Verification of the Cutting Force Equation Under High-Speed Conditions

A. M. Rozenberg and L. A. Khvorostukhin

Professor A. M. Rozenberg developed a formula for calculating the force in metal cutting, relating the cutting force to the physico-mechanical characteristics of the metal being machined, to the geometry of the cutting tool, and to the parameters of the cutting process:

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

Here \(P_1\) is the cutting force in the direction of the cutting speed,

\(\sigma_0\) — the conventional yield point of the metal being machined in compression,

\(n\) — the hardening exponent in the stress polytrope under compression,

\(e\) — the base of natural logarithms,

\(\zeta\) — chip shrinkage,

\(\gamma\) — the rake angle of the cutter,

\(\eta\) — the angle of friction between the chip and the cutter,

\(a,b\) — the thickness and width of the removed chip layer.

The above formula was verified in the work of A. M. Rozenberg and A. N. Eremin in cutting steel, aluminum, and lead [1]. The verification was carried out in free cutting and mainly at such low cutting speeds that the possible influence on the chip-deformation process of the temperature in the chip shear zone is excluded. This was of substantial importance because the physico-mechanical characteristics of the metal being machined, \(\sigma_0\) and \(n\), were determined by compression of specimens at room temperature, and a strict verification of formula (1) had to be performed by carrying out the cutting process under the same conditions.

It was of considerable theoretical as well as practical interest to verify formula (1) under high-speed conditions, when sufficiently high temperatures develop on the tool rake face and in the chip shear zone, which may affect the physico-mechanical characteristics of the metal being machined. Such a verification is of practical interest, since it answers the question of whether formula (1) can be applied to high-speed cutting conditions, provided that the characteristics of the metal being machined, \(\sigma_0\) and \(n\), are determined by compression tests at room temperature. The theoretical interest of such a verification lies in the fact that it was to reveal the influence of temperature in the shear zone on the stresses in the cutting process. In addition to all the above, it was necessary to decide whether formula (1) is applicable to the case of non-free cutting.

Formula (1), by its physical essence, makes it possible to calculate the force applied to the front face of the cutter.

In addition to the forces applied to the front face and producing the process of chip deformation, there are also forces located on the rear face, which are the result of elastic-plastic contact of the rear face, in a narrow region of it adjoining the cutting edge, with the machined surface. The forces on the rear face do not depend on the thickness of the layer being cut or on the cutting angle, and their magnitude is all the less noticeable, relative to the forces on the front face, the greater the thickness of the layer being cut and the greater the cutting angle. When determining the cutting force analytically, the force located on the rear face in the direction of the cutting speed should be added to the value of the force \(P_1\) calculated from equation (1). But in most cases, with feeds and chip thicknesses used in practice, the magnitude of this addition is very small. It was of interest to determine from what feeds onward the force on the rear face may be neglected because of its relative smallness.

Experimental procedure

The experiments were carried out on a lathe. The following grades of steels were taken for the experiments:

1. Steel 10.
2. Steel 4.
3. Steel ShKh15.
4. Steel 30KhGS.
5. Steel 9KhS.

The standard chemical composition of these steels is as follows (Table 1).

Table 1

Thus, steels of different chemical composition were taken for the experiments. Steels 10 and 4 are ordinary carbon steels; ShKh-15 is ball-bearing steel; 9KhS is tool steel; 30KhGS is alloy structural steel.

The indicated steels had the following Brinell hardnesses (Table 2).

Table 2

All steels, before the experiments, were subjected to annealing or normalizing in order to equalize their structure. The hardness was measured on the blanks both along and across the section, in order to verify the uniformity of the mechanical properties within each given blank. From each

a piece was cut off from each bar, from which specimens for compression were made.

Cutting was performed with tools fitted with carbide inserts of grade T15K6. The tools had the following geometry:

Front rake angle \(20^\circ, 10^\circ, 0^\circ, -10^\circ\).
Clearance angles \(10^\circ\).
Cutting-edge inclination angle \(0^\circ\).
Principal cutting-edge angle in plan \(75^\circ\).
Auxiliary cutting-edge angle in plan \(15^\circ\).
Nose radius \(0.5\) mm.
The rake face was flat.

