ON THE STRENGTH CALCULATION OF THE CUTTING PART OF A CUTTER

M. F. POLETIKA, M. K. UTESHEV

(Presented by Prof. Dr. A. M. Rozenberg)

Questions concerning the strength of the cutting part of cutters are becoming topical at the present time, on the one hand, owing to the predominant spread of high-productivity cutting materials: hard alloys and mineral ceramics, whose durability is often limited by premature brittle wear; on the other hand, this is also promoted by the ever wider use in machine building of hard and superhard materials.

One of the causes of brittle wear is the nonuniformity of the distribution of stresses on the working faces of the tool.

The study of the distribution of stresses on the working faces of a cutter is possible both by theoretical and by experimental means.

Among the experimental methods for determining stresses, two methods are known: the split-cutter method and the polarization-optical method.

We carried out investigations to determine stresses in the cutting part of a cutter under different cutting conditions by the polarization-optical method.

The investigations were carried out on a model 1K62 lathe. The polarization-optical setup was mounted directly on the machine. During cutting, the optical patterns were photographed and the cutting forces were recorded.

The experiments were conducted under conditions of free orthogonal cutting. The cutters were made of an optically sensitive material based on epoxy resin, and the material being machined was lead.

As a result of processing the optical patterns (the method is described in detail in [5]), the stressed state of the points of contact of the cutter with the chip was fully determined.

In Fig. 1 the Mohr circles are given for various points of contact of a cutter with \(\gamma = 10^\circ\), \(\alpha = 7^\circ\), constructed from the experimental results. The arrangement of the points along the length of contact is shown in Fig. 2, where

\[ \tau_{F_i} \]
— the corresponding tangential stress at various points of contact;

\[ \sigma_{N_i} \]
— the normal stresses at the same points of contact;

\[ \sigma_{X_i} \]
— the normal stress in sections perpendicular to the rake face of the cutter;

$\alpha_i$ is the angle of inclination of the plane of the front-face surface at the point under consideration to the principal plane.

As is seen from Fig. 1, at points of the cutter located on the surface of contact with the chip we have a linear state of stress; only at the tip of the cutter, evidently, will the state of stress be plane. The magnitudes of the stresses at the cutter tip are difficult to determine experimentally; therefore the Mohr circle numbered 12 for the cutter tip is shown conditionally.

In this way all stress components were determined for various cutting conditions. In Figs. 2–4 stress-distribution diagrams are presented for the tangential and normal stresses as functions of feed, rake angle, and cutting speed.

Analysis of the stress diagrams obtained shows that the normal stresses are distributed extremely nonuniformly, from zero to a maximum, varying approximately according to an exponential law; therefore any approximation of the stress distribution by a triangular law, and still more by a rectangle or trapezoid, is doubtful and may lead to large errors.

The tangential stresses are distributed more uniformly, and the portion of the curve adjacent to the tip is almost flattened. The flat portion corresponds to the length of plastic contact, while the remaining portion of the branch corresponds to elastic contact.

It is interesting to note that, for different rake angles of the cutter, the values of the tangential stresses on the cutting edge are almost equal to one another; for different feeds these stresses differ somewhat from one another. When the cutting speed is changed, the stresses at the tip differ sharply in their value owing to the influence of the strain-hardening factor.

It should be pointed out that the curves of the distribution of tangential stresses obtained by us agree well with the experimental results of M. B. Gordon.

Knowing the normal and tangential stresses at each point, it is not difficult to determine the coefficient of friction as the ratio

\[ \mu_i = \frac{\tau_{Fi}}{\sigma_{Ni}} . \]

In Figs. 2–4 curves are given for the variation of the coefficient of friction along the length of contact of the cutter with the chip.

As is seen from the figures, the curve of the coefficient of friction increases continuously from the cutter tip to the end of contact; moreover, on some curves characteristic points of “break” of the curve are visible. In all probability these points are the boundary between the plastic and elastic zones of contact.

It should be noted that in the elastic zone we do not see constancy of the coefficient of friction, although here we have kinetic friction.

Of particular interest are the stresses adjacent to the cutter tip. Knowledge of the stresses at the tip is extremely important from two points of view: first, these stresses are the actual stresses in the near-tip point of the chip-formation zone; second, the strength of the cutting edge of the cutter depends on the magnitudes of these stresses.

In Figs. 5 and 6 experimental plots are presented for the variation of these quantities as functions of the rake angle and feed; some interesting relations are also given there. As is seen from the plot, with an increase in the rake angle the maximum normal stress $\sigma_M$ decreases, while $\tau_M$ remains constant.

Fig. 1.

