FORCE EQUATION IN CUTTING A BRITTLE METAL (cast iron)

A. M. ROZENBERG and Yu. A. ROZENBERG

1. Preliminary Considerations

The equation for the cutting force in cutting a plastic metal was derived on the basis of the equality of the specific works of plastic deformation under equal deformation [1]. This proposition is a consequence of the fourth (energy) theory of strength. Naturally, this proposition cannot be used to derive the force equation when cutting a brittle metal, whose fracture is not accompanied by any appreciable plastic deformations.

A study of the cutting process of such a brittle metal as gray cast iron shows that here the process of chip separation proceeds by brittle shear (cleavage) at an angle, definite for each particular case, to the direction of motion of the cutter. This permits us to assume that the cutting force will be determined by the resistance of the metal to cleavage (brittle shear) in the plane where the shear stress is greatest.

Moreover, in contrast to the case of cutting a plastic metal, where the stress in the shear zone was a function of the degree of plastic deformation and, in a number of cases, could also depend on the rate of deformation, in cutting brittle metals the shear stress must be constant for the given metal being machined.

Thus, in deriving the force equation for cutting a brittle metal, we have taken as the basis the condition of constancy of the maximum tangential stress and its independence of the geometry of the tool, the dimensions of the layer being cut, and the cutting speed.

This condition, which was taken as the basis of the derivation, naturally required experimental verification first of all. Such verification was carried out in the work of Yu. A. Rozenberg [2]. In that work, when cutting gray cast irons of various hardnesses from $H_B = 140$ to $H_B = 217$, the cutting conditions were varied over a wide range: cutting speed from 2 to 550 m/min, feed from 0.0482 to 0.93 mm/rev, depth of cut from 0.5 to 5 mm, and front rake angle of the cutter from $+20^\circ$ to $-10^\circ$. In the experiments the cutting forces were measured; moreover, by using a special microscope, the position of the shear plane during cutting was determined, along which the chip separated from the workpiece, i.e., the plane of maximum tangential stresses. The experiments performed made it possible to determine the forces on the front face of the cutter and to calculate the values of the tangential stresses in the shear plane.

As a result of a large number of experiments, the proposition was confirmed that the tangential stress $\tau$ in the shear plane is independent of the cutter front angle, the dimensions of the layer being cut, and the cutting speed for cast iron of constant hardness. This proposition is illustrated by Figs. 1 and 2, which present the dependences of the tangential stress $\tau$ on the front angle $\gamma$, the hardness of the cast iron $H_B$, and the temperature on the front face of the cutter, and

therefore, the cutting speed. Figs. 1 and 2 show that, disregarding a certain scatter of the experimental points—which is the result, on the one hand, of experimental errors and, on the other, of the inevitable variation in the hardness of cast iron even within a single blank—one may speak of the constancy of the shear stresses in the shear plane. Moreover, these same experiments establish a direct proportionality between the shear stress and the hardness of the cast iron being machined.

Fig. 1

In the same work it was shown that the shear stresses are equal in cutting and in compression at the moment of fracture. The compression process was taken for determining the shear stresses because, in compression of a brittle metal, fracture is usually effected by shear, i.e., similarly to what occurs in cutting a brittle metal, whereas in tension fracture by separation is possible. Specimens with a ratio of height to diameter

\[ \frac{h_0}{d_0}=2 \]

were subjected to compression; the load \(Q\) at the moment of the occurrence of brittle shears was recorded. From this load, taking into account the cross section \(f\) of the specimen at the moment of fracture, the stress \(\sigma\) was determined, from which the shear stress was calculated, with

Fig. 2

\[ \sigma=\frac{Q}{f}, \]

\[ \tau=\frac{\sigma}{2}. \]

A comparison of the shear stresses in cutting and in compression is given in Fig. 3, which indicates the equality of these stresses, as well as

Fig. 3

their direct proportionality to the hardness of cast iron. According to Fig. 3, the relation between $\tau$ and $H_B$ can be expressed by the dependence

\[ \tau = 0.174 H_B . \tag{1} \]

2. Derivation of the cutting-force equation

Having shown the validity of our assumptions, let us proceed to derive the equation for the cutting force that carries out the shearing and deformation of the chip, i.e., the force applied to the rake face of the cutter.