The tools were lapped with boron carbide paste before each experiment and were used only in a sharp condition, without noticeable traces of blunting or chipping of the cutting edges.

The experiments were carried out at cutting speeds high enough that no built-up edge formed on the tool.

In the experiments, the three components of the cutting force \(P_z, P_x, P_y\) were measured, as well as the longitudinal chip shrinkage \(\zeta\) and the chip widening \(\dfrac{b_1}{b}\), where \(b\) is the width of cut and \(b_1\) is the width of the cut chip.

From these quantities, for each experiment the values of the relative displacement \(\varepsilon\), the angle \(\beta_1\) determining the position of the plane of maximum shear stresses, the shear stress \(\tau\), the coefficient of friction \(\mu\), and the friction angle \(\omega\) were calculated.

The calculation was made using the following relationships (Fig. 1):

Fig. 1

\[ \tg \beta_1 = \frac{\cos \gamma}{\dfrac{b}{b_1}\zeta_1 - \sin \gamma}, \]

\[ \varepsilon = \ctg \beta_1 + \tg(\beta_1 - \gamma), \]

\[ \mu = \tg(\omega + \gamma), \]

\[ \tg \omega = \frac{R_{xy}}{P_z}, \]

\[ R_{xy}=\sqrt{P_x^2+P_y^2}, \]

\[ \eta=\operatorname{arctg}\mu, \]

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

\[ P_t=\frac{P_z}{\cos\omega}\cdot\cos(\omega+\beta_1). \]

The chip shrinkage \(\zeta\) for substitution into equation (1) was calculated from the expression

\[ \zeta=\zeta_1\frac{b}{b_1}, \]

where \(\zeta_1=\dfrac{l_0}{l}\) is the longitudinal shrinkage,

\(b\) is the width of the cut,

\(b_1\) is the width of the removed chip.

Forces on the flank face were not taken into account and were not excluded from the dynamometer readings. The experiments on the lathe were carried out at a depth of cut \(t=3\) mm and feeds \(s=0.30\) mm/rev, 0.15 and 0.07 mm/rev. The ratio of the depth of cut to the feed was chosen to be large, so as not to obtain significant distortion of the cross-sectional shape of the removed chip in comparison with the section of the layer being cut.

To determine the physico-mechanical characteristics of the material being machined, specimens were compressed under a Gagarin press. From the compression results, the values \(\sigma_0\) and \(n\) were obtained. To reduce the influence of friction at the ends of the compressed specimens on the stressed state, the ratio of the initial height of the compressed specimen \(h_0\) to its diameter \(d_0\) was taken as

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

and compression was carried out in a special guide sleeve.

The results of the cutting experiments were plotted as experimental points on graphs of \(\tau\varepsilon=f(\varepsilon)\).

On these same graphs, using the compression experiments, the curve

\[ A_{wt}=\frac{\sigma_0}{n}\left\{e^{\frac{n\varepsilon}{1.5}}-1\right\}. \tag{2} \]

was constructed.

The deviations of the experimental points from the cutting experiments from the curve drawn according to equation (2) characterized the deviation of the experimentally obtained values of the cutting force \(P_z\) from the results of the theoretical calculation of the force by equation (1), and made it possible to judge the accuracy and suitability of equation (1). Since, in calculating the shear stress in cutting \(\tau\), the forces on the flank face were not excluded from the forces measured by the dynamometer, the position of the experimental points on the graph \(\tau\varepsilon=f(\varepsilon)\) made it possible to judge whether, in calculating the cutting force by equation (1), the forces on the flank face of the tool could be neglected.

Experimental results

Compression experiments were used to determine the physico-mechanical characteristics of the steels—the conventional yield strength \(\sigma_0\) and the polytropic stress exponent \(n\). The values of these quantities for the steels tested were as follows (Table 3).

