STUDY OF THE CUTTING PROCESS BY THE POLARIZATION-OPTICAL METHOD

M. F. Poletika, M. Kh. Uteshev

(Presented by Professor Doctor A. M. Rozenberg)

One of the most difficult problems in the theory of metal cutting is the problem of the distribution of stresses in the cutting zone. This is explained, on the one hand, by the complexity of the deformed state in the cutting process, and, on the other hand, by the fact that in the shear zone and in the contact layer of the chip the deformation reaches an extremely high degree.

When attempts are made to solve this problem analytically by the methods of the mathematical theory of plasticity, great difficulties arise, since for deep plastic deformations this theory has not yet been sufficiently developed. It was therefore natural to try to solve this problem experimentally; for this purpose the most suitable and, perhaps, the only more or less accurate method should be considered the polarization-optical method.

For this purpose, in the metal-cutting laboratory of TPI a polarization-optical setup was developed and manufactured, shown in Fig. 1. The setup is mounted on a model 1K62 lathe. It consists of an optical system for observing and photographing the cutting process in polarized light and a dynamometer for measuring the cutting force.

A mercury-quartz lamp DRSh-250 was used as the light source. To obtain monochromatic light, either a green light filter ZhZS-9 or an aqueous solution of nickel sulfate was used. Circular polarization of the light was provided by means of mica plates.

Polaroids 45 mm in diameter were fastened in special bushings, which could be rotated through any angle read from a limb; this made it possible to determine the directions of the principal stresses with crossed polarizers. Photographs were taken with a 9×12 camera.

To measure the cutting forces, a highly sensitive elastic-electric two-component dynamometer equipped with strain gauges was made. The use of foil gauges in this instrument instead of wire gauges made it possible to dispense with an amplifier, connecting the oscillograph vibrators directly into the measuring diagonals of the bridges. As the oscillograph, an OT-24-51 instrument with a GEMZ-53 type II vibrator was used. The role of the object under study was performed by

Fig. 1. Optical system of the setup.

either a transparent cutter made of optically active material (when investigating stresses in the cutter), or a transparent disk (when investigating stresses in the workpiece).

The experiments were carried out under conditions of free orthogonal cutting with transverse feed, both at a relatively high speed and at microspeeds, for which purpose the machine was driven by a special reduction gear.

In investigating the stressed state in the cutter, the latter was made of a transparent optically sensitive material. The geometric parameters of the cutter were: rake angle—\(10^\circ\), clearance angle—\(15^\circ\).

The material being machined was a lead disk 187 mm in diameter and 3 mm wide. The disk was turned with transverse feed \(S = 0.475\) mm/rev and cutting speed \(v = 1.6\) mm/min. Each experiment for the optical determination of stresses was repeated twice: the first time with a cutter made of optically low-sensitivity material (Plexiglas) and with plane polarization of light for recording the isoclines; the second time with a cutter made of optically active plastic (based on ED-6 epoxy resin) with circular polarization of light for recording the fringe pattern.

It had been established in advance that changing the cutter material (while, of course, preserving its geometric parameters) does not have any appreciable effect on the cutting process.

In our case of free cutting, with a sufficiently large value of the ratio \(b/a\) of the width of cut to its thickness, it may be assumed that the cutting zone is in a plane state of stress. The stresses \(\sigma_N\) and \(\tau_F\) at a certain point on the contact surface of the chip with the cutter, which, in accordance with generally accepted terminology, are more conveniently denoted by \(\sigma_y\) and \(\tau_{xy}\), will then be expressed by the formulas

\[ \tau_{xy} = \tau_F = \frac{\sigma_1 - \sigma_2}{2}\sin 2\alpha, \]

\[ \sigma_y = \sigma_N = \sigma_1 \cos^2 \alpha + \sigma_2 \sin^2 \alpha, \]

where \(\sigma_1\) and \(\sigma_2\) are the principal stresses at the point under consideration \((\sigma_1 > \sigma_2)\); \(\alpha\) is the angle of inclination of the calculated plane to the principal plane.

By the fringe method the difference of the principal stresses was determined experimentally, and from the isocline pattern—the directions of the principal stresses. The procedure for calculating the stresses is described in detail in [3]. The results obtained are presented as graphs in Figs. 2 and 3. The first of these gives the values of the difference of the principal stresses \((\sigma_1 - \sigma_2)\) and the angle of inclination of the principal stress at various points along the contact length, while the second gives the distribution of the normal and tangential stresses on the rake face. The curve of the change in the coefficient of friction is also given there.

The character of the stress-distribution curves obtained by us is the same as that obtained in earlier investigations (including by other methods), and therefore raises no doubts. In this connection it is necessary to emphasize certain very important consequences following from the analysis of Fig. 3:

1) the normal stress decreases continuously with increasing distance from the cutting edge of the cutter, and the law of its variation differs substantially from a triangular one;

2) the tangential stress over a certain portion adjacent to the cutting edge remains approximately constant, which confirms the existence of a zone of plastic contact between the chip and the cutter;

Fig. 2.

Fig. 3. Distribution of normal and tangential stresses on the front face of the cutter.

3) the coefficient of friction varies along the length of the contact, including on its elastic portion. This latter circumstance once again indicates that the coefficient of friction in metal cutting is not an independent physical characteristic.

For a quantitative assessment of the accuracy of the results obtained, after processing the dynamometer readings we determined the forces on the front and rear faces of the cutter and calculated the average stresses on the contact surface of the chip with the cutter.

In Table 1 these quantities are compared with those calculated from the data of the optical method (by graphical integration of the diagrams of normal and tangential stresses) for a cutting speed \(v = 1.6\ \text{mm/min}\).

