THEORETICAL DERIVATION OF THE CUTTING FORCE EQUATION

A. M. ROSENBERG and A. N. EREMIN

Until now, empirical relationships have been used to calculate the forces on the cutter during the cutting of metals. These relationships are obtained experimentally and make it possible to calculate cutting forces with sufficient accuracy for practice. In order to obtain empirical relationships, it is necessary, for each given material being machined, to conduct special experiments which, in view of the large number of factors affecting the cutting forces, are quite numerous and laborious, and also require special measuring apparatus.

The need has arisen to relate the cutting force, as the force producing deformation of the chip metal, to the strength and plasticity characteristics of the metal being machined, and also to the geometrical parameters of the tool and the dimensions of the layer of metal being removed. Since the cutting process is a process of metal deformation, the possibility of obtaining such a relationship is quite probable. If a reliable equation is available that relates the cutting forces to the physico-mechanical characteristics of the metal being machined, then an experiment for the direct determination of cutting forces—laborious and requiring special apparatus—will become unnecessary; determination of the physico-mechanical characteristics of the metal being machined presents no substantial difficulty and is feasible in an ordinary factory laboratory.

The process of cutting ductile metals is a process of plastic deformation. The work expended on the cutting process is consumed, on the one hand, in deforming the metal of the chip and, on the other hand, in deforming the surface layer of the machined part. The deformation of the chip is produced by forces located on the rake face of the tool, and these forces depend on the dimensions of the layer being removed, the degree of its deformation, the temperature–velocity conditions of deformation, and, undoubtedly, the physico-mechanical characteristics of the metal being machined. The deformation of the surface layer of the machined part is partly connected with the deformation of the chip and is partly carried out by forces located on the flank face of the tool, which are the forces of elastic-plastic contact of the cutting surface with the portion of the flank face immediately adjacent to the cutting edge. These forces do not depend on the thickness of the layer being removed or on the degree of its deformation and, as shown by numerous experiments by the authors, by Yu. P. Zimin on steel [1], by Yu. A. Rosenberg on cast iron [2] (the work was carried out under the direction of Prof. A. M. Rosenberg), and also by N. N. Zorev [3], are very small in magnitude, so that they become noticeable in comparison with the forces on the rake face only for extremely thin chips. They may be neglected for those thicknesses of the removed layer which are usually used in production when working with cutters, drills, and counterbores.

The conception of the cutting process as a process of plastic deformation of metal transformed into chip makes it possible to assume that this process must obey the same laws that govern

other types of plastic deformation—compression, tension, torsion, rolling, drawing, etc. Many experiments have shown that in the cutting process, just as in other types of plastic deformation, strain hardening of the deformed metal is observed, along with an increase in its hardness and strength, a decrease in ductility, and an increase in stresses and cutting forces as the degree of plastic deformation of the chip increases, etc. But whereas for other types of deformation, quantitative relations between the factors of the deformation process have by now been established to one degree or another, for the cutting process such quantitative relations have not yet been established.

As the basis for deriving a theoretical equation of the cutting forces, which should relate the cutting forces to the physicomechanical characteristics of the metal being machined, it is necessary to adopt those laws of plastic deformation that are common to various types of plastic deformation. In the theory of metal forming, the law of independence of the potential energy of change of body shape from the mechanical scheme of deformation is widely used [4]. This law may be formulated as follows: “Irrespective of the mechanical scheme of deformation, under a given temperature–velocity regime the degree of deformation is uniquely related to the specific work of changing the shape of the body.” This means that in different types of deformation, to attain identical (or, better to say, “equivalent”) degrees of plastic deformation, equal specific works of plastic deformation must be expended. Thus, if we take two deformation processes: on the one hand, the cutting process, and on the other hand, the process of compression, tension, torsion, etc., then we may write the condition of equality of the specific works for equal equivalent deformations. Into this equality there will inevitably enter the forces, stresses, dimensions of the deforming volume, and the physicomechanical characteristics of the material. This will make it possible to relate the degree of deformation of the metal in the cutting process, the forces and stresses in cutting, to the physicomechanical characteristics of the metal determined in other types of deformation.

In deriving the equation of the cutting forces, we shall compare the cutting process with the process of plastic compression, since only in compression is it possible to attain such considerable plastic deformations as are observed in the cutting process.

Both the cutting process and the compression process are processes of plastic deformation occurring by shear. Therefore, the equivalent deformations in these two processes must be established through the equality of relative shears. The relative shear in cutting is expressed by the equation

\[ \varepsilon = \operatorname{ctg}\beta_1 + \operatorname{tg}(\beta_1-\gamma), \tag{1} \]

where \(\beta_1\) is the angle determining the position of the shear plane,

\(\gamma\) is the rake angle of the cutter.

The relative shear in cutting can be expressed through the chip shrinkage \(\zeta\), if it is taken into account that

\[ \operatorname{tg}\beta_1=\frac{\cos\gamma}{\zeta-\sin\gamma} \]

and the corresponding substitutions are made in equation (1). In this case, for the relative shear we obtain the expression

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

Relative shear in compression of a specimen

\[ \varepsilon = 1.5 \ln \frac{h_0}{h}, \tag{3} \]

where \(h_0\) is the initial height of the specimen before compression,
\(h\) is the height of the specimen at the given degree of deformation.

Equating the relative shears in cutting and in compression, we obtain

\[ \operatorname{ctg}\beta_1 + \operatorname{tg}(\beta_1-\gamma) = 1.5 \ln \frac{h_0}{h} \]

or

\[ \frac{\zeta^2 - 2\zeta \sin\gamma + 1}{\zeta \cos\gamma} = 1.5 \ln \frac{h_0}{h}. \]

Hence it is not difficult to obtain the equation of equivalent deformations in cutting and in compression, if the degree of deformation in compression is expressed through the ratio \(\frac{h_0}{h}\), and in cutting is characterized by the chip shrinkage \(\zeta\).

\[ \frac{h_0}{h} = e^{\frac{\varepsilon}{1.5}}, \]

\[ \frac{h_0}{h} = e^{\frac{\operatorname{ctg}\beta_1+\operatorname{tg}(\beta_1-\gamma)}{1.5}}, \tag{4} \]

\[ \frac{h_0}{h} = e^{\frac{\zeta^2 - 2\zeta \sin\gamma + 1}{1.5\,\zeta\cos\gamma}}. \tag{5} \]

Here \(e\) is the base of natural logarithms.

Equation (5) makes it possible, for values of \(\zeta\) and \(\gamma\) known for a particular cutting case, to determine the magnitude of the degree of deformation in compression, \(\frac{h_0}{h}\), characterized by the same value of relative shear as the cutting process.

For the derivation of the cutting-force equation on the principle of equality of the specific works of deformation at equal degrees of deformation, it is necessary to have analytical expressions for the specific works of deformation in cutting and in compression. We shall now proceed to derive these equations.

Specific work of shear deformation in the cutting process

Figure 1 gives the diagram of the forces acting on the rake face of the tool and carrying out the process of chip deformation.

Here: \(N\) is the force normal to the rake face,
\(F\) is the friction force,
\(R\) is the resultant force on the rake face,
\(P_1\) is the force acting in the direction of the cutting speed,
\(P_2\) is the force acting in the direction perpendicular to the cutting surface,
\(P_t\) is the force acting in the shear plane,
\(a\) is the thickness of the layer being cut,
\(b\) is the width of the layer being cut,
\(\eta\) is the friction angle.

If the tangential stress in the shear plane is \(\tau\), then

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

\[ P_t=R\cos(\omega+\beta_1), \]

where

\[ \omega=\eta-\gamma, \]

\[ R=\frac{P_1}{\cos\omega}=\frac{P_2}{\sin\omega}. \]

Making the simplest substitutions, we obtain:

\[ \tau=\frac{P_t\sin\beta_1}{ab} =\frac{P_1\sin\beta_1\cos(\omega+\beta_1)}{ab\cos\omega} =\frac{P_2\sin\beta_1\cos(\omega+\beta_1)}{ab\sin\omega}. \]

Fig. 1

Here the tangential stress \(\tau\) in cutting is given as a function of the forces \(P_1\) and \(P_2\), which can be determined experimentally, and of the angles \(\beta_1\) and \(\omega\), where the angle \(\beta_1\) can be calculated from the shrinkage of the chip, measured directly on the chip, while the angle \(\omega\) is determined through the expression

\[ \tg \omega=\frac{P_2}{P_1}. \]

Fig. 2

In Fig. 2 we give the velocity diagram.
Here \(v\) is the cutting speed,
\(v_t\) is the shear velocity,
\(v_F\) is the friction velocity.

