STRESS STATE IN THE PLASTIC DEFORMATION ZONE DURING METAL CUTTING

L. M. SEDOKOV

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

Determination of the components of the stress state is a necessary preliminary condition for calculating the magnitude of the acting forces in any deformation process.

The assumption that, in the cutting process, the stress state corresponds to the deformed state—as is done by L. S. Murashkin [1] and by some other researchers—is unfounded, since the process of plastic deformation in metal cutting is a process of complex loading, for which correspondence of the stress state to the deformed state is not obligatory.

It is therefore expedient to determine the components of the stress state independently of the deformed state of the metal being transformed into chip.

In Fig. 1a an element is singled out, one face of which is parallel to the rake face of the cutting tool, the second face is perpendicular to it, and the third coincides with the shear plane. On the face of the selected element that is parallel to the rake face of the cutting tool, normal stresses $\sigma_N$ and tangential stresses $\tau_F$ act. On the perpendicular face of the selected element, tangential stresses $\tau_F$ and normal stresses $\sigma_F$ will act. In Fig. 1, $\beta_1$ is the shear angle, $\gamma$ is the rake angle of the cutter, and $\eta$ is the friction angle.

Fig. 1. Equilibrium of an elementary volume of metal being transformed into chip.

Tangential stresses in the shear plane can be determined from the equilibrium condition of the element under consideration

\[ \tau=\sigma_N\left\{\frac{1-\alpha}{2}\sin(\beta_1-\gamma)+\mu\cos 2(\beta_1-\gamma)\right\}. \tag{1} \]

Normal stresses

\[ \sigma=\sigma_N\{\sin^2(\beta_1-\gamma)+\mu\sin 2(\beta_1-\gamma)+\alpha\cos^2(\beta_1-\gamma)\}. \tag{2} \]

In equations (1) and (2) the following notation has been adopted:

\[ \alpha=\frac{\sigma_F}{\sigma_N}; \qquad \mu=\frac{\tau_F}{\sigma_N}. \]

Since the zone of plastic deformation of the metal transformed into chip is, under ordinary cutting conditions, taken to be very narrow and, for practical calculations, is replaced by a single plane, while the deformed state is regarded as homogeneous, the stressed state of the metal along the shear plane may be taken as constant.

On the rake face of the cutting tool there will act normal stresses \(\sigma_N\) and tangential stresses \(\tau_F\), the ratio between which will be equal to the mean value of the coefficient of friction \(\mu\). On the perpendicular face of the elementary volume, by virtue of the law of equality of tangential stresses, tangential stresses \(\tau_F\) of the same magnitude will act.

It remains to determine the ratio of the normal stresses acting on the two mutually perpendicular faces of the elementary volume, i.e., the coefficient \(\alpha\), knowing which one can calculate the magnitude of the normal and tangential stresses acting in the shear plane.

The equilibrium condition of the elementary volume requires that the resultant of all forces on its two mutually perpendicular sides be equal to the total force acting in the shear plane.

The projections of all forces acting on the two mutually perpendicular sides of the element under consideration onto the \(x\) and \(y\) axes will be, respectively,

\[ \Sigma X=\tau_F\,dz\,dx+\sigma_N \operatorname{tg}(\beta_1-\gamma)\,dz\,dx, \]

\[ \Sigma Y=\sigma_F\,dz\,dx+\tau_F \operatorname{tg}(\beta_1-\gamma)\,dz\,dx. \]

It is not difficult to see that, with an unchanged stressed state along the shear plane, the ratio of these projections to each other will be equal to the ratio of the normal and tangential forces acting on the rake face of the cutting element. Consequently,

\[ \frac{\sigma_F+\operatorname{tg}(\beta_1-\gamma)\tau_F} {\tau_F+\operatorname{tg}(\beta_1-\gamma)\sigma_N} =\frac{F}{N}=\mu. \]

The expression obtained makes it possible to establish the relationship between the normal stresses in two mutually perpendicular areas of the elementary volume

\[ \alpha=\frac{\sigma_F}{\sigma_N}=\mu^2. \tag{3} \]

Substituting into equations (1) and (2) the value of the coefficient \(\alpha\) according to condition (3), one can obtain the relation between the tangential and normal forces acting in the shear plane, which, on the other hand,

on the other hand, must be equal to $\operatorname{tg}[(\beta_1-\gamma)+\chi]$. On the basis of what has been said, we obtain

\[ \frac{\sigma}{\tau} = \frac{\operatorname{tg}^2(\beta_1-\gamma)+\mu\,2\operatorname{tg}(\beta_1-\gamma)+\mu^2} {(1-\mu^2)\operatorname{tg}(\beta_1-\gamma)+\mu-\mu\operatorname{tg}^2(\beta_1-\gamma)} = \frac{\operatorname{tg}(\beta_1-\gamma)+\mu} {1-\mu\operatorname{tg}(\beta_1-\gamma)}. \]

This expression is an identity, which testifies to the validity of relation (3).

