Characteristics of Plastic Deformation in Metal Cutting

L. M. Sedokov

(Presented by the scientific seminar of the Department of Strength of Materials)

In studying the cutting process as a process of plastic deformation, it is necessary to have reliable characteristics of the deformation taking place. V. D. Kuznetsov (1943), V. A. Krivoukhov (1945), L. S. Murashkin (1958), P. A. Gorezko (1959), R. B. Martirosyan (1963), and other investigators take chip compression as the characteristic of plastic deformation, which is then equated to the amount of upsetting of a cylindrical specimen under compression.

Meanwhile, it is well known [1—4] that chip compression cannot unambiguously characterize the degree of deformation that has occurred. Thus, for example, when chip compression is equal to unity, i.e., when, in the opinion of the above-mentioned investigators, plastic deformation should be absent, in reality considerable deformation will take place (if the rake angle is negative, the relative shear will be greater than 2 units).

E. Merchant (1945), V. V. Kuz’yushin (1951), N. N. Zorev (1952), S. S. Nekrasov (1953), A. M. Rosenberg and A. N. Eremin (1954), and other investigators proposed that, as a characteristic of plastic deformation in metal cutting, one should take the value of the relative shear, which is determined by the equation:

\[ \varepsilon = \operatorname{ctg}\beta_1 + \operatorname{tg}(\beta_1 - \gamma), \tag{1} \]

where \(\beta_1\) is the angle determining the direction of the shear plane;
\(\gamma\) is the rake angle of the cutter.

However, as subsequent studies have shown, the value of the relative shear determined by equation (1) also cannot be recognized as a reliable characteristic of the degree of plastic deformation during chip formation.

V. I. Apraksin [5] points out the erroneousness of equation (1); A. M. Rosenberg and G. L. Kufarev [6, 7, 8] note that determination of the degree of plastic deformation by equation (1) cannot be sufficiently reliable, since it was obtained on the basis of a conventional scheme of chip formation with a single shear plane.

Assuming that the direction of the texture in the chip, which is sufficiently clearly visible in microphotographs of chip roots, coincides with the direction of greatest elongation, A. M. Rosenberg and G. L. Kufarev [6] proposed determining the maximum shear from the texture angle,

without tying oneself to the adoption of a definite chip-formation scheme.

However, in this case, in order to obtain calculation formulas it is necessary to adopt a definite scheme for the formation of texture in the chip [9].

In cutting metals, simple shear takes place. Under simple shear, a circle drawn on a surface normal to which there is no elongation will become an ellipse, and a square—a parallelogram.

It has been proved analytically [11, 12] that in a circle drawn before deformation one can find two mutually perpendicular fibers which, after deformation, coincide with the direction of the greatest elongations while preserving mutual perpendicularity. However, the orientation of these fibers depends on the degree of deformation, which introduces considerable uncertainty.

In Fig. 1a the usual scheme of simple shear is presented. The fiber \(ON'\), which received the greatest elongation, before deformation occupied the position \(ON\), which made an angle \(\Theta\) with the direction of shear. When the degree of deformation changes, another fiber will receive the greatest elongation, and its initial position will be determined by another value of the angle \(\Theta\).

Fig. 1, a, b. Scheme of simple shear.

Using the method of P. O. Pashkov [10], it is easy to establish the initial position of the fiber which, after deformation, coincides with the direction of the texture. In Fig. 1b a scheme of simple shear is presented which differs from the scheme in Fig. 1a in that a square is taken as the initial element. The analytical relations obtained from both schemes will be the same.

According to scheme 1b, the fiber that will receive the greatest elongation occupied the position \(OD\) before deformation. The tangent of the angle of inclination of the fiber \(OD\) was determined by the quantity \(n\). In order to obtain uniform deformation not only with respect to the amount of shear but also with respect to elongation, it is necessary that all fibers before deformation have a direction coinciding with \(OD\). In other words, as the initial shape of the element before deformation it is necessary to take not a square, but a parallelogram, the angle between the sides of which would be determined by the quantity \(n\). Since \(n\) depends on the degree of deformation, the uncertainty in the simple-shear scheme remains.

In Fig. 2a the root of the chip that has formed is shown. The volume of metal between the rake face of the tool and the shear plane \(OBA\) will pass into the chip and occupy the position \(OB'A\). Consequently, for analysis of the process of simple-shear deformation in metal cutting, it is expedient to single out the volume of metal which, before deformation, was enclosed between the direction of the cutting velocity (the machined sur-

surface) \(AB\), the shear plane \(OA\), and the front face of the cutting element \(OB\).

An arbitrary point \(C\), lying on the surface \(AB\) machined by the preceding pass of the cutter, will move in the direction of the shears and will take the position \(C'\) on the outer side of the chip that has formed.

Fig. 2, a, b. Chip root (a) and diagram of simple shear in metal cutting (b).

On the basis of the constructions carried out in Fig. 2b, we obtain:

\[ \frac{\rho}{\rho_0}=\frac{\sin \alpha_0}{\sin \alpha} \tag{2} \]

\[ \tg \alpha=\frac{x_0}{x_0+s}\tg \alpha_0 . \tag{3} \]

With an increase in the angle \(\alpha_0\), \(x_0\) will decrease and the shear path \(s\) will increase. The angle \(\alpha\) will increase, but its growth will be slower than that of \(\alpha_0\). Therefore the ratio \(\rho:\rho_0\) will increase and will attain its greatest value at \(\alpha_0=90^\circ-(\beta_1-\gamma)\). Consequently, the direction \(OB'\) will characterize the direction of the greatest elongations, the direction of the texture in the chip, and the arrangement of material particles in the chip which, before deformation, lay on one straight line parallel to the front face of the cutting element [13].