Fig. 4

Figure 4 presents the forces and angles characterizing the mechanics of the cutting process. Here

$F$ — friction force on the rake face,
$N$ — normal force on the rake face,
$R$ — resultant of all forces on the rake face,
$P_t$ — component producing brittle shear,
$\beta_1$ — angle determining the position of the shear plane,

$P_1$ — principal cutting force,
$\eta$ — angle of friction of the chip on the tool,
$a$ — thickness of the layer being cut,
$b$ — width of the layer being cut.

In all cases of cutting a brittle metal, for the force producing shear the following expression may be written:

\[ P_t=\tau \frac{ab}{\sin \beta_1}. \]

On the other hand,

\[ P_t=P_1 \frac{\cos(\omega+\beta_1)}{\cos \omega}, \]

whence

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

and since $\omega=\eta-\gamma$, then

\[ P_1=\tau ab \frac{\cos(\eta-\gamma)}{\sin \beta_1 \cos(\eta+\beta_1-\gamma)} . \tag{2} \]

In the same way, for the component $P_2$ we find

\[ P_2=\tau ab \frac{\sin(\eta-\gamma)}{\sin \beta_1 \cos(\eta+\beta_1-\gamma)} . \tag{3} \]

Equations (2) and (3) show that, in order to determine the components $P_1$ and $P_2$ for known $\tau, a, b, \gamma$, it is necessary to know the angles $\beta_1, \eta$. As applied to a ductile metal, the angle $\beta_1$ could be determined by measuring chip shrinkage. As applied to a brittle metal this is impossible. Direct measurement of the angle $\beta_1$ during the cutting process is difficult and requires special apparatus. Therefore we shall take another route to the determination of the unknowns in equations (2) and (3).

The work of Yu. A. Rozenberg [2] showed that the angle of friction $\eta$ is a function of the temperature on the rake face and of the rake angle $\gamma$,

\[ \eta=f(t^\circ,\gamma). \]

On the other hand, the angle $\beta_1$ in turn depends on $\eta$ and $\gamma$. Thus, the angles $\eta$ and $\beta_1$ depend on the rake angle $\gamma$ and the temperature on the rake face, and, at their constant values, do not depend on the cutting speed, the dimensions of the layer being cut, or the hardness of the cast iron. This has been proved by a large number of experiments carried out over a wide range of variation of cutting conditions. Therefore it is possible to express the trigonometric multipliers in equations (2) and (3) by certain coefficients, which will vary with changes in the temperature on the rake face of the tool and with changes in the rake angle.

In addition, the tangential stress $\tau$ can be expressed through hardness according to relation (1). Then

\[ P_1=0.174\,H_B\,ab k_1, \]

\[ P_2=0.174\,H_B\,ab k_2. \]

Instead of the thickness and width of the cut, one may substitute the feed $s$ and the depth of cut $t$.

\[ P_1=0.174\,H_B\,s\,t\,k_1, \tag{4} \]

\[ P_2=0.174\,H_B\,s\,t\,k_2. \tag{5} \]

Here

\[ k_1=\frac{\cos(\eta-\gamma)} {\sin\beta_1\cdot \cos(\eta+\beta_1-\gamma)}, \]

\[ k_2=\frac{\sin(\eta-\gamma)} {\sin\beta_1\cdot \cos(\eta+\beta_1-\gamma)}. \]

For the coefficients \(k_1\) and \(k_2\), graphs are given in Figs. 5 and 6, constructed from the results of numerous experiments. Figs. 5 and 6 show that the coefficients \(k_1\) and \(k_2\) depend on the temperature on the rake face of the

