ON THE CHARACTERIZATION OF THE DEGREE OF METAL DEFORMATION DURING CUTTING

G. L. KUFAREV, G. D. DEL, M. G. GOLDSHMIDT

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

The degree of final deformation of the metal in the cutting process is assessed by most researchers in terms of the magnitude of the relative shear. The known relations for determining the degree of deformation, as well as the expressions characterizing the direction of the chip texture, ultimately differ from one another to the extent that the views of various authors on the mechanism of the chip-formation process differ. The investigation of deformations in the cutting zone was carried out by the method of division grids. Grids with a base of 0.2 mm were applied to a previously lead-coated specimen by the technology used to obtain typographic screened clichés [1]. Such grids constitute a square system of etched points close to circles. Steel 15, brass L62, and zinc were used as the work materials. The cutting speed varied from 23 mm/min to 12 mm/min, the rake angle \(\gamma\)—from \(+20^\circ\) to \(+40^\circ\), and the cut thickness varied within the range \(0.8 \div 2\) mm.

Specimens were made as assemblies of two plates, so that the division grid was located approximately along the width of the specimen section. In this case the grid was oriented in such a way that one of the families of straight lines formed by the centers of the circles was parallel to the direction of cutting speed. The recorded chip roots were photographed and then processed on a projector at a total magnification of 400 times.

Experiments showed that, in all the cases studied, the direction of the principal tensile deformation, determined by the position of the major axis of the ellipse, coincides with the lines that before deformation were perpendicular to the direction of tool displacement.

Fig. 1. Comparison of the pattern of distortion of lines perpendicular before deformation to the direction of cutting speed with the direction of the principal elongations. Cutting of zinc \((V = 23.5\ \text{mm/min},\ \gamma = +30^\circ,\ a = 1\ \text{mm})\).

This proposition had to be carefully verified, since even a small error in determining the angle at relatively large deformations may entail substantial inaccuracies when attempting to determine the position, in the undeformed body, of the fibers forming this angle. Such verification was carried out by comparing the deformation pattern obtained by means of oriented lines, produced in various ways before deformation, with the texture of the metal. The lines were scratched on the surface of the specimen, in one case so that they were perpendicular to the direction of the cutting speed, and in the other parallel to the rake face of the tool.

The results of these experiments are presented in Figs. 1 and 2, where the dashed lines show the field of directions of the principal elongation strain, sketched from a photograph of a microsection of the chip root.

Fig. 2. Comparison of the distortion pattern of lines parallel, before deformation, to the rake face with the direction of the principal elongations. Cutting of zinc \((v = 23.5\ \text{mm/min},\ \gamma = +30^\circ,\ a = 1\ \text{mm})\).

It is clear from Figs. 1 and 2 that only those lines which before deformation were perpendicular to the direction of the cutting speed coincide with the direction of the principal elongations.

According to the relations obtained in [1] for determining the principal strains, the magnitude of the principal elongation strain \(e_1\)* during cutting can be determined as the logarithm of the ratio of the chip thickness, measured in the direction of greatest elongation of the grains, to the thickness of the cut. Or, in the absence of a transition zone,

\[ e_1 = \ln \frac{\zeta}{\sin \psi_э}, \tag{1} \]

where \(\zeta\) is the chip shrinkage, \(\psi_э = 90^\circ - \beta_1 - \psi + \gamma\) (Fig. 4) is the angle formed by the chip texture with the rake face of the cutter; this angle can be measured directly on the chip section as the angle between the direction of the texture and the tangent to the tool-side surface of the chip [2].

Earlier, G. L. Kufarev obtained the relation [3]

\[ e_1 = \ln \sqrt{ \frac{\zeta^2}{2\sin^2 \psi_э} + \sqrt{ \frac{\zeta^4}{4\sin^4 \psi_э} - \operatorname{ctg}^2 \psi_э } }. \tag{2} \]

Formula (2) was derived on the basis of reliable experimental facts: a circle drawn on the side surface of a specimen turns, after deformation, into an ellipse whose major axis coincides with the chip texture; lines which before deformation were parallel to the motion of the cutter are, after deformation, arranged parallel to the rake face. Therefore relation (2) may be considered sufficiently rigorous.

