Analysis of the Krivoukhov–Belousov Theoretical Equation for Calculating Cutting Force

G. L. Kufarev

(Presented by Professor Doctor A. M. Rozenberg)

In recent years, in the literature on the theory of metal cutting, more and more new theoretical equations have appeared for calculating the principal component of the cutting force. The desire to obtain such an equation is entirely natural, since, if it is possible to calculate theoretically the force \(P_z\), there is no need to carry out tens of thousands of complex experiments involving force measurements in order to obtain standard reference materials.

Many of the proposed equations qualitatively correctly reflect the physical phenomena accompanying the cutting process and thereby make it possible to understand more deeply the regularities of the cutting process and, with a greater or lesser degree of accuracy, to determine by calculation the force \(P_z\).

Alongside these, there are also such “theoretical” equations for calculating the force that are completely inconsistent with experiment, which indicates the complete untenability of the assumptions and hypotheses underlying the calculation.

One such equation is the Krivoukhov–Belousov equation [1]. It has the following form:

\[ P_z = 0.112 \cdot t \cdot s \cdot \gamma_1 \cdot c \cdot \Theta_{\mathrm{pl}} \cdot \left( 1 + e^{-0.041 \cdot s \cdot \sin \varphi \sqrt{\frac{v}{a \cdot D}}} \right) \times \]

\[ \times \left( 1 - 0.177 \frac{\operatorname{tg}\gamma}{\mu} \right) + F_1,\ \mathrm{kg}. \]

Here:
\(t\)—depth of cut, mm;
\(s\)—feed, mm/rev;
\(\gamma_1\)—specific weight of the machined metal, g/cm\(^3\);
\(c\)—heat capacity, cal/g at \(0^\circ\)C;
\(\Theta_{\mathrm{pl}}\)—melting temperature, \(^\circ\)C;
\(a\)—thermal diffusivity, cm\(^2\)/s;
\(\varphi\)—principal cutting-edge angle in plan;
\(\mu\)—coefficient of friction on the rake face;
\(D\)—diameter of the workpiece being machined, cm;
\(F_1\)—friction force on the flank face, kg.

The coefficient of friction \(\mu\) is taken by the authors as constant and equal to 0.5. The friction force on the rear face is determined by the equation

\[ F_1 = (0.1 H_B - 15)\cdot l,\ \text{kg}. \]

Here: \(H_B\)—the Brinell hardness of the metal being machined;
\(l\)—the active length of the cutting edge.

Let us denote:

\[ 0.112 \cdot t \cdot s \cdot \gamma \cdot c \cdot \Theta_{\mathrm{pl}} = k, \]

\[ \left(1 + e^{-0.041 \cdot s \cdot \sin \varphi}\sqrt{\frac{v}{a \cdot D}}\right)=A, \]

\[ \left(1 - 0.177 \frac{\tg \gamma}{\mu}\right)=B. \]

Then the equation given above for calculating the force \(P_z\) will take the form

\[ P_z = k \cdot A \cdot B + F_1,\ \text{kg}. \tag{1} \]

Let us analyze this equation.

It is well known, and confirmed by tens of thousands of experiments carried out by various investigators, that the force \(P_z\), from the moment the built-up edge on the tool disappears, decreases as cutting speed increases. The reduction of the force in this range of cutting speeds is explained by the fact that an increase in cutting speed leads to an increase in temperature on the front face of the tool; as this temperature rises, the coefficient of friction of the chip on the tool decreases. The reduction of the coefficient of friction, in turn, leads to a decrease in chip compression and (at a constant rake angle \(\gamma\) of the tool) in the degree of plastic deformation of the chip metal, characterized by relative shear. The reduction in the degree of plastic deformation also explains the corresponding decrease in cutting force. It is precisely for this reason that in the latest standards for cutting conditions with carbide tools [2], which apply exactly in that range of speeds where the built-up edge is already absent, the following empirical dependence has been adopted for calculating the force \(P_z\):

\[ P_z = C_{pz} \cdot t^{x_{pz}} \cdot s^{y_{pz}} \cdot v^n \cdot k_{pz}. \]

For the case of longitudinal turning of steels, the exponent of the speed is given as negative and equal to \(-0.15\). Thus, changing the cutting speed from 50 to 200 m/min reduces the force \(P_z\) by 23%, and to 500 m/min—by 41%.

