11

0.097 94 1.00 76.9 −1.06 71.2 1.00

0.110 103 2.68 84.4 −2.53 75.0 −1.98

0.120 103 −2.94 93.7 −0.05 84.4 2.21

0.130 112 0.40 103.1 2.42 90.0 2.26

= 38.5 +562fF

= 11 +690fF

= 20 +518f

f (mm/rev) ν = 1.4 m/s ν = 1.2 m/s ν = 0.9 m/s

(N) P

(N) F

(N) P

(N) F

(N) P

(N)

0.070 0.070 −2.81 54.4 2.11 56.2 3.98

0.074 0.074 4.29 58.1 3.40 56.2 1.54

0.084 0.084 2.37 61.9 1.05 52.5 −8.32

0.097 0.097 −1.25 71.2 2.48 76.9 8.11

0.110 0.110 2.63 75.0 −1.74 80.6 3.92

0.120 0.120 −3.04 78.7 −4.07 88.1 5.30

0.130 0.130 4.41 93.7 4.82 96.6 6.69

= 10 +567fF

= 9.5 +611fF

= 9.5 +611f

Energy Partition in the Cutting System 111

considered) is a straight line F

(

)

= C

f . Constants C

and C

were determined

using the experimental data and so the position of the sine wave means were determined

for each test (Table 2.6). Then, using Eq. (2.81), the sinusoidal component of F

was

calculated as

(

)

= F

(

)

−F

(

)

(2.84)

The experimental points for each cutting (deforming) speed were placed in the “f −F

”

coordinate system. Then a sine wave was found using a specially developed curve-ﬁtting

program for the best approximation of the experimental points. Some results are shown

in Fig. 2.20. As such, for the cutting speeds in Table 2.6, the wavelengths of the energy

waves (deformation and thermal) were determined. The results are as follows: 6.3, 6.5,

7.0, 7.6 and 7.8 µm. To conﬁrm the validity of Eq. (2.83), the initial cutting speed was

selected to be ν

= 0.9m/s and thus l

ν1

= 0.56 and l

= 7.8 µm. Using Eq. (2.83), the

wavelengths for other cutting speed were calculated with the following results: 6.336,

6.723, 7.033, 7.348 and 7.780 µm. A fairly good agreement between the calculated and

the experimentally obtained results proves that in solids, energy expands into waves. The

wavelength of these waves depends on the velocity of deformation and can be determined

experimentally. Moreover, a comparison of these results with the frequency of chip

formation obtained for the same conditions (Fig. 1.43) shows that the chip formation

process generated these waves.

To prove that the parameters of the waves do not correlate with the rotational speed of

the workpiece or with other velocities of the moving parts of the machine tool, a special

−10

0.07 0.09 0.11

Feed, f (mm/rev)

−10

Force, Ps (N)

n=2.9m/s

n=2.3m/s

n=1.8m/s

n=1.4m/s

n=1.2m/s

n=0.9m/s

0.13

Fig. 2.20. Sine wave approximation of the experimental data in the coordinate system “f – F

”

(after Astakhov [8]).

112 Tribology of Metal Cutting

test was carried out. It was found from the previous experiments that parameters F

and

are very sensitive to even small changes in the cutting process [2]. On the contrary,

parameters F

and l

depend only on the cutting regime. Accounting for these facts, the

methodology of the current experiment was developed as follows. A stepped workpiece

having shoulders of different diameters was ﬁrst turned on at the smallest diameter using

a cutting speed of 2.4 m/s and the parameters F

and l

ν1

were determined for this

case. Then, the next shoulder was turned keeping the same setup and cutting speed by

reducing the spindle rotational speed (n (rpm)). As such, the parameters F

and l

ν2

were

calculated using, experimental results. Then, the third shoulder of greater diameter was

machined keeping the same conditions including the cutting speed (thus smaller rpm).

Again, the parameters F

and and l

ν3

were determined. Because these parameters were

found the same for all the three runs, it was concluded that they depend only on the

cutting speed and do not correlate with testing particularities.

Proposed methodology of evaluation of cutting force measurements. In order to apply

the ﬁndings discussed in the practical determination of the cutting force, the following

steps are recommended:

Step 1: The ﬁrst series of tests is to be conducted varying the cutting speed ν and

measuring the cutting force F

. As a result, a number of points

(

)

are obtained.

Step 2: The experimental points are entered into a computer and are placed in the

coordinate system ν − F

. A curve-ﬁtting program is used to approximate them with

a sine wave with reasonable accuracy. It is accomplished numerically and/or graphically.

Equation (2.75) describes this sine wave as

= F

sin



2π



ν +ν





, (2.85)

where F

is the mean of the sine wave

z−max

z−min

, (2.86)

where F

is its amplitude

z−max

−F

z−min

, (2.87)

where l

and ν

are its wavelength and initial phase, respectively.

