COMPONENTS OF CUTTING WORK IN HIGH-SPEED MACHINING OF STEELS

A. M. ROZENBERG

General Considerations

The work expended on the process of cutting metals is spent on the deformation of the chip and of the machined surface, and also on overcoming the friction forces on the front and rear faces of the tool. In addition, an entirely imperceptible part of the work is spent on the formation of a new surface. This part of the work is associated with the surface energy of the metal being machined, constitutes a vanishingly small part of the total cutting work, and may be disregarded. Deformations in the cutting process may be elastic, plastic, and brittle; by brittle deformations we mean those in which a chip is formed that is removed from a brittle metal. In this case, essentially speaking, we are dealing with elastic and insignificant plastic deformations that lead to the destruction of the metal being converted into chip.

All the cutting work may be divided into the work of deformation in the chip shear zone \(A_z\), the work of deformation of the machined surface \(A'_z\), the work of friction on the front face \(A_F\), and the work of friction on the rear face \(A_{F1}\). The deformation work includes both the work of elastic and of plastic deformation; moreover, it makes no sense to separate the work of elastic deformation as an independent term, on the one hand because of the difficulty of accounting for it, and on the other hand because of its small magnitude in comparison with the work of plastic deformation, especially in the cutting of ductile metals. The deformation of the machined surface (the cut surface) is in part the result of a wave of plastic deformation, preceding the cutting edge of the tool, propagating from the shear plane forward and somewhat inward into the region of the machined surface; it is in part the result of the work of the friction forces on the rear face of the tool during the interaction of the rear face with the cut surface. This latter part is performed by the work of friction on the rear face. Therefore it is advisable not to take the work of deformation of the machined surface (more precisely, of the cut surface) separately into account; rather, part of it should be considered included in the work of chip deformation, while another part should be considered expressed by the work of the forces on the rear face of the tool.

Thus it may finally be considered that the entire cutting work \(A\) is summed from the following essential and appreciable components: the work of deformation \(A_z\), performed in the chip shear zone, the work of friction on the front face \(A_F\), and the work of friction on the rear face \(A_{F1}\), that is,

\[ A = A_z + A_F + A_{F1}. \tag{1} \]

It should be noted that the work of friction on the front face of the tool, expended on overcoming the friction forces between the chip and the front face, also carries out deformation of the chip in the zone of the built-up ...

layer, and from this point of view it could have been included in the work of chip deformation; however, it is better to leave it as an independent part, since the deformation of it that takes place in the near-tool layer of the chip is secondary, superimposed on the primary deformation acquired by the chip in the shear zone. Thus, the deformation work \(A_t\) is expended on deformation of the chip in the shear zone, and the result of this work is the release of deformation heat and heating of the metal in the shear zone. The work of friction on the rake face \(A_F\) is expended on secondary deformation of the near-tool layer, and its result is additional heating of the near-tool layer of the chip, which, together with the heating of the chip in the shear zone, creates one or another temperature on the rake face of the tool. The work of friction on the flank face \(A_{F_1}\) gives rise to heat release in the contact zone of the flank face with the cut surface and, together with heating of the cutting part of the tool from its rake face, creates the temperature on the flank faces.

Wear of the tool on the rake and flank faces is undoubtedly associated with the friction works \(A_F\) and \(A_{F_1}\), performed on these faces by the friction forces \(F\) and \(F_1\), and therefore the friction works are of independent interest as applied to questions of tool life and tool wear.

Existing Studies

The division of cutting work into its component parts is of considerable interest. This is indicated by the fact that already in the works of Briks [1], and then of Rudnik [2], the cutting force, and hence also the cutting work, is presented as consisting of two heterogeneous parts, one of which is proportional to the cross section of the layer being cut, while the other is proportional to the semiperimeter of the cross section of the cut.

The first approximate percentage ratios between the individual components of cutting work were proposed by Prof. Glebov [3]. According to his data, these ratios are as follows:

This division was proposed by Glebov on the basis only of hypothetical assumptions, since he did not have at his disposal the materials necessary for a more or less accurate calculation of such essential elements as the friction force and the work of friction. We already pointed out at the time that Glebov clearly overestimated the role of cutting work, which should be vanishingly small [4].

The first scheme showing the change in the individual components of cutting work with changes in cutting speed was given by Prof. Kasyan [5]. This scheme is cited in a number of works and is presented by us in Fig. 1.

From Prof. Kasyan’s scheme it follows that, when cutting steel, the work of deformation may vary within very wide limits, from 3 to 84%, while the work of friction on the rake and flank faces varies from 16 to 97%, with-

whereas the work of friction on the flank face by itself may amount to from 8 to 90%. The share of the work of friction on the rake face is comparatively small and amounts to from 8 to 30% of the total cutting work.

As we see, Kassyan’s scheme is quantitatively incomparable with Glebov’s data.

Fig. 1

The calculation of the separate components of cutting work for steels on the basis of experimental data was carried out by Klyushin [6]; moreover, within the range of variation of the cutting speed from 3 to 143 m/min he obtained the following variations:

Here the extreme limits of variation are given for two steels; therefore the sum may not amount to 100%.

Klyushin’s data are completely incomparable with Kassyan’s data and differ very substantially from the values proposed by Glebov with respect to the work of friction, which is estimated by Glebov at 10—15%, whereas by Klyushin it is 34—54%.

Noteworthy data on the work of cutting are available from Zorev [7]. On the basis of experimental materials, Zorev calculated the specific works of deformation and the specific works of friction on the rake face of the tool when cutting steel 40, and related them to the temperature on the rake face. Specific work is the quotient obtained by dividing the total work by the volume of metal converted into chip. Therefore, at a constant cross section of the layer being removed, the ratios of the total and specific works are identical, and it is immaterial whether one compares total or specific works. In Fig. 2 we present Zorev’s graph showing the change in the specific work of friction on the rake face, and in Fig. 3 a graph showing the change in the specific work of deformation.

