Experimental Verification of the Main Hypotheses on Stresses in the Cutting Zone

G. L. Kufarev, M. G. Goldshmidt, V. A. Govorukhin

(Presented by the scientific seminar of the Departments of Machine Tools and Metal Cutting and Mechanical-Engineering Technology)

For a complete solution of questions concerning the dynamics of the cutting process, it is necessary to have data on the stressed state at various points of the chip-formation zone, which, as studies have shown, has appreciable dimensions even at cutting speeds of up to 250 m/min [1]. In solving certain particular problems, owing to the absence of detailed information, researchers have analyzed the stresses on a notional shear plane, assuming one or another law of their distribution [2, 3, etc.]. The validity of such solutions can be assessed after analyzing the hypotheses on which they are based, in particular, the nature of the variation of tangential and normal stresses in the notional shear plane.

As a result of extensive experiments, N. N. Zorev came to the conclusion that the shear resistance in cutting may be regarded as a constant of the material being machined [4]. It is also known that the microhardness on the notional shear line is approximately constant. These circumstances led many researchers to the idea that a uniform distribution of tangential stresses in the shear plane is the most probable. However, direct experimental confirmation of this hypothesis was obtained comparatively recently in works [5, 6, 7].

If the uniform distribution of tangential stresses is generally accepted, then different opinions exist regarding the pattern of normal stresses. Many authors adopt a uniform law of stress distribution [2, 3, etc.]. Thus, M. I. Klushin [3], using the equilibrium equations for plane deformation, proves the constancy of hydrostatic pressure along the notional shear plane. Evidently, the analysis of stresses on the basis of a scheme with a single shear plane, without taking the deformation zone into account, gave him grounds to assume

\[ \frac{\partial \tau_{xy}}{\partial x}=\frac{\partial \tau_{xy}}{\partial y}=0, \]

i.e., the absence of strain hardening of the material in the chip-formation zone. The conclusions of M. I. Klushin are therefore valid only for a non-hardening material and only in the case when the notional shear plane coincides with the slip line.

The nonuniformity of the distribution of normal stresses is indicated in their analytical studies by Oxley [8] and Albrecht [9].

As a result of the method developed by G. D. Delem for studying plastic deformation by measuring hardness [10] and the application of this method to the experimental study of the cutting process, it became possible to assess the validity of the hypotheses set forth above. Calculation of stresses along the conventional shear line, performed according to the procedure described in [5], shows the following.

Regardless of the conditions of the cutting process—the tool rake angle, cutting speed, chip thickness—and of the type of chip formation, the tangential stresses along the conventional shear line are distributed almost uniformly.

The magnitude of the tangential stresses $\tau_{xy}$ depends on their intensity $k$, determined from the microhardness of the deformed metal: $\tau_{xy} = k \cos 2\Theta$, where $\Theta$ is the angle between the positive direction of the $x$ axis and the slip line.

Table 1

Note: the angle $\Theta'$ was calculated in the system of axes $x'y'$.

The stresses in the shear plane are determined in the system of axes $xy$; $\tau_{xy} = k$ when $\cos 2\Theta = 1$, i.e., when the shear plane coincides with the slip line.

To clarify the question of whether the conventional shear plane coincides with the slip line, values of $\Theta'$ were calculated for 17 different cases of the cutting process. The results of this calculation are presented as the value $\Theta'_{\mathrm{avg}}$ for each case and are compared with the angle $\Phi$ in Table 1.

As can be seen from Table 1, the difference between the angles $\Theta'_{\mathrm{avg}}$ and $\Phi$, measured by the angle $\Delta$, occurs in almost all calculated cases, and may be either positive or negative. Thus, it may be considered that identifying the concepts “conventional shear line” and “slip line” is incorrect, although in individual cases the slip line comes very close to the conventional shear line.

The normal stresses in all calculated cases are distributed nonuniformly. On the basis of the data obtained on the distribution of stresses along the conventional shear plane, it proves possible to analyze the influence of the factors of the cutting process on the character of the epure of normal stresses. In view of the fact that the conventional

the shear plane makes a small angle with the slip line (in our calculations no more than \(10^\circ\)); the normal stress along \(AB\) (Fig. 1) does not differ very substantially from the hydrostatic pressure \(\sigma\).

On the free surface of the chip-formation zone the hydrostatic pressure is \(\sigma=-k\). The hardness, and consequently the intensity of the tangential stresses in the conventional shear plane, change insignificantly when the rake angle and the contact conditions on the rake face of the tool are changed. Thus, in cutting L62 brass, changing the rake angle from \(42^\circ\) to \(8^\circ\) led to an 8% change in microhardness along the conventional shear line.

Fig. 1

Consequently, to a first approximation it may be assumed that the hydrostatic pressure at point A does not depend on the conditions of the cutting process. In this case the character of the \(\sigma\) diagram will depend only on the position of the resultant cutting force applied to the rake face of the tool, \(R\). This follows from Fig. 1: rotation of the resultant toward an increase in the positive value of the action angle \(\omega\) increases the normal force \(P_N\); at \(\sigma_A=\mathrm{const}\), a different character of variation of \(\sigma\) is required to balance the force; the better the lubrication conditions and the larger the rake angle of the tool, the more the law of variation of \(\sigma\) must approach the scheme shown in Fig. 1, б. With a decrease in the rake angle of the tool and an increase in the mean coefficient of friction on the rake face, the \(\sigma\) diagram, passing through all intermediate stages, will approach the scheme in Fig. 1, а. Only in a special case can the distribution of normal stresses on the conventional shear plane be uniform.

REFERENCES

1. G. L. Kufarev, V. A. Govorukhin, M. G. Goldshmidt. Deformation zone in metal cutting (article in the present collection).

2. L. M. Sedokov. Stress state in the zone of plastic deformation in the process of metal cutting. Izv. TPI, vol. 114, 1964.

3. M. I. Klushin. Calculation of the cutting part of a tool for strength. Stanki i instrument, No. 2, Moscow, 1958.

4. N. N. Zorev. Problems of the mechanics of the metal-cutting process. Mashgiz, Moscow, 1956.

5. G. D. Del, M. G. Goldshmidt. Determination of stresses in the cutting zone. Izvestiya TPI, vol. 162, 1967.

1. G. L. Kufarev, V. A. Govorukhin. Stress state in cutting metals at high speed. Izvestiya TPI, vol. 183, 1968.

2. M. G. Goldschmidt, G. D. Del, G. L. Kufarev. Stress state during the formation of a sliding chip. Izvestiya TPI, vol. 139, 1966.

3. P. Oxley. The mechanism of metal cutting with a variable yield stress. Design and Technology of Mechanical Engineering. “Mir,” Series B, No. 4, Moscow, 1963.

4. Albrecht. New propositions in the theory of metal cutting. Design and Technology of Mechanical Engineering. “Mir,” Series B, No. 3, IL, Moscow, 1961.

5. G. D. Del. Investigation of plastic deformation by measuring hardness. Izvestiya TPI, vol. 138, 1965.