EFFECT OF STRAIN RATE ON STRESS DURING THE CUTTING OF A DUCTILE METAL

A. M. ROZENBERG and A. N. EREMIN

General Considerations

The influence of speed on the cutting process may be considered from various points of view.

At very low cutting speeds, a change in speed directly affects the coefficient of friction between the chip and the tool.

An increase in cutting speed here is usually accompanied by an increase in the coefficient of friction. This is the result of the fact that, as the cutting speed increases, the contact time of the chip with the rake face of the tool is reduced, owing to which the manifestation of the adsorption and oxidizing effects of the surrounding medium—effects that usually lower the coefficient of friction—is diminished. As the speed of chip movement along the tool increases, the friction process takes place between surfaces less covered with adsorbed films or oxide films, i.e., between cleaner surfaces. The friction process thereby comes closer to the process of dry friction, which is always associated with an increase in the coefficient of friction. This is especially noticeable at very low cutting speeds (microspeeds), where an increase in speed is not accompanied by a noticeable rise in temperature, and the reduction of the adsorption effect appears in its pure form. An increase in speed in this range, through an increase in the coefficient of friction, causes an increase in chip upsetting, in the degree of its deformation, and an increase in stress in the deformation zone; and, as a result of the combined influence of the increased upsetting and stress, it inevitably leads to an increase in the cutting forces. The increase in stress and cutting forces here is not the result of the direct influence of the strain rate, but only a consequence of the increase in the coefficient of friction and the resulting increase in the degree of plastic deformation of the chip.

In that range of cutting speeds where an increase in cutting speed entails a rise in temperature both in the deformation zone and in the zone of friction of the chip along the tool, the strain rate may affect the cutting process in two ways. On the one hand, the rise in temperature in the zone of chip friction along the tool causes a change in the coefficient of friction, proceeding according to a quite definite regularity, which in turn affects the degree of deformation, strain hardening, and stress in the chip deformation zone, the process of formation and geometry of the built-up edge, and the associated change, once again, in the degree of chip deformation and the stress in the deformation zone.

On the other hand, the rise in temperature in the deformation zone may act in the direction of lowering stresses, if the duration of the action of heat proves sufficient at least for the partial occurrence of the recovery process in the metal being deformed.

Thus, for example, when cutting steel, the rise in temperature as a result of increasing the cutting speed is accompanied by a change in the coefficient of fri-

... along a curve that has a maximum. Up to this maximum, the rise in temperature is associated with the appearance and gradual expression of built-up edge, with an initial decrease in the degree of chip deformation and a decrease in stress, and then with an increase in the degree of chip deformation and in stress. Beyond the maximum value of the coefficient of friction, a further increase in speed and temperature causes a decrease in the coefficient of friction, the degree of deformation, and the stress.

In addition to these changes in stress and cutting forces, which are connected with a change in the degree of chip deformation, in the cutting process, as in processes of other types of deformation, there must exist a direct effect of the rate of deformation on the stress.

In the present article we wish to show that, in the cutting process, the rate of deformation directly affects the stress, and that this effect is essentially the same as in other types of plastic deformation.

The effect of strain rate on the mechanical properties of metals

The question of the effect of strain rate on the mechanical properties of metals is set forth with sufficient completeness by V. D. Kuznetsov in volumes II and V of the monograph Physics of the Solid State. We shall present the basic propositions on this question.

If deformation is carried out at room temperature, then all metals, with respect to the influence of strain rate on them, may be roughly divided into two groups: refractory and low-melting. The influence of strain rate on the mechanical properties of refractory metals is considerably lower than on those of low-melting metals. If, however, refractory metals are tested at high temperatures, their sensitivity to strain rate increases, and in this respect they approach the low-melting metals.

Conversely, low-melting metals at reduced temperatures exhibit lower sensitivity, i.e., they approach refractory metals. Thus, the division of metals into two groups is purely conventional. It may be said that the higher the melting temperature of a metal, the smaller, at room temperature, is the effect of strain rate on its mechanical properties.

The stress during plastic deformation is the result of two simultaneously acting processes—the process of hardening, which raises the stress, and the process of recovery, which removes the hardening and therefore lowers the stress. Hardening is an athermal process, that is, one independent of the temperature at which deformation takes place. Recovery, on the contrary, is a thermal process dependent on temperature. Recovery proceeds the more intensely, removes hardening more significantly, and lowers the stress the more, the higher the temperature at which deformation takes place and the closer it is to the melting temperature of the given metal. The recovery process occurs over time and proceeds the more fully the longer the recovery time, that is, the lower the strain rate.