The work of friction on the flank face is not presented by Zorev; therefore, from Zorev’s data one can judge two components, \(A_t\) and \(A_F\). Figs. 2 and 3 show that, if the sum \(A_t + A_F\) is taken as 100%, the ratios of these separate components to their sum may be as follows:

Fig. 2

In the figure: vertical axis \(H_{wF}\), kg·mm/mm³; horizontal axis—Temperature, \(t^\circ C\). Legend: \(o\, S=0.125\), \(\triangle S=0.39\); \(+\, S=0.195\), \(\bullet S=0.49\); \(\times S=0.285\), \(\odot S=0.78\).

Fig. 3

In the figure: vertical axis \(H_{wF}\), kg·mm/mm³; horizontal axis—Temperature, \(t^\circ C\).

If it is assumed that the work of friction on the flank face amounts to approximately 10% of the total work, then the ratios of the deformation work and the work of friction to the total cutting work will be somewhat lower. Thus, according to Zorev’s data, in those cases when the work of friction on the flank face is relatively small (this will occur with thick cuts), the share of deformation work in the total cutting work may be estimated at 60–66%, and the share of the work of friction on the rake face at 30–24%. These values, as we see, approximately correspond to those obtained by Klushin and are completely incomparable with Kas’yan’s scheme.

According to Zorev, with an increase in the temperature on the rake face, i.e., with an increase in cutting speed, the share of deformation work changes only slightly and always remains greater than the work of friction on the rake face. The same follows from Klushin’s data. According to Kasyan’s scheme, however, the share of deformation work, as the cutting speed changes, may vary by more than 20 times, and it may be either several times greater than, or several times less than, the work of friction on the rake face of the tool.

Thus, the brief materials cited show that there are obvious contradictions in this question. We have a large amount of experimental material collected in the research work of the Department of Machine Tools and Metal Cutting, which makes it possible not only to establish the actual ratios of the individual components of cutting work, but also to draw certain generalized conclusions. All our conclusions will apply, in cutting steels, to that range of medium and high cutting speeds in which built-up edge on the tool is absent.

Initial relationships and notation

Both the total cutting work \(A\) and its component parts \(A_{\tau}\), \(A_F\), and \(A_{F_1}\) can be expressed analytically. If the work per minute is determined, the following expressions are obtained:

\[ A = P_z \cdot V, \tag{2} \]

\[ A_F = F \cdot V_F = F \frac{V}{\zeta}, \tag{3} \]

\[ A_{F_1} = F_1 \cdot V, \tag{4} \]

\[ A_{\tau} = P_{\tau} \cdot V_{\tau}. \tag{5} \]

Fig. 4

Here (Figs. 4 and 5):

\(V\) — cutting speed,
\(V_F\) — speed of motion of the chip along the rake face of the tool,
\(V_{\tau}\) — speed of motion in the shear plane of the chip,
\(\zeta\) — chip shrinkage,
\(P_z\) — principal component of the cutting force,
\(F\) — friction force on the rake face,
\(F_1\) — friction force on the flank face; it acts in the direction of the cutting speed \(V\),

$P_{\tau}$ — the force in the shear plane, which effects the shear of the metal during chip formation.

Fig. 5

It is convenient to carry out all subsequent derivations and reasoning using specific works, i.e., works referred to a unit volume of metal converted into chip. Let us call: the specific cutting work $A_w$,

the specific work of deformation in the chip shear zone $A_{w\tau}$,

the specific work of friction on the rake face $A_{wF}$,

the specific work of friction on the flank face $A_{wF_1}$.

Then:

\[ A_w=\frac{A}{abv}=\frac{A}{stv}=\frac{P_z}{ab}=\frac{P_z}{st}. \tag{6} \]

\[ A_{wF}=\frac{A_F}{abv}=\frac{A_F}{stv}=\frac{F}{ab\zeta}=\frac{F}{st\zeta}. \tag{7} \]

\[ A_{wF_1}=\frac{A_{F_1}}{abv}=\frac{A_{F_1}}{stv}=\frac{F_1}{ab}=\frac{F_1}{st}. \tag{8} \]

\[ A_{w\tau}=\frac{A_\tau}{abv}=\frac{A_\tau}{stv}=\frac{P_\tau V_\tau}{abv}=\frac{P_\tau V_\tau}{stv}. \tag{9} \]

Here $a,b$ are the thickness and width of the cut,

$s,t$ are the feed and depth of cut.

In addition, the specific cutting work $A_w$ may be represented as consisting of two parts—the specific work performed by the rake face of the tool $A_{wn}$ and the specific work on the flank face $A_{wz}$.

In this case

\[ A_w=A_{wn}+A_{wz}, \]

\[ A_{wn}=A_{w\tau}+A_{wF}, \]

\[ A_{wn}=\frac{P_1v}{abv}=\frac{P_1}{ab}=\frac{P_1}{st}, \]

\[ A_{wz}=A_{wF_1}. \]

Here $P_1$ is the part of the force $P_z$ applied on the rake face,

\[ P_1=P_z-F_1. \]

The specific works on the rake and flank faces of the tool have different laws of variation when the cutting parameters are changed.

This is quite natural, since the first of them is connected with the process of chip deformation and therefore must depend on chip shrinkage, the degree of its deformation, the tool rake angle, friction on the rake face, and wear of the rake face. The work on the flank face, however, is not explicitly connected with the processes occurring on the rake face of the tool. The force \(F_1\), which determines the work on the flank face, does not depend to any noticeable degree on the tool rake angle, on the degree of chip deformation, or on the thickness of the layer being cut, and is to a considerable extent connected with the amount of dulling on the flank face. Let us first determine the relative magnitude of the specific work on the flank face.

