STRAIN RATE DURING THE CUTTING OF DUCTILE METAL

G. L. KUFARЕV

(Presented by Professor, Doctor A. M. Rozenberg)

At the present time, opinions on the question of the influence of the strain rate on the process of cutting ductile metal are extremely varied and contradictory. Along with this, the experimental theory of plasticity gives a number of definite indications as to how the strength characteristics of metals change at different rates of their deformation. In order to extend the experimental conclusions of the theory of plasticity to the cutting process, it is necessary to know the actual strain rates of the metal in this process.

V. D. Kuznetsov [1] showed that, of all existing methods for estimating the strain rate, the most correct is the one that estimates it by the rate of increase of the true relative deformation (relative shear). If the dependence of the relative shear \((\gamma_k)\) on time is known, then the strain rate is determined as follows:

\[ v_g=\frac{d\gamma_k}{dt}. \]

The deformation of ductile metal during the formation of continuous chip proceeds by successive shears along curvilinear surfaces (Fig. 1), enclosing a certain volume \(ABC\). Each elementary volume of the removed layer of metal begins to undergo plastic deformation on the surface \(AB\), the tangential stresses on which are equal to the yield limit in shear of the initial work material. With further motion in the zone \(ABC\), the degree of deformation increases and reaches a maximum (corresponding to the deformation of the chip metal) on a certain surface \(AC\).

Fig. 1

Consequently, in order to determine the strain rate in the cutting process, it is necessary to know the character of the increase of the relative-

nous shear as a function of time or of the path of motion of a metal particle in zone ABC. In other words, it is necessary to know the degree of deformation of the metal at any point of zone ABC. An exact mathematical calculation proves in this case to be futile because of the extreme complexity of the process. Experimental work carried out in this direction, both by Soviet and foreign researchers, makes it possible to judge the character of the deformation occurring here only qualitatively and very incompletely.

Our experiments were designed to obtain the most complete picture of the deformation taking place in the cutting of ductile metals.

Cutting was carried out by free planing of copper specimens at a constant speed \(v = 19\ \text{mm/min}\), with cutters having rake angles \(\gamma = 17^\circ,\ 27^\circ\), and \(37^\circ\). The specimen consisted of two plates measuring \(50 \times 35 \times 3\ \text{mm}\), rubbed together by their large faces. On the inner side of one of the plates a system of small circles was applied (Fig. 2), the diameter of which was \(0.03\ \text{mm}\). In Fig. 2 the black dots denote small circles \(0.015\ \text{mm}\) in diameter,

Fig. 2

arranged, in turn, on circles \(0.2\ \text{mm}\) in diameter. From the distortion of the circles \(0.2\ \text{mm}\) in diameter it was possible to judge the magnitude of the average deformation of the metal in the chip. The small circles were obtained by pressing a corundum needle under a static load of 100 and 50 g on a PMT-3 instrument. The impressions thus obtained are deformed in exactly the same way as circular marks applied to the surface would be deformed. This is confirmed by the fact that the deformation in the chip, calculated from the distortion of the circular impressions, coincides with the average deformation calculated from the distortion of the large circles. The specimen was clamped in a vise in such a way that the direction of the rows of circles was parallel to the motion of the cutter. After this, the surface of the specimen was leveled by removing extremely thin shavings. The thickness of cut in the course of the experiment was chosen such that, when it was removed, a continuous chip was formed. Cutting was carried out until the part of the material with the applied grid of circles had undergone its final deformation. After this the process was stopped. In order that, by the moment the cutter was stopped, the cutting process would finally

but it was found that the grid of circles was applied at a distance of 20 mm from the edge of the specimen along its length.

The obtained chip roots were photographed and sketched under a microscope by the coordinate method, with a measurement accuracy of 0.005 mm. Such sketches of the deformed grid of circles for cases of operation with tool rake angles \(\gamma = 27^\circ\) and \(37^\circ\) are given in Figs. 1 and 2 in the article “Calculation of the Magnitude of Relative Shear in the Cutting of Metals,” included in the present collection. The crosses in these

Fig. 3

figures indicate the centers of the small circular impressions. The deformation in the plastically deforming zone was evaluated from the distortion of the circular impressions, which, because of their small size, are close to ellipses, despite the fact that in this case there is nonuniformity of deformation through the thickness of the cut. The major semiaxis of these ellipses was measured under a microscope at a magnification of 500 times. The pseudomaximum relative shear was calculated by the formula

\[ g' = 2 e_1 = 2 \ln \frac{2a}{d}, \]

where \(e_1\) is the principal strain in the direction of the major semiaxis of the ellipse,
\(a\) is the magnitude of the major semiaxis of the ellipse,
\(d\) is the initial diameter of the circular impression \((d = 0.03\ \text{mm})\).