From the standpoint of the effect on the stress during deformation of the simultaneously acting processes of hardening and recovery, all metals—both low-melting and refractory—change their mechanical qualities in fundamentally the same way when the strain rate changes. At very low temperatures, close to absolute zero, recovery does not occur for all metals; deformation is accompanied only by hardening, and the effect of strain rate on stress should be absent. At very high temperatures, close to the melting temperature of the metal, recovery proceeds so intensely that it removes all hardening. In this case, at broad...

over a broad range of changes in the strain rate, the stress may not depend on the strain rate; and only at very high strain rates, at which recovery will not have time to remove all the hardening, will the stress again begin to increase with increasing strain rate. At room temperature, in the deformation of refractory metals, recovery will play a scarcely noticeable role, since the intensity of recovery in refractory metals at room temperature is altogether insignificant. Therefore the strain rate should have only a slight influence on the mechanical properties and on the stress.

At the same room temperature, recovery in low-melting metals proceeds with great intensity, considerably reducing the stress. The time factor here plays a major role. The higher the strain rate, the less recovery will remove the hardening, and an increase in strain rate should substantially increase the stress.

All these propositions, developed by the school of the Siberian Physico-Technical Institute, assume that the temperature of the deforming volume of metal remains constant when the strain rate is increased. If, however, when the rate is increased, the temperature rises as a result of the heat released during deformation, as occurs in the cutting process at high rates, then the influence of the increasing strain rate on the stress should decrease owing to the influence of temperature on the intensity of recovery.

Numerous experiments by various investigators show that, under the influence of an increase in strain rate, the mechanical characteristics of a metal increase: the yield point, the ultimate strength, and the true stress. However, this question has been least fully investigated as applied to the true strength, that is, precisely to the characteristic that is of greatest interest. The influence of strain rate on the mechanical characteristics of metals has been studied by various investigators both in tension and in compression. Let us present the results of some experiments.

With regard to the influence of tensile rate on the mechanical characteristics, there is no complete agreement among the various experiments. Thus, for example, Charpy [1] carried out fracture tests on steel specimens at a rate of up to 30 m/sec, and it turned out that the work of dynamic fracture at such a high rate exceeds the work of static fracture by only 11%.

According to the data of Stanton and Bairstow [2], in the fracture of steel specimens the work of dynamic fracture exceeded the work of static fracture by 38%. Blount, Kirkaldy, and Sankey [3], for boiler iron and steel, obtained an excess of dynamic work over static work averaging 60%. According to the data of Körber and Sack [4], the ratio of the works of dynamic and static fracture increases with increasing carbon content in steel from 1.22 at 0.08% carbon to 1.69 at 0.86% carbon. I. A. Oding and S. P. Shikhobalov [5], in tension of chromium-nickel-manganese steel at rates from 0.06 mm/sec to 5.5 mm/sec, obtained no change in true strength and elongation, that is, in the work of tension. A. V. Bobylev and A. I. Chipizhenko [6], when changing the fracture rate of copper specimens by a factor of 300, obtained a quite insignificant increase in ultimate strength.

The strain rate has a considerably stronger effect on the mechanical characteristics of low-melting metals. Experiments by V. D. Kuznetsov and L. A. Shvirk with tin and lead, experiments by M. A. Bolshanina, N. A. Bolshanina, and I. K. Gorelov with tin [6], experiments by Sachs with zinc [7], and others showed that the ultimate strength for these metals increases substantially with increasing strain rate.

The question of the influence of strain rate on the mechanical characteristics of a metal in compression has also been investigated by many authors. Here, essentially the same situation was found as in tension. In compression of refractory metals, the influence of rate proved to be significantly less substantial than in compression of low-melting metals. If the ratio of the stress under dynamic compression to the stress under static compression (at identical degrees of deformation) is called the dynamic coefficient, then, according to a number of experiments, the dynamic coefficient increases as the melting temperature of the metal decreases. Thus, according to the experiments of Kunin and Nizamova [8], the dynamic coefficient had the following values (Table 1).

Table 1

Similar values were also obtained in the experiments of Seehase [9].

Table 2

L. D. Sokolov calls the ratio of stresses at different compression rates the rate coefficient [10]. If the rate coefficient for a change in rates from \(0.01\ \mathrm{mm/sec}\) to \(1\ \mathrm{mm/sec}\) is denoted by \(z_1\), and for rates from \(1\ \mathrm{mm/sec}\) to \(2000\ \mathrm{mm/sec}\) by \(z_2\), then, according to Sokolov, the values of these coefficients for compression under room-temperature conditions will be as follows (Table 3).

Table 3

This table expresses the same tendency as in Table 1 for the experiments of Kunin and Nizamova; however, the velocity coefficient here acquired somewhat different values.

For ShKh-15 steel, L. D. Sokolov and S. Z. Yudovich [11], when varying the compression rate from 0.01 mm/sec to 2000 mm/sec, i.e., by a factor of 200,000, obtained a velocity coefficient of only 1.16.

The experiments of many authors have also shown that the velocity coefficient increases considerably with an increase in the temperature at which the test is carried out.

We shall limit ourselves here to this enumeration of the investigations performed and turn to the question of the influence of the strain rate on the stress in the cutting process.