Fig. 5

tool and on the rake angle of the tool, and are completely independent of the cross section of the layer being cut and of the hardness of the cast iron. In order to take the values \(k_1\) and \(k_2\) from Figs. 5 and 6, it is necessary to know the temperature on the rake face of the tool. The temperature

Fig. 6

depends on the cutting speed, the front angle of the tool, and the dimensions of the cross-section of the layer being cut. But studies carried out by Yu. A. Rozenberg showed that cutting conditions of equal tool life when working cast iron with alloy VK-8 correspond to approximately identical temperatures, as illustrated by Table 1. In this table the cutting conditions of equal tool life are taken from the data of the handbook Cutting-Speed Conditions [Mashgiz, 1950], while the temperatures were measured experimentally.

Table 1

| $H_B$ | $s$, mm/rev | $t$, mm | \multicolumn{2}{c}{\(T=30\) min} | \multicolumn{2}{c}{\(T=90\) min} | \multicolumn{2}{c}{\(T=240\) min} |
|---|---:|---:|---:|---:|---:|---:|---:|---:|
| | | | $v$, m/min | Temperature | $v$, m/min | Temperature | $v$, m/min | Temperature |
| 207—217 | 0,2 | 3 | 79 | 620 | 63 | 560 | 52 | 490 |
| 207—217 | 0,6 | 3 | 58 | 610 | 47 | 520 | 38,5 | 495 |
| 183—192 | 0,2 | 3 | 92 | 560 | 74 | 500 | 60 | 460 |
| 183—192 | 0,6 | 3 | 68 | 580 | 55 | 510 | 45 | 470 |
| 160—170 | 0,2 | 3 | 112 | 590 | 90 | 520 | 73 | 470 |
| 160—170 | 0,6 | 3 | 82 | 595 | 66 | 540 | 54 | 490 |
| 140—150 | 0,2 | 3 | 140 | 590 | 112 | 550 | 92 | 505 |
| 140—150 | 0,6 | 3 | 103 | 620 | 83 | 537 | 68 | 485 |
| 220 | 0,24 | 4 | 60,5 | 570 | 49 | 510 | 40 | 450 |
| 220 | 0,24 | 3 | 64 | 576 | 51,5 | 520 | 42 | 460 |
| 220 | 0,24 | 2 | 67 | 580 | 55 | 515 | 45 | 475 |
| 220 | 0,24 | 1 | 74,5 | 575 | 60 | 510 | 48,5 | 460 |
| Average temperature | | | — | 590 | — | 525 | — | 475 |

On this basis, Figs. 5 and 6 give the temperature straight lines corresponding, when working with alloy VK-8, to tool lives of 30, 90, and 240 minutes. Thus, by taking the desired tool life as a guide, the corresponding values of the quantities $k_1$ and $k_2$ can be taken from Figs. 5 and 6.

Figs. 5 and 6 show that, within the range of tool lives from 30 to 240 minutes, the values of $k_1$ and $k_2$ vary only very slightly, which makes it possible to take certain average values for them. These averages for different front angles will be as follows (Table 2).

Table 2

If these average values are used as a guide, the error in calculating the forces will not exceed $\pm 5\%$, but it should be kept in mind that the further development of hard alloys and the introduction into service of such alloys as will make it possible to machine cast iron at cutting speeds higher than those allowed by alloy VK-8, and therefore at higher temperatures, will shift the lines of constant tool life into the region of higher temperatures, which will lead to a decrease in the coefficients $k_1$ and $k_2$ and to a reduction in cutting forces.

Usually, all the cast irons most commonly used in machine building, with hardness from 140 to 220 units, are machined with a hard alloy with a front angle $\gamma = 10^\circ$. Studies carried out by A. K. Baikalov [3],

showed that a rake angle of \(+10^\circ\) in high-speed cutting of cast irons is optimal from the standpoint of tool life. For this optimal rake angle we shall have particular values of the equations for the cutting forces

\[ P_1 = 0.6\,H_B st, \tag{6} \]

\[ P_2 = 0.322\,H_B st. \tag{7} \]

3. Influence of the forces on the flank face of the cutter on the cutting forces

All the equations given above for the cutting forces, both of the general form (2), (3), (4), (5), and the particular ones (6) and (7), make it possible to calculate the components of the force on the rake face of the cutter.