Experimental data obtained in the cutting laboratory of the Tomsk Polytechnic Institute, presented in Table 1, also indicate the same change in the force \(P_z\) with increasing cutting speed under conditions in which no built-up edge is present. These experiments were carried out at a feed \(s = 0.291\) mm/rev and \(\varphi = 70^\circ\).

How, then, does formula (1) reflect the influence of cutting speed on the force \(P_z\)? When the cutting speed changes in equation (1), only the quantity \(A\) will change, since \(k\) and \(F_1\) do not depend on speed. The quantity \(B\) also does not depend on the cutting speed, since the coefficient of friction is taken by the authors as constant. As for the change in the quantity \(A\), when the speed is increased from 20 to 500 m/min (at \(s = 0.291\) mm/rev and \(D = 100\) mm), it decreases by only 10–12%.

Table 1

(for different steels, the value of thermal conductivity changes). Consequently, the force $P_z$, calculated by formula (1), will decrease when the speed changes from 20 to 500 m/min by less than 10—12%, which is in no way consistent with experiment.

Even if the quantity $B$ is calculated taking into account the actual coefficient of friction, in this case as well the result obtained is just as unsatisfactory.

Figure 1 presents a graph expressing the dependence of the quantity $B$ in formula (1) on the coefficient of friction for various rake angles of the cutter. It follows from the graph that a decrease in the coefficient of friction, which is always accompanied by an increase in cutting speed in the range of interest to us, reduces the quantity $B$, and consequently also the force $P_z$, only for positive rake angles. Taking, for example, actual coefficients of friction ($\mu = 0.9$—$0.65$, respectively, for $\gamma = 20$—$0^\circ$ and $v = 20$—200 m/min), we obtain a decrease in $B$ of only 2—3%.

If, for positive rake angles, the adopted dependence of the quantity $B$ and the force $P_z$ on the coefficient of friction is confirmed at least

qualitatively, then for \(\gamma < 0\) even this is not obtained. With a negative rake angle, the cutting force calculated by formula (1), as the coefficient of friction decreases, should increase, which is not consistent with elementary notions of the cutting process. Moreover, this increase in force will be very substantial, since the average level of the coefficient of friction at negative rake angles is much lower than at positive ones. At small values of the coefficient of friction, the quantity \(B\) becomes more sensitive to its change.

The erroneousness of the adopted dependence of the force \(P_z\) (the quantity \(B\)) on the coefficient of friction also follows from the fact that the quantity \(B\) may take negative values, whereas in reality the component of the force \(P_z\) acting on the rake face of the cutter is never negative.

Thus, according to the experiments of B. M. Orlov [3], when cutting 20Kh steel \((\gamma = 30^\circ)\) with carbon tetrachloride lubrication, the coefficient of friction was equal to 0.1. The force \(P_z\) in this case, for different chip thicknesses, varied from 12 to 75 kg. In the experiments of L. A. Khvorostukhin [4], when cutting copper \((\gamma = 40^\circ)\) with soapy-water lubrication, the coefficient of friction was equal to 0.08, and the force \(P_z = 21\) kg \((s \cdot t = 0.9\ \text{mm}^2)\).

Fig. 1. Dependence of the quantity \(B\) in equation (1) on the coefficient of friction for different rake angles of the cutter.

If for these experiments the quantity \(B\) is determined from the graph shown in Fig. 1, then in the first case \(B = 0\), and for the second experiment \(B = -0.85\); that is, the cutting force calculated for each of these experiments by equation (1) proves to be negative, which indicates the complete inadmissibility of this equation.

It is known that the cutting force depends substantially on the strength characteristics of the metal being machined. How does formula (1) reflect this circumstance?

Table 2 gives the characteristics, necessary for calculating the force \(P_z\), of steels, copper, and aluminum, which differ greatly in their strength indices \((H_B, \sigma_B)\).