Strain Rate in Cutting

Before proceeding to a direct study of the influence of strain rate on the cutting process, let us point out that the cutting speed is not the strain rate; however, comparison of strain rates in cutting can be carried out through comparison of cutting speeds. This will become clear from the following considerations. First of all, let us clarify what should be understood by strain rate. As applied to tensile, compression, shear, etc. deformations, this question has been most fully treated by V. D. Kuznetsov [12] and M. A. Bolshanina [13]. At present, different authors and investigators understand different concepts by strain rate: 1) the rate of increase of load in kg/sec; 2) the rate of rise of stress in kg/mm² sec; 3) the rate of increase of deformation in mm/sec; 4) the rate of increase of true relative deformation,

$$ \frac{\Delta l}{l \Delta t}\ \frac{\%}{\text{sec}}; $$

5) a quantity equal to the interval of time during which the specimen ruptures.

V. D. Kuznetsov and M. A. Bolshanina show that it is most correct to express the strain rate by the rate of increase of true relative deformation. In addition, V. D. Kuznetsov indicates that in all cases the velocity gradient can serve as a measure of the strain rate.

“Strictly speaking, every plastic deformation consists in the fact that one part or one layer of the body being deformed moves with a definite velocity relative to another part or another layer. If the distance between the layers is equal to $dx$, and the difference in velocities is equal to $dv$, then the velocity gradient

$$ \frac{dv}{dx} $$

is in all cases a measure of the strain rate” [12].

The strain rate may also be expressed by the rate of change of the shear angle,

$$ \frac{d\gamma}{dt}, $$

and it can be proved that the velocity gradient in the shear process is equal to the rate of change of the shear angle.

As applied to such deformation processes as uniform tension, compression, and torsion, determination of the strain rate presents no difficulty, since here the intervals of time during which definite absolute or relative deformations are attained are usually known.

In the cutting process, determination of the strain rate encounters a number of difficulties.

We imagine that, ahead of the cutting tool removing a chip, there is a certain plastically deformed volume of metal. In Fig. 1 this volume in longitudinal section is represented by the hatched area.

In this deformed volume, on its outer contour \(abcd\) the degree of plastic deformation is equal to zero, while at the boundary with the chip \(ae\), which we call the shear plane, the degree of plastic deformation reaches its greatest value, equal to the degree of deformation of the removed chip.

If we imagine that some microvolume of metal at point \(g\) enters the zone of plastic deformation and, with a certain velocity equal to the cutting speed, traverses the segment \(g h\), then during this time it passes through all degrees of plastic deformation from zero to the degree of deformation of the chip. If over the segment \(g h\) the degree of deformation increased uniformly, then, knowing the segment \(g h\) and the cutting speed, it would not be difficult to determine the strain rate. But observations and experiment show that along \(g h\) the degree of deformation increases nonuniformly. The entire deformed volume has such an insignificant degree of deformation that it is usually impossible to observe distortion of the original

Fig. 1                Fig. 2

structure of the metal. Only in the immediate vicinity of the line \(ae\) does the original structure of the metal abruptly change into the texture of the deformed chip. Thus, almost all of the deformation of the metal during cutting takes place in a narrow zone near the line \(ae\). Measurement of microhardness likewise shows that the hardening of the metal, which characterizes the degree of deformation, occurs in the immediate vicinity of the line \(ae\), and that with a slight distance from this line the hardness rapidly decreases to the hardness of the undeformed metal.

All this indicates that, in the cutting process, the plastically deformed zone located ahead of the cutter and the chip is characterized by a variable degree of deformation, the degree of deformation in this zone increasing, as it approaches the shear plane \(ae\), nonuniformly and not gradually.

Presumably, the curve of the distribution of the degree of deformation along the line \(g h\) may be represented as shown in Fig. 2.

Here the degrees of deformation are plotted along verticals from the line \(g h\). A microvolume of metal, moving from point \(g\) to point \(h\) with some velocity, at each given instant has its own degree of deformation and its own strain rate. Initially, at point \(g\), the strain rate is equal to zero; then it increases at first very slightly, and, on directly approaching point \(h\), extremely strongly. In order to be able to calculate the strain rate at each point of the deformed volume, it would be necessary to know the law of distribution of the degree of deformation, which we do not know, and therefore we also cannot determine the strain rates in the cutting process.

The strain rate is undoubtedly connected with the cutting speed, since the cutting speed determines the magnitude of those intervals of time during which the volumes of metal pass through all degrees of deformation. Therefore

with an increase in cutting speed, the strain rate will also increase. If the strain pattern remained unchanged with a change in cutting speed, it would be possible to consider the strain rate exactly proportional to the cutting speed. But since, with a change in cutting speed, some change in the strain pattern is also possible, one can speak only of an approximate proportionality between the cutting speed and the strain rate.