Specific work on the flank face of the tool

The magnitude of the specific work on the flank face can be calculated if the force \(F_1\) is known. The force \(F_1\) can be obtained only experimentally by special methods for determining forces on flank faces [7, 8]. Equation (8) shows that \(A_{WF1}\) depends on \(F_1\), \(a\), and \(b\). On the basis of a number of investigations carried out [9, 10], it may be assumed that, with ordinary chips, when the width is considerably greater than the thickness of the layer being cut, the friction force on the flank face does not depend on the chip thickness or on the tool rake angle and is directly proportional to the width of cut. Therefore, in accordance with equation (8), it may be concluded that the specific work on the flank face will not change when the width of cut \((b, t)\) changes and will be inversely proportional to the chip thickness.

For very small chip thicknesses, which may occur in the operation of milling cutters, reamers, broaches, the specific work \(A_{WF1}\) may be very large. In grinding operations, when especially thin chips are removed, the work on the flank face of the cutting element constitutes the principal and largest part of all the work expended on the cutting process. Conversely, with large chips, in the operation of turning tools, drills, counterbores, and progressive broaches, the specific work on the flank faces will be very small and may constitute an entirely insignificant part of the total cutting work.

Fig. 6

The forces on the flank face of a sharp cutting element are very small; in cutting steels they are usually measured in several kilograms per \(1\ \text{mm}\) of cutting-edge length. With increasing cutting speed and temperature they usually increase somewhat. This is illustrated by Fig. 6, which shows the change in the force \(F_1\) in high-speed face milling with end mills of steels 10 and 40X, with feed per tooth \(0.105\ \text{mm}\) and milling depth (width of cut) \(3\ \text{mm}\). Fig. 6 shows that per \(1\ \text{mm}\) of cutting-edge length there are from 2.0 to 4 kg in milling steel 10 and from 5.0 to 6.0 kg in milling steel 40X [11].

Using such experimental materials, it is not difficult to calculate the specific work on the flank face. In Fig. 7 we have presented a graph of the change in the total specific work \(A_W\) and the specific work on the flank face \(A_{WF1}\) as a function of speed.

of cutting and feed per tooth in high-speed milling of steel 10 with an end mill [11]. This graph shows that, with an increase in the feed per tooth \(S_z\), both the total specific work expended on the entire cutting process and the specific work on the flank face decrease considerably.

It is of interest to show the specific work on the flank face as a percentage of the entire specific cutting work. This is done in Fig. 8, which is constructed from Fig. 7; here the change in the specific work on the flank face is shown as a percentage of the total specific work, as a function of the feed per tooth \(S_z\), for two cutting speeds. Fig. 8 shows that with very thin chips the specific work on the flank face may amount to more than 50% of the total, whereas with thick chips its share in the total work becomes

Fig. 7

negligible. Thus, already at \(S_z = 0.2\) mm, \(A_{WF_1}\) amounts to 1–7% of the total work; if the thickness of cut is increased further, then the specific work on the flank face will continue to decrease both in absolute value and relative to the total work.

The same is shown in Fig. 9 for the case of high-speed turning of steel ShKh15 [10], with the only difference that here the graph is plotted not for a constant cutting speed, but for conditions of constant chip shrinkage, i.e., constant temperature on the rake face.

Fig. 8

It follows from Fig. 9 that, beginning with a feed \(S = 0.15\text{–}0.2\) mm/rev, the specific work on the flank face becomes so insignificant in the total work

in cutting work, that in the case of machining steel with a sharp cutting tool it may be neglected in calculating cutting forces, power, and cutting work.

This had already been experimentally proved by us earlier for the case of high-speed cutting of 8 different grades of carbon and alloy steels [12].

Fig. 9

As Figs. 8 and 9 show, in milling and turning approximately the same ratios are retained between the specific works on the rear face and the total specific cutting works. This is entirely natural, since the nature of the forces on the rear face in all cutting processes must be the same, for the forces on the rear face are elastic-plastic forces of contact of the rear-face region in contact with the cutting surface.

As the experiments of Yu. P. Zimin on steel and Yu. A. Rosenberg on cast iron have shown, the forces on the rear face increase approximately in proportion to the hardness of the metal being machined. In the same way, the specific work on the rear face \(A_{WF_1}\) will also increase. But its ratios to the total specific cutting work will, for all steels, be approximately the same as is shown in Figs. 8 and 9, because with an increase in the hardness of the metal being machined not only the forces and the work on the rear face will increase, but, in parallel with this, also the forces and work on the front face. Therefore, the conclusion that the share of the specific work on the rear face in the total cutting work is insignificant when removing chips of considerable thickness may be extended to all steels.

Taking this into account, the erroneousness of Kasyan’s scheme becomes obvious: in it the share of work on the rear face reaches 70% of the entire cutting work, all the more so because Kasyan’s scheme was constructed for the case of operation with a fairly large feed \((S=0.466\ \text{mm/rev})\).

On the basis of the fact that, with the usually employed feeds and depths of cut, the share of the work on the rear face for a sharp tool is very small, in what follows we shall consider separately the works of deformation and friction on the front face, and establish their mutual ratios without taking into account the work on the rear face. This is all the more convenient because, for the work on the front face and their ratios, we shall be able to draw certain generalized conclusions and connect these works also with the physicomechanical characteristics of the metal being machined, which we cannot do with respect to the work on the rear face.