On the basis of the foregoing, analytical dependences can be obtained for calculating all characteristics of the plastic deformation occurring in the cutting process.

Conditional relative elongation

\[ \bar{\delta}=\frac{\rho-\rho_0}{\rho_0}=\frac{\rho}{\rho_0}-1 =\frac{\cos(\beta_1-\gamma)}{\sin(\beta_2-\beta_1)}-1 . \tag{4} \]

True elongation

\[ \delta=\ln \frac{\cos(\beta_1-\gamma)}{\sin(\beta_2-\beta_1)} =\ln \frac{2\cos\gamma}{\sin \beta_2}. \tag{5} \]

True maximum shear:

\[ g=\delta_1-\delta_3=2\delta_1 \]

\[ g=2\ln \frac{\cos(\beta_1-\gamma)}{\sin(\beta_2-\beta_1)} =2\ln \frac{2\cos\gamma}{\sin \beta_2}. \tag{6} \]

The conventional relative shear can be calculated using equation (6)

\[ \varepsilon' = 2 \operatorname{sh} \frac{g}{2} \]

\[ \varepsilon' = 2 \operatorname{sh} \ln \frac{\cos(\beta_1-\gamma)}{\sin(\beta_2-\beta_1)} = 2 \operatorname{sh} \ln \frac{2\cos\gamma}{\sin\beta_2}. \tag{7} \]

The texture angle in the chip, \(\beta_2\), entering equations (4) and (7), is related to the chip contraction and the tool geometry by the following dependence, which follows from the constructions in Fig. 2:

\[ \operatorname{tg}\beta_2=\frac{2\cos\gamma}{\xi-2\sin\gamma}. \tag{8} \]

The angle between the direction of the texture in the chip and the shear plane can be determined from the formula:

\[ \operatorname{ctg}\Psi=\operatorname{ctg}(\beta_2-\beta_1) =\operatorname{ctg}\beta_1+2\operatorname{tg}(\beta_1-\gamma) \tag{9} \]

or, taking (1) into account,

\[ \operatorname{ctg}\Psi=\varepsilon+\operatorname{tg}(\beta_1-\gamma). \tag{10} \]

Fig. 3 gives a graph for determining the true maximum shear in metal cutting with the formation of a continuous chip, calculated from equation (6), when the chip contraction and the tool rake angle are known.

Fig. 3. Dependence of true shear on chip contraction and the tool rake angle.

The fact that chip contraction cannot be numerically equal to the greatest elongation is clearly seen from the graph presented in Fig. 4.

The conventional relative shear, determined from equation (1), is equal (if \(\beta_1=\gamma\)) to, or somewhat greater than, the value determined from equation (7), as illustrated in Fig. 5. It is interesting to note that equation (1) sometimes leads to a paradox. At large contractions, the value of the relative shear \((\varepsilon)\) for one and the same contraction will be...

will increase, and, consequently, the cutting force will also increase if a sharper cutter is used, i.e., a cutter with a larger rake angle \(\gamma\), as is evident from the graph given for \(\xi = 4\) and \(\gamma > 25^\circ\). This paradox is also seen in the corresponding graphs given by V. V. Kuzyushin [1], A. M. Rozenberg and A. N. Eremin [4], but no attention has been paid to it up to now, although it definitely indicates a fundamental shortcoming of equation (1).

Fig. 4. Dependence of the greatest elongation on the rake angle of the tool at equal chip shrinkages.

Fig. 5. Dependence of the conditional shears, determined from equations (1) and (7), on the rake angle of the tool at constant chip shrinkages.

Thus, if the direction of the texture in the chip (angle \(\beta_2\)) is known, then the greatest elongation is determined by formula (5), the greatest maximum shear by formula (6), and the conditional relative shear by formula (7). In the case where the texture angle in the chip is not recorded in the experiments, it can be calculated from the chip shrinkage (8).

References

1. V. V. Kuzoshin. Plastic deformation in cutting steel. Stanki i instrument, No. 4, 1951.

2. N. N. Zorev. Investigation of elements of the mechanics of the cutting process. Mashgiz, 1952.

3. A. V. Molochkov. Some questions of the deformation of steel chips. Collection of Scientific Works of the Belarusian Polytechnic Institute. Issue V. Publishing House of the Academy of Sciences of the BSSR, Minsk, 1953.

4. A. M. Rozenberg, A. N. Eremin. Plastic deformation of chips in metal cutting. Izv. TPI, vol. 75, 1954.

5. V. I. Apraksin. Kinematics of the cutting zone. Scientific-technical information bulletin No. 2. Leningrad Polytechnic Institute. 1962.

6. G. L. Kufarev. Experimental study of plastic deformation in metal cutting. Studies in the Physics of Solids. Academy of Sciences of the USSR, 1957.

7. A. M. Rozenberg, G. L. Kufarev. Determination of the degree of plastic deformation of metal during cutting. Vestnik mashinostroeniya, No. 6, 1958.

8. Cutting of Metals and Tools. Mashinostroenie, 1964.

9. L. M. Sedokov. Force, deformation, and stress in metal cutting. Izv. TPI, vol. 133, 1965.

10. P. O. Pashkov. Plasticity and fracture of metals. Sudpromgiz, 1950.

11. A. Nadai. Plasticity and Fracture of Solids. IL, 1954.

12. Ya. B. Fridman, T. K. Zilova, N. I. Demina. Study of plastic deformation and fracture by the method of rolled grids. Oborongiz, 1962.

13. L. M. Sedokov. Kinematic analysis of the process of formation of continuous chips. Izv. TPI, vol. 85, 1957.