It follows from Table 2 that, without making a significant error, the quantities \(\gamma_1\), \(c\), and \(\Theta_{\text{pl}}\) may be regarded as constant for all steels. Thus, for all steels the quantity \(k\) in equation (1) remains practically unchanged. Consequently, with a change in steel grade in equation (1), only the quantity \(A\), which depends on thermal conductivity, will change. Calculation shows that when the thermal conductivity \(a\) changes from 0.224 to 0.042, i.e., by more than a factor of 5, the quantity \(A\) changes by less than 8%. Moreover, from the standpoint of the influence of the strength characteristics of metals on the force \(P_z\), formula (1) contains qualitative contradictions. The greater the thermal—

Table 2

conductivity, the greater the value of $A$ and the force $P_z$. Consequently, when calculating by formula (1), the force in cutting steel 10 ($a=0.224$) is obtained as greater than in cutting steel ShKh15 ($a=0.095$), although the hardness of the latter is almost twice as high as the hardness of steel 10. Such a result is undoubtedly incorrect.

Because the value $k$ is constant for all steels, the force $P_z$ in machining them at speeds below 2.5 m/min, when calculated by formula (1), turns out to be practically unchanged, since the value of $A$ in this case changes only within 1%. This contradicts experiments conducted at microspeeds, at which no built-up edge is present and to which, consequently, formula (1) should apply.

For the same reasons, the force $P_z$, calculated by formula (1), remains constant when working at any speeds in the case where the diameter of the workpiece is large and, all the more so, if a plane is being machined ($D=\infty$), which also contradicts experience.

The force $P_z$ in cutting steel also remains unchanged in the case when the angle $\varphi=0$. Such a process can be carried out in longitudinal cutting of a twist drill, and also in transverse turning. In this case the value of $A$ remains unchanged and equal to 2.

As for the force $F_1$, the formula proposed by the authors is purely empirical and, moreover, can be used only for metals with a very limited range of hardness, since the force $F_1$ becomes negative when $H_B>150$.

To determine the discrepancies given by formula (1) in comparison with experiment, tests were carried out on several representative steels, copper, and aluminum [4].

The results of these tests are presented in Table 3.

As was to be expected, the discrepancies between the calculated data and the experimental data proved to be very substantial: from +65 to −52% when working at high speeds, or from +440 to −52% if the tests at microspeeds are also taken into account, where formula (1) proves to be completely untenable.

When working on steel 40, with which the authors [1] operated, formula (1) does indeed give satisfactory results, but only at high cutting speeds.

Thus, formula (1) cannot be used either for calculating the force \(P_z\), since it gives very large errors, or for analyzing the cutting process, since the qualitative influence of the individual process parameters on the cutting force is reflected incorrectly by formula (1).

From the work [1] available to us, it is not possible to identify all the errors in the derivation of the equation under analysis that have made it unsuitable. We shall point out only some of them.

The authors believe that ... “The effective cutting power consists of four principal component parts:

Table 3

\[ w = w_1 + w_F + w_N + w_{F_1}, \]

where \(w_1\) is the power of the primary deformation in the cutting zone,

\(w_F\) is the power of the secondary deformation developed by the friction force \(F\) on the rake face of the cutter,

\(w_N\) is the power of the secondary deformation developed by the normal force \(N\) on the rake face of the cutter,

\(w_{F_1}\) is the power of friction on the flank face.”

From our point of view, the work of the normal force, and consequently \(w_N\), are equal to zero. The force \(N\) is the force of interaction of the chip with the rake face of the cutter. In order for the work of this force to be different from zero, it is necessary to have a velocity of displacement of the chip relative to the rake face of the cutter in the direction normal to the rake face. Obviously, such a velocity, as in any other friction process, is absent in the present case.