To this it should be added that the strain rate at a constant cutting speed will depend on the degree of strain, so that chips removed at identical cutting speeds but having different degrees of strain will be obtained, despite the constancy of the cutting speed, at different strain rates. In the case where the degree of strain is higher, the strain rate will also be higher, since in the same intervals of time a higher degree of strain will be attained.

Strictly speaking, the thickness of the layer being cut also affects the strain rate during cutting. Experiments by A. M. Rosenberg have shown that, with an increase in the thickness of the cut, the plastically deformed volume of metal lying ahead increases [14]. Thus, at constant cutting speed and degree of chip strain, with an increase in the thickness of the cut the segments \(gh\) (Fig. 1) will increase, i.e., the intervals of time during which the final degree of strain is attained will increase, i.e., the strain rate will decrease.

Thus, in the cutting process the strain rate depends on the cutting speed, on the nature of the strain pattern, on the degree of strain, and on the thickness of the layer being cut. The influence of cutting speed is primary and basic, since it can vary within very wide limits, whereas the influence of the cut thickness and of the degree of strain on the strain rate is considerably less significant.

We showed above that even for low-melting metals the influence of strain rate on mechanical properties is such that changing the strain rate by hundreds and thousands of times causes a change in mechanical properties of only tens of percent. From this point of view, the influence of changes in the degree of strain and in the cut thickness—through a change in strain rate—on the stress in the cutting process can in most cases be neglected. Therefore, in the further exposition we shall assume that, in the cutting process, the strain rate changes only with a change in cutting speed, and approximately in proportion to the latter.

Influence of Strain Rate on Stress in the Cutting Process

To determine the influence of strain rate on stress in the cutting process, we selected three different metals having different melting temperatures: steel 40, aluminum, and lead. It could be expected that, for these three metals, the influence of strain rate would differ sharply.

It was necessary to conduct all experiments at a strictly constant temperature, since, in different temperature ranges, strain rate affects the mechanical characteristics of one and the same metal in different ways, as was stated above. In the cutting process, this condition could be satisfied only by experiments carried out in the range of very low speeds (microspeeds), where it was possible to increase the cutting speed considerably without any noticeable rise in temperature in the chip-deformation zone. At cutting speeds usually employed in production, even a comparatively small increase in speed causes a substantial rise in temperature due to the heat released in cut—

as a result of deformation, which would inevitably have led to a distortion of the effect of speed on stress. For these reasons, all our experiments were carried out at cutting speeds varying within the range from \(0.0003\ \mathrm{m/min}\) to \(0.76\ \mathrm{m/min}\), which made it possible, without a noticeable increase in temperature in the deformation zone, to vary the speed over sufficiently wide limits. Such low cutting speeds were convenient for two further reasons. First, at these speeds no built-up edge forms on the rake face of the tool, which in itself would alter the course of deformation during cutting, would change the geometry of the tool, and would change the degree of deformation of the chip. Second, such low speeds made it possible for us to compare more reliably, quantitatively, the effect of strain rate in the cutting process with its effect in other deformation processes.

The effect of speed on the mechanical characteristics of the metal in the cutting process can be detected either from the change in the cutting force with a change in cutting speed, or from the change in the tangential stress in the shear zone of the chip. From the change in cutting force with a change in speed one can judge the pure effect of strain rate only in the case where, at different speeds, identical coefficients of friction of the chip on the rake face of the tool and identical chip shrinkages are obtained, since the coefficient of friction and chip shrinkage, as independent factors, affect the cutting force.

Fig. 3

This follows directly from the equation relating the tangential stress \(\tau\) in the shear plane to the cutting force \(P_1\) (Fig. 3).

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

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

\(a, b\) are the thickness and width of cut,

\(\eta\) is the angle of friction, \(\tan \eta=\mu\), where \(\mu\) is the coefficient of friction of the chip on the tool,

\(\beta_1\) is the angle determining the position of the shear plane, which determines the magnitude of chip shrinkage.

It follows from this expression that one can judge a change in \(\tau\) from a change in \(P_1\) only in the case where \(a\), \(b\), \(\gamma\), \(\eta\), and \(\beta_1\) retain constant values.

In order to judge the effect of strain rate on the basis of a comparison of stresses in the shear zone, it is necessary to carry out this comparison at identical degrees of chip deformation, since the degree of deformation has an independent effect on the stress.

The experiments were carried out by us using the process of free cutting, in order to simplify the process as much as possible and to make the determination of the coefficient of friction and the degree of chip deformation as reliable as possible. In all cases, the comparison was made of cutting forces in their part applied on the rake face of the tool; the forces on the flank face, as not participating in the process of chip deformation, were subtracted from the forces measured by the dynamometer.

To determine the forces applied on the flank face, a method developed by us and verified by a large number of experiments [15] was used. The cutting forces (two components) in the experiments were measured by a very sensitive and accurate elastic dynamometer.

Effect of Speed on the Mechanical Characteristics in Cutting Steel

For the experiments, steel 40 was used; it possessed good homogeneity and, in mechanical tests in tension, compression, and torsion, showed good constancy of its physicomechanical characteristics.