The vector of the friction velocity is directed parallel to the front face of the cutter. It is not difficult to obtain the following expressions for the velocities \(v_t\) and \(v_F\)

\[ v_t = v \frac{\cos\gamma}{\cos(\beta_1-\gamma)}, \]

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

But

\[ \frac{\cos(\beta_1-\gamma)}{\sin\beta_1}=\zeta, \]

therefore

\[ v_F=\frac{v}{\zeta}. \]

Specific work of shear deformation in cutting

\[ A_{w_t}=\frac{P_t\cdot v_t}{abv} =\frac{P_1\cos(\omega+\beta_1)\cdot\cos\gamma}{ab\cos\omega\cdot\cos(\beta_1-\gamma)}. \tag{7} \]

If we express the specific work through the shear stress

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

then we obtain

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

In this expression

\[ \frac{\cos\gamma}{\cos(\beta_1-\gamma)\cdot\sin\beta_1} = \frac{1+\tg^2\beta_1}{\tg\beta_1+\tg^2\beta_1\tg\gamma}. \]

Substituting

\[ \tg\beta_1=\frac{\cos\gamma}{\zeta-\sin\gamma}, \]

we obtain

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

Thus, the specific work of shear is

\[ A_{w_t}=\tau\cdot\varepsilon. \tag{8} \]

Specific work of friction

\[ A_{w_F}=\frac{Fv_F}{abv} = \frac{P_1\sin\eta\sin\beta_1}{ab\cos\omega\cos(\beta_1-\gamma)}, \]

where the friction angle \(\eta=\omega+\gamma\).

If we add the two specific works, we obtain

\[ A_{w_t}+A_{w_F} = \frac{P_1}{ab} \left\{ \frac{\cos(\omega+\beta_1)\cos\gamma+\sin(\omega+\gamma)\sin\beta_1} {\cos\omega\cos(\beta_1-\gamma)} \right\}. \]

The factor in braces becomes unity, and therefore

\[ A_{w_t}+A_{w_F} = \frac{P_1}{ab} = \frac{P_1v}{abv} = A_w. \]

Thus, the sum of the specific works of shear deformation and friction is equal to the total specific work of cutting. Hence, for the specific work of shear one can obtain the following expression

\[ A_{w_t}=A_w-A_{w_F}=\frac{P_1}{ab}\left\{1-\frac{\sin\eta\sin\beta_1}{\cos\omega\cos(\beta_1-\gamma)}\right\} \]

and since

\[ \frac{\cos(\beta_1-\gamma)}{\sin\beta_1}=\zeta, \]

then

\[ A_{w_t}=\frac{P_1}{ab}\left\{1-\frac{\sin\eta}{\zeta\cos\omega}\right\}. \tag{9} \]

This expression will be convenient to use subsequently for certain conclusions.

The total specific work \(A_w\) in cutting is the sum of the specific works of plastic shear deformation \(A_{w_t}\) and friction \(A_{w_F}\). Thus, the total work of cutting is the sum of the works of shear and friction

\[ A=A_t+A_F. \]

The work of shear \(A_t\) produces the process of chip shear and creates one or another magnitude of relative shear \(\varepsilon\) and the corresponding chip shrinkage \(\zeta\). The work of friction \(A_F\) is expended in overcoming the friction forces between the chip and the rake face of the cutter. The work of friction also deforms the chip, but this deformation is secondary; it embraces only a thin layer of the chip adjacent to the rake face of the cutter, does not extend to the entire volume of chip metal, and does not change the chip shrinkage or the magnitude of the relative shear in it. Therefore, the degree of chip deformation, calculated from its longitudinal shrinkage, is created by the work of shear \(A_{w_t}\).

For the specific work of shear deformation we have obtained the expression

\[ A_{w_t}=\tau\varepsilon. \]

This expression is of considerable interest in itself. If it is assumed that, in the cutting process under a given temperature–speed regime, the tangential stress \(\tau\) for a work-hardening metal must be uniquely related to the deformation \(\varepsilon\), so that, independently of the external cutting conditions (different rake angles, different conditions of chip friction along the rake face), each value of \(\varepsilon\) corresponds to a definite value of \(\tau\), then the specific work of deformation \(A_{w_t}\) will also have a definite value for each \(\varepsilon\). Thus, for the cutting process, the law of independence of the specific work of shear deformation from the external cutting conditions proves to be valid. This proposition, of course, requires experimental verification, which will be given by us in the subsequent exposition. The experimental verification of this proposition is extremely simple. If the stated proposition is correct, then all experimental points of tests satisfying the condition of constancy of the temperature–speed regime must, in the coordinates \(A_{w_t}\)—\(\varepsilon\), give one single curve, independently of the conditions under which the experiment is carried out.

Let us now turn to the specific work of plastic compression.

Specific work of plastic compression

The total work of compression deformation of a specimen can be determined by measuring the area of the diagram “compression force—compression deformation.” For analytical calculation it is necessary to solve the integral

\[ A_g = \int_{h_0}^{h} Q\,dh, \]

where \(Q\) is the compressive force at the given moment.

To solve it, it is necessary to know the law governing the increase of stress with increasing deformation.

If one assumes a power law for the dependence of the true stress on deformation, then the stress–deformation curve may be represented by the well-known compression polytrope

\[ \sigma'_1 = \sigma_0 \left( \frac{h_0}{h} \right)^n , \tag{10} \]

which is not absolutely exact, but in practice gives good results [5].

Fig. 3

Here \(\sigma\) is the true stress at the given moment for the given degree of deformation \(\dfrac{h_0}{h}\),

\(\sigma_0\) is the conditional yield strength, the stress that would have occurred at the initial moment of deformation at \(h = h_0\), if the polytropic law had been valid from the very initial stage of deformation,

\(n\) is the exponent of the stress polytrope, a quantity constant for the given temperature–speed conditions of deformation.

Equation (10) is shown in Fig. 3 in logarithmic coordinates

\[ \lg \frac{h_0}{h} \;-\; \lg \sigma . \]

Dependence (10) is valid only for the rectilinear part. The straight line, if extended to the stress axis, gives the intercept \(\sigma_0\)—the conditional yield strength. The rectilinear part of the curve begins with the deformation \(\dfrac{h_0}{h_p}\). Up to this deformation there is located the additional polytropic part of the curve. Professor Doctor M. A. Bolshanina [5], on the basis of Kunin’s experiments, gives the equation for the specific work of deformation in compression

\[ A_{w\,cж} = \frac{\sigma_0}{n} \left\{ \left( \frac{h_0}{h} \right)^n - (1 - x) \left( \frac{h_0}{h_p} \right)^n - x \right\}. \tag{11} \]

Here \(K\) is the completeness coefficient, which takes into account the character of the additional-tropic part of the curve; it is the ratio of the true work in the additional-tropic part to the work that would have been required if the polytropic relation had been valid from the very beginning of deformation. The values of \(x\) and \(\dfrac{h_0}{h_p}\) for various metals and their states (temperature, strain hardening, structure) can be obtained only experimentally.

If it is assumed that the law of the polytrope begins to operate from the very beginning of deformation, then \(x = 1\) and

\[ A_{w,\mathrm{comp}}=\frac{\sigma_0}{n}\left\{\left(\frac{h_0}{h}\right)^n-1\right\}. \]

The specific work according to equation (12) is always greater than according to equation (11), but the difference between them will be the less noticeable, the greater the degree of deformation \(\dfrac{h_0}{h}\). The chip shrinkages usually observed in cutting are equivalent to such considerable degrees of deformation in compression that the difference between (12) and (11) proves to be less than \(1\%\). Thus, if for the rake angle of the tool \(\gamma = 0^\circ\) one takes the value \(\zeta = 2\) (usually the shrinkage is always greater than two), then this will correspond to \(\dfrac{h_0}{h}=5.4\), and the difference between (12) and (11) will be less than \(0.5\%\). Therefore, for the subsequent derivations we shall take equation (12).