In addition to the forces on the rake face, in the cutting process there are also forces on the flank face of the tool, and in practice we are interested in the resultant forces

\[ P_z = P_1 + P'_1 \quad \text{and} \quad R_{xy} = P_2 + P'_2, \]

where \(P'_1\) and \(P'_2\) are the forces applied to the flank face of the cutter in the directions, respectively, of the forces \(P_1\) and \(P_2\), and the force \(R_{xy}\) is the geometric sum of the components \(P_x\) and \(P_y\).

The force \(P'_2\) is the normal force, \(P'_1\) the friction force on that portion of the flank face which is in contact with the machined surface.

On a sharp cutter the forces on the flank face are comparatively small, but if, in machining a ductile metal, they are quite insignificant in comparison with the forces on the rake face and in a number of cases may be neglected,

Fig. 7

then in cutting a brittle metal, and in particular cast iron, their share in the total cutting forces is rather considerable. This also follows from consideration of Figs. 7, 8, 9, and 10, which present experimental regularities in the variation of the forces on the rake and flank faces for different feeds as a function of temperature.

Fig. 8

Vertical axis: Cutting force on the front face, \(Q_1\)

Horizontal axis: Temperature on the front face

In-figure labels:

• \(S = 0.93\)
• \(S = 0.6\)
• \(S = 0.364\)
• \(S = 0.2\)
• \(S = 0.12\)
• \(S = 0.091\)
• \(S = 0.045\)
• Gray cast iron
  \(H_B = 207\text{–}217\)
  Cutter BK-8
  \(\gamma = 0^\circ;\ t = 3\ \mathrm{mm}\)

Fig. 9

Vertical axis: Cutting force on the rear face, \(Q_1\)

Horizontal axis: Temperature on the front face

In-figure labels:

• Gray cast iron
  Cutter BK-8
  \(\gamma = 0^\circ;\ \varphi = 75^\circ;\ t = 3\ \mathrm{mm}\)
• \(H_B = 207\text{–}217\)
• \(H_B = 183\text{–}192\)
• \(H_B = 160\text{–}170\)
• \(H_B = 140\text{–}150\)

Fig. 10

Vertical axis: Cutting force on the rear face, \(Q_2\)

Horizontal axis: Temperature on the front face

In-figure labels:

• \(H_B = 207\text{–}217\)
• \(H_B = 183\text{–}192\)
• \(H_B = 160\text{–}170\)
• \(H_B = 140\text{–}150\)
• Gray cast iron
  Cutter BK-8
  \(\gamma = 0^\circ;\ \varphi = 75^\circ;\ t = 3\ \mathrm{mm}\)

As in the cutting of ductile metals, here the forces on the flank face do not depend on the rake angle of the cutter or on the thickness of the layer being cut, and are directly proportional to the width of the layer being cut and to the hardness of the metal being machined, since they are forces of elastoplastic contact of a narrow strip of the flank face with the cutting surface.

Using Figs. 9 and 10, one can represent the dependence of the forces on the flank face on the hardness of cast iron at constant temperature. This is done in Figs. 11 and 12. Here the forces are given per unit length of the cutting edge.

Fig. 11

Figs. 11 and 12 show that, despite the fairly complex law governing the variation of the forces on the flank face with temperature, at constant temperature the forces on the flank face are directly proportional to the hardness of cast iron.

Fig. 12

The forces on the flank face may be expressed by the following regularities:

\[ P_1' = C_1 H_b b = C_1 H_b \frac{t}{\sin \varphi}, \tag{8} \]

\[ P_2' = C_2 H_b b = C_2 H_b \frac{t}{\sin \varphi}. \tag{9} \]

In these equations the thickness of the layer being cut is absent, since it has no influence on the forces on the flank face. \(C_1H_B\) and \(C_2H_B\) are the specific forces per 1 mm of length of the contact area of the flank face and the cutting surface. These values, as well as the coefficients \(C_1\) and \(C_2\),

Fig. 13

depend on temperature. The values of \(C_1\) and \(C_2\), obtained from a large number of experiments carried out at different feeds, rake angles, depths of cut, and cutting speeds, are given in Table 3 and in Fig. 13.