The steel had the following mechanical characteristics:

conditional yield point in compression $\sigma_0 = 75.5\ \mathrm{kg/mm^2}$,
compression polytropic exponent $m = 1.226$,
ultimate tensile strength $\sigma_b = 55.5\ \mathrm{kg/mm^2}$,
elongation at rupture $\delta = 24.6\%$,
yield point $\sigma_s = 25\ \mathrm{kg/mm^2}$,
reduction of area at rupture $\psi = 52\%$,
Brinell hardness $H_B = 170$.

Cutting was carried out with a width of cut $b = 3\ \mathrm{mm}$, with various thicknesses of cut “$a$,” with various rake angles from $40^\circ$ to $-1^\circ$, using various lubricants and without lubricant. The cutting speeds were varied from $0.00088\ \mathrm{m/min}$ to $0.76\ \mathrm{m/min}$. Such a change in speed (by a factor of 864) was sufficient to reveal the influence of the rate of deformation.

The results of experiments without the use of lubricant, with a cutter having a rake angle of $30^\circ$, are given in Fig. 4, where, in the coordinates “thickness of cut—cutting forces,” the dependences for the two components of the cutting force at various cutting speeds from $0.0011\ \mathrm{m/min}$

Fig. 4

to $0.760\ \mathrm{m/min}$ are plotted through the experimental points. The upper lines refer to the main component of the cutting force $P_z$, located in the direction of the cutting speed; the lower ones refer to the component $P_x$, located perpendicular to the cutting surface. The cutting forces are expressed by straight lines, which indicates the independence of the coefficient of friction on the rake face from the thickness of cut at constant cutting speed. The straight lines do not pass through the origin, cutting off on the ordinate axis segments equal to the forces on the rear face of the cutter.

The same graphs were also obtained by us for cutters with other rake angles.

When considering Fig. 4, one gets the impression that the cutting forces depend significantly on the cutting speed, increasing as the latter increases. The increase in the forces is very considerable. Thus, in going from \(v = 0.0011\ \text{m/min}\) to \(v = 0.760\ \text{m/min}\), the force \(P_z\) increases by 33%, and the force \(P_x\) by 56%. However, a detailed investigation of this question shows that, when the cutting speed is increased, the chip shrinkage and the coefficient of friction of the chip on the rake face of the cutter increase. The dependence of the shrinkage \(\zeta\) and of the coefficient of friction \(\mu\) on the cutting speed, obtained by us as a result of these same experiments, is presented in Fig. 5, which shows that an increase in

Fig. 5

speed significantly increases the coefficient of friction and noticeably increases the chip shrinkage, especially in the region of the very lowest cutting speeds.

Therefore Fig. 4 shows the influence on the cutting forces not only of the speed itself, as the strain rate, but also of the influence of the coefficient of friction and of the chip shrinkage.

In other words, from Fig. 4 it is impossible to draw correct conclusions about the influence of the strain rate itself on the cutting forces, and hence on the stresses in the chip-deformation zone.

In order to isolate the influence of the speed itself, it was necessary to carry out the experiments in such a way that, at different speeds, constant values of the coefficients of friction and of chip shrinkage would be obtained. Assuming that, under the condition that a constant temperature is maintained on the rake face, the coefficient of friction increases with increasing cutting speed because of the reduction in the adsorptive and oxidizing action of the surrounding medium on the friction surfaces, we came to the conclusion that, by selecting a suitable lubricant, constancy of the friction regime and of the coefficient of friction could be achieved. We tested various lubricants (vegetable oil, mineral oil, grease, a 1% solution of soap in water). The best results were obtained with the use of a 1% solution of soap in water. The results of the experiments with this lubricant are given in Fig. 6 for a cutter rake angle of 30° and in Fig. 7 for a rake angle of 20°. In these figures, in addition to the cutting forces, the upper part shows the change in chip shrinkage.

Figs. 6 and 7 show quite clearly that, when applying

with lubrication by a 1% solution of soap in water, the effect of cutting speed on shrinkage and on the cutting forces became entirely imperceptible. The coefficient of friction in this case also does not depend on the cutting speed, since, at a constant rake angle of the cutter, it is determined by the ratio

\[ \frac{P_x}{P_z}, \]

which, for values of \(P_x\) and \(P_z\) independent of speed for each given value of the chip thickness \(a\), remains constant.

Fig. 6

Here it should be pointed out that lubrication may exert a surface-active effect on the metal being machined, but if such an effect in our experi-

Fig. 7

had taken place, then with an increase in cutting speed this influence should have decreased, which should have led to an increase in the cutting forces, something we do not observe.

Since the change in cutting speed in this case entailed no change either in the cutting forces or in the coefficient of friction and chip shrinkage, naturally the shear stress in the deformation zone of the chip proves to be independent of the speed.