Specific works of deformation and friction on the front face of the tool

For the specific work of deformation in the chip shear zone, equation (9) was introduced. In this equation, according to Figs. 4 and 5,

\[ P_{\tau}=P_{1}\frac{\cos(\omega+\beta_{1})}{\cos\omega}, \]

\[ \frac{V_{\tau}}{V}=\frac{\cos\gamma}{\cos(\beta_1-\gamma)}, \]

\[ A_{w\tau}=\frac{P_1}{ab}\cdot\frac{\cos(\omega+\beta_1)}{\cos\omega}\cdot\frac{\cos\gamma}{\cos(\beta_1-\gamma)}. \]

If it is taken into account that the tangential stress \(\tau\) in the shear plane is

\[ \tau=\frac{P_{\tau}\sin\beta_1}{ab} =\frac{P_1}{ab}\cdot\frac{\cos(\omega+\beta_1)}{\cos\omega}\sin\beta_1, \]

then

\[ A_{w\tau}=\tau\frac{\cos\gamma}{\cos(\beta_1-\gamma)\sin\beta_1}. \]

Here

\[ \frac{\cos\gamma}{\cos(\beta_1-\gamma)\sin\beta_1} = \frac{\zeta^2-2\zeta\sin\gamma+1}{\zeta\cos\gamma} =\varepsilon, \]

where \(\varepsilon\) is the relative shear of the metal during chip formation. Therefore

\[ A_{w\tau}=\tau\varepsilon. \tag{10} \]

Thus, the specific work of deformation in the shear zone during cutting is equal to the product of the tangential stress and the relative shear.

In addition, we have shown experimentally that the specific work of deformation in cutting plastic metals is equal to the specific work of deformation in compression \([13, 12]\). Owing to this, we can express the specific work of deformation in cutting through the physico-mechanical characteristics of the metal being machined and obtain the relation

\[ A_{w\tau}=\frac{\sigma_0}{n}\left\{e^{\frac{n\varepsilon}{1.5}}-1\right\}. \tag{11} \]

Here \(e\) is the base of natural logarithms, and \(\sigma_0\) and \(n\) are the parameters of the stress polytrope in compression, with \(\sigma_0\) being the conditional yield strength in compression and \(n\) the strain-hardening exponent.

The parameters of the stress polytrope in compression relate the stress to the degree of deformation \(\frac{h_0}{h}\) in compression according to

\[ \sigma=\sigma_0\left(\frac{h_0}{h}\right)^n, \]

where \(h_0\) is the height of the specimen before compression, and
\(h\) is the height of the deformed specimen for which the stress \(\sigma\) is determined.

\(\sigma_0\) and \(n\) can be determined by a compression test of a specimen made of the metal being machined \([14]\). In compression, the initial dimensions of the specimen should be taken as \(h_0=2.5d_0\), in order to reduce the influence of friction at the ends on the stressed state.

Equation (11) was verified by numerous experiments on 9 different grades of steel, copper, and aluminum.

According to equation (11), the specific work of deformation in cutting depends on the degree of deformation \(\varepsilon\) and on the physicomechanical characteristics of the metal, \(\sigma_0\) and \(n\). As all these quantities increase, the specific work of deformation increases. Since the relative shear \(\varepsilon\) is a function of the chip shrinkage \(\zeta\) and the rake angle \(\gamma\), \(A_{w\tau}\) is also dependent on these quantities.

If we carry out the cutting process for metals for which \(\sigma_0\) and \(n\) depend on the rate of deformation, then for these metals the cutting speed itself will have an effect on the specific work of deformation. An example of such a metal is aluminum.

Equation (11) may be rewritten in a more expanded form as

\[ A_{w\tau}=\frac{\sigma_0}{n} \left\{ e^{\frac{n}{1.5}\cdot \frac{\zeta^2-2\zeta\sin\gamma+1}{\zeta\cos\gamma}} -1 \right\}. \tag{12} \]

Equation (12) clearly shows on what factors the specific work of deformation in cutting depends.

Naturally, the factors of the cutting process which, in one way or another, affect the chip shrinkage \(\zeta\) will correspondingly affect the specific work of deformation in cutting as well.

Equation (12) shows that the specific work of deformation does not depend on the dimensions and shape of the cross section of the cut—on the depth of cut, feed, width and thickness of the cut, or the angle in plan. With unchanged shrinkages, a change in the dimensions of the cut section will not change the specific work of deformation \(A_{w\tau}\), while the total work of deformation \(A_{\tau}\) will in this case be directly proportional to the dimensions of the cross section of the cut:

\[ A_{\tau}=ab\,\frac{\sigma_0}{n} \left\{ e^{\frac{n}{1.5}\cdot \frac{\zeta^2-2\zeta\sin\gamma+1}{\zeta\cos\gamma}} -1 \right\}. \tag{13} \]

As is known, we obtain an unchanged shrinkage when the cross section of the cut is varied under conditions of constant temperature on the rake face of a tool of given geometry, since in this case the coefficient of friction on the rake face, the direction of the resultant force on the rake face, the position of the shear plane, and the chip shrinkage remain unchanged. Thus, regimes of constant temperature are characterized not only by constant coefficients of friction, by the direction of the resultant, by the position of the shear plane, and by chip shrinkage, but also by a constant value of the specific work of deformation in the chip shear zone. This follows not only from equation (12), but also from equation (10), since with constant \(\zeta\) and \(\gamma\), the relative shear \(\varepsilon\) is also constant, and with it the shear stress \(\tau\) in the shear plane is constant, which leads to constancy of \(A_{w\tau}\).

According to equation (12), with known \(\sigma_0\), \(n\), \(\gamma\), and \(\zeta\), the specific work of deformation can be calculated. This will be the specific work calculated from the physicomechanical characteristics of the metal being machined. If, on the other hand, cutting tests are carried out with measurement of the forces and chip shrinkage, then from the results of the tests one can calculate the shear stress \(\tau\) and the relative shear \(\varepsilon\). In this case the specific work can be determined from equation (10), and this will be an experimentally obtained value of the specific work.

Figure 10 shows the effect of chip shrinkage on the specific work of deformation in cutting steel 12KhN3A, which has the physicomechanical characteristics \(\sigma_0=78\ \text{kg}/\text{mm}^2\) and \(n=0.165\). In Fig. 10 the solid curve is constructed from equation (12), i.e., from physicomechanical characteri-

tics of the metal obtained from compression tests. The points located on the graph were obtained experimentally, and their arrangement confirms the correctness of equation (12).