Labels in the diagram: “conditional”; “Direction of $\tau_F$ and $\sigma_x$”; $\sigma_2=\sigma_m(t?)$; $\sigma_{x_1}$, $\sigma_{x_2}$, $\sigma_{x_3}$, $\sigma_{x_4}$, $\sigma_{x_5}$, $\sigma_{x_6}$, $\sigma_{x_7}$, $\sigma_{x_8}$, $\sigma_{x_9}$; $\sigma_{y_2}$, $\sigma_{y_3}$, $\sigma_{y_4}$, $\sigma_{y_5}$, $\sigma_{y_6}$, $\sigma_{y_7}$, $\sigma_{y_8}$, $\sigma_{y_9}$, $\sigma_{y_{12}}$; $\tau_F$, $\tau_{F_1}$, $\tau_{F_2}$, $\tau_{F_3}$, $\tau_{F_4}$, $\tau_{F_5}$, $\tau_{F_6}$, $\tau_{F_7}$, $\tau_{F_8}$, $\tau_{F_{9,12}}$; $O_1$, $O_2$, $O_3$, $O_4$, $O_5$, $O_6$, $O_7$, $O_8$, $O_9$, $O_{12}$; $\alpha$; $1,2,3,4,5,6,7,8,9,12$.

Fig. 2.

Labels visible in the graphs:

• $\tau_F,\ \mathrm{kg/cm^2}$
• $\sigma_N,\ \mathrm{kg/cm^2}$
• $\mu$
• $S = 0.475$
• $S = 0.35$
• $S = 0.185$
• $S = 0.097$
• Distance from the cutter tip, mm

Fig. 3.

In the upper graph:

\[ q_F,\ \frac{\mathrm{kg}}{\mathrm{cm}^2} \]

\[ \gamma = 10^\circ,\qquad \alpha = 7,\qquad S = 0.475 \]

Curves:

\[ V = 12\ \frac{\mathrm{m}}{\min} \]

\[ V = 0.250\ \frac{\mathrm{m}}{\min} \]

\[ V = 0.0152\ \frac{\mathrm{m}}{\min} \]

In the middle graph:

\[ \sigma_n,\ \frac{\mathrm{kg}}{\mathrm{cm}^2} \]

Curves:

\[ V = 12\ \frac{\mathrm{m}}{\min} \]

\[ V = 0.250\ \frac{\mathrm{m}}{\min} \]

\[ V = 0.0152\ \frac{\mathrm{m}}{\min} \]

In the lower graph:

\[ M \]

Curves:

\[ V = 0.015\ \frac{\mathrm{m}}{\min} \]

\[ V = 12\ \frac{\mathrm{m}}{\min} \]

\[ V = 0.250\ \frac{\mathrm{m}}{\min} \]

Distance from the cutter tip

Fig. 3

Fig. 5.

Fig. 6.

With an increase in feed, \(\sigma_M\) increases very strongly, whereas \(\tau_M\) increases only slightly.

The graphs give curves relating the maximum stresses to the mechanical characteristics of the metal being machined. As the mechanical characteristic, the tensile strength \(\sigma_b\) of lead was taken. It is interesting to note that the ratio \(\dfrac{\tau_M}{\sigma_M} \simeq 0.5\) remains constant with increasing rake angle and feed; this is possible in the present case if \(\tau_M = \tau_T\), where \(\tau_T\) is the yield point of the material, while for normal stresses the ratio is

\[ \frac{\sigma_M}{\sigma_b} = 1.5 \div 2. \]

The extent to which average stress values cannot characterize the degree of loading of the cutting edge is evident from the ratio of maximum stresses to average stresses; thus, for normal stresses this ratio is \(\dfrac{\sigma_M}{\sigma_N} \simeq 4\), and for tangential stresses \(\dfrac{\tau_M}{\tau_F} \simeq 2\).

Thus, the maximum stresses are related with sufficient accuracy to the mechanical characteristics of the material being machined at a constant cutting temperature.

\[ \sigma_M \simeq (1.5 \div 2)\sigma_b;\qquad \tau_M = 0.5\sigma_b. \tag{1} \]

We investigated the stress distributions on the flank face of the cutter; for this purpose a wear land was artificially created on the flank face. The stress state of a point on the wear land of the flank face will be as shown in Fig. 7, \(a\).

The stress distribution here is likewise nonuniform; moreover, the epure of distribution of the normal and tangential stresses was found to be the same, and therefore the coefficient of friction at every point is a constant quantity and equal to unity, i.e., in this case the coefficient of friction obeys the Amontons–Coulomb law.

Comparing the data on the coefficient of friction on the rake and flank faces, it should be said that such a difference in the coefficients of friction is evidently explained by different contact conditions and by a different loading scheme.

Thus, in the cutting process we have, on the working faces of the tool, a nonuniform distribution of contact stresses.

The extension of the results of experiments performed with a plastic cutter and with lead as the machined material to ordinary cutting conditions often meets objections, the principal ones of which may be formulated as follows:

1. Lead is a material that does not strain-harden, whereas under ordinary cutting conditions we are dealing with strain-hardening metals.

2. The regularities of chip formation for a plastic cutter may differ substantially from those for a metal cutter.

3. The cutting conditions at low speeds, at which polarization-optical investigations are usually carried out, are different from those at ordinary cutting speeds.