It may be considered that this equation gives the magnitude of the specific work of plastic deformation in compression and does not include the work of friction, since in compression, although friction at the end faces does occur, the work of friction is almost absent, because there are no (or almost no) displacements in the direction of the friction forces. That the work of friction in compression may be neglected is indicated by compression experiments carried out without lubricant and with lubricant. Thus, Corresponding Member of the Academy of Sciences of the USSR V. D. Kuznetsov writes concerning the effect of lubrication on the compression polytrope of steel U-8:

“Compression experiments were carried out with the lubricant used in cutting and without lubricant. No noticeable effect of the lubricant on \(m\) and \(\sigma_0\) was found” [6].

The same was shown by Ivanov’s experiments on compression of copper specimens with and without lubricant [7].

Completely analogous results were also obtained in the work of Bolshanina and Studenok in compressing aluminum without lubricant and with kerosene lubricant, as well as with alcohol containing \(0.1\%\) stearic acid [8].

In cutting, on the contrary, the work of friction constitutes a noticeable part of the total work, since there we have both very high coefficients of friction (they are 2–5 times higher than in compression) and continuous motion at a considerable velocity in the direction of the friction force. Therefore, we shall equate the specific work in compression not to the whole specific work in cutting, but only to that part of it which is equal to the specific work of shear deformation \(A_{w,t}\):

\[ A_{w,\mathrm{comp}} = A_{w,t}. \]

Writing the equality between these specific works, we obtain

\[ \frac{\sigma_0}{n}\left\{\left(\frac{h_0}{h}\right)^n-1\right\} = \frac{P_1}{ab} \left\{ 1-\frac{\sin \eta}{\zeta \cos \omega} \right\}. \tag{13} \]

Here, as before,

\[ \left(\frac{h_0}{h}\right)^n=e^{\frac{n\varepsilon}{1.5}}, \]

and therefore

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

or, in a more expanded form, since

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

then

\[ P_1=\frac{\sigma_0}{n}ab\, \frac{ e^{\frac{n}{1.5}\,\frac{\zeta^2-2\zeta\sin\gamma+1}{\zeta\cos\gamma}}-1 } {1-\dfrac{\sin\eta}{\zeta\cos(\eta-\gamma)}} . \tag{15} \]

This equation can also be represented in a somewhat different form if, for the equality of the specific works, equation (7) is used:

\[ \frac{\sigma_0}{n}\left\{e^{\frac{n\varepsilon}{1.5}}-1\right\} = \frac{P_1}{ab}\, \frac{\cos(\omega+\beta_1)\cos\gamma} {\cos\omega\cdot\cos(\beta_1-\gamma)} . \]

Here the factor on the right-hand side can be reduced to the expression

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

then

\[ P_1= \frac{\sigma_0}{n} \left\{ e^{\frac{n}{1.5}\,\frac{\zeta^2-2\zeta\sin\gamma+1}{\zeta\cos\gamma}}-1 \right\} \frac{ab\zeta} {\zeta-\sin\gamma-\cos\gamma\,\tg(\eta-\gamma)} . \tag{16} \]

Of course, both equation (15) and (16) give numerically the same results.

Both of these equations show that the force \(P_1\) is determined by the physical characteristics of the material being machined, \(\sigma_0\) and \(n\), by the dimensions of the removed layer \(a\) and \(b\), by the front angle of the cutter \(\gamma\), by the angle of friction \(\eta\), and by the chip compression \(\zeta\).

Naturally, the force \(P_1\) will also depend on those factors which, to one degree or another, influence the values of \(\sigma_0\) and \(n\). Thus, for example, temperature and deformation rate for different materials affect \(\sigma_0\) and \(n\) differently; through \(\sigma_0\) and \(n\) they will correspondingly also affect the force \(P_1\). It may be predicted in advance that, for example, for refractory metals the influence of cutting speed, as the deformation rate, should have a less substantial effect on the cutting force than for low-melting metals, since this is precisely the effect of deformation rate on \(\sigma\) and \(n\). Of course, if, when the cutting speed changes, the coefficient of friction (the angle of friction) or the chip compression, or both together, also change, this will inevitably cause a change in the cutting force; but this change will no longer be the result of a change in the deformation rate.

From the equations for the force \(P_1\), it is not difficult to pass to the equation for \(P_2\), since according to Fig. 1

\[ P_2=P_1\tg\omega \]

or

\[ P_2=P_1\tg(\eta-\gamma). \]

Therefore, for the force \(P_2\) the following equation may be written:

\[ P_2=\frac{\sigma_0}{n}\,ab\sin(\eta-\gamma)\, \frac{ e^{\frac{n}{1.5}\cdot\frac{\zeta^2-2\zeta\sin\gamma+1}{\zeta\cos\gamma}}-1 }{ \cos(\eta-\gamma)-\frac{\sin\eta}{\zeta} }. \tag{17} \]

Thus, the force \(P_2\) will depend on the same quantities as \(P_1\), which is quite logical, since both these forces are components of one resultant \(R\).

The forces \(P_1\) and \(P_2\), which can be calculated from equations (15) and (17), are forces located on the rake face of the tool and producing deformation of the layer of metal being cut.

For the further analysis of the equations obtained, and for comparing them with experimental results, it is convenient to have a relation for the shear stress in cutting expressed in terms of the physical constants of the metal being machined. We shall obtain such an equation by equating (8) and (12):

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

Hence the shear stress in cutting is

\[ \tau=\frac{\sigma_0}{n\varepsilon}\left\{e^{\frac{n}{1.5}}-1\right\}. \tag{18} \]

For a non-hardening metal, for which an increase in deformation is not accompanied by an increase in stresses,

\[ n=0 \]

and

\[ \tau=\sigma_0\cdot\frac{0}{0}. \]

Resolving this indeterminacy by using L’Hôpital’s rule, we obtain

\[ \tau=\frac{2}{3}\sigma_0=\text{const} \]

and the force \(P_1\) for a non-hardening metal will have the expression

\[ P_1=\frac{2}{3}\sigma_0ab\, \frac{\zeta^2-2\zeta\sin\gamma+1} {\cos\gamma\left(\zeta-\sin\gamma-\cos\gamma\tan\omega\right)}. \tag{19} \]

Experimental verification of the cutting-force equations

The experimental verification was carried out by us on three metals: steel, lead, and aluminum. The greatest number of experiments was performed on steel. This metal was of interest to us because it has the greatest practical application. In addition, steel is characterized by the fact that, having a high melting temperature, it exhibits little sensitivity of stress to the rate of deformation both in cutting and in other methods of plastic deformation, which was confirmed by us through special experiments. Therefore, in experiments under a constant temperature regime we could disregard the influence of speed on the stress during cutting. In the deformation of aluminum, the rate of deformation has a small but already noticeable influence on the stress; in the deformation of lead this influence is even

more significantly. Therefore, for these metals the influence of speed cannot be neglected, and it must be taken into account. Lead was of interest to us because, under conditions of constant deformation rate, at room temperature it is already a metal that does not harden, i.e., its stress does not depend on the degree of deformation \((n = 0)\).

For an experimental verification of the cutting-force equations, we had to determine the parameters of the cutting process entering into the equations \((\omega, \eta, \gamma, \beta_1)\), and the parameters of the compression process \((\sigma_0, n)\). Here, first of all, it is necessary to discuss how we determined the friction angle \(\eta\) and the shear angle \(\beta_1\) in cutting.

To determine the friction angle \(\eta\) and the coefficient of friction of the chip on the rake face of the tool, it was necessary to separate, from the forces measured by the dynamometer, the forces acting on the rake face of the tool. For this purpose we used the method described by us in [9].