* Here and below, as the characteristic of deformation, the logarithm of the ratio of the major semiaxis of the ellipse to the radius of the original circle is adopted.

Calculations show that \(e_1\), determined by formulas (1) and (2), differs very little in all practically possible cases. For example, at \(\zeta = 2\) and \(\psi_{\mathrm{e}} = 45^\circ\), the difference in the result of calculating \(e_1\) by (1) and (2) does not exceed \(0.5\%\). At larger \(\zeta\) and \(\psi_{\mathrm{e}}\) (regardless of their combination), this difference is still smaller. Consequently, formulas (1) and (2) may be considered to give practically identical results.

Thus, relation (1) makes it possible, by a sufficiently simple experimental method, to determine in cutting the magnitude of the principal elongation, knowledge of which permits cutting to be compared with other types of plastic deformation without resorting to equivalent strains.

Since chip-formation deformation is plane deformation \((e_1 = -e_2,\ e_3 = 0)\) [4], formula (1) can also be used to calculate the magnitude of the true maximum shear:

\[ q = e_1 - e_2 = 2e_1 = 2 \ln \frac{\zeta}{\sin \psi_{\mathrm{e}}}. \tag{3} \]

Most researchers, when deriving formulas for determining the degree of deformation in the cutting zone, base their derivations on the scheme of simple shear.

The relations characterizing the degree of deformation under simple shear are obtained on the basis of the scheme (Fig. 3), according to which the direction of the greatest strains changes at each given moment, while the direction of shear remains constant.

Fig. 3

For simple shear, a relation can be obtained that connects the magnitude of the principal strain \(e_1\) with its direction, which is determined by the position of the most elongated element relative to an axis perpendicular to the direction of shear, i.e., by the angle \(\gamma_n\) (Fig. 3).

According to P. O. Pashkov [5], an infinitely small element in the form of a square with side \(dx_0\) is transformed, as a result of flow, into a parallelogram. The curvature of the parallelogram into a curvilinear quadrilateral may be neglected because of the smallness of the dimensions. Point \(A\) on the side of the square, after deformation, will occupy the position of point \(A_1\). The direction of the greatest deformation is characterized by the parameter \(n\), which depends on the magnitude of deformation [5]:

\[ n = -\frac{1}{2}\tan \alpha + \sqrt{\frac{\tan^2 \alpha}{4} + 1}. \tag{4} \]

The true elongation is expressed by the relation [5]

\[ e_1=\frac{1}{2}\ln\frac{1+(n+\operatorname{tg}\alpha)^2}{1+n^2}. \tag{5} \]

From Fig. 3 it follows that

\[ \operatorname{tg}\gamma_n=\frac{d x_0\operatorname{tg}\alpha+n d x_0}{d x_0} =\operatorname{tg}\alpha+n, \tag{6} \]

whence

\[ \operatorname{tg}\alpha=\operatorname{tg}\gamma_n-n. \tag{7} \]

Substituting (7) into (4), we obtain

\[ n=\frac{1}{\operatorname{tg}\gamma_n}, \tag{8} \]

and from equation (5), taking account of (7) and (8), we obtain

\[ e_1=\frac{1}{2}\ln \operatorname{tg}^2\gamma_n. \tag{9} \]

On the basis of experimental data, by means of relation (9), it is possible to analyze to what extent the cutting process corresponds to the deformation scheme of simple shear in the form in which it is defined by A. Nadai, P. O. Pashkov, and V. G. Osipov, and for which the deformation characteristics adopted in cutting are valid.