Thus, the experiments carried out on steel 40 showed that the cutting speed can affect the cutting force through a change in the coefficient of friction and chip shrinkage, but the cutting speed, as a quantity expressing the rate of deformation, does not exert an independently noticeable influence on the cutting force and the tangential stress during cutting.

It was of considerable interest to determine, for comparison, the influence of the rate of deformation for this same steel under other types of deformation. We did this by means of torsion tests. For torsion, specimens were made from the same steel 40 with the following dimensions:

diameter . . . . . . . . . . . . . . . . . . . . . . 10 mm
distance between heads (length of the cylindrical
part between fillets) . . . . . . . . . . . . . . . 60 mm.

Torsion was carried out on a special apparatus which made it possible, during the experiment, to measure and automatically record on a moving paper tape the magnitude of the torque and its change with the angle of rotation of the rotating grip. This measurement and recording were performed by a hydraulic dynamometer with a self-recording manometer. The specimen was very accurately centered in the grips and, during torsion, was brought to fracture. The torsion speed could be varied over wide limits. In our experiments we

Fig. 8

took two torsion speeds—0.523 radians per minute and 220 radians per minute. Thus, the rate of deformation was changed by a factor of 420. In Fig. 8 the dependence of the torque \(M\) on the specimen on the twisting angle \(\varphi\), expressed in radians, is presented.

Fig. 8 shows that, in torsion of steel 40, no noticeable influence of the rate of deformation on the torque, and hence also on the stress in torsion, is found. Thus, manifestation of the influence of the speed

deformation in torsion tests of steel 40 turned out to be just as insignificant as in cutting tests of this steel.

Effect of Speed on the Mechanical Characteristics in Cutting Lead

Lead was chosen by us for the experiments as a metal with a low melting temperature, for which the influence of the strain rate should have manifested itself to a considerable degree. The lead-cutting experiments were carried out over cutting speeds from \(0.00025\) to \(0.106\ \text{m/min}\), with measurement of the two components of the cutting force by an elastic dynamometer. For the further presentation we used not only our own experiments, but also the experiments on lead by A. M. Rozenberg and Yu. P. Zimin. All the experiments were carried out by the method of free cutting. The rake angles on the cutters were: \(10^\circ\), \(15^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\). Cutting was performed both without lubrication and with the use of various lubricants.

The results of measuring the forces in cutting with a cutter having \(\gamma = 15^\circ\) are presented in Fig. 9.

In the upper part of the figure the change in chip compression with the change in cut thickness and cutting speed is shown. Fig. 9 shows that, with an increase in cutting speed, the forces \(P_z\) and \(P_x\) increase. \(P_z\) increases especially considerably. With an increase in speed from \(0.0014\) to \(0.106\ \text{m/min}\), i.e., by only 76 times, the component \(P_z\) increases almost twofold.

Fig. 9

With increasing speed, in parallel with \(P_z\), \(P_x\) also increases, but less significantly than \(P_z\), which indicates a change in the coefficient of friction; chip compression also increases. Similar results were obtained at other rake angles of the cutter as well.

We attempted, by using various lubricants, to eliminate the influence of speed on the coefficient of friction and on chip shrinkage, as had been achieved in cutting steel. But this attempt was not successful. The most effective lubricant for lead proved to be kerosene. It considerably reduced friction, chip shrinkage, and the absolute values of the cutting forces, but did not eliminate the effect of speed on the coefficient of friction and on shrinkage. The results of experiments with kerosene lubrication are presented in Fig. 10, which shows that the absolute values of the cutting forces and shrinkage became considerably smaller, but the influence of cutting speed remained no less pronounced.

Fig. 10

Therefore, in order to resolve the question of the effect of strain rate on the mechanical characteristics in cutting lead, it was necessary to proceed to calculating the shear stress \(\tau\) in the shear plane. The angle \(\beta_1\), determining the position of the shear plane, was calculated from the chip shrinkage \(\zeta\)

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

The shear stress \(\tau\) was calculated from the relation

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

The force \(P_1\) was determined by subtracting from the force \(P_z\) measured by the dynamometer the magnitude of the force applied to the rear face of the cutter in the direction of the cutting speed.

The friction angle \(\eta\) was determined from the relation

\[ \operatorname{tg}\eta=\operatorname{tg}(\omega+\gamma);\qquad \operatorname{tg}\omega=\frac{P_2}{P_1}, \]

where \(P_2\) and \(P_1\) are the components of the force applied to the front face of the cutter. \(P_2\) was determined by subtracting from \(P_x\) the force applied to the rear face in the direction normal to the cutting surface.

The shear stress \(\tau\) was related to the relative shear \(\varepsilon\), which was determined from the equation

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

From the results of the calculations we constructed a graph of the dependence of the shear stress \(\tau\) on the magnitude of the relative shear \(\varepsilon\), shown in Fig. 11.