Thus, having measured with the dynamometer the forces \(P_z\) and \(P_x\) (in free cutting) and subtracting from them the forces on the flank face, we obtained the force \(P_1\), applied on the rake face and acting in the direction of the cutting speed, and the force \(P_2\), also acting on the rake face of the tool in the direction perpendicular to the cutting surface. If we conventionally assume all forces to be applied to the cutting edge, then it is not difficult to obtain the following relations:

\[ \tg \omega = \frac{P_2}{P_1}, \]

\[ \eta = \omega + \gamma . \]

Thus, to determine the friction angle \(\eta\), it is necessary to determine the angle \(\omega\) and to know the rake angle \(\gamma\) of the tool. Of course, in this way we can determine only a certain mean friction angle, whereas in reality the coefficient of friction and the friction angle over the contact area of the chip with the rake face of the tool are undoubtedly variable. We have no data for determining the law of distribution of the coefficient of friction over the contact area, and therefore it is necessary to use mean values.

Since in order to determine \(\eta\) it is necessary to know \(\gamma\)—the rake angle of the tool—the experiments had to be carried out in those ranges of cutting speeds within which no built-up edge forms on the rake face, since, as is known, it changes both the geometry of the tool and the actual rake angle over wide limits.

In accordance with this, in cutting steel the experiments were carried out by us either at very low speeds from \(0.0003\) to \(0.76\ \text{m/min}\) (we call these speeds microspeeds), or at such high speeds at which formation of a built-up edge was likewise not observed. Experiments with microspeeds had the further undoubted advantage that, within the above-indicated limits, the influence of temperature on the cutting process was excluded, since all the heat that is the result of the work expended, at these speeds, is freely dissipated and does not heat to any noticeable extent either the chip or the metal from which the chip is removed. This was also important because our compression tests were carried out at room temperature, and a strict comparison of the cutting process with them required that the latter likewise be performed without noticeable heating of the chip and of the metal from which it was removed.

To determine the angle \(\beta_1\), the longitudinal shrinkage of the chip \(\zeta\) and its width \(b_1\) were measured; the angle \(\beta_1\) was calculated from the relation

\[ \tg \beta_1 = \frac{\cos \gamma}{\dfrac{b}{b_1}\,\zeta - \sin \gamma}. \]

Free cutting at microspeeds was carried out by planing on a milling machine; in this case the piece of metal being machined, from which the chip was removed, was clamped in special vises on the table of the milling machine and, from the feed mechanism, received the motion and cutting speed. The tool was clamped in a special, extremely accurate and sensitive elastic two-component dynamometer, which in turn was bolted to the crosshead of the machine.

At high cutting speeds the experiments were carried out on a lathe; the chip was removed by a tool having an entering angle in plan \(\varphi = 90^\circ\), from the end of a cup by the longitudinal feed of the tool. In this case the cutting forces were measured by a hydraulic dynamometer.

When working at microspeeds, the chip was removed from a strip of metal \(60\text{–}80\) mm long, and thus, for calculating shrinkage, the path traversed by the tool was known. When working on the lathe a continuous chip was removed, and the path traversed by the tool in removing the chip segment taken for calculating shrinkage was determined by weighing the chip segment.

Before removing each chip, the tool was invariably lapped on all faces with GOI paste, so that there were absolutely no defects on the cutting edge.

1. Experiments on cutting steel

The basic experiments carried out by us to verify the equations were performed on steel 40 at microspeeds.

The experiments were carried out with a wide range of variation of the rake angles from \(-1^\circ\) to \(40^\circ\), with and without lubrication, and with a wide range of variation of cutting speed. In all experiments only a purely continuous chip was removed, without any signs of cracking or separation into elements, since observations showed that when these signs appear the cutting process is distorted, chip shrinkage cannot be determined accurately, the coefficient of friction decreases, etc.

Figure 4 presents one of the graphs of cutting forces and chip shrinkages obtained by us at microspeeds. Each point of this graph is the mean value of 5–15 readings made visually over the entire period of removal of the given chip. Through the points obtained in this way, average curves were drawn giving the dependence of the cutting forces on chip thickness (depth of cut). Through the shrinkage points, curves were drawn for the dependence of shrinkage on depth of cut.

From each of these curves we took the values of cutting forces and shrinkages for a definite depth of cut chosen by us, and from these values calculated all the quantities of interest to us \((\omega, \eta, \beta_1, \varepsilon, \tau)\). These quantities are given in Table 1. Thus, each of the values entered in Table 1 is the average result of a large number of experiments.

Table 2 gives the results of experiments carried out at high cutting speeds, without built-up edge, by the method of free cutting.

To determine the parameters of the compression polytrope, we compressed cylinders with a ratio of height \(h_0\) to diameter \(d_0\) (their initial values before compression)

\[ \frac{h_0}{d_0} = 2.5. \]

Such a considerable ratio was chosen by us in order to reduce the influence of friction at the end faces on the stressed state. Compression was carried out in a special guiding sleeve, to prevent the possibility of skewing of the specimens being compressed.

Table 1

Rake angle	Cutting speed, m/min, $v$	$a$	$b$	$P_1$	$P_2$	$\zeta$	$\varepsilon$	$\omega$	$\beta_1$	$\tau$	Test conditions
$40^\circ$	0,0048—0,76	0,1	3,0	30	−10	1,62	1,24	−18°25′	38°10′	60,6	Lubricant: soapy water
$40^\circ$	0,0048	0,1	3,0	25	−12	1,53	1,16	−25°40′	40°50′	60,4	Vegetable oil
$40^\circ$	0,035	0,1	3,0	29,5	−11	1,6	1,24	−20°50′	38°50′	62,2	Vegetable oil
$40^\circ$	0,098	0,1	3,0	33,5	−9	1,6	1,24	−15°	38°50′	66	Vegetable oil
$40^\circ$	0,76	0,1	3,0	38	−6,5	1,73	1,33	−9°30′	35°20′	67,5	Vegetable oil
$40^\circ$	0,0009	0,2	3,0	70,5	−10	1,82	1,41	−8°	33°10′	58,7	Without lubrication
$40^\circ$	0,0048	0,2	3,0	78,5	−7	1,82	1,41	−5° 5′	33°10′	63,5	Without lubrication
$40^\circ$	0,035	0,2	3,0	83,5	−3,5	1,82	1,41	−2° 30′	33°10′	65,3	Without lubrication
$40^\circ$	0,098	0,2	3,0	87,5	0	1,82	1,41	0	33°10′	67,0	Without lubrication
$40^\circ$	0,760	0,1	3,0	46	3,5	1,82	1,41	4° 20′	33°10′	66,5	Without lubrication
$30^\circ$	0,0009—0,76	0,2	3,1	74,5	−11,7	1,9	1,64	−8° 55′	31°50′	59	Soapy water
$30^\circ$	0,0009—0,76	0,1	3,1	40	−5,5	1,94	1,69	−7° 50′	31°	61,5	Soapy water
$30^\circ$	0,0003—0,098	0,2	3,05	95	+2,5	2,12	1,84	1° 30′	28°10′	62,0	Machine oil
$30^\circ$	0,0003—0,0048	0,2	3,05	62,5	−16,0	1,8	1,56	−14°45′	33°45′	56,0	Vegetable oil
$30^\circ$	0,035—0,098	0,2	3,05	65	−16,5	1,8	1,56	−14°15′	33°45′	58,0	Vegetable oil
$30^\circ$	0,0011	0,2	3,05	92,5	+5,2	2,2	1,91	+3° 15′	27°	60,0	Without lubrication
$30^\circ$	0,0048	0,2	3,05	100	9,0	2,3	1,99	+5° 10′	25°45′	62,0	Without lubrication

Continuation of Table 1.