Table 3

For orientation with respect to high-speed cutting conditions, Fig. 13 marks the lines of tool lives of 30, 90, and 240 minutes. For the conditions corresponding to these tool lives, the coefficients \(C_1\) and \(C_2\) take the following values (Table 4).

Table 4

As Figs. 9 and 10 and Table 3 show, the forces on the flank face undergo considerable changes with changes in temperature (i.e., cutting speed). If, however, we consider high-speed cutting conditions within the limits of tool lives from 30 to 240 minutes, the changes here will be very slight. Therefore, for high-speed cutting conditions one may adopt certain average values of the coefficients:

\[ C_1 = 0.014 \quad \text{and} \quad C_2 = 0.027 . \]

4. Equations of the Total Cutting Forces

The cutting forces \(P_z\), located in the direction of the cutting speed, and \(R_{xy}\)—in the direction perpendicular to the cutting surface—are the sum of the forces located on the rake and flank faces of the tool:

\[ P_z = P_1 + P'_1, \]

\[ R_{xy} = P_2 + P'_2, \]

therefore, in cutting gray cast iron,

\[ P_z = 0.174 H_B s t k_1 + C_1 H_B \frac{t}{\sin \varphi}, \tag{10} \]

\[ R_{xy} = 0.174 H_B s t k_2 + C_2 H_B \frac{t}{\sin \varphi}. \tag{11} \]

Here \(k_1\), \(k_2\) are functions of the rake angle and temperature, and \(c_1\), \(c_2\) are functions of temperature. In order to be able, from the cutting conditions \((v, s, t, \varphi)\), to pass to the temperature, we give graphs by which it is possible

Fig. 14

to determine the temperature on the rake face (Figs. 14–17). These graphs are given for a rake angle of \(10^\circ\), as the optimum.

In equations (10) and (11), the first terms on the right-hand side have a strict physical meaning and are derived from the condition of constancy of the tangential

stresses in the shear plane. The second terms have been derived purely experimentally, since the physical basis for calculating the forces on the flank face is still insufficiently clear. It should be noted that the experimentally obtained dependences for the forces on the flank face can, to some extent, be physically explained and justified. Thus, for example, it is quite logical that there is no dependence of the forces on the flank face on the tool rake angle and on the thickness of the layer being cut, since these forces do not participate in the process of separation and deformation of the layer being cut.

Fig. 15

Fig. 16

If equations (10) and (11) are rewritten for the case of applying speed-cutting conditions for cast iron with the optimum rake angle \(\gamma = 10^\circ\), then they take the form

\[ P_z = 0.6 H_B s t + 0.014 H_B \frac{t}{\sin \varphi}, \tag{12} \]

\[ R_{xy}=0.322\,H_B st+0.027\,H_B\,\frac{t}{\sin\varphi}. \tag{13} \]

In application to these particular relations, it is not difficult to estimate the comparative magnitude of the forces on the clearance face and to decide the question of the feeds at which their magnitude may be neglected. The ratio of the forces on the clearance and rake faces is expressed as follows:

\[ \frac{P_1'}{P_1}=\frac{0.0234}{s\cdot\sin\varphi}=\frac{0.0234}{a}, \]

\[ \frac{P_2'}{P_2}=\frac{0.084}{s\cdot\sin\varphi}=\frac{0.084}{a}. \]

Fig. 17

With an increase in feed, the share of the forces on the clearance face decreases, while with a decrease in the plan angle \(\varphi\) it increases.