This can also be proved analytically. Adopting the position of the authors of the equation, let us assume that the velocity \(v_N\) is different from zero and is equal to

\[ v_N = \frac{v}{\xi}\cdot \tg\gamma . \]

Then the scheme of velocities in the cutting process will have the form shown in Fig. 2. Here: \(v_F\) is the velocity of displacement

of the cutter relative to the chip, equal to \(v_F=\dfrac{v}{\xi}\), \(v_\tau\)—the shear velocity, i.e., the velocity of displacement of the chip relative to the workpiece during deformation, \(v\)—the cutter velocity. From this diagram one may write the following relation

\[ OA=\frac{v\cdot \sin\beta_1}{\sin[(90^\circ+\gamma)-(\beta_1-\gamma)]} =\frac{v_F}{\cos\gamma} =\frac{v}{\xi\cdot \cos\gamma}. \]

Here

\[ \xi=\frac{\cos(\beta_1-\gamma)}{\sin\beta_1}. \]

Inverting the fraction and transforming the sine of the difference of angles, we obtain

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

It follows from this that

\[ \sin\gamma\cdot \sin(\beta_1-\gamma)=0. \]

This equality, and consequently also the adopted velocity diagram, will be valid only in two cases: when \(\gamma=0\) and when \(\beta_1=\gamma\). In the first case \(v_N=0\), since \(\tan\gamma=0\); in the second case the velocity \(v_N\) coincides in direction with the velocity \(v_\tau\) and must be considered only as part of the latter. In other words, the adopted velocity diagram will be valid only when \(v_N=0\), which is what had to be proved.

Fig. 2. Velocity diagram in the cutting process.

The second essential error in the derivation of the equation is that the average excess temperature of the cutting zone, taken by the authors as equal to:

\[ \Theta_1 = 0.215\,\Theta_{\mathrm{pl}} + 0.215\,\Theta_{\mathrm{pl}}\cdot e^{-a_1\sqrt{\frac{v}{a\cdot D}}} \cos\left( 2\pi\frac{\tau}{\tau_1} - a_1\sqrt{\frac{v}{a\cdot D}} \right), \]

does not depend on the degree of deformation of the metal in the cutting process. The aver-

the excess temperature determines the power of the primary deformation:

\[ w_1 = J \cdot a_1 \cdot b_1 \cdot v \cdot \gamma_1 \cdot c \cdot \Theta_1. \]

Here \(J\) is the mechanical equivalent of heat;
\(a_1, b_1\) are the thickness and width of the cut;
\(\gamma_1\) is the specific weight of the metal being machined;
\(c\) is the heat capacity.

Thus, the authors believe that the power of primary plastic deformation does not depend on the degree of plastic deformation. This is fundamentally incorrect.

In addition, the authors write that they were able to exclude from the equation the friction force \(F\) and the chip shrinkage on the grounds that “...the work of friction forces on the rake face of the cutter amounts to approximately 21.5% of the total cutting work...” and “...chip shrinkage is proportional to the specific friction force on the rake face \(\dfrac{F}{s \cdot t}\).”

Table 4 gives the values of the total specific cutting work \(A_w\) and the specific work of friction \(A_{wF}\) for experiments on various metals, obtained experimentally [4]. It follows from the table that, for these experiments, the work of friction amounts to from 13 to 41% of the cutting work, and not 21.5%, as the authors assert.

As for the proportionality between chip shrinkage and the specific friction force, it holds only within the limits of machining one and the same material. In the general case, as shown by calculations performed on the basis of the experiments of Khvorostukhin [4] and a number of other authors, the ratio \(\dfrac{F}{s \cdot t} : \xi\) varies by more than a factor of 6 (from 13 to 79).

Table 4

These are the principal errors in the derivation of equation (1), which, in the author’s opinion, deprive it of both practical and theoretical interest.

References

1. Krivoukhov V. A., Belousov A. I. Determination of the cutting force on the basis of the physical characteristics of the metals being machined. “Investigations in the Physics of Solids,” Academy of Sciences of the USSR, 1957.

2. Cutting conditions for ferrous metals with tools equipped with hard alloy. NIIBTIN, Mashgiz, 1958.

3. Orlov B. M. The influence of lubrication on the process of cutting steel at microspeeds. “Investigations in the Physics of Solids,” Academy of Sciences of the USSR, 1957.

4. Khvorostukhin L. A. Calculation of forces in high-speed cutting on the basis of the physicomechanical characteristics of metals. Dissertation, Tomsk, 1953.