Step 3: The second series of cutting tests is conducted. In this series, N points

(

N = 1 ...n

)

corresponding to different cutting speeds are used. In each point, a number

of tests using different cutting feeds are conducted measuring the cutting force F

Step 4: The experimental results of Step 3 are entered into the computer. Then, N

coordinate systems “f−F

” (one for each cutting speed used in the tests) are formed and

the experimental points corresponding to different cutting feeds are placed in each of

Energy Partition in the Cutting System 113

these systems. Using Eqs. (2.81) and (2.84), one can ﬁnd the wavelength of the energy

waves for each of the cutting speed used in the tests.

Step 5: Using the experimental results of Step 4 for N = 1 and l

determined in Step 1,

and, using Eq. (2.83), the wavelengths of the energy waves for the cutting speed used at

Points 2 ... N are calculated.

Step 6: The wavelengths obtained at Steps 4 and 5 are compared. The parameters of

sine waves obtained in Step 4 are corrected in an iteration procedure until a fairly good

agreement of the wavelengths obtained in Steps 4 and 5 is achieved.

Importance of the interaction of deformation and thermal waves in metal cutting. The

high energy rate and cyclic nature of the chip formation process in metal cutting result

in the generation of deformation and thermal waves. Because these waves are generated

by the same source, namely, the chip formation process (cutting tool), they are coherent

and their interference takes place in the cutting process. This interference affects the

amount of external energy required since, according to the von Mises’ criterion of failure

with physical meaning given by Hencky, the critical value of the distortion energy (the

total strain energy per unit volume) is constant for a given workpiece material. The

revealed existence of interference explains the unexplained phenomena of the metal

cutting process:

• Great scatter in the reported data on cutting force. Even under similar cutting

conditions and with extraordinary care while performing the experiments, scatter in

cutting force measurements exceeds 50% (for example, [53]).

• Foundation of high-speed machining. When the cutting speed increases, the volume

of work material removed per unit time also increases so that the energy spent in

cutting should increase. Moreover, an increase in the cutting speed leads to the

corresponding increase in the strain rate in the chip formation zone. According to

Oxley [54], this rate is in the range from 10

to 10

−1

, or even higher in metal

cutting (discussed in Chapter 1). The available data on materials testing at high

strain rate (for example, [55]) show that the shear ﬂow stress increases dramatically

for many common materials when the rate of strain exceeds 10

. Knowing these

facts, one might expect a signiﬁcant increase in the cutting force when the cutting

speed increases. Particularly, the difference should be very signiﬁcant at high cutting

speeds in the so-called high-speed machining. The practice, however, shows that

opposite is the case. Zorev [12] studied a number of work materials (from low- to

high-carbon steel, low and high alloyed steels) at low and high cutting speeds and

conclusively proved that the cutting force decreases (at different rates for different

work materials) with an increase in the cutting speed. Moreover, the results of

multiple studies on the cutting force in high-speed machining (for example, [15,16])

show a signiﬁcant decrease (30–40%) in the cutting force at high cutting speed and

thus rates of strain. The results presented in this chapter resolve this contradiction.

When the cutting speed increases, the time interval between the two successive tool

positions (Points 1 and 2 in Fig. 2.15) decreases. As such, higher thermal energy

adds to the total energy needed for the fracture of the layer being removed. As a

result, the cutting force decreases. Poor reproducibility of the high-speed machining

114 Tribology of Metal Cutting

results and the inability to reproduce the results obtained by other researchers [56]

can easily be explained by the wave interference phenomena described.

• Inconsistency in tool life. The resource of the cutting tool is deﬁned later in Chapter 4

as the amount of energy that can be transmitted through the cutting wedge (deﬁned

as a part of the tool located between the rake and the ﬂank contact areas) until it

fails [6]. Great inconsistency in tool life, known to the specialists in the ﬁeld, can be

easily explained by the wave phenomena described because the energy transmitted

through the cutting wedge depends on the interaction of the deformation and thermal

waves in the machining zone.

The results obtained offer a novel approach in the selection of optimal cutting regime

in machining, particularly high-speed machining. This regime should be selected so that

the constructive interference of energy waves results in their maximum reinforcement

to reduce the energy needed to accomplish the process and to increase the tool life.

At higher strain rates and cutting speeds, the beneﬁts of the proposed approach are more

signiﬁcant.