Fig. 10

In Fig. 11, for this same steel, we show the change in chip shrinkage and in the specific work of deformation with change in cutting speed. Similar graphs could be presented from the experimental material available to us for 9 different steels, copper, and aluminum.

Fig. 11

Let us now proceed to the derivation of an equation for the specific work of friction on the rake face of the tool, \(A_{WF}\). In one of our works it was proved [13] that the sum of the specific works of deformation and friction on the rake face is equal to the specific work \(A_{Wn}\) performed by the rake face of the tool.

Therefore

\[ A_{w\tau}+A_{wF}=A_{wn}, \]

that is,

\[ A_{wF}=A_{wn}-A_{w\tau}, \]

\[ A_{wF}=\frac{P_1}{ab}-A_{w\tau}. \tag{13a} \]

Here \(A_{wn}\) is the specific work expended on the formation and removal of the chip.

For the force \(P_1\), applied on the rake face, we have derived and confirmed experimentally the equation

\[ P_1=\frac{\sigma_0}{n}\,ab\, \frac{e^{\frac{n\varepsilon}{1.5}}-1} {1-\dfrac{\sin\eta}{\zeta\cos(\eta-\gamma)}} . \tag{14} \]

Here \(\eta\) is the angle of friction between the chip and the rake face. Substituting (11) and (14) into (13a), we obtain

\[ A_{wF}=\frac{\sigma_0}{n}\, \frac{e^{\frac{n\varepsilon}{1.5}}-1} {1-\dfrac{\sin\eta}{\zeta\cos(\eta-\gamma)}} -\frac{\sigma_0}{n}\left(e^{\frac{n\varepsilon}{1.5}}-1\right), \]

\[ A_{wF}=\frac{\sigma_0}{n}\left(e^{\frac{n\varepsilon}{1.5}}-1\right) \frac{\sin\eta}{\zeta\cos(\eta-\gamma)-\sin\eta}. \tag{15} \]

Equation (15) shows that \(A_{wF}\) depends on the physico-mechanical characteristics of the metal, the rake angle, the chip compression ratio, and the friction angle \(\eta\) between the chip and the rake face; moreover, the dependence of the specific work of friction on the physico-mechanical characteristics is exactly the same as that of the specific work of deformation.

It is possible, in this expression, to introduce, instead of the friction angle \(\eta\), the value of the coefficient of friction \(\mu\).

After transformations we obtain that

\[ \frac{\sin\eta}{\zeta\cos(\eta-\gamma)-\sin\eta} = \frac{\tan\eta}{\zeta(\cos\gamma+\tan\eta\sin\gamma)-\tan\eta}, \]

since

\[ \tan\eta=\mu, \]

then

\[ A_{wF}=\frac{\sigma_0}{n}\left(e^{\frac{n\varepsilon}{1.5}}-1\right) \frac{\mu}{\zeta(\cos\gamma+\mu\sin\gamma)-\mu}. \tag{16} \]

If we compare equation (16) with (11), we obtain the relation between the specific works of friction and deformation

\[ A_{wF}=A_{w\tau}\, \frac{\mu}{\zeta(\cos\gamma+\mu\sin\gamma)-\mu}. \tag{17} \]

Equation (17) shows that the relation between \(A_{wF}\) and \(A_{w\tau}\) is determined by the coefficient of friction on the rake face \(\mu\), the chip compression ratio, and the rake angle of the tool. It is not possible to consider the influence of each of these factors separately on the ratio of the specific works, since in

in the cutting process the changes in these factors are interrelated—changing \(\gamma\) always leads to a change in \(\mu\) and \(\zeta\), and changing \(\mu\) inevitably changes \(\zeta\).

Equations (15) and (16) show that the specific work of friction does not depend directly on the dimensions and shape of the layer being cut. A change in the dimensions and shape of the cross section of the cut will not change the specific work of friction on the rake face, provided that \(\mu\), \(\zeta\), and \(\gamma\) do not change. This condition is satisfied by constant-temperature regimes for a tool of constant geometry.

Equations (15) and (16) are general relationships for the specific work of friction on the rake face of the tool. Let us show how satisfactorily they are confirmed by experiments. Figure 12 shows the change in \(\mu\) with the change in chip contraction recorded in experiments. From these experimental values of \(\mu\) and \(\zeta\), taking for steel 12KhN3A the values \(\sigma_0 = 78\ \mathrm{kg/mm^2}\) and \(n = 0.165\), determined by compression tests, one can, using equation (16), calculate the specific work of friction \(A_{WF}\). This specific work, calculated from the physicomechanical characteristics of the metal, is represented in Fig. 12 by the solid curve. In addition, from the experimental values of the cutting forces one can directly determine the friction force on the rake face and, from it, calculate the specific work of friction, bypassing the physicomechanical characteristics of the metal. These values of the specific work are represented in Fig. 12 by crosses.

Fig. 12

As can be seen from Fig. 12, the position of the crosses relative to the curve is quite satisfactory, and the greatest deviations do not reach 10%.

Figure 13 shows the change in \(A_{WF}\) as a function of cutting speed. Here again the solid curve is drawn according to equation (16); the crosses are the result of calculations from experimental data. Figures 12 and 13 show that the specific work of friction on the rake face, over a wide range of cutting speeds and with substantial change in chip contraction and

Fig. 13

the coefficient of friction changes very little, increasing with an increase in chip shrinkage, because this is accompanied by a considerable increase in the coefficient of friction, and decreasing with an increase in cutting speed, since in this case \(\mu\) and \(\zeta\) decrease. The change in the specific work of friction with a one-and-a-half-fold change in the coefficient of friction and a twofold change in shrinkage does not exceed 20%.