If the condition of accuracy in calculating the forces is set within 10%, then at a plan angle \(\varphi \ge 45^\circ\) the forces on the clearance face may be neglected in calculating the force \(P_z\) at feeds exceeding \(s=0.33\) mm/rev, and in calculating the force \(R_{xy}\) at feeds exceeding \(s=1.2\) mm/rev. In these cases one may use only the first terms in equations (12) and (13).

5. Components \(P_x\) and \(P_y\) in cutting gray cast iron

In the process of free cutting, the component \(P_x\) in the direction of feed motion and \(P_y\) in the direction of the radius of the workpiece being machined may be obtained as

\[ P_x=R_{xy}\cdot\sin\varphi, \]

\[ P_y=R_{xy}\cdot\cos\varphi. \]

In the process of non-free cutting, these relations do not give correct results, since in this case the component \(R_{xy}\) is not located exactly in the plane normal to the main cutting edge, but deviates from it owing to the influence of the non-free chip flow and owing to the presence of the nose radius of the cutter, which has a variable plan angle.

The following approximate method may be proposed for taking into account the indicated features of the non-free cutting process.

The presence of a rounding radius at the nose of the cutter creates a variable angle in plan for it, as a result of which the actual mean value of the angle in plan, which determines the position of the component \(R_{xy}\), turns out to be less than the nominal angle in plan \(\varphi\).

Fig. 18

From Fig. 18 it can be seen that the cutting edge of the cutter may be divided into two portions: portion \(AB\)—rectilinear and inclined to the feed direction at the angle in plan \(\varphi\), and portion \(BD\)—curvilinear, the mean angle in plan of which may be taken as \(\dfrac{\varphi}{2}\). Hence

\[ \varphi_{\mathrm{av}} = \frac{AB\varphi + BD\,\dfrac{\varphi}{2}}{AB+BD}; \]

\[ AB = AK - BK; \qquad AK=\frac{t}{\sin\varphi}; \]

\[ BK=\frac{BE}{\sin\varphi}; \qquad BE=CD=OD-OC=r(1-\cos\varphi), \]

where \(r\) is the rounding radius.

Then

\[ AB=\frac{t-r(1-\cos\varphi)}{\sin\varphi}, \]

\[ BD=r\varphi, \]

\[ \varphi_{\mathrm{av}} = \frac{ \left(\dfrac{t}{r}+\cos\varphi-1\right)\dfrac{1}{\sin\varphi}+\dfrac{\varphi}{2} }{ \dfrac{\dfrac{t}{r}+\cos\varphi-1}{\varphi\sin\varphi}+1 }. \tag{14} \]

As follows from equation (14), the mean value of the angle in plan depends on the ratio \(\dfrac{t}{r}\), and the smaller this ratio is, the smaller \(\varphi_{\mathrm{av}}\) is for a given \(\varphi\).

The influence of the constrained chip flow on the direction of \(R_{xy}\) can be estimated as follows. If only one cutting edge \(AB\) (Fig. 19) took part in the cutting process, then the direction of chip flow, for a small angle \(\lambda\) of inclination of the cutting edge, would coincide with \(v_{AB}\), perpendicular to \(AB\). Conversely, when cutting with one cutting edge \(BC\), the direction of chip flow would coincide with \(v_{BC}\), perpendicular to \(BC\). It may be assumed that, under the simultaneous action of both cutting edges

Fig. 19

\(AB\) and \(BC\), the direction of chip flow, and hence also the direction of the force \(R_{xy}\), will coincide with \(v_{TR}\), the resultant of \(v_{AB}\) and \(v_{BC}\). In this case it may be admitted that the following relation will hold here:

\[ \frac{v_{AB}}{v_{BC}}=\frac{AB}{BC}, \]

where

\[ AB=\frac{t}{\sin \varphi_{cp}}. \]

By introducing \(\varphi_{cp}\), we thereby take into account the influence of the rounding radius on the direction of chip flow. \(BC\) may be taken equal to \(\dfrac{s}{2}\), in view of the smallness of the angle \(\varphi_1\) and taking into account the rounding radius on the main and auxiliary cutting edges.