If the deformation of chip formation does not occur in a single plane, the direction of the shears for different particles located in the plastic region is not the same. This circumstance alone makes it possible to regard the cutting process as not being simple shear, since the latter is characterized by constancy of the shear direction throughout the entire deformation process, and, consequently, one can speak only of the degree to which the deformation of chip formation corresponds to the scheme of simple shear.

Figure 4 shows a comparison of the experimental data obtained in cutting with the relation between \(e_1\) and \(\gamma_n\) for simple shear.

The experimental points corresponding to cutting at microspeed of steel 15 and brass L62 were obtained as a result of processing the distortion pattern of the dividing grid. The direction of the shears was taken as coinciding with the conventional shear plane; the shear angle (\(\gamma_n\) in Fig. 4) was taken equal to

\[ \gamma_n=90-\psi. \tag{10} \]

The points denoted by identical symbols correspond to a single row occupying, before deformation, a position parallel to the direction of motion of the cutter and thus, as it were, describe the entire deformation history of a particle passing through the chip-formation zone.

In cutting at microspeed, as experiments have shown, the angle \(\gamma_n\) changes from a value approximately equal to the angle \(\beta_1\) to a final value. Consequently, when \(\beta_1<45^\circ\), the initial stages of deformation do not proceed according to the scheme of simple shear (Fig. 4). Subsequently, beginning with \(e_1\approx0.2\), the deformation proceeds according to a scheme very close to simple shear (the points are located equidistantly from the curve). The situation does not change if, for the direction of the shears, as noted above, assumed constant for purposes of comparison with the simple-shear scheme, another conventional shear plane is adopted. Thus, if the angle \(\beta_1\) is decreased by \(7\)—\(8^\circ\), the bulk of the points will lie on curve (9). However, even in this case, in the initial stages the deformation will not correspond to simple shear.

The results presented confirm the assumption stated above that, with a finite chip-formation zone, the cutting process is not simple shear.

Nevertheless, the schemes of the chip-formation process adopted as a working hypothesis as schemes of simple shear are quite realistic. This is shown by an analysis of experimental data obtained as a result of processing the chip roots, given in the works of T. N. Loladze and

Fig. 4

B. M. Orlova, recorded when cutting steels at speeds exceeding 70 m/min. The magnitude of the principal elongation strain in the chip was determined by formula (1), or, in the case of a substantial transition surface, as the logarithm of the ratio of the chip thickness measured in the plane coinciding with the direction of the texture to the thickness of the cut; the shear angle was determined by formula (10).

As can be seen from Fig. 4, the experimental points are located sufficiently close to dependence (9), which confirms the satisfactory approximation of the cutting process as a process of simple shear. The deviation of the experimental points toward an overestimation of the angle \(\gamma_n\) (or, under the given conditions, an underestimation of \(\beta_1\)) is probably explained by the fact that, by assigning the entire chip-formation deformation to a single shear plane, some preliminary deformation of the metal being transformed into a chip remains unaccounted for.

REFERENCES

1. G. D. Del, M. G. Goldshmidt. Study of plastic deformations on models made from typographic halftone clichés. Factory Laboratory, No. 4, 1966.

1. A. M. Rozenberg, A. N. Eremin. Elements of the Theory of the Process of Cutting Metals. Moscow, Mashgiz, 1956.

2. Cutting of Metals and Tools, edited by Prof. A. M. Rozenberg. Moscow, Mashinostroenie, 1964.

3. G. L. Kufarev. Studies in Solid-State Physics. Publishing House of the Academy of Sciences of the USSR, 1957.

4. P. O. Pashkov. Plasticity and Fracture of Metals. Sudpromgiz, 1950.

5. T. N. Loladze. Chip Formation in Metal Cutting. Moscow, Mashgiz, 1952.

6. B. M. Orlov. The Influence of Cutting Fluids on the Process of Cutting Steel. Dissertation. Tomsk, 1958.