2.7.2 Method 2: using the appropriate state of stress

As discussed above, the state of stress in the layer being removed is triaxial even in the

simplest method of machining – orthogonal cutting. As known [57], the state of stress in

the body, which undergoes plastic deformation, affects the fracture strain. For example, it

is known that hydrostatic compression increases the fracture strain and tension decreases

the fracture strain [58]. However, this general knowledge is not sufﬁcient to control the

deformation process. Therefore, a certain generalized parameter characterizing the state

of stress should be selected to study the correlation between the state of stress and the

fracture strain.

One of the most versatile and physically sound criteria for the characterization of the

state of stress in plastic deformation of engineering materials is Π − a criterion deﬁned

as [57]

Π =

(

)



(

)

−3I

(

)

, (2.88)

where I

(

)

and I

(

)

are the ﬁrst and the second stress invariants that is expressed in

terms of the principal stresses σ

, σ

and σ

= σ

+σ

(2.89)

=−

(

+σ

)

(2.90)

Figure 2.21 shows the relationships between the strain at fracture and the state of stress

represented by Π factor. As shown, the strain at fracture and thus the energy needed

for the separation of the layer being removed signiﬁcantly depend on the state of stress

for a wide variety of work materials. It is also shown in Fig. 2.21 that different work

Energy Partition in the Cutting System 115

3.0

2.0

1.0

0.5

0.2

0.1

−8 −8 0

Fig. 2.21. Effect of stress triaxiality represented by the Π criterion on the strain at fracture:

1 – niobioum, 2 – iron, 3 – tungsten, 4 – molybdenum, 5 – beryllium, 6 – magnesium, 7 – zinc,

8 – tin alloy, 9 – brass, 10 – brass alloy, 11 – strain-hardened and 12 – cast lead (afterAstakhov [2]).

materials have different sensitivity to the stress triaxiality. This fact is well known in

material testing [59,60].

The state of stress in the deformation zone depends primarily on the tool geometry so

that, by proper selection of the tool geometry according to the properties of the work

material, one can achieve signiﬁcant reduction in the energy spent in machining. In this

respect, the importance and signiﬁcance of the cutting tool geometry (Appendix A) can

be appreciated for the ﬁrst time as having deﬁned the physical sense as this geometry

forms this state of stress.

Since the tool forms a certain hydrostatic pressure in the layer being removed, it is of

interest to analyze its inﬂuence on the strain at fracture. It was mentioned in [2] that Lode

[61] investigated the validity of the yield criteria using some thin-walled tubes made

of steel, copper and nickel subjected to various combinations of uniaxial tension and

internal hydrostatic pressure. In doing this, he devised a sensitive method to determine

the effect of the intermediate principal stress on yielding. This may be explained as

follows.

Tresca [62] suggested that yielding occurs when the maximum value of the extreme shear

stress in the material, equal to half the difference between the algebraic maximum (σ

)

and minimum (σ

) principal stresses, i.e.

max

= τ

=±

(

−σ

)

(2.91)

116 Tribology of Metal Cutting

attains a critical value. As shown, the Tresca yield criterion requires the maximum and

minimum principal stresses to be known in advance. When applied for yielding in uniaxial

tension where σ

= σ

, which is the uniaxial yield stress of the material, σ

= σ

= 0,

the criterion gives

−σ

= σ

(2.92)

which, if σ

≥ σ

, yields

(

−σ

)

= 1 (2.93)

The intermediate principal stress (σ

) can thus vary from its maximum value σ

= σ

its minimum value σ

= σ

without apparently affecting the yield criterion expressed by

Eq. (2.93). To characterize the inﬂuence of the intermediate principal stress (σ

) Lode

introduced the parameter

2σ

−σ

−(σ

+σ

)/2

(σ

−σ

)/2

(2.94)

which is known as the Lode stress parameter [21].

Equation (2.94) can be rearranged so that

+σ

+µ

−σ

(2.95)

The von Mises yield criterion in terms of the principal stresses (Eq. (2.12)) can be

written as

(

−σ

)

(

−σ

)

(

−σ

)

= 2σ

(2.96)

and if σ

from Eq. (2.95) is substituted, after rearranging and simplifying, the von Mises

yield criterion becomes

(

−σ

)



3 + µ



(2.97)

When σ

= σ

, Eq. (2.94) shows that µ

=−1 and when σ

= σ

, µ

=+1. Because

≥ σ

, it follows that −1 ≤ µ

≤+1. When µ

=−1, the principal stresses

are σ

,σ

= σ

which is the uniaxial tension

(

−σ

)

with hydrostatic stress σ

. When

=+1, the principal stresses are σ

= σ

,σ

, which is the uniaxial compression.

In addition to the stress parameter (µ

) deﬁned by Eq. (2.94), Lode also introduced the

plastic strain parameter, υ

, deﬁned as

2dε

−dε

dε

−dε

dε

−1





dε

+dε







dε

−dε



, (2.98)