It follows from Fig. 11 that, in cutting lead, the speed has a very strong effect on the shear stress \(\tau\). It is noteworthy that at low speed (\(0.0014\ \mathrm{m/min}\)) the stress does not depend on the degree of deformation and, for markedly varying values of the relative shear, remains constant. At higher speeds the stress increases with increasing degree of deformation, and the more significantly, the higher the speed.

Fig. 11

In our opinion, this is explained as follows. At a constant cutting speed, an increase in the degree of deformation in itself apparently leads to an increase in the rate of deformation, since each particle of metal, passing through the deformation zone, attains a higher degree of deformation in the same interval of time, so that the deformation rate in percent per unit time increases with increasing degree of deformation. This must lead, with the cutting speed constant, to an increase in stress as the degree of deformation (relative shear) increases. At very low cutting speeds this is less noticeable because recovery can proceed more completely; as the cutting speed is increased, recovery reduces the increase in stress less significantly, and it grows more noticeably with increasing degree of deformation.

We compared the change in stress in the cutting process under the influence of speed with the same change in the compression process. Compression was carried out by us at various speeds, and the relation between the stress in the specimen and the degree of deformation was determined. In each experiment we carried out compression at a constant deformation rate; for this purpose, during each experiment the speed of displacement of the planes compressing the specimen was decreased inversely proportional to the degree of deformation of the specimen. We did not have the possibility of continuously and gradually decreasing, during a single experiment, the speed of displacement of the planes compressing the specimen. Therefore we changed the speed stepwise after every \(20\text{–}25\%\) of deformation of the specimen. Thus, in each experiment the deformation rate did not remain strictly constant. It fluctuated about a certain mean value for the given experiment within the limits of \(\pm 10\%\text{—}\pm 12.5\%\). Such a fluctuation in deformation rate could not have had any substantial influence on the results, as will be clear from the following discussion.

The compression specimens had the following dimensions:

diameter $d_0 = 10\ \mathrm{mm}$,

height $h_0 = 25\ \mathrm{mm}$.

Compression was carried out both without lubricant and with kerosene as lubricant. In Figs. 12, 13, and 14 the results of the compression experiments are presented on logarithmic plots. Along the abscissa is plotted the relative deformation

\[ \frac{h_0}{h}, \]

where $h_0$ is the initial height of the specimen and $h$ is the height at the given instant; along the ordinate, the compressive force.

We calculated the strain rate in the experiments as a percentage of relative deformation per second.

Fig. 12

Fig. 13

Fig. 14

In the experiments this rate varied from \(0.13\%\,\dfrac{1}{\sec}\) to \(40\%\,\dfrac{1}{\sec}\), i.e., by a factor of 307.

The graphs presented make it possible to draw the following conclusions. With increasing strain rate, the compressive force, and hence also the stress, increase appreciably.

Thus, when the strain rate was increased 307-fold, the compressive force and the stress increased by 70%.

The points of each experiment lay, with a sufficient degree of accuracy, on a straight line, which indicates that the polytropic law is obeyed. Only at the very largest degrees of deformation—at \(\dfrac{h_0}{h} > 4\)—do the points of some experiments deviate upward from the straight lines, which apparently is the result of the influence of friction on the end faces of the specimens on the state of stress.

All the straight lines are inclined at an angle of \(45^\circ\) to the axis of abscissas. This indicates the circumstance that lead at room temperature, under conditions of constant strain rate, does not harden (the polytropic exponent \(m = 1\), \(n = m - 1 = 0\)); the stress remains constant throughout the entire compression.

The pre-polytropic part of the compression curve in these figures is given only for the very lowest strain rate, since it was not needed for the question under investigation.

For comparison of the influence of strain rate on the stresses in cutting and in compression of lead, we constructed the graph shown in Fig. 15, where the strain rate is plotted along the axis of abscissas and the stress along the axis of ordinates. The strain rate for compression is plotted in \(\%\,\sec^{-1}\). For cutting, we do not know the absolute values of the strain rate; therefore, for cutting, the cutting speeds in \(\mathrm{mm}/\mathrm{min}\) are plotted here, since

Fig. 15

it may be assumed that the strain rate is proportional to the cutting speed at one and the same degree of deformation. On the axis of ordinates for the cutting process the shear stress “\(\tau\)” in the shear plane is plotted at a constant value of the relative shear \(\varepsilon = 1.4\). The results of compression experiments are shown by circles, and those of cutting experiments by crosses; the solid line is drawn as the mean for the cutting experiments, and the dashed line for the compression experiments. The fact that the lines (solid and dashed) did not coincide with one another is of no significance, since it is the result of our not knowing the absolute values of the strain rates in cutting and of our having, purely conventionally, equated on the axis of abscissas \(1\%\,\sec^{-1}\) in compression and \(1\ \mathrm{mm}/\mathrm{min}\) in cutting. It is important to note that both lines in the logarithmic graph were obtained as straight lines and have one and the same slope, which indicates an entirely identical influence of strain rate on the stress both in cutting and in compression. If

express analytically the effect of strain rate on stress, we obtain

\[ \tau = \tau_0 \cdot V^{0.102}. \]