In this connection, the following should be noted. The absence of strain hardening in lead, as is known, is explained by the prevailing role of recovery phenomena during deformation of lead at a low rate. With a considerable increase in the deformation rate, recovery does not have time to occur, and lead, in the process of deformation, begins to behave in the same way as strain-hardening metals. The appli-

…relative to cutting, this has been convincingly shown by the investigations of A. M. Rosenberg and A. N. Eremin [4]. Therefore, the circumstance that an increase in cutting speed does not change the fundamental nature of the stress-distribution diagrams (Fig. 4) we consider sufficient to answer both the first and, in essence, the last of the objections noted.

As for the influence of the tool material on the chip-formation process, in this connection one could cite a number of works showing that a change in the tool material does not introduce

Fig. 7.

fundamental changes into the chip-formation process, and sometimes has no effect at all on the parameters of this process. Nevertheless, for greater persuasiveness we carried out a special experiment with a bimetallic cutter.

A thin steel plate of thickness equal to 0.08 mm was glued to the rake face of a plastic cutter. The experiment showed that in this case the nature of the stress distribution did not change; the cutting forces changed in the direction of increase, since the contact conditions changed.

Investigations were carried out on the cutting of harder metals—cadmium, aluminum, copper—by cutters with glued-on steel plates, and by cutters made of optically sensitive…

glass. The character of the stress-distribution curves proved to be analogous to the curves obtained when cutting lead at a cutting speed \(V = 7.2\ \text{m/min}\).

On the basis of processing the experimental results, empirical equations were derived for the contact-stress distribution curves.

The stress-distribution curves are expressed by exponential functions.

For normal stresses the equation has the form

\[ \sigma_{N_x}=\sigma_M e^{-b_1 c_x}, \tag{2} \]

for tangential stresses

\[ \tau_{F_x}=\tau_M e^{-3\left(\frac{c_x}{c}\right)^2}, \tag{3} \]

where \(\sigma_{N_x}, \tau_{F_x}\) are the current stresses,

\[ \begin{aligned} \sigma_M,\ \tau_M &\text{ are the maximum stresses at the tip,}\\ c_x &\text{ is the current coordinate of the contact point under consideration,}\\ b_1 &\text{ is a certain coefficient.} \end{aligned} \]

Equation (2), in semilogarithmic coordinates, represents the equation of a straight line, where the tangent of the angle of inclination will be equal to the coefficient “\(b\).”

Thus, the distribution of normal stresses according to a triangular law will be in semilogarithmic coordinates.

Formulas (2) and (3) were checked in all cases and showed satisfactory results.

It is interesting to note that equation (3) was first proposed by N. A. Reznikov as a result of his investigations and the investigations of M. B. Gordon; moreover, these investigations were carried out by the split-tool method in cutting copper and steel.

As was indicated above, from the standpoint of the strength of the cutting edge, the maximum stresses at the tool tip are of greatest interest.

As a result of certain transformations, these stresses can be obtained from formulas (2) and (3):

\[ \sigma_M=-\frac{c\cdot \sigma_N \cdot b_1}{\left(e^{-b_1 c}-1\right)}, \tag{4} \]

\[ \tau_M=\frac{\tau_F\cdot c}{\displaystyle \int_0^c e^{-3\left(\frac{c_x}{c}\right)^2}\,dc_x}, \tag{5} \]

where \(\sigma_N, \tau_F\) are the mean contact stresses,

\[ c \text{ is the length of contact of the tool with the chip.} \]

The integral in formula (5) is taken approximately; then, after integration, we obtain

\[ \tau_M \cong \frac{\tau_F}{0.55}. \tag{5a} \]

In other words, the area of the epure of tangential stresses is approximately equal to the area of the triangle.

Thus, determining the stresses acting at the tip under different cutting conditions presents no difficulty.

For a complete determination of the stressed state of the cutting edge, the stresses \(\sigma_M\) and \(\tau_M\) are insufficient; it is also necessary to know the normal stress in the section perpendicular to the rake face \(C\) (Fig. 7).

This stress can easily be determined with the aid of Airy’s stress function.

To evaluate the strength of the cutting edge, it is necessary to invoke one or another strength criterion. In our opinion, such a suitable criterion for brittle materials is the generalized Mohr theory; in this case, undoubtedly, it is necessary to know the mechanical characteristics of the material of the cutting part of the cutter.

We do not dwell on these questions here, since they are the subject of an independent investigation.

REFERENCES

1. N. N. Zorev. On the interdependence of processes in the chip-formation zone and in the contact zone of the tool rake face. Vestnik mashinostroeniya, No. 12, 1963.

2. M. B. Gordon. Distribution of friction forces on the rake face of a cutter in the zone of contact with the chip. Vestnik mashinostroeniya, No. 5, 1953.

3. M. I. Klushin. Metal Cutting. Mashgiz, 1958.

4. A. M. Rozenberg, A. N. Eremin. Elements of the Theory of Metal Cutting. Mashgiz, 1956.

5. Metal Cutting and Tools. Edited by Prof. Dr. A. M. Rozenberg. Mashgiz, 1964.

6. A. H. Uorsing, D. Geffner. Methods for Processing Experimental Data. Moscow, 1953.