ON THE EQUILIBRIUM OF FORCES ON A CUTTING TOOL WITH A REDUCED RAKE FACE

M. F. POLETIKA

A common drawback of regimes of intensive built-up-edge formation is the poor quality of the machined surface. Under conditions of a stable built-up edge, the roughness of this surface is associated, first of all, with friction and deep plastic deformations produced by the “rear” surface of the built-up edge as it slides over the cutting surface.

All these are very complex processes, the analysis of which requires a prior, thorough study of the general regularities of built-up-edge formation on tools with a shortened rake face, in particular, of force interactions. Many results of this study were set forth in the author’s works [1, 2, 3]. In the present article we shall consider the question of the static equilibrium of forces in the system tool—built-up edge—chip—workpiece.

The real picture of the forces and moments acting on a tool with a reduced rake face is very complex because of the curvilinear outlines of the built-up edge, and also because of the known uncertainty in the position of the shear surface near the apex of the built-up edge. Simplifying this picture by the same devices as we use in analyzing the general case of chip formation, we arrive at the equivalent scheme shown in Fig. 1.

This scheme shows the interaction forces between four contacting bodies: the tool, the built-up edge, the chip, and the workpiece. In contrast to the usual scheme with a single shear plane, here, between the tool and the workpiece, in addition to the chip, there is also another intermediate body—the built-up edge. The forces \(N_{\gamma}=N'_{\gamma}\) and \(F_{\gamma}=F'_{\gamma}\), acting on the front “face” \(OB\) of the built-up edge, play the role of “internal” forces with respect to the tool—workpiece system.

For the forces “external” with respect to the indicated system, the following equations of static equilibrium may be written:

\[ \left. \begin{aligned} \sum X &= N_{\beta}\sin\beta_{1}+F_{\beta}\cos\beta_{1}+F_{h}-Pz=0,\\ \sum Y &= N_{\beta}\cos\beta_{1}-F_{\beta}\sin\beta_{1}+N_{h}-R_{XY}=0 \end{aligned} \right\}\ldots \tag{1} \]

\[ \sum M_{0}=l_{\beta}N_{\beta}-l_{h}N_{h}-l_{f}N_{f}\cos\gamma+(OO_{1}+l_{f}\sin\gamma)R_{XY}=0. \tag{2} \]

In accordance with Fig. 1, here

Fig. 1. Diagram of the forces acting on the cutter, built-up edge, and chip

$N_\beta,\ F_\beta$ — forces acting on the shear plane;
$N_f,\ F_f$ — forces applied to the rake face (land) of the cutter;
$N_{\mathrm{н}},\ F_{\mathrm{н}}$ — forces acting on the “rear” face of the built-up edge;
$P_z,\ R_{XY}$ — resultant forces (the usual technological components of the cutting force);
$l_\beta,\ l_{\mathrm{н}},\ l_f$ — distances of the points of application of the acting forces from the tip of the cutter or built-up edge (coordinates of the “centers of pressure”).

If we introduce the notation:

\[ l_\beta=\psi_\beta\cdot AO=\frac{a\psi_\beta}{\sin\beta_1}; \]

\[ L_f=\psi_f\cdot f; \]

\[ l_{\mathrm{н}}=\psi_{\mathrm{н}}\cdot OO_1= \frac{\psi_{\mathrm{н}} f\sin(\gamma_{\mathrm{н}}-\gamma)}{\cos\gamma_{\mathrm{н}}}, \]

in which

\[ \psi_\beta,\ \psi_f,\ \psi_{\mathrm{н}} \]
are the relative coordinates of the “centers of pressure,”

$f$ is the length of the rake face (land) of the cutter,
$a$ is the chip thickness,
$\beta_1,\ \gamma_{\mathrm{н}},\ \gamma$ are the angles shown in Fig. 1,

then equation (2) takes the following form:

\[ \sum M_0= \frac{a\psi_\beta}{\sin\beta_1}N_\beta - \frac{f\psi_{\mathrm{н}}\sin(\gamma_{\mathrm{н}}-\gamma)}{\cos\gamma_{\mathrm{н}}}N_{\mathrm{н}} - f\psi_f\cos\gamma_0\,P_z+ \]

\[ +f\left[\operatorname{tg}\gamma_{\mathrm{д}}\cos\gamma-(1-\psi_f)\sin\gamma\right]R_{XY}=0. \tag{3} \]

Only two forces, \(P_z\) and \(R_{XY}\), among those entering equations (1) and (3), can be determined experimentally with a dynamometer. To compute the remaining forces, for known geometric parameters of chip formation \((\beta_1,\gamma_d)\), the equations of static equilibrium are insufficient. In particular, the forces \(F_n\) and \(N_n\), responsible for the formation of the undercut surface, cannot be found experimentally.

Let us try to estimate the forces \(N_n\) and \(F_n\) on the “rear” surface of the built-up edge by an indirect method. To this end we shall assume that the average shear stress \(\tau_{\rm av}\) in the shear zone is known in advance and that, consequently, the tangential force acting in the shear plane can be calculated from the formula

\[ F_z=\frac{ab\tau_{\rm av}}{\sin\beta_1}, \tag{4} \]

where \(b\) is the width of cut.