Figure 3 shows the pattern of increase of the relative shear of certain elementary volumes of metal, located at the level of rows 19, 16, 13, and 9 of the impressions \((\gamma = 37^\circ)\), along their path of motion \(s\) in the process of deformation. The path length for constructing these graphs was determined by direct measurement on the drawing of the chip root. The somewhat overestimated values of the final relative shear in the upper and lower layers of the chip, observed on the graphs, are explained by some work hardening from the preceding pass of the tool and by secondary deformation of the metal adjacent to the rake face of the tool.

Knowledge of the nature of the increase of relative shear along the path of displacement in the deformation process makes it possible, for a known cutting speed (in the present case \(v = 19\ \text{mm/min}\)), to calculate the magnitude of the veloc-

…of deformation at any point of the deforming volume. For this purpose, let us represent the expression for the strain rate in the following form:

\[ v_g=\frac{d\gamma_k}{dt}=\frac{d\gamma_k}{ds}\cdot\frac{ds}{dt}=v_g'\cdot v_i . \]

Here \(v_g'\) is the rate of change of the relative shear along the path of motion of the elementary deforming volume,

\(v_i\) is the velocity of motion of the elementary deforming volume.

Thus, in order to calculate the strain rate during cutting, it is necessary to know two dependences: relative shear—deformation path \((\gamma_k—s)\) and deformation path—time \((s—t)\). Differentiating the first of them with respect to the path, and the second with respect to the time of motion in the deforming zone, we obtain the values of the velocities \(v_g'\) and \(v_i\), and therefore also the strain rate \(v_g=v_g'\cdot v_i\).

For the experiments carried out, it proved possible to find the dependences \(\gamma_k—s\) and \(s—t\) that are of interest to us graphically, and therefore the velocities \(v_g'\) and \(v_i\) were determined by their graphical differentiation by the chord method.

Let us illustrate the method for determining the strain rate by calculating the latter for a metal particle located at the level of the 19th row of circular imprints when removing a chip with a cutter having a rake angle \(\gamma=37^\circ\) (Fig. 2 in the article “Calculation of the Magnitude of Relative Shear in Metal Cutting”).

According to the method indicated above, a graph of the dependence \(g'—s\) is constructed (Fig. 3). Using this graph, we construct the dependence \(\gamma_k—s\) (Fig. 4), where the maximum relative shear is determined by the formula [2]:

\[ \gamma_k=2\,\operatorname{sh}\frac{g'}{2}. \]

Knowing the cutting speed \((v=19\ \text{mm/min})\) and the distances between any two imprints of the 19th row of circles, we construct a graph of the path of the deforming elementary volume (Fig. 5). By graphically differentiating the dependences shown in Figs. 4 and 5, we obtain the values of the velocities \(v_g'\) and \(v_i\) for any portion of the deformation path. The product of these velocities determines the magnitude of the strain rate.

In Fig. 6 are shown graphs of the change in strain rate as a function of the path of deforming volumes located at the level of the 19th and 9th rows of imprints \((\gamma=37^\circ)\). Analysis of these graphs makes it possible to draw the following conclusions:

Fig. 4

1. The strain rate during cutting of a ductile metal at different points of the deforming zone is not constant.

2. The strain rate along the path of motion of the deforming elementary volume first increases, reaching a certain maximum, and then rapidly decreases to zero.

1. The chip layers adjacent to the cutter are deformed at much higher rates than its outer layers.

2. When cutting copper at a speed \(v = 19\ \mathrm{mm/min}\) with a cutter having \(\gamma = 37^\circ\), the maximum rate of deformation proved to be \(v_g = 1.6\ \mathrm{s}^{-1}\).

