PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
OF THE ORDER OF THE OCTOBER REVOLUTION AND THE ORDER
OF THE RED BANNER OF LABOR
NAMED AFTER S. M. KIROV

Volume 188
1974

ON THE RELATIONSHIP BETWEEN THE CHIP–TOOL CONTACT LENGTH AND CHIP SHRINKAGE

M. F. POLETIKA

In the present article we set forth the results of an experimental study of certain contact characteristics in the cutting of various metals and alloys.

All experiments were carried out under conditions of orthogonal free cutting in longitudinal turning. Several steels, copper, beryllium bronze, brass, aluminum, some titanium alloys, and also soft metals—tin, lead, cadmium—were used as the materials being machined; the steels had been heat-treated to different hardnesses.

The contact length \(c\) was measured from the wear trace on the rake face of the tool. Only when machining soft metals was it sometimes necessary, in order to obtain a clear boundary of the contact area, first to apply a thin layer of ink to the rake face. The components of the cutting force were determined with a dynamometer. To separate the forces acting on the rake and flank faces of the tool, a method was used of extrapolating the force dependences to zero chip thickness.

A number of researchers have studied the dependence of the chip–tool contact length on various factors [1, 2, 3, etc.]. It has been established that the contact length \(c\) depends primarily on the chip thickness and the cutting speed, and that there is a definite relationship between the magnitude \(c\) and chip shrinkage.

In what follows, instead of the contact length \(c\), we everywhere consider the relative contact length \(c/a\). This offers known advantages, since it eliminates the direct influence of chip thickness on the contact length.

The typical form of the dependence of the quantity \(c/a\) on chip shrinkage is shown in Fig. 1. The dependence is unambiguous when the cutting speed and feed are varied over fairly wide limits. The rake angle of the tool has only a slight effect on the ratio \(c/a\) (at \(\varsigma=\mathrm{const}\)).

Figure 2 gives data for the machining of beryllium bronze that had undergone different heat treatments and had Brinell hardnesses ranging from 110 to 360. We see that the mechanical properties of the work material also have practically no effect on the dependence \(c/a=f(\varsigma)\).

Finally, in Fig. 3 we attempted to collect the results of experiments with various work materials, as well as with various tool-

Fig. 1. Relationship between the relative contact length and chip shrinkage at different rake angles

...with different materials. Here, a single curve no longer results. However, the deviations from it are not very large.

The relationship between chip shrinkage and the relative contact length, illustrated by Figs. 1, 2, 3, is an expression of the conditions of static equilibrium in the tool—chip—workpiece system. Indeed, let us turn to an equivalent scheme of chip formation with a single shear plane (Fig. 4), on which the forces and moments acting in the cutting zone are shown.

Let us denote by $\psi_0$ the relative coordinate of the point of application of the normal force $N_\gamma$ to the rake face of the tool (the center of chip pressure on the tool).

\[ \psi_0=\frac{OO_\gamma}{OB}=\frac{l_\gamma}{c}. \tag{1} \]

The point $O_\beta$ of application of the force $N_\beta$, normal to the shear plane, we shall call, by analogy, the center of pressure in the shear zone, and denote its relative coordinate by $\chi_0$:

\[ x_0=\frac{OO_\beta}{OA}=\frac{l_\beta\sin\beta_1}{a}. \tag{2} \]

The other notation used by us is clear from consideration of Fig. 4. Equating to zero the sum of the moments taken with respect to point \(O\), we obtain

Fig. 2. Influence of the hardness of the material being machined on the dependence
\[ c/a=f(\zeta) \]

\[ \frac{c}{a}=\frac{N_\beta}{N_\gamma}\cdot\frac{x_0}{\psi_0\sin\beta_1}. \tag{3} \]

On the other hand, the normal forces \(N_\beta\) and \(N_\gamma\) can be expressed through the resultant \(R\) as follows:

\[ N_\beta=R\sin(\omega+\beta_1); \]

\[ N_\gamma=R\sin(\omega+\gamma). \]

Substitution of these expressions into equality (3) leads to the formula

\[ \frac{c}{a}=\frac{x_0(1+\operatorname{tg}\omega\,\operatorname{ctg}\beta_1)} {\psi_0(\cos\gamma+\sin\gamma\,\operatorname{tg}\omega)}. \tag{4} \]

In our studies [4] it was shown that, when the cutting conditions are varied over wide limits, the quantity \(\psi_0\) changes only slightly \((\psi_0=0.15\div0.2)\). As a first approximation one may set \(\psi_0=\mathrm{const}\). With still greater justification, an analogous assumption may be made for the quantity \(x_0\) \((x_0=\mathrm{const})\), and then it will follow from equation (4) that the relative contact length is a function of two quantities: the inclination angle of the shear plane \(\beta_1\) and the action angle \(\omega\).

Fig. 3. Dependence \(c/a=f(\xi)\) over a wide range of variation in cutting conditions

\[ \frac{c}{a}=f_1(\beta_1,\omega). \tag{5} \]

At a constant rake angle, for a given work material the angles \(\beta_1\) and \(\omega\) are related to one another unambiguously [3]. Therefore, for these conditions the relative contact length must be a function of only one of these angles, for example the angle \(\beta_1\), or, equivalently, a function of the chip shrinkage.

\[ \frac{c}{a}=f(\zeta). \tag{6} \]

The graph in Fig. 1 is an illustration of dependence (6).

Fig. 4. Diagram of the forces acting on the chip

REFERENCES

1. N. G. Abuladze. The nature and length of the plastic contact of the chip with the rake face of the tool. In: Machinability of Heat-Resistant and Titanium Alloys. Kuibyshev, 1962.

2. A. I. Belousov. On the contact area of the chip with the cutter during free cutting. In: New Research in the Field of Machining by Cutting. Mashgiz, Moscow, 1957.

3. N. N. Zorev. Normal forces and friction forces in oblique free cutting. Proceedings of TsNIITMASH, book 15, 1948.

4. M. F. Poletika. Contact phenomena in metal cutting. Izvestiya TPI, vol. 133, 1965.