Solving equations (1) and (3) jointly and taking expression (4) into account, we easily obtain

\[ N_n=\frac{a^2\psi_z b\tau_{\rm av}-f\psi_f\sin\beta_1\cos\beta_1\,P_z+A R_{XY}}{B}, \tag{5} \]

\[ F_n=P_z-ab\tau_{\rm av}\tan\beta_1-R_{XY}\tan\beta_1+ \]

\[ +\frac{\tan\beta_1}{B} \left(a^2\psi_z b\tau_{\rm av}-f\psi_f\sin\beta_1\cos\beta_1\cdot P_z+A R_{XY}\right)\ldots, \tag{6} \]

where

\[ A=a\psi_z+f\sin\beta_1\cos\beta_1\tan\gamma_d\cos\gamma_d -(1-\psi_f)f\sin\beta_1\cos\beta_1\sin\gamma_0; \]

\[ B=a\psi_z+f\psi_n \frac{\sin(\gamma_d-\gamma_0)\sin\beta_1\cos\beta_1}{\cos\gamma_d} \Bigg\}. \tag{7} \]

Further, since an estimate of the forces of interest to us with an accuracy of \(\pm 20\%\) may be considered quite sufficient, we shall take

\[ \begin{aligned} \psi_z&=0.5;\\ \psi_n&=0.25;\\ \psi_f&=0.2\div 0.5. \end{aligned} \tag{8} \]

The first of these assumptions, as applied to an ordinary tool (without a built-up edge), has already been used earlier [4, 5]. The second is based on the study, from microphotographs of the cutting zone, of the nature of the junction between the built-up-edge contour and the undercut surface. This assumption is very approximate. The third assumption is based on an analysis of diagrams of contact stresses.

The value of \(\psi_f\) adopted in the third of assumptions (8) approximately corresponds to the entire possible range of variation of this quantity for different cases of chip formation on a tool with a reduced rake face. Here the lower limit \((\psi_f=0.2)\) refers to the moment of transition from restricted contact to ordinary cutting (a small ratio \(a/f\)), while the upper limit \((\psi_f=0.5)\) corresponds to maximum values of the factor \(a/f\).

The value of \(\tau_{\rm av}\) was taken from experiments with an ordinary tool (without a land) on the same work material, which is fully justified, since \(\tau_{\rm av}\) varies only weakly with cutting conditions.

The results of calculations by formulas (5) and (6) are collected in Table 1. Their analysis makes it possible to draw a number of interesting conclusions.

The forces \(N_{\mathrm{n}}\) and \(F_{\mathrm{n}}\), acting on the built-up edge from the side of the cutting surface, are very large; moreover, the normal force \(N_{\mathrm{n}}\) most often exceeds in magnitude the horizontal component \(R_{XY}\). This means that the second component entering into the force \(R_{XY}\) and equal to the integral sum of stresses on the “front” surface of the built-up edge is either small or directed in the direction opposite to \(R_{XY}\). This is precisely what should be expected, since the rake angle on the built-up edge \((\gamma_{\mathrm{n}})\) is very large.

Fig. 2. Diagram of the influence of chip thickness on the chip-flow angle

With an increase in the ratio \(a/f\), the forces \(F_{\mathrm{n}}\) and \(N_{\mathrm{n}}\) increase. Study of microphotographs of the cutting zone shows that this is a consequence of a change in the outline of the built-up edge. The latter begins to interact more intensively with the cutting surface, and stabilization of the built-up edge occurs.

In conclusion, let us consider the question of the relationship between the chip-flow angle \(\gamma_{\mathrm{d}}\) and the ratio \(a/f\). From experimental investigations [1, 3] it follows that, under all circumstances, with an increase in the ratio \(a/f\), the chip-flow angle \(\gamma_{\mathrm{d}}\) increases. We shall show that this is a necessary consequence of the conditions of static equilibrium.

Let us turn to Fig. 2, which repeats the diagram in Fig. 1 for the case \(\gamma_{0}=0^\circ\), and suppose that the chip thickness has suddenly increased significantly, becoming equal to \(a' > a\) (the corresponding contours of the cutting zone are shown by dashed lines). Since the stresses \(\sigma_N\) and \(\tau_{\mathrm{sr}}\) in the shear zone and the relative coordinate of the point of application of the resultant in the shear plane depend only weakly on the chip thickness, an increase in chip thickness will cause an approximately proportional increase in the force \(R_3\) and in the coordinate

Table 1

\(l_\beta\), without changing the direction of \(\vec{R}_\beta\). The new values of the forces in the shear plane are denoted in Fig. 2 by \(F'_\gamma\), \(N'_\gamma\), and \(R'_\beta\).

A simple examination of Fig. 2 leads to the conclusion that it is impossible to equilibrate the new force \(R'_\beta\) solely by a corresponding increase in the resultant acting on the chamfer. Nor will equilibrium be restored by increasing the forces \(F_{\mathrm{n}}\) and \(N_{\mathrm{n}}\) on the built-up edge within the limits determined by the data of Table 1. The force \(R'_\beta\) creates a substantial unbalanced moment.

On an ordinary cutter, such a situation leads to an increase in the length of contact between the chip and the cutter and to a corresponding increase in the arm \(L_\gamma\), on which the force \(R\) acts, as a result of which equilibrium is restored. In the present case, the increase of \(l_\gamma = l_f\) is impeded by the limited size of the rake face. Therefore, the only “natural” way to restore equilibrium will be rotation of the shear plane \((\beta'_1 > \beta_1)\) and the inevitably associated increase in the chip-flow angle.

REFERENCES

1. A. M. Rosenberg, M. F. Poletika. Features of the cutting process with a chamfered tool in high-speed turning. “Izv. TPI,” vol. 75, 1954.

2. M. F. Poletika. Microstructure and microhardness in the cutting zone during operation of a chamfered cutter. “Izv. TPI,” vol. 85, 1957.

3. M. F. Poletika. On built-up-edge formation on a cutter with a shortened rake face. “Izv. TPI,” p. 107, 1963.

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