Table 3

In Fig. 2, for steel 10, the theoretical curve according to equation (2) is plotted; along the abscissa axis are plotted the values of the relative shear, and along the ordinate axis—the specific work of shear deformation $A_{wt}$ in kgm/mm$^3$. The relative shear in compression was calculated from the equation

\[ \varepsilon = 1.5 \ln \frac{h_0}{h}. \]

Experimental points from cutting tests are plotted in the same figure. For each test the quantities $\tau$, $\varepsilon$, and $A_{wt} = \tau \varepsilon$ [1] were calculated, which made it possible to arrange all the results of the cutting tests in the coordinates $\varepsilon$—$A_{wt}$.

Fig. 2

Inscriptions in the figure: ordinate axis—“Specific work $A_{wt}$”; abscissa axis—“Relative shear $\varepsilon$”; “Steel 10”; $\sigma_0 = 56,4$; $n = 0,28$; $v = 30 \div 200$ m/min; $\gamma = 20^\circ$, $\gamma = 10^\circ$, $\gamma = 0^\circ$; $S = 0,29$ mm/rev; $S = 0,145$ mm/rev.

The tests were carried out at three different rake angles and two feeds and at cutting speeds varying from 30 to 200 m/min. The graph shows that the experimental points are well positioned relative to the theoretical curve, which confirms the correctness of equation (1) for calculating the cutting force.

In Fig. 3 the same graph is given for steel 4, which shows an equally good arrangement of the experimental points relative to the curve. In Figs. 4, 5, and 6 the same results are presented for steels 30KhGS (Fig. 4), ShKh15 (Fig. 5), and 9KhS (Fig. 6). For these three steels, which have higher mechanical properties than the first two and are characterized by

Fig. 3

Fig. 4

with significantly higher specific cutting work, the experimental points were located on average 4–7% below the theoretical curves, while at the same time completely copying their outlines.

Fig. 5

Vertical axis: specific work \(A_{\mathrm{sh}}\).
Horizontal axis: relative shear \(\varepsilon\).

Annotation in the figure:

Steel ShKh-15
\(\delta_0 = 83.8;\ n = 0.27\)
\(v = 25\text{–}140\ \mathrm{m/min}\)

\(\Delta\) — \(\gamma = +10^\circ\)
\(O\) — \(\gamma = 0^\circ\)
\(x\) — \(\gamma = -10^\circ\)
\(S = 0.291\ \mathrm{mm/rev}\)

All the experiments indicated were carried out in free cutting, and the graphs presented show that equation (1) satisfies this type of cutting process sufficiently well.

Fig. 6

Vertical axis: specific work \(A_{\mathrm{sh}}\).
Horizontal axis: relative shear \(\varepsilon\).

Annotation in the figure:

Steel EKhS
\(\delta_0 = 94.5;\ n = 0.155\)
\(v = 25\text{–}140\ \mathrm{m/min}\)

\(\Delta\) — \(\gamma = +10^\circ\)
\(O\) — \(\gamma = 0^\circ\)
\(x\) — \(\gamma = -10^\circ\)
\(S = 0.291\ \mathrm{mm/rev}\)

If we compare the graphs obtained with the results of the work of A. M. Rozenberg and A. N. Eremin, carried out at microspeeds on steel 40 [1], we may note the following circumstance. The experimental points for cutting at microspeeds were located somewhat above (by 3–4%) the theoreti-

...the theoretical curve (Fig. 7), whereas at high cutting speeds we obtained results that gave experimental points which, for mild steels, lay on the curve, and for harder steels somewhat below (by 4–7%) the theoretical curve. It may be assumed that a certain lowering of the experimental points here is connected with the influence of temperature in the shear zone on the mechanical characteristics of the metal being machined. At microspeeds the temperature in the shear zone is equal to room temperature, since all the heat released as a result of deformation has time to dissipate; at high speeds, however, heat dissipation is the less possible the higher the cutting speed, and the amount of heat released per unit volume of the metal being deformed will be the greater, the greater the specific work of deformation \(A_{sh}\). We also observe that for the softest steel, 10, for which the specific works are the smallest, and hence the temperature in the shear zone is lower than for

Fig. 7

harder steels, characterized by more substantial specific works, the experimental points are located directly on the curve; for steel 4 they are already shifted somewhat downward relative to the curve, and for still stronger steels the experimental points deviate downward from the curve still more appreciably. This, apparently, is the manifestation of the influence of the temperature in the shear zone of the chip on the mechanical characteristics of the metal and on the shear stress during cutting. This influence is very small, despite the fact that the temperatures in the shear zone may be quite high and reach 500–700°. If the influence of temperature were significant, we could not obtain a close arrangement of the results, obtained at high speeds, relative to a curve constructed from static-compression tests at room temperature.