Fig. 14

The same circumstance is confirmed by Fig. 14, which gives a graph of the change in the specific works \(A_{W\tau}\) and \(A_{WF}\) as functions of the temperature on the rake face when cutting ShKh15 steel, according to the experiments of Yu. P. Zimin [10]. With an increase in the temperature on the rake face, i.e., with an increase in cutting speed, the specific work of deformation decreases, while the specific work of friction remains constant. It should be noted that Fig. 14 contains points for five different feeds; moreover, at a constant temperature the magnitude of the specific work does not depend on the feed, as was already indicated earlier in the analysis of equations (12) and (16).

Thus we have shown that equations of general form, relating the specific works of deformation \(A_{W\tau}\) and of friction on the rake face \(A_{WF}\) to the physicomechanical characteristics of the metal being machined and to the parameters of the cutting process, are satisfactorily confirmed by experiment.

Ratios of the Individual Components of the Work on the Rake Face of the Tool

It is not difficult to express the ratios of the works of deformation and friction to the total work performed by the rake face of the tool. From the general equations given earlier we obtain:

\[ \frac{A_{W\tau}}{A_{Wn}} = 1 - \frac{\sin \eta}{\zeta \cos(\eta-\gamma)}, \tag{18} \]

\[ \frac{A_{WF}}{A_{Wn}} = \frac{\sin \eta}{\zeta \cos(\eta-\gamma)} . \tag{19} \]

Equations (18) and (19) show that the relative magnitudes of the specific works do not depend on the physicomechanical characteristics of the material being machined and are completely determined by three parameters: the friction angle \(\eta\) (or the coefficient of friction \(\mu\)), the rake angle \(\gamma\), and chip shrinkage \(\zeta\). For known values of these parameters, equations (18) and (19) can be used to determine what part of the work on the rake face of the tool is made up by the work of deformation and the work of friction. Since, with changes in cutting speed and in the temperature on the rake face during the cutting process, will—

will change $\zeta$, $\mu$, and $\eta$, then the ratios of the specific works determined by equations (18) and (19) will also change. These ratios can be made dependent only on the chip reduction according to the following considerations. The chip reduction $\zeta$, for a given rake angle $\gamma$, is completely determined by the position of the shear plane, which depends on the direction of the resultant force on the rake face of the tool, determined in turn by the coefficient of friction $\mu$ or the friction angle $\eta$. Thus it follows that, for definite values of $\gamma$ and $\zeta$, the angle $\eta$ (or the coefficient of friction $\mu$) must have a definite value, and the ratios of the works must likewise have a definite magnitude.

At the same time, in the region of high-speed cutting (without built-up edge on the rake face of the tool), the mechanical characteristics and grade of the steel being machined should not noticeably affect the values of $\mu$ and $\eta$ for given $\gamma$ and $\zeta$. Experiments confirm these assumptions. Figure 15 gives experimentally obtained plots of the coefficients of friction in high-speed cutting of various steels with tools having different rake angles. The plots show that, regardless of the type of steel, for a given $\gamma$ the coefficient of friction $\mu$ (and hence also the friction angle $\eta$) can be related to the chip reduction. Figure 16 presents the dependence

Fig. 15

\[ 1-\frac{\sin \eta}{\zeta \cos(\eta-\gamma)}=f(\zeta,\gamma), \]

which was obtained on the basis of Fig. 15.

Naturally, Fig. 16 gives a graphical representation of equation (18); therefore the ratio

\[ \frac{A_{w\tau}}{A_{wn}} \]

is placed on the ordinate axis.

From Fig. 16, knowing $\zeta$ and $\gamma$, one can determine the indicated ratio, which will indeed be sufficiently accurate for any steel in the region of high-speed cutting.

Figure 17 presents the plot

\[ \frac{A_{wF}}{A_{wn}}=f(\zeta,\gamma). \]

It can be constructed using Fig. 16, since

\[ \frac{A_{wF}}{A_{wn}}=1-\frac{A_{w\tau}}{A_{wn}}. \]

Figures 16 and 17 give a clear picture of the change in the component parts of the cutting work on the rake face of the tool. With an increase in the rake angle $\gamma$, the share of deformation work decreases and the share of friction work increases. With an increase in chip reduction, the share of deformation work increases and the share of friction work decreases. Within the limits of chip-reduction values encountered in practice, and at rake angles from $20^\circ$ to $-10^\circ$, the deformation work

may amount to from 60 to 86%, while the work of friction is from 14 to 40% of the total work performed by the rake face of the tool. Under these conditions the work of deformation always exceeds the work of friction.

If the regularity of the change in chip shrinkage with change in cutting speed is known or has been obtained experimentally, then on the basis of

Fig. 16

In the figure: vertical axis—ratio \(A_{\mathrm{pl}}/A_{N\gamma}\), “Ratio”; horizontal axis—chip shrinkage \(\xi\). Curves are marked \(\gamma=-10^\circ\), \(\gamma=0^\circ\), \(\gamma=+10^\circ\), \(\gamma=+20^\circ\). Inside the plot: “High-speed cutting of steels.”

Fig. 17

In the figure: vertical axis—ratio \(A_{M\gamma}/A_{N\gamma}\), “Ratio”; horizontal axis—chip shrinkage \(\xi\). Curves are marked \(\gamma=20^\circ\), \(\gamma=10^\circ\), \(\gamma=0^\circ\), \(\gamma=-10^\circ\). Inside the plot: “High-speed cutting of steels.”