Rake angle	Cutting speed, m/min, $v$	$a$	$b$	$P_1$	$P_z$	$\zeta$	$\varepsilon$	$\omega$	$\beta_1$	$\tau$	Test conditions
30°	0,035	0,2	3,05	112	15,0	2,4	2,1	7°40′	24°30′	66,0	Without lubrication
"	0,098	"	"	121	19,0	2,55	2,25	8°55′	22°55′	66,7	"
20°	0,00088—0,76	0,3	3,1	112	— 8,0	1,9	1,65	—4°45′	31°45′	55,5	Soapy water
20°	0,00088—0,76	0,2	3,0	82	— 8,5	2,1	1,99	—5°55′	28°10′	60,2	Soapy water
"	"	0,1	3,0	45	— 4,0	2,2	2,1	—5°5′	26°45′	63,0	"
"	"	0,27	3,0	106,5	—11,5	2,0	1,93	—6°10′	29°30′	59,7	"
"	0,00088	0,1	"	60	9,5	2,8	2,64	+ 9°	20°55′	62	Without lubrication
"	0,0048	"	"	64	12,5	2,85	2,68	11°	20°30′	64,6	"
"	0,035	"	"	72,5	18	3,1	2,92	14°	18°50′	67,6	"
"	0,098	"	"	76,5	21	3,1	2,92	15°20′	18°50′	70,4	"
"	0,280	0,11	"	89,0	27	3,25	3,05	18°45′	17°50′	69,8	"
22°	0,0011– 0,0045	0,15	3,0	58,5	— 7,5	2,1	2,02	—7° 20′	27°30′	57	Vegetable oil
"	0,033	"	"	61,0	— 7,5	2,1	2,02	—7°	27°30′	59	"
"	0,098—0,28	"	"	64,0	— 7,5	2,1	2,02	—6°40′	27°30′	61,5	"
14°	0,09	0,1	3,0	47,5	1,5	2,27	2,3	2°	25°30′	60,6	Lubrication
8°30′	0,0008—0,76	0,15	3,0	84	7,5	2,55	2,68	5°	22°25′	63,7	Soapy water
"	"	0,1	"	61	8,5	2,8	2,89	7°55′	20°25′	63,2	"
"	"	0,05	"	34	4,5	2,95	3,04	7°35′	19°20′	68,0	"
—1°	0,035	"	"	43	12	3,4	3,74	15°30′	16°15′	71	Soapy water
"	"	0,064	"	49	13	3,19	3,51	14°50′	17°25′	67,0	"
"	"	0,04	"	34	9,5	3,5	3,8	15°40′	16°	68,9	"

Table 2

$\gamma$	$v$	$a$	$b$	$P_1$	$P_2$	$\zeta$	$\varepsilon$	$\omega$	$\beta_1$	$\tau$	$b_1$
$30^\circ$	41	0,214	3,0	100	20,6	1,8	1,47	$11^\circ40'$	$36^\circ25'$	63	3,15	Without lubrication
$30^\circ$	52	0,214	3,0	100	20,4	2,08	1,41	$11^\circ30'$	$30^\circ20'$	59,7	3,15	Without lubrication
$20^\circ$	51,5	0,214	3,0	112	39,5	2,29	2,11	$19^\circ26'$	$26^\circ40'$	58	3,15	Without lubrication
$20^\circ$	67,3	0,214	3,0	130	57	2,63	2,16	$23^\circ40'$	$26^\circ$	63,4	3,2	Without lubrication
$9^\circ$	26,6	0,214	3,0	210	150	4,95	3,7	$35^\circ40'$	$15^\circ35'$	68,7	4	Without lubrication
$0^\circ$	39	0,214	3,0	232	202	5,24	4,0	$41^\circ$	$15^\circ$	69,5	4,2	Without lubrication

NOTE: In calculating $\beta_1$, $\varepsilon$, and $\tau$, the widening of the chip was taken into account. The value of $\beta_1$ was determined from the relation

\[ \operatorname{tg}\beta_1 = \frac{\cos\gamma}{\dfrac{b}{b_1}\zeta - \sin\gamma}. \]

Figure 5 shows the compression diagram. From this diagram the constants entering into the equation of the compression polytrope were obtained:

\[ \sigma_0=75.5;\quad m=1.226;\quad n=0.226. \]

In order to check how significantly friction at the end faces during compression affected the constants obtained, we used the method proposed by Shofman [10]. For this purpose, specimens with different ratios were subjected to compression

Fig. 4

\[ \frac{d_0}{h_0}: 4.5;\ 2.5;\ 1.5;\ 1.0;\ 0.4. \]

Figure 6 presents the stress curves obtained for these different specimens. Through the points of the curves corresponding to equal relative deformations

\[ \left(\lambda = 1-\frac{h}{h_0}\right), \]

curves were drawn which were smoothly extended to

Fig. 5

intersections with the ordinate axis, on which they cut off the stresses \((\sigma)\) corresponding to a definite degree of deformation at the ratio

\[ \frac{d}{h}=0, \]

i.e., in the absence of friction at the end faces.

Fig. 6

From these stresses \((\sigma_{\mathrm{ist}})\) the curve in Fig. 7 is constructed, which gives the compressive stresses under the condition that there is no friction at the end faces. If we compare this curve with the curve in Fig. 5, obtained for specimens with the initial ratio

Fig. 7

\[ \frac{h_0}{d_0}=2.5, \]

then we can see that, at so large a ratio, friction at the end faces has almost no effect on the magnitude of the stresses.

Fig. 8 gives the curve of deforming forces in compression.

On the basis of Tables 1 and 2, in Fig. 9 we have plotted, in \(\varepsilon—\tau\) coordinates (shear—stress), the points obtained from cutting experiments; the results obtained at microspeeds are denoted by circles, and those at high cutting speeds by crosses. In the same figure, a solid line shows

Fig. 8

the dependence \(\tau=f(\varepsilon)\), obtained in compression and expressed by the equation

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

Fig. 9 shows that the curve for this dependence is well positioned relative to the experimental points, being only lower than them on average by 3–4%. In Fig. 9 the mean discontinuous curve through the experimental points is drawn; it proves to be these same 3–4% higher than the curve drawn according to the equation.

Fig. 9

Thus, the theoretically derived equation for the cutting force corresponds satisfactorily to the results of experiments.

\[ P_1=\frac{\sigma_0}{n}\,ab\, \frac{ e^{\frac{n}{1.5}\cdot\frac{\zeta-2\zeta\sin\gamma+1}{\zeta\cos\gamma}}-1 }{ 1-\frac{\sin\eta}{\zeta\cos(\eta-\gamma)} }, \]

where
\(\sigma_0\) is the conventional yield point in compression,
\(n\) is the polytropic exponent of stresses in compression,
\(\zeta\) is the chip shrinkage,
\(a,b\) are the dimensions of the cut section,
\(\gamma\) is the rake angle of the cutter,
\(\eta\) is the angle of friction of the chip along the rake face of the cutter.

In Fig. 9, the crosses show the results of experiments carried out at high cutting speeds, beyond the region of built-up edge formation. Despite the completely different thermal regime of deformation, the results of these experiments fell well on the same mean line that corresponds to the experiments carried out at microspeeds. Thus, the experiments carried out at high speeds satisfy the same cutting-force equation.

We previously indicated that, in the cutting process, the specific work of deformation of plastic shear \(A_{w_t}\) is equal to the product of the shear stress \(\tau\) and the relative shear \(\varepsilon\)

\[ A_{w_t}=\tau\varepsilon. \tag{8} \]

We noted that this dependence is of substantial interest in itself, since it shows that, for the cutting process, the law of independence of the specific work of shape change from the deformation scheme proves to be valid. We indicated that this proposition requires experimental verification.

For experimental verification, the results of the experiments must be represented in the coordinates \(A_{w_t}\)—\(\varepsilon\). Since \(\tau=f(\varepsilon)\), in this graph all experiments carried out at different values of \(\gamma\), with different coefficients of friction \(\mu\) and chip shrinkages \(\zeta\), must, for the given temperature–velocity deformation regime, fall on a single curve \(A_{w_t}=f(\varepsilon)\). The curve \(A_{w_t}=f(\varepsilon)\) must satisfy the equation

\[ A_{w_t}=\frac{\sigma_0}{n}\left\{e^{\frac{n\varepsilon}{1.5}}-1\right\} \tag{20} \]

and can be constructed from the values of \(\sigma_0\) and \(n\) determined from compression experiments.

For the case of cutting steel, the points of all experiments carried out at various microspeeds must also fall on this same curve, since for steel the deformation rate has an imperceptible influence on the stress, provided the temperature regime of deformation remains constant.