Fig. 5

When cutting at higher speeds, the picture of deformation becomes somewhat different. Fig. 7 gives a photograph of the chip root obtained in free cutting of a copper specimen with a cutter having \(\gamma = 37^\circ\) at a speed \(v = 234\ \mathrm{m/min}\). The marks drawn on the surface of the specimen along an Archimedean spiral with a pitch of \(0.1\ \mathrm{mm}\) make it possible to estimate the magnitude of the zone of primary plastic deformation. As follows from the photograph, the zone of simultaneous deformation is in this case considerably smaller; the phenomenon of the so-called “build-up” of metal above the machined surface is almost absent.

Fig. 6

Because of the small dimensions of the deformed zone (\(0.1\ \mathrm{mm}\)), the above-described method for determining the rate of deformation proves unsuitable—it is not possible to reveal with sufficient accuracy the pattern of the increase of deformation in such a narrow zone. However, knowing the character of the increase of deformation from experiments with a low cutting speed, the final degree of deformation (as applied to the chip in Fig. 7, \(\gamma_k = 1.15\)), and also the cutting speed (\(v = 234\ \mathrm{m/min}\)), one can approximately estimate the magnitude of the rate of deformation in this case as well.

Let us calculate the magnitude of the rate of deformation for the process of formation of the chip shown in Fig. 7.

• In Fig. 8 a presumptive graph of the increase of shear along the path of motion is presented. Here the path of the primary deformation \((s_g = 0.13\ \mathrm{mm})\) is determined directly from the photograph; the magnitude of the final shear is calculated from the chip shrinkage:

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

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

where \(\zeta\) is the chip shrinkage.

Fig. 7

The initial degree of deformation was taken to be 0.18, since, as previous experiments showed, such a small deformation cannot be detected from the distortion of the grid. The dashed line in Fig. 8 shows

Fig. 8

the case in which the deformation along the path of motion of the deforming elementary volume increases uniformly. The velocity \(v_g''\) was determined by graphical differentiation of the graph shown in Fig. 8. Under the condition of a uniform increase of deformation along the path of motion of the cutter, the velocity \(v_g'\) proved to be equal to \(7.45\ 1/\mathrm{mm}\).

Fig. 9

In Fig. 9 the solid line shows the assumed graph of the change in the velocity of the deforming volume along the path of its motion, while the dashed line shows the same graph for the case in which the motion of the deforming elementary volume is uniformly retarded. In this graph the initial velocity is equal to the cutting speed,

and the final one is the velocity of motion of the chip along the rake face of the tool, which can be readily determined when the chip compression ratio \((\zeta)\) is known,

\[ v_{\text{chip}}=\frac{v}{\zeta}, \]

where \(v\) is the cutting speed.

The strain rate will still be determined by the product of the velocities \(v'_g\) and \(v_i\).

Figure 10 presents, obtained by such a calculation, the graph of the strain rate along the path of displacement of an elementary deforming volume. Here the solid line illustrates the increase in strain rate in the case where the laws governing the change in the magnitude of the relative shear and in the velocity of motion of the elementary volume in the deformed zone, obtained at low speeds, remain valid also at high cutting speeds. The dashed line characterizes the change in strain rate in the idealized case where the degree of deformation in the zone of interest to us increases uniformly, while the deforming volume moves with uniform deceleration. As follows from Fig. 10, even under these ideal conditions, which ensure the minimum value of the strain rate, its magnitude for the example considered reaches \(2.9 \cdot 10^4\ \text{s}^{-1}\). Under the more realistic deformation conditions that we have assumed, the maximum value of the strain rate proved to be equal to \(7.15 \cdot 10^4\ \text{s}^{-1}\).

Fig. 10

The calculation carried out, as well as the results of experiments at low cutting speed, make it possible to draw the following conclusions.

1. Evaluating the assumptions adopted in the calculation, one may assert that the magnitude of the strain rate in the process of chip formation shown in Fig. 7 reaches, at the very least, values on the order of \(5 \cdot 10^4\ \text{s}^{-1}\) and cannot be lower than the value obtained for idealized deformation conditions \((2.9 \cdot 10^4\ \text{s}^{-1})\).

2. When the cutting speed is changed by a factor of 12.3 thousand (from \(19\ \text{mm/min}\) to \(234\ \text{m/min}\)), the strain rate increases by a factor of 45–90 thousand. Thus, the strain rate grows with increasing cutting speed substantially faster than the latter.

References

1. V. D. Kuznetsov, Physics of the Solid State, vol. V, Tomsk, 1951.
2. V. G. Osipov, “On the characteristics of finite deformations,” in Problems of Metallurgy, Academy of Sciences of the USSR, 1958.