Figs. 16–17 make it simple to determine the ratio of the specific works on the rake face of the tool. In Fig. 18 are presented the experimentally obtained curves of chip shrinkage in cutting steel 10 with \(\gamma=-10^\circ\) and \(\gamma=0^\circ\). According to these

curves, graphs were constructed of the balance of work on the rake face of the tool. The main part is the work of deformation in the chip shear zone, which at $\gamma=-10^\circ$ amounts to from 80 to 87%, and at $\gamma=+20^\circ$ to from 69 to 76%. With a change in cutting speed, the change in the specific proportion of the deformation and friction works is comparatively small and does not exceed 10%. With increasing cutting speed the specific proportion $A_{w\tau}$ decreases somewhat; the specific proportion $A_{wF}$, on the contrary, increases somewhat. It may be assumed that, if we carried out experiments with a further increase in cutting speed, we would subsequently have a constant ratio of the specific works, since the shrinkage and the coefficient of friction, with further increase of speed, cease to decrease and their values stabilize. Thus, in the region of very high cutting speeds (300–400 m/min), one may expect an approximately constant ratio between the specific works $A_{w\tau}$ and $A_{wF}$. Since at these speeds, for any tool geometry, shrinkages usually have the order $\zeta=1.8$–$2.5$, the work of deformation will amount to approximately 65–80%, and the work of friction to 35–20%.

Fig. 18

If the ratios of the specific works presented in Figs. 16 and 17 are referred not to the chip shrinkage $\zeta$, but to the relative shear $\varepsilon$, connected with chip shrinkage by the relation

\[ \varepsilon=\frac{\zeta^2+2\zeta\sin\gamma+1}{\zeta\cos\gamma}, \]

then the graph shown in Fig. 19 is obtained.

Here the relative shear $\varepsilon$ is plotted along the abscissa axis, and percentages along the ordinate axis. For different rake angles, differently designated points are plotted; moreover, it can be seen that if some mean curve is drawn through these points, the points will deviate from it by no more than 3–4%. Therefore it may be considered that the curve will satisfy all rake angles from $+20$ to $-10^\circ$ sufficiently well. This curve shows what proportion of the work on the rake face $A_n$ is constituted by the deformation work $A_\tau$, in percent. Naturally, the complement to 100% belongs to the work of friction on the rake face. Thus, for example, at relative shear $\varepsilon=2$ the work of deformation amounts to 68.5%; the remainder, i.e., 31.5%, is the work of friction on the rake face. At $\varepsilon=6$, respectively, the shares of these works are 88 and 12%.

Fig. 20 shows that, in ordinary cases of high-speed cutting, when $\varepsilon > 1.5$, the work of deformation always exceeds the work of friction on the rake face, and by an amount that is greater the larger the degree of chip deformation expressed by the relative shear $\varepsilon$. With an increase in cutting speed the degree of deformation decreases, and in the work performed by the rake face the share of the work of friction will increase and the share of the work of deformation will decrease. The absolute values of these works will decrease with increasing cutting speed; moreover, the work of deformation $A_c$ will decrease very substantially, so that its magnitude may change by a factor of 2 or more, while the work of friction $A_f$ will decrease comparatively little.

Fig. 19

On the basis of the material presented, the dependences of a general character that have been obtained, and the graphs reflecting them, we can give a summary diagram of the division of the cutting work into its component parts. This is done in Fig. 20 for the case of cutting ShKh15 steel with rake angle $\gamma = 10^\circ$ and feed $S = 0.43\ \mathrm{mm/rev}$. At the top is shown the chip-shrinkage curve obtained experimentally. The presence of this curve is quite sufficient to determine, from Figs. 16 and 17, the ratios of the works of deformation and friction on the rake face. Since a fairly large feed has been taken ($S = 0.43\ \mathrm{mm/rev}$), the share of the work on the flank face will be very small and, using the materials presented above,

Fig. 20

will decrease approximately as significantly as the specific work \(A_{W\tau}\).

The temperature in the friction zone on the rake face of the tool depends on the sum of the specific works of deformation and friction, i.e., on the specific work performed by the rake face \(A_{Wn}\). However, despite the fact that with increasing cutting speed the magnitude \(A_{Wn}\) decreases significantly, the temperature on the rake face invariably rises with increasing cutting speed; moreover, at high speeds it exceeds the temperature in the shear zone by a factor of 2–3, although the specific work of friction \(A_{WF}\) amounts to only 30–40% of the work of deformation \(A_{W\tau}\). This is explained by the following reasons.

Heating of the chip in the shear zone is effected by the heat released as a result of the work of deformation. Here it is heated through to a temperature proportional to the specific work of deformation \(A_{W\tau}\). The heated metal of the chip, in the process of friction with the rake face, is additionally heated by the heat released as a consequence of those additional deformations that arise as a result of friction against the rake face. Friction exerts a braking action on the contact layer of the chip, owing to which a layer of a certain thickness proves to be additionally deformed to a very considerable degree. The thickness of this layer of secondary deformation is very small and usually amounts to 5–10% of the thickness of the deformed chip. The work of friction is expended mainly on the deformation of this layer, and heat is released which heats the chip, with this heat being equivalent to the work of friction.

The temperature of the layer of secondary deformation can be estimated if, to the temperature acquired by the chip in the shear zone, one adds the temperature calculated from the specific work of friction \(A_{WF}\). But we obtained the value \(A_{WF}\) by dividing the work of friction \(A_F\) by the entire volume of metal converted into chip. Such a value \(A_{WF}\) was convenient for comparing the component parts of the cutting work. In principle, however, for calculating the specific work of friction, i.e., the work referred to a unit volume of the metal deformed by friction, it is more correct to divide the work of friction \(A_F\) by the volume of that layer of the chip which, under the influence of the friction force, undergoes additional deformation. Since the thickness and volume of this layer are 10–20 times smaller than the thickness and volume of the removed chip, the magnitude of the specific work of friction, calculated from the volume of this layer, will prove to be 10–20 times higher than that calculated from the entire volume of the chip [7]. Earlier, for steel 12KhN3A, we presented graphs of the specific works of deformation and friction (Figs. 11 and 13), where \(A_{W\tau}\) varied within the limits from 375 to 170 \(\mathrm{kg/mm/mm^3}\), and \(A_{WF}\) from 58 to 53 \(\mathrm{kg/mm/mm^3}\), decreasing with increasing cutting speed from 40 to 200 m/min. The values of the specific work of friction \(A_{WF}\) are substantially smaller than the specific work of deformation. If the considerations set out above are taken into account, these values of the specific work of friction must be increased by a factor of 10–20. Then the actual specific work of friction in the layer of secondary deformation of the chip will prove to be 2–3 times greater than the specific work of deformation in the shear zone, and it will become clear why the temperatures on the rake face of the tool are so high and exceed the temperature in the shear zone by a factor of 2–3.