Such a graph for steel 40 is presented by us in Fig. 10. Here the solid curve is constructed according to equation (20), using the values of \(\sigma_0\) and \(n\) obtained from compression experiments. As the graph shows, all the experimental points are located, on average, somewhat above the curve, but they reproduce its character quite distinctly, which serves as good evidence of the correctness of equation (8). In Fig. 10, the large circles and crosses show the results of experiments carried out at high cutting speeds. As we see, these experiments also fall into the general regularity.

A quite natural question may arise as to why experiments carried out at high cutting speeds give stresses and specific works the same as in experiments carried out at microspeeds, although at high speeds the temperature in the chip-deformation zone is undoubtedly higher than at microspeeds.

In this connection two assumptions may be put forward. One assumption is that the heat liberated in the deformation zone (and as a result of the deformation itself) at high speeds does not have time to create conditions for the occurrence of the relaxation process, owing to the fact that each microvolume of metal is in the process of deformation under the action of this heat for an extremely short interval of time (thousandths and ten-thousandths of a second), passing through the deformation zone at high speed. The other assumption is that the temperature in the deformation zone, which is high at considerable cutting speeds, could reduce the stress, but at high temperatures the influence of the deformation rate, which increases the stress, begins to make itself felt, thereby balancing to one degree or another the influence of the temperature that lowers the stress.

Fig. 10

It is of considerable interest to compare the influence of the individual variables of the cutting process on the cutting force according to experimental data and according to the equation.

Equation (15) shows that, provided \(\sigma_0, n, \gamma, \eta, \zeta\) remain constant, the cutting force is directly proportional to the width \(b\) and the thickness \(a\) of the layer being cut. The constancy of \(\sigma_0, n, \gamma, \eta, \zeta\) in the case of cutting hard-to-machine metals, and in particular steels, is satisfied, as we have already shown earlier, by regimes of constant temperature on the rake face of the tool. In this case, a change in “\(a\)” and “\(b\)” should, according to the equation, cause the same change in the forces \(P_1\) and \(P_2\). Figure 11 shows the influence of the thickness \(a\) of the layer being cut on the cutting forces under the condition that the temperature on the rake face of the tool remains constant. We see that the cutting force, in simple proportional coordinates, is expressed as a straight line as a function of “\(a\).” True, this straight line does not pass through the origin, but cuts off a certain segment on the ordinate axis at \(a=0\). But this occurs only because in Fig. 11 the forces \(P_z\) and \(P_x\), measured by a dynamo-

by the meter and include both the forces on the front face and the forces on the rear face of the cutter. Equation (15), however, gives only the forces on the front face, which under the conditions indicated above should be proportional to the chip thickness \(a\). The intercept on the ordinate axis at \(a=0\) gives the magnitude of the force on the rear face.

Fig. 11

Therefore Fig. 11 gives the same influence of the chip thickness \(a\) on the cutting forces as equation (15).

Figure 12 shows the influence of the rake angle of the cutter on the force \(P_1\). The experimental results are plotted on the graph with circles, and the results of calculations by equation (15) with crosses. Figure 13 shows the influence of the coefficient of friction on the force \(P_1\). Naturally, the latter figure shows

Fig. 12

the influence on the force \(P_1\) not only of the coefficient of friction itself, but also of the chip compression \(\xi\), since with a change in the coefficient of friction in the real cutting process the chip compression inevitably changes as well. Figures 12 and 13 show

produce good agreement between the calculated and experimental values of the forces \(P_1\) and, thus, also confirm the correctness of equation (15).

Figs. 14, 15, 16, 17, and 18 show the influence of the friction coefficient \(\mu\) itself on the force \(P_1\), if this friction coefficient did not cause a change in the chip compression \(\zeta\). The curves plotted in these figures were calculated

Fig. 13

Visible annotations in the figure: \(P_1\); \(\mu\); “with lubrication”; “without lubrication”; \(\gamma = 20^\circ\); \(\gamma = 30^\circ\); Steel 40; \(a = 91\); \(b = 3.0\ \text{mm}\); \(o\)—from experiment; \(x\)—from the equation.

according to equation (15). The figures show that, as the friction coefficient increases, the cutting force increases, and the more intensively, the smaller the chip compression. At a constant friction coefficient, the force \(P_1\) is usually greater the more significant the compression \(\zeta\), but there may be exceptions to this usual and quite understandable situation.

Fig. 14

Visible annotations in the figure: \(P_1\); \(\mu\); \(\gamma = 10^\circ\); \(\zeta = 2\); \(\zeta = 3\); \(\zeta = 4\); \(\zeta = 5\).

Fig. 15

Visible annotations in the figure: \(P_1\); \(\mu\); \(\gamma = 0^\circ\); \(\zeta = 1\); \(\zeta = 2\); \(\zeta = 3\); \(\zeta = 4\); \(\zeta = 5\).

Thus, from the figures it is seen that, at shrinkage \(\zeta = 1\) and at considerable coefficients of friction, the force \(P_1\) may be greater than at more considerable shrinkages. Practically, in the cutting process, such a case—in which, with a smaller shrinkage, a more considerable cutting force would be required—cannot occur, since this would contradict the principle that the deformation process proceeds with the least expenditure of work. Theoretically, however, such a case, which appears paradoxical, can be readily explained.

Fig. 16

Fig. 17

Fig. 18

In Fig. 19 we have presented such a case for \(\gamma = 0\), \(\mu = 0.5\), \(\zeta = 1\) and \(\zeta = 2\). \(P_{t1}\) is the shear force at \(\zeta = 1\). \(P_{t2}\) is the shear force for \(\zeta = 2\). For simplicity, \(P_{t1}\) and \(P_{t2}\) have been taken in the graph as equal to the segments \(AB\) and \(AC\), i.e., the influence of relative shear on the shear stress has not been taken into account, owing to the smallness of this influence. Figure 19 shows that, despite the fact that \(P_{t2} > P_{t1}\), \(P_1\) for \(\zeta = 1\) turned out to be greater than for \(\zeta = 2\).

Figs. 14–18 show that in real cutting processes, in which considerable coefficients of friction cause considerable chip shrinkage, the cutting force \(P_1\) increases as the shrinkage increases.

It is very important to note that, for considerable chip shrinkages \((\zeta > 2)\) observed in the cutting process, the influence of the friction coefficient \(\mu\) itself (at \(\zeta = \mathrm{const}\)) on the cutting force is very insignificant, so that even a substantial error in determining the coefficient of friction when calculating by equation (15) gives only an insignificant error in the calculated cutting force \(P_1\).

Fig. 19

For example, if we take \(\gamma = 0\) and \(\zeta = 3\), then in passing from \(\mu = 0.5\) to \(\mu = 0.75\), i.e., with an increase of \(\mu\) by 50%, the force \(P_1\) changes from 76 kg to 85 kg, i.e., only by 11.8%. Thus, it turns out that, for a practically sufficiently accurate calculation of the force \(P_1\) by equation (15), there is no need to know with great accuracy the value of the coefficient of friction \(\mu\) and the friction angle \(\eta\); it is only necessary to know the order of magnitude of these quantities with an accuracy of \(\pm 15\text{–}20\%\).

2. Experiments on Cutting Aluminum

In Fig. 20 we present a check of equation (15) by our experiments on cutting aluminum at a speed \(V = 0.0075\ \mathrm{m/min}\). The cutting was carried out in the same manner as in the experiments on steel described above. In Fig. 20 the circles represent the experimentally obtained values of the cutting forces at various thicknesses of the cut layer “\(a\).” The crosses show the values of the cutting forces obtained by calculation from equation (15). To make it possible to calculate the cutting forces, we compressed aluminum specimens (with the ratio \(\dfrac{h_0}{d_0} = 2.5\)), obtained the compression polytrope, and from it determined the characteristics \(\sigma_0 = 13.8\) and \(n = 0.235\).

The chip shrinkage \(\zeta\) and the friction angle \(\eta\) during cutting were determined in cutting experiments.

Fig. 20 shows that the calculated values of the cutting force (crosses) are located somewhat below the experimental ones (circles). It is possible that

this is the result of the difference in deformation rates during cutting and compression, since, in the deformation of aluminum, as a comparatively low-melting metal, a noticeable effect of the deformation rate on the stress is manifested. In any case, here too one may note good agreement between the calculated and experimental values.