Table 1

It is not difficult to see that the agreement of the results is very good, which indicates the high accuracy of the photoelastic method as applied to the study of the stressed state of the cutter.

Equally convincing evidence of this is provided by comparing the angles of action \(\omega\) found by different methods. According to the dynamometer readings,

\[ \omega_d = \operatorname{arctg}\frac{P_y}{P_z} = 20^\circ 30'. \]

The same angle, calculated by the graphoanalytic method on the basis of analysis of the optical picture of the stressed state in the cutter head, proved to be equal to

\[ \omega_0 = 20^\circ 50'. \]

To investigate the stresses in the material being machined in front of the cutter, transparent disks of celluloid were prepared. Celluloid was chosen because, in its mechanical properties, it imitates plastic metals, produces a continuous chip during cutting, and, which is very important, retains sufficient transparency and elastic-optical properties under small plastic deformations. The mechanical characteristics of the celluloid we used were: \(\sigma_B = 483\ \text{kg}/\text{cm}^2\), \(\delta = 30\%\), \(E = 2 \cdot 10^4\ \text{kg}/\text{cm}^2\), Poisson’s ratio \(\mu = 0.3\).

Figure 4 shows the pattern of isochromatic fringes and isoclinics obtained at a cutting speed of \(0.8\ \text{m/min}\) and a feed of \(0.475\ \text{mm/rev}\). From this pattern it is evident that the elastic region beneath the cutter is divided by a neutral (zero) line into two zones: a zone of compressive stresses in front of the cutter and a zone of tensile stresses behind the cutter. All the isochromatic fringes emerge from the cutter tip, which is thus a point of stress concentration.

On the line \(AB\), elastic deformations become plastic; that is, the stresses along this line, which may be called the boundary line, reach the yield point of the material. Since this line corresponds to a fringe of one order, all points lying on it have a constant difference of principal stresses

\[ \sigma_1 - \sigma_2 = \text{const on } AB, \tag{1} \]

![Figure 4]

Fig. 4.

Labels in the figure: “Region of tensile stresses”; “Zero line (neutral)”; “Region of compressive stresses.”

![Figure 5]

Fig. 5.

which is an expression of the Tresca–Saint-Venant plasticity condition

\[ \sigma_1-\sigma_2=2K=2\tau_{\max}. \tag{2} \]

Turning to Fig. 5, which shows the trajectories of the principal stresses, we note that these lines may practically be regarded as straight. This means that, in our case, the second principal stress is negligibly small and may be taken equal to zero. Then from (2) we shall have

\[ \tau_{\max}=\frac{\sigma_1}{2} \]

and

\[ \tau_{\max}=\tau_T=K=\frac{\sigma_T}{2}\quad \text{on } AB, \]

where \(\sigma_T\) is the yield point of the original (unhardened) material.

The values of the stress \(\sigma_1\), calculated for points located on the line \(AB\), agree quite well with the value of the yield point of celluloid in a tensile test, which confirms the considerations set forth above. As for the shape and position of the line \(AB\) itself, it is important to note that, in all the cases we investigated, it is a curve with its convexity directed downward. As the cutting speed increases, the volume of the plastic-deformation zone decreases and the line \(AB\) shifts to the right. This is illustrated in Fig. 6, which shows the positions of the boundary line at different cutting speeds.

Fig. 6.

In addition to the trajectories of the principal stresses (Fig. 5), we calculated and plotted the trajectories of the maximum shear stresses (Fig. 7). On both of the diagrams presented, the experimentally obtained epures of the stress distribution along their trajectories are plotted. It is not difficult to see that both the shear and the normal

Fig. 7.

Fig. 8.

the stresses, as the distance from the boundary line \(AB\) increases, decrease continuously: at first very sharply, and then considerably more slowly.

It also follows from Fig. 7 that the line \(AB\), which separates the elastically stressed and plastic regions, does not coincide with the trajectory of maximum shear stresses and therefore is not a slip line (characteristic), as some investigators are inclined to believe. This means that the maximum shear stress at any point lying on the line \(AB\) itself does not coincide in direction with the tangent to \(AB\) at that point.

The angle between the trajectory of the maximum shear stress and the boundary line continuously decreases in the direction from the cutter tip to point \(B\), which lies on the free surface. Since, as was established above, the magnitude of the maximum shear stress is constant at all points of the line \(AB\), the shear stresses acting along \(AB\), and the normal stresses acting perpendicular to it, must also vary continuously. This is illustrated in Fig. 8, which gives the diagrams of shear and normal stresses on the line \(AB\) and indicates their direction at each point. We find that in the direction from point \(A\) to point \(B\) the normal stresses decrease continuously, while the shear stresses increase.

The results presented above of the study of the cutting process by the polarization-optical method make it possible to draw the following conclusions:

1. The elastic-optical method of stress investigation, when applied to the process of metal cutting, gives accurate quantitative results.

2. The elastic-optical method shows that there exist two stress zones: a zone of compressive stresses ahead of the cutter and a zone of tensile stresses behind the cutter.

3. The boundary line where elastic deformations pass into plastic deformations is curvilinear, and the stresses on it reach the yield point of the unworked material. This line is not a trajectory of maximum shear stresses and therefore cannot be called a slip line or a characteristic.

References

1. Andreev G. S. Investigation of stresses in the working part of a cutter on a polarization-optical apparatus with the use of cinematography. Vestnik mashinostroeniya, No. 5, 1958.

2. Kamskov L. F. On external friction in the cutting of plastic metals. Vestnik mashinostroeniya, No. 5, 1959.

3. New studies in the field of metal cutting. Ed. by Prof. Dr. A. M. Rozenberg. Mashgiz. In press.

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

5. Frocht M. M. Photoelasticity, Vols. I–II, 1948–1950.