With increasing cutting speed, the temperature on the rake face rises, which is established by objective measurement. All investigators believe that this rise is a consequence of the reduction in the time interval of contact between the chip and the rake face of the tool as the cutting speed increases, which leads to a decrease in heat removal from the layer of secondary deformation into the body of the chip and the body of the cutter. Without denying the possible influence of the time factor on heat conduction, we assume that here there is also another reason for the increase in temperature in the friction zone with increasing cutting speed.

This reason consists in the fact that the thickness of the layer of secondary deformation decreases with increasing cutting speed. A decrease in the thickness of the layer of secondary deformation was observed by T. N. Loladze [15]. When machining steel with \(\sigma_{v}=63.5\ \text{kg}/\text{mm}^2\), with feed \(s=0.54\ \text{mm/rev}\), an angle in plan \(\varphi=60^\circ\), and a rake angle \(\gamma=-10^\circ\), the author measured, on longitudinal sections of the removed chip, the thickness of the layer of secondary deformation on the side of the chip facing the rake face of the cutter. The values of cutting speed, chip shrinkage \(\zeta\), thickness of the removed chip \(a_1\), thickness of the layer of secondary deformation \(a_2\), and the ratio \(\dfrac{a_2}{a_1}\) are given by us in the table.

It follows from the table presented that, with increasing cutting speed, the ratio of the thickness of the layer of secondary deformation to the thickness of the removed chip decreases markedly. If at a cutting speed of \(51\ \text{m/min}\) the layer of secondary deformation amounts to \(8.7\%\) of the thickness of the removed chip, then at a speed of \(159\ \text{m/min}\) it is only \(4.8\%\). The actual value of the specific work of friction can be obtained if the work of friction is referred to the layer of secondary deformation. Therefore, in accordance with equation (15), which gave the value of the specific work of friction \(A_{WF}\) referred to the entire volume of the removed chip, the expression for the actual specific work of friction, which we shall denote by the symbol \(A'_{WF}\), may be given in the following form:

\[ A'_{WF}=\frac{\sigma_0}{n}\left(e^{\frac{n\varepsilon}{1.5}}-1\right) \frac{\sin\eta}{\zeta\cos(\eta-\gamma)-\sin\eta}\cdot\frac{a_1}{a_2}. \tag{21} \]

We showed earlier in Fig. 13 that the specific work of friction \(A_{WF}\), referred to the entire volume of the chip, decreases only slightly with increasing cutting speed. Since the ratio \(\dfrac{a_1}{a_2}\) increases substantially with increasing cutting speed, the actual specific work of friction \(A'_{WF}\) will also increase. This will lead to an increase in temperature in the layer of secondary deformation, and consequently also in the friction zone of the chip along the rake face of the tool.

LITERATURE

1. Briks. Cutting of Metals. St. Petersburg, 1896.
2. Rudnik S. S. Theory of Metal Cutting. ONTVU, Mashbudvidav, 1932.
3. Glebov S. F. Theory of the Most Advantageous Cutting of Metals. Gosmashmetizdat, 1933.
   Glebov S. F. The work of cutting and its component parts. Bulletin of the Metal Industry, Nos. 10, 11, 1936.
   Glebov S. F. The work of cutting and its component parts. Collection of Summarized Reports of the Conference on Metal Cutting. Academy of Sciences of the USSR, 1937.
4. Rosenberg A. M. Dynamics of Milling. Soviet Science Publishing House. Moscow, 1945.

1. Avakov A. A. Some questions concerning tool life. Proceedings of the Tbilisi Institute of Railway Transport Engineers. Issue XXIV, p. 46, fig. 17, 1951.

Besprozvannyi I. M. Fundamental theories of metal cutting. P. 147, fig. 209. Mashgiz, 1948.

1. Klushin M. I. Metal cutting. Mashgiz, 1953.

2. Zorev N. N. Investigation of elements of the mechanics of the cutting process. Mashgiz, 1952.

3. Rozenberg A. M. Determination of forces on the rear face of a cutter. Collection of works of the West Siberian Branch of the Academy of Sciences of the USSR, “Problems of Mechanical Science and Machine Building.” Novosibirsk, 1951.

4. Rozenberg A. M. and Rozenberg Yu. A. The force equation in cutting brittle metal (cast iron). Bulletin of the Tomsk Polytechnic Institute, vol. 75. Tomsk, 1954.

5. Zimin Yu. P. Investigation of the process of high-speed cutting of steels. Dissertation. Tomsk, 1947.

6. Promitov A. I. Dynamics of high-speed milling of steels. Author’s abstract of dissertation. Tomsk, 1954.

7. Rozenberg A. M. and Khvorostukhin L. A. Verification of the cutting-force equation under high-speed conditions. Bulletin of the Tomsk Polytechnic Institute. Vol. 75, Tomsk, 1954.

8. Rozenberg A. M. and Eremin A. N. Theoretical derivation of the cutting-force equation. Bulletin of the Tomsk Polytechnic Institute, vol. 75, 1954.

9. Kuznetsov V. D. Physics of the Solid State. Vol. II. Tomsk, 1941.

10. Loladze T. N. Chip formation in metal cutting. Mashgiz, 1952.