It should be noted that, although in Fig. 15 we quite conventionally equated, on the abscissa axis, a strain rate in compression of \(1\%\,\text{s}^{-1}\) with a cutting speed of \(1\ \text{mm/min}\), nevertheless, apparently, the strain rates in cutting and in compression for these values were very close to each other. This may be concluded from the fact that, at a strain rate in compression of \(1\%\,\text{s}^{-1}\), and with the height of the specimen changing during deformation from 25 to 6 mm, the speed of displacement of the planes compressing the specimen varied in the experiment from 15 mm/min to 3.6 mm/min. Thus, the speeds of displacement of the deforming bodies in cutting and in compression differ only slightly.

Effect of Speed on the Mechanical Characteristics in Cutting Aluminum

As follows from the material presented above, the effect of strain rate on stress in cutting steel proved to be completely imperceptible; in cutting lead, on the contrary, it proved to be very substantial. The effect of strain rate in twisting steel and in compressing lead proved to be exactly the same as in cutting. The different effect of strain rate on stress for steel and lead is explained by the fact that these two metals have entirely different melting temperatures. Steel has a very high melting temperature, and when it is deformed under room-temperature conditions, the recovery process proceeds to such an insignificant degree that it cannot affect the stress. Deformation of lead under room-temperature conditions, on the contrary, is accompanied by intense recovery, and the strain-rate factor has a significant effect on the completeness of the course of recovery and on the stress in the deformation process.

Aluminum has a melting temperature higher than that of lead, but considerably lower than that of steel. Therefore, one might expect that, for aluminum, the effect of speed on the stress during cutting should be less substantial than for lead, but already more clearly manifested than for steel. This is exactly what we obtained as a result of experiments carried out on cutting aluminum at various microspeeds. We varied the cutting speeds within the range from 0.0048 m/min to 0.76 m/min.

The results of experiments on cutting aluminum without the use of lubricant are shown in Fig. 16; Fig. 17 presents the results of experiments with kerosene used as a lubricant.

Fig. 16 shows a considerable scatter of the experimental points, which is the result of aluminum sticking to the rake face of the cutter during cutting without lubricant. Despite the considerable scatter of the points, nevertheless a significant increase in both components of the cutting force with increasing cutting speed is quite clearly manifested.

With lubrication by kerosene, a better arrangement of the experimental points was obtained in Fig. 17. The effect of speed on the cutting forces became considerably smaller than without lubricant, but was not eliminated entirely.

In Fig. 18, on the basis of the results of these same experiments, the change in chip shrinkage and the coefficient of friction with changing cutting speed is shown.

This figure shows that, in cutting without lubricant, an increase in speed is accompanied by a significant increase in chip shrinkage (and therefore also in the degree of its deformation) and in the coefficient of friction.

When cutting with kerosene lubrication, chip shrinkage remains unchanged; the coefficient of friction increases quite insignificantly with a substantial increase in speed.

It may be concluded from this that the increase in forces when cutting aluminum without lubricant, shown in Fig. 16, is not only the result of an increase in stress under the influence of speed, but also the result of an increase in the degree of deformation and in the coefficient of friction.

Fig. 16

Since, when cutting aluminum with lubricant, the coefficient of friction and the chip shrinkage remain unchanged, the influence of speed on the cutting forces, shown in Fig. 17, is the influence of the deformation rate itself on the stress. Therefore, from the experiments with lubricant, we calculated the shear stresses $\tau$ and plotted them as a function of cutting speed on the logarithmic graph in Fig. 19.

Fig. 17

Fig. 19 shows that, when the cutting speed is changed by a factor of 158, the stress increased by 18.5%. Thus, if it is assumed that the stress

...since the stress on a logarithmic plot varies according to a straight-line law, the influence of the strain rate on the stress during cutting of aluminum can be expressed by the relation

\[ \tau=\tau_0 \cdot V^{0.034}. \]

We did not carry out experiments to determine the influence of strain rate on stress for other types of deformation of aluminum. There is hardly any need for this, since for two characteristic metals (steel

Fig. 18

and lead) we showed that the effect of strain rate is entirely the same in cutting and in other types of deformation.

The influence of rate on the stress during cutting of aluminum proved not to be as significant as in cutting lead. This, as was indicated above, was to be expected.

Fig. 19

Conclusions

1. In the cutting process, the stress is affected not only by the strain rate, but also by the degree of deformation of the metal in the shear zone of the chip.

2. By means of specially conducted experiments over a wide range of variation in cutting speed, under the condition that the temperature remained constant, it was possible to isolate the effect of the strain rate itself on the stress during cutting.

1. Experiments have shown that the degree of influence of the strain rate on the stress during cutting depends on the melting temperature of the metal and manifests itself in exactly the same way as in other types of deformation.

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