In Fig. 21 we present experiments on cutting aluminum in the coordinates \(A_{wt} - \varepsilon\). This figure gives the results of experiments carried out at various cutting speeds—from \(0.002\) to \(0.76\ \text{m/min}\), with rake angles from 20 to \(50^\circ\), when cutting both without lubrication and with lubrication by kerosene and vaseline oil. Since, in the deformation of aluminum, the deformation rate has a noticeable effect on the stress, in order to construct the graph we reduced the results of all the experiments to the lowest speed, using the equation previously obtained by us, which reflects the effect of speed on the stress in cutting aluminum,

\[ \tau = \tau_0 v^{0.034}. \]

Fig. 20

Fig. 21

The solid curve in Fig. 21 is drawn according to equation (20), with substitution into it of the values \(\sigma_0\) and \(n\) obtained from experiments on the compression of aluminum. Fig. 21 shows that, despite the large scatter, the experimental points are located near the curve, and by their arrangement repeat its character. The considerable scatter of the experimental points is explained by the nonuniformity of the aluminum, which was at our disposal in the form of cast billets, and also by the circumstance that, when cutting aluminum, a metal buildup forms on the rake face of the cutter, hindering the movement of the chip along the rake face and creating considerable fluctuations in the values of the cutting forces and of the chip compression.

3. Experiments on the Cutting of Lead

Lead was of interest to us as a metal that does not strain-harden, for which, under a constant temperature–speed regime of deformation, the stress does not depend on the degree of deformation and is a constant quantity. Therefore, for lead, under a constant temperature–speed

under the condition that the specific work of deformation in cutting \(A_{wt}\) must be directly proportional to the degree of deformation

\[ A_{wt} = \tau \varepsilon = \text{const.}\,\varepsilon \]

and the dependence of the specific work of deformation on the relative shear must be expressed on the graph by a straight line passing through the origin. We decided to verify this proposition by experiments carried out at microspeeds.

The experiments were conducted at cutting speeds from \(0.00025\) to \(0.106\ \text{m/min}\), with rake angles of 15, 30, 45, and 60°, both with lubrication and without lubrication.

We have shown that, when cutting lead, the rate of deformation has a significant effect on the cutting stress. This effect was expressed by the dependence

\[ \tau = \tau_{0}\, v^{0.102}. \]

Using this dependence, we recalculated the stresses obtained in cutting at different speeds and reduced them to the lowest speed. The stresses reduced to one speed enabled us, from a large number of experiments, to construct the graph

\[ A_{wt} = f(\varepsilon). \]

This graph is presented in Fig. 22.

The points of numerous experiments, carried out over a wide range of variation of rake angles, speeds, and degree of deformation, were well located relative to a straight line passing through the origin. Thus,

Fig. 22

our assumptions were fully justified; the experiments on lead confirm the correctness of our conclusions and the possibility of extending them also to a non-work-hardening plastic metal.

4. Verification of Our Conclusions by the Work of Other Researchers

The establishment of the circumstance that the coefficient of friction in itself has an insignificant effect on the cutting force makes it possible for us to verify the correctness of the equation for the cutting force not only by our own experiments, but also by experiments conducted at the Siberian Physico-Technical Institute (SPhTI) under the direction of Corresponding Member of the Academy of Sciences V. D. Kuznetsov.

In a number of works carried out at SFTI, the principal component of the cutting force, chip shrinkage, and, in parallel with this, the constants of the compression polytrope \(\sigma_0\) and \(n\) were determined for the same metal. The experiments were carried out at a cutting speed of \(0.45\ \text{m/min}\), with free cutting and without built-up edge. In view of the fact that only one component of the cutting force \((P_z)\) was measured, it is not possible to determine the coefficient of friction directly from the experiments. Therefore, in calculating the cutting force we took the values of the coefficients of friction for analogous cutting conditions \((v, a, \gamma,\) work material, lubricant) from our experiments, which can lead only to minor errors in view of the circumstance indicated above.

Tables 3 and 4 give the data of the experiments of Karpova and Zhdanova [11] on the cutting and compression of U8 steel: in Table 3, annealed steel, and in Table 4, steel quenched and tempered to sorbite.

In both cases the cutting was performed with a tool having a rake angle \(\gamma = 37^\circ\), using lubricant.

Table 3

Annealed U8 steel

\(\gamma^\circ\)	\(b\), mm width of cut	\(v\), m/min	\(a\), mm thickness of cut	\(\zeta\)	\(P_z\)	\(\sigma_0\)	\(m = n + 1\)
\(37^\circ\)	1.52	0.45	0.043	2.1	12.3	\(92\ \text{kg/mm}^2\)	1.1
\(37^\circ\)	1.52	0.45	0.065	2.05	17.0	\(92\ \text{kg/mm}^2\)	1.1
\(37^\circ\)	1.52	0.45	0.098	2.01	24.9	\(92\ \text{kg/mm}^2\)	1.1
\(37^\circ\)	1.52	0.45	0.123	2.01	31.2	\(92\ \text{kg/mm}^2\)	1.1
\(37^\circ\)	1.52	0.45	0.157	2.09	42.0	\(92\ \text{kg/mm}^2\)	1.1
\(37^\circ\)	1.52	0.45	0.168	2.06	43.0	\(92\ \text{kg/mm}^2\)	1.1
\(37^\circ\)	1.52	0.45	0.197	2.09	51.0	\(92\ \text{kg/mm}^2\)	1.1
\(37^\circ\)	1.52	0.45	0.22	2.2	54.4	\(92\ \text{kg/mm}^2\)	1.1
\(37^\circ\)	1.52	0.45	0.248	2.1	62.1	\(92\ \text{kg/mm}^2\)	1.1

Table 4

U8 steel quenched and tempered to sorbite

\(\gamma^\circ\)	\(b\), mm	\(v\), m/min	\(a\), mm	\(\zeta\)	\(P_z\)	\(\sigma_0\)	\(m = n + 1\)
\(37^\circ\)	1.52	0.45	0.046	1.62	24	121	1.15
\(37^\circ\)	1.52	0.45	0.062	1.76	27	121	1.15
\(37^\circ\)	1.52	0.45	0.095	2.2	39	121	1.15
\(37^\circ\)	1.52	0.45	0.123	2.71	58	121	1.15
\(37^\circ\)	1.52	0.45	0.15	2.68	62	121	1.15
\(37^\circ\)	1.52	0.45	0.181	2.56	70	121	1.15
\(37^\circ\)	1.52	0.45	0.194	2.68	75	121	1.15
\(37^\circ\)	1.52	0.45	0.202	2.42	75	121	1.15
\(37^\circ\)	1.52	0.45	0.232	2.84	91	121	1.15

To calculate the cutting force by equation (15), we took from our experiments at microspeeds, for analogous conditions, \(\mu = 0.4\).

Figure 23 gives the graph of the coefficient of friction, which is the result of our experiments on cutting steel 40 \((H_B = 170)\) at microspeeds with lubrication. It is seen from the graph that the angle \(\gamma = 37^\circ\) corresponds to \(\mu = 0.4\). Of course, U8 steel differs from steel 40, and it is possible that the values of \(\mu\) for it will be somewhat different; but if we turn to Fig. 18, we shall see that even for \(\gamma = 30^\circ\) and \(\zeta = 2\) (such an average value of shrinkage occurred in the SFTI experiments on U8 steel), when the value of \(\mu\) varies from 0.3 to 0.5, the cutting force

varies only within the limits of 8%; for \(\gamma = 37^\circ\) this variation will be still smaller. In addition, at \(\gamma = 37^\circ\), in our experiments with steel 40 the shrinkage was less than in the experiments of SFTI with steel U8, which permits the assumption that, by taking \(\mu = 0.4\), we may make a certain error in the direction of decreasing \(\mu\).

Fig. 23

A comparison of the force calculated from the equation with that obtained experimentally is given in Figs. 24 and 25, in which circles indicate the experimentally obtained values, and crosses indicate the calculated force values.