Then

\[ \frac{v_{AB}}{v_{BC}}=\frac{2t}{s\sin \varphi_{cp}}. \]

On the other hand,

\[ \frac{OE}{\sin \angle OKE}=\frac{KE}{\sin \angle KOE}. \]

Here

\[ OE=v_{AB} \]

\[ \angle OKE=\varphi_{cp}+\varphi_1-\psi \]

\[ \angle KOE=\psi. \]

Therefore,

\[ \frac{v_{AB}}{v_{BC}}= \frac{\sin(\varphi_{cp}+\varphi_1-\psi)}{\sin\psi} = \frac{2t}{s\cdot \sin \varphi_{CD}} . \]

Hence,

\[ \tan \psi = \frac{\sin(\varphi_{cp}+\varphi_1)} {\dfrac{2t}{s\sin \varphi_{cp}}+\cos(\varphi_{cp}+\varphi_1)} . \tag{15} \]

Having determined from equations (14) and (15) the angles \(\varphi_{cp}\) and \(\psi\), and knowing the magnitude \(R_{xy}\), one can determine the components \(P_x\) and \(P_y\) from the relations

\[ P_x=R_{xy}\cdot \cos(\psi+90^\circ-\varphi_{cp}) =R_{xy}\sin(\varphi_{cp}-\psi). \tag{16} \]

\[ P_y=R_{xy}\cdot \sin(\psi+90^\circ-\varphi_{cp}) =R_{xy}\cos(\varphi_{cp}-\psi). \tag{17} \]

Naturally, equations (16) and (17) give an approximate calculation of the forces \(P_x\) and \(P_y\), since in our derivations we did not take into account the influence of a number of additional conditions: the influence of forces arising on the auxiliary cutting edge, the variability of the rake angle and of the chip thickness on the rounded portion of the cutting edge, etc.

Nevertheless, equations (16) and (17) agree well with the result of direct measurement of the forces \(P_x\) and \(P_y\).

Fig. 20

In Fig. 20, for two ratios \(\dfrac{t}{s}\), both the experimental and calculated values of the forces are given; the difference between them does not exceed 5%, which may be considered quite permissible.

6. Influence of tool dulling on cutting forces

Dulling of carbide tools in high-speed cutting of cast iron proceeds mainly by wear of the flank face, with a gradual widening of the wear land. The rake face undergoes almost no change; therefore, with dulling, the cutting forces increase mainly due to the forces on the flank face, which grow with increasing wear-land width. It is precisely for these reasons that, as dulling progresses, the components \(P_x\) and \(P_y\) increase more significantly than \(P_z\).

since in the components $P_x$ and $P_y$ there is a larger share of the forces on the flank face than in the component $P_z$. For the same reasons, when cutting with small feeds, the forces increase with blunting relatively more significantly than when cutting with large feeds. We do not present here the results of studies on the influence of blunting on cutting forces, since these results are very diverse. Experiments show that the increase in cutting forces with increasing wear of the tool on the flank face depends on the magnitude of the wear, on the tool rake angle, on the magnitude of the feed, and on the cutting speed. At present it does not seem possible to give sufficiently simple regularities by means of which one could take into account the influence of all the factors listed above on the increase in cutting forces during wear and blunting of the tool. Therefore one can only indicate that, at optimal wear on the flank face with a wear land width of 1 mm, the component $P_z$ increases by 15–30%, and the increase is the greater, the smaller the feed; the components $P_x$ and $P_y$ increase by 20–100%, again the more, the smaller the feed. At smaller values of the wear land, there is a proportionally smaller increase in the cutting forces.

REFERENCES

1. Rozenberg A. M. and Eremin A. N. Theoretical equation of the cutting force. The article is included in the present collection.

2. Rozenberg Yu. A. Investigation of the process of cutting gray cast iron. Author’s abstract of dissertation. Tomsk, 1952.

3. Baikalov A. K. Optimal geometry of cutting tools in high-speed turning of gray cast iron. Author’s abstract of dissertation. Tomsk, 1953.