The overall influence of temperature on the stress in cutting steels may be estimated at a 10% reduction in shear stress, which completely refutes the notion of high-speed cutting as the cutting of a metal softened by temperature in the chip-deformation zone. The insignificance of the influence of temperature makes it possible to use equation (1) for calculating cutting forces at high speeds, with determination of the physicomechanical characteristics of the metal being machined, \(\sigma_0\) and \(n\), from static-compression tests at room temperature. The deviation of the experimental points from the theoretical curve by 4–7% may be considered entirely permissible for practical calculations.

Experiments on steel 10 were carried out by us with three feeds: 0.07 mm/rev, 0.15 mm/rev, and 0.29 mm/rev. Different feeds were chosen for the special purpose of revealing the influence of the forces on the flank face on the force \(P_z\) measured by the dynamometer. Since, by their nature, the forces on the flank face do not depend on the chip thickness, that is, in this case, on the feed, then when plotting the experimental points for cutting on the graph \(A_{ud} = f(\varepsilon)\), if for calculating \(A_{ud}\) we use the force \(P_z\) measured by the dynamometer (without taking into account the forces on the flank face), we should, in essence, obtain for each feed its own arrangement of experimental points; moreover, the smaller the feed, the relatively higher the experimental points should be located for the same values of the relative displacement. The same may be said with respect to the influence of the rake angle. The experimental points for larger rake angles should be located higher than for smaller ones, since, as the rake angle increases, the forces on the flank face, which do not depend on the rake angle, become more noticeable in comparison with the forces on the rake face, which decrease as the rake angle increases.

Fig. 2 shows that, at feeds of 0.145 and 0.29 mm/rev, the indicated arrangement of the experimental points cannot be observed. This circumstance indicates that the forces on the flank face are so small that already at a feed of

Fig. 8

0.15 mm they may be neglected in calculating the cutting force. Of course, this is true as applied to a sharp cutter, not worn on the flank face, since with increasing wear on the flank face the forces on the flank face will also increase in their absolute values.

In Fig. 8 are presented the results of experiments on steel 10 with feeds \(S = 0.07\) mm/rev. Here the experimental points are already located considerably above (by 20–15%) the theoretical curve. This was the result of the fact that, at a very small feed, the forces on the flank face became noticeable in comparison with the forces on the rake face, which also created an excess of the experimental points, since in calculating the specific works we

of the forces measured by the dynamometer did not exclude the forces on the rear face of the tool.

All these experiments show that, at feeds greater than \(0.15\ \text{mm/rev}\), the cutting force may be calculated from equation (1) without taking into account the forces on the rear face; at smaller feeds, however, this calculation will give a value of the cutting force that is the more underestimated the smaller the feed. This is not a shortcoming of equation (1), since, by its physical nature, it cannot take into account the forces on the rear face, for the nature of the forces on the front and rear faces is entirely different. For an approximate allowance for the forces on the rear face, at feeds of the order of \(0.05\)—\(0.15\ \text{mm/rev}\), it may be recommended to increase the force obtained from equation (1) by 15–20%.

General conclusions

The investigation carried out showed:

1. Equation (1) can be used to calculate the cutting force both in free and in non-free cutting, and both at low and at high cutting speeds.

2. At feeds exceeding \(0.15\ \text{mm/rev}\), there is no practical need to take into account the forces on the rear face when calculating the main component of the force for a sharp cutter.

3. When calculating the force by equation (1), the chip shrinkage should be calculated taking into account chip widening.

\[ \zeta = \zeta_1 \cdot \frac{b}{b_1}. \]

1. The temperature in the shear zone has a very insignificant effect on the stress in the process of metal cutting.

Literature

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