In making the comparison it is necessary to take the following into account. The experiments determined the force \(P_z\), which includes in its value the sum of the forces on the front and rear faces of the cutter.

Fig. 24

The arrangement of the circles in Figs. 24 and 25 shows that through them one can draw, with sufficient reliability, a certain mean straight line which, when extended to its intersection with the ordinate axis, cuts off on this axis the value of the force on the rear face. Equation 15 makes it possible to calculate the force only on the front face. Therefore, in order to compare the experimental and calculated values, to the magnitude of the force obtained

by calculation, we added an intercept on the ordinate axis at \(a = 0\), and the values obtained in this way were shown in Figs. 24 and 25 by crosses. The comparison shows that a sufficiently close agreement was obtained between the experimental and calculated values of \(P_z\). For annealed U8 steel, the calculated values are 7–8% lower than the experimental ones; for U8 steel (sorbite), the calculated and experimental points were intermingled. The scatter of the points for U8 sorbite steel is apparently the result of experimental difficulties, to which the authors point: “The cutting of U8 steel treated to sorbite proved especially difficult... The readings of the dial gauge were very unstable, and the range of oscillations reached 20 mm, which corresponded to 40 kg. In the calculations, the average value of the cutting force was taken” [12].

Fig. 25

Fig. 25 shows that the scatter of the experimental values of chip shrinkage is also very considerable.

Tables 5 and 6 give the experimental data of M. A. Bolshaninova and Yu. A. Studenok on the cutting and compression of aluminum without lubricant and with kerosene lubricant [13].

For calculating the cutting force, we adopted from our experiments with aluminum the coefficients of friction: for cutting without lubricant \(\mu = 1.15\), and for cutting with kerosene lubricant \(\mu = 0.71\).

Table 5

Cutting aluminum without lubricant

\(\gamma^\circ\)	\(b\), mm	\(v\), m/min	\(a\), mm	\(\xi\)	\(P_z\)	\(\sigma_0\)	\(m = n + 1\)
45°	3.6	0.45	0.2	2.54	27.1	15.2	1.09
45°	3.6	0.45	0.4	2.48	56.1	15.2	1.09
45°	3.6	0.45	0.6	2.34	77.3	15.2	1.09
45°	3.6	0.45	0.8	2.39	95.7	15.2	1.09
45°	3.6	0.45	1.0	2.43	120.7	15.2	1.09
45°	3.6	0.45	1.2	2.47	137.7	15.2	1.09
45°	3.6	0.45	1.4	2.47	161.2	15.2	1.09
45°	3.6	0.45	1.6	2.37	186.2	15.2	1.09

Cutting of aluminum with lubrication by kerosene

Table

$\gamma^\circ$	$b$, mm	$v$, m/min	$a$, mm	$\xi$	$P_z$	$\sigma_0$	$m=n+1$
$45^\circ$	3.6	0.45	0.2	2.31	21.8	15.2	1.09
$45^\circ$	3.6	0.45	0.4	2.12	44.3	15.2	1.09
$45^\circ$	3.6	0.45	0.6	2.26	65.4	15.2	1.09
$45^\circ$	3.6	0.45	0.8	2.23	85.8	15.2	1.09
$45^\circ$	3.6	0.45	1.0	2.19	96.2	15.2	1.09
$45^\circ$	3.6	0.45	1.2	2.25	113.6	15.2	1.09
$45^\circ$	3.6	0.45	1.4	1.95	129.1	15.2	1.09
$45^\circ$	3.6	0.45	1.6	1.94	151.0	15.2	1.09

The results of comparisons of the experimental and calculated values of the cutting forces are presented in Figs. 26 and 27. The agreement proved to be very close.

The calculated values turned out to be 5–10% lower than the experimental ones. It should be noted that aluminum is a metal for which the rate of

Fig. 26. Axes: $P_z$ versus $a$, mm. In-figure inscription: “Bolshanin and Studenok. Aluminum. Without lubricant. $\gamma=45^\circ$; $b=3.6$ mm; $v=0.45$ m/min.”

Fig. 26

deformation noticeably affects the stress. Bolshanin and Studenok carried out compression of aluminum to determine $\sigma_0$ and $n$ at the usual low rate of deformation, whereas their cutting was carried out at a cutting speed $v=0.4$ m/min, so that the rate of deformation during cutting was undoubtedly

Fig. 27. Axes: $P_z$ versus $a$, mm. In-figure inscription: “Bolshanin and Studenok. Aluminum. Lubricant—kerosene. $\gamma=45^\circ$; $b=3.6$ mm; $v=0.45$ m/min.”

Fig. 27

higher than in compression. As the experiments of Kunin and Nizamova show, in the compression of aluminum, with increasing deformation rate, the exponent of the compression polytrope \(\eta\) remains constant, while the conventional yield point \(\sigma_0\) increases [14].

According to the experiments of Kunin and Nizamova, in the transition from static compression to dynamic compression (by a falling load), \(\sigma_0\) increases for aluminum by 35%. According to the experiments of L. D. Sokolov [15], in the transition from a compression speed of \(1\ \text{mm/sec}\) to a speed of \(2000\ \text{mm/sec}\), the stress in the compression of aluminum increases by 28%. Thus, if we knew the ratio of the deformation rates in cutting and in compression in the experiments of Bolshanina and Studenok, then, in calculating the cutting force, we would have to introduce into equation 15 an increased value of \(\sigma_0\), which would also increase the value of the force obtained by calculation. In Figs. 26 and 27 the calculated values of the forces (crosses) would be located higher and would probably somewhat exceed the values of the forces obtained experimentally.

Thus it may be stated that, for U8 steel (in its two states) and for aluminum, in cutting with lubrication and without lubrication, we obtained satisfactory agreement of the forces calculated from equation 15 with the forces obtained experimentally.

Brief Conclusions

The theoretical derivation of equations for the cutting forces and the experimental verification of the equations make it possible to draw the following conclusions:

1. For the process of cutting plastic metals, the law of independence of the specific work of change of shape from the mechanical deformation scheme proves to be valid. In application to the cutting process, this law may be formulated as follows:

“For a given temperature–rate regime, the specific work of deformation in the shear zone of the chip is uniquely related to the degree of deformation.”

1. The application of this law enabled us to derive equations for the cutting forces that include the physico-mechanical characteristics of the material being machined.

2. A thorough experimental verification of the derived equations for the cutting forces showed their sufficient accuracy.

References

1. Zimin Yu. P. Investigation of the process of high-speed cutting of steels. Abstract of dissertation, Tomsk, 1947.

2. Rozenberg Yu. A. Investigation of the process of cutting gray cast iron. Abstract of dissertation, Tomsk, 1952.

3. TsNITMASH. Book 15. New Investigations in the Field of Metal Cutting. Mashgiz, 1948.

4. Gubkin S. I. Theory of Metal Working by Pressure. Metallurgizdat, 1947.
   Smirnov-Alyaev. Resistance of Materials to Plastic Deformations. Mashgiz, 1949.
   Smirnov-Alyaev and Rozenberg V. M. Technological Problems of the Theory of Plasticity. Lenizdat, 1951.

5. Kuznetsov V. D. Physics of the Solid State. Vol. II. Tomsk, 1940.

6. Kuznetsov V. D. Physics of the Solid State. Vol. III, p. 711. Tomsk, 1944.

7. Ibid. P. 684, Fig. 519.

8. Ibid. P. 656, Fig. 510.

9. Rozenberg A. M. Determination of forces on the rear face of the cutter. In: Problems of Mechanical Engineering and Machine Building, Western Siberian Branch of the Academy of Sciences of the USSR, 1951.

10. Shofman. Experimental investigation of cold and hot upsetting. TsNIITMASH. Book 32, collection New Investigations in the Field of Forging Technology. Mashgiz, 1950.

11. Kuznetsov V. D. Physics of the Solid State. Vol. III, pp. 701–710. Tomsk, Poligrafizdat.

12. Ibid. Chapter XXVII, § 18.

13. Ibid. P. 708.

14. Kuznetsov V. D. Physics of the Solid State. Vol. II, p. 622.

15. Journal of Technical Physics, issue 5, pp. 543–548, 1947.