CALCULATION OF THE RESULTANT CUTTING FORCE BASED ON THE CHARACTERISTICS OF THE STATIC COMPRESSION PROCESS

G. L. KUFAREV, V. A. GOVORUKHIN, Yu. P. OLEINIK

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

In order to calculate the magnitude of the resultant cutting force, it is necessary, in addition to the magnitude of the tangential stress at the upper boundary of the plastic-deformation zone—the methods for determining which are set forth in works [1, 2]—also to know the value of the force \(P_N\) (Fig. 1).

The force \(P_N\) depends on the magnitude of the hydrostatic pressure. As investigations have shown [3], the magnitude of the hydrostatic pressure varies over the thickness of the chip, and to assume it constant—which would substantially simplify the solution of the problem before us—is in no case permissible.

Fig. 1

It does not appear possible to determine the magnitude of the hydrostatic pressure from the deformation of the material in the chip, as was done for tangential stresses, since there is no direct relationship between hydrostatic pressure and deformation. The magnitude of the hydrostatic pressure is determined only by the boundary conditions on the rake face of the tool and on the free surface of the chip.

For this reason it is proposed to determine the magnitude of the force \(P_N\) without operating with hydrostatic pressure. In accordance with the diagram in Fig. 1, one may write:

\[ F = P_N \cdot \cos(\Phi-\gamma) - P_\tau \cdot \sin(\Phi-\gamma). \]

Solving this equation for the force of interest to us, we obtain:

\[ P_N = \frac{F}{\cos(\Phi-\gamma)} + P_\tau \cdot \operatorname{tg}(\Phi-\gamma). \tag{1} \]

If the degree of deformation of the chip metal \(e_i\) is assumed to be known and, consequently, also the value of the angle \(\Phi\), as well as the characteristics of the material flow curve under static compression, then the only unknown quantity in equation (1) will be the magnitude of the friction force \(F\). It can be expressed as follows:

\[ F = q_F \cdot b \cdot c, \tag{2} \]

where \(q_F\) is the specific friction force on the rake face of the cutter;
\(c\) is the length of the contact area of the chip with the cutter;
\(b\) is the width of the layer being cut.

The quantity \(c\) can be determined from the equation proposed by N. G. Abuladze [4]:

\[ c = a[\zeta(1-\operatorname{tg}\gamma) + \sec\gamma]. \tag{3} \]

Here \(a\) is the thickness of the layer being cut.

The validity of this relationship for determining the length of the plastic contact of the chip with the cutter has been confirmed by the experiments of N. N. Zorev, V. F. Bobrov, B. Chao, and K. Trigger, as well as by the results of a large number of experiments carried out by the authors of the present article. The experiments were conducted in cutting the following steels: St. 20, St. 60, St. 2Kh13, St. 30KhGSA, St. U10A, St. 4Kh13, St. Kh12, St. 9Kh18, St. 3Kh2V8, St. 45Kh, and Armco iron.

The value of \(q_F\) in these experiments was determined from the friction force, calculated from the measured components of the cutting force, and from the length of the plastic contact of the chip with the cutter, calculated by formula (3). It was found that the magnitude of the specific friction force is proportional to the intensity of deformation of the chip metal; however, the degree of proportionality depends on the grade of the material being machined. The increase in the value of \(q_F\) with increasing degree of deformation has been qualitatively confirmed by numerous experiments carried out by N. N. Zorev [5].

A comparison of the dependence of \(q_F\) on the deformation intensity with the flow curves under compression (in Fig. 2, such results are presented, as an example, for four of the eleven materials machined in the experiments) showed that the lines drawn through the points of the experimental values of \(q_F\) are practically parallel to the polytropic portion of the flow curve; that is, the intensity of increase of \(q_F\) with increasing deformation of the chip metal coincides with the intensity of increase of the quantity \(\frac{\sigma_i}{2}\) in the compression process. At the same time, the magnitude of the specific friction force is substantially less than the stress intensity under compression.

Because of the indicated circumstances, it proved possible to approximate the line of variation of \(q_F\) with increasing deformation intensity of the chip metal by the equation:

\[ q_F = \frac{1}{2}(A \cdot e_i^k) - \Delta. \tag{4} \]

Fig. 2

Fig. 3

Here the first term is nothing other than \(\dfrac{\sigma_i}{2}\) at the given degree of deformation of the metal in the compression process and, consequently, is determined by the flow curve obtained in compression. The quantity \(\Delta\) is the difference between the quantities \(\dfrac{\sigma_i}{2}\) and \(q_F\).

Studies have shown that the quantity \(\Delta\) depends on the properties of the metal being machined, increasing in direct proportion to the quantity \(\sigma_{0.2}\) (Fig. 3). In this case the following relationship is found:

\[ \Delta = 0.278 \cdot \sigma_{0.2}, \tag{5} \]

where \(\sigma_{0.2}\) is the stress intensity at a strain intensity \(e_i = 0.2\) in the compression process.

This relationship is not accidental. In the work of M. F. Poletika [6], a graph is presented on which the quantity \(q^F\) is also related by an almost linear dependence to the true ultimate strength of the material being machined.

Thus, having the flow curve in compression and knowing the final degree of deformation of the chip metal, one can determine, using equations 2, 3, 4, and 5, the magnitude of the friction force, and then, from equation (1), the magnitude of the force \(P_N\).

Then the resultant cutting force is determined by the equation

\[ R = \sqrt{P_N^2 + F_\tau^2}. \]

The information obtained in these calculations proves sufficient also for determining the technological components of the cutting force.

It should be emphasized that the proposed approach to solving the problem of determining the resultant cutting force, unlike other analogous solutions, does not require knowledge of the magnitude of the coefficient of friction between the chip and the rake face of the cutter.

LITERATURE

1. G. L. Kufarev, M. G. Goldshmidt, V. A. Govorukhin, F. K. Pleshakov. Determination of the shear force in cutting at microspeeds. Present collection.

2. G. L. Kufarev, V. A. Govorukhin. Resistance to deformation in metal cutting. Present collection.

3. G. L. Kufarev, M. G. Goldshmidt, V. A. Govorukhin. Experimental verification of hypotheses on stresses in the cutting zone. Present collection.

4. N. G. Abuladze. Character and length of the plastic contact of the chip with the front surface of the tool. In: Machinability of Heat-Resistant and Titanium Alloys. KuAI, Kuibyshev, 1957.

5. N. N. Zorev. Problems in the Mechanics of the Metal-Cutting Process. Mashgiz, Moscow, 1956.

6. M. F. Poletika. Contact phenomena in metal cutting. Proceedings of TPI, vol. 133, 1965.