The principal stresses for an element of metal being transformed into a chip are determined from the stresses acting on two mutually perpendicular planes

\[ \sigma_{3,1}=\frac{\sigma_N}{2} \left\{(1+\alpha)\pm\sqrt{(1-\alpha)^2+4\mu^2}\right\}. \]

Replacing the coefficient $\alpha$ by the coefficient of friction according to condition (3), we find the principal stresses in the zone of plastic deformation during metal cutting

\[ \sigma_3=\sigma_N(1+\mu^2);\quad \sigma_1=0. \tag{4} \]

Consequently, the stressed state in the zone of plastic deformation during metal cutting, in all cases, taking into account the generally known simplifications of the scheme of this process, is linear (uniaxial) compression.

It is important to note here that relation (3) satisfies well two characteristic particular cases. When the coefficient of friction $\mu=0$, the cutting process is in no way different from ordinary uniaxial compression

\[ \tau_F=0;\quad \sigma_F=0;\quad \sigma_3=\sigma_N;\quad \tau=\sigma=\frac{\sigma_N}{2}. \]

When the coefficient of friction $\mu=1$, then, as experiments show, the difference of the angles $(\beta_1-\gamma)$ is close to zero, $\sigma_F=\tau_F$; the angle between the shear plane and the direction of the resultant force will be close to $45^\circ$, which indicates the equality of the normal and tangential stresses in the shear plane, $\sigma=\tau$.

Obviously, it is possible to find a value of the coefficient $\alpha$, different from $\mu^2$, that will satisfy these two characteristic cases of chip formation; however, in order to satisfy the equilibrium condition of the isolated element, it is then necessary to assume variability of the ratio $\tau_F:\sigma_F$ along the shear plane, which is possible only with other schemes of chip formation and other schemes of the acting forces.

Thus, adopting the chip-formation scheme with a single shear plane as a scheme that approximately correctly reflects the mechanism of plastic deformation during metal cutting, we must accept $\alpha=\mu^2$, which leads to the recognition of the stressed state during metal cutting as linear, although the deformed state, as investigations [2] show, will be different—shear adjacent to compression. The fact that in two characteristic cases (3) corresponds to reality once again confirms the practical possibility of replacing the zone of plastic deformation, under ordinary cutting conditions, by a single shear plane.

In Fig. 2 a construction of the stress circle for the metal-cutting process is given. Using the concept of the pole of the stress circle [3], it is easy to obtain the direction of the principal stress, coinciding with the direction of the resultant of all forces on the rake face of the cutting tool $R$. On the basis of the construction carried out, one may conclude that the direction of the plane of maximum tangential

stresses in the general case does not coincide with the plane of shear; there is an angle \(\rho\) between them.

The foregoing makes it possible to resolve the question of the magnitude of the tangential stresses in the shear plane as that of stresses on an inclined plane. In doing so, however, it is necessary to bear in mind that plastic

Fig. 2. Determination of the direction of the principal stresses and of the magnitude of the tangential stresses acting in the shear plane.

deformation, according to the adopted scheme of chip formation, begins and ends in one and the same plane. Consequently, the tangential stresses in the shear plane will be average over the entire period of deformation, i.e., they will be average between the yield point and the stresses corresponding to the final stage of plastic deformation.

In addition, it is necessary to take into account that, for one and the same value of the maximum shear, the greatest elongation in cutting will amount to \(\frac{3}{4}\) of the greatest elongation under axial plastic compression. Therefore it may be assumed that the stress in cutting will be \(\frac{4}{3}\) times greater than the corresponding compressive stresses, if the magnitude of the maximum shear is the same.

Consequently, the tangential stresses in the shear plane during cutting may be determined from the formula

\[ \tau = \frac{4}{3}(\tau_{\max})_{\mathrm{av}}\cdot \cos 2\varphi . \tag{5} \]

Here \((\tau_{\max})_{\mathrm{av}}\) is the average magnitude of the maximum stresses in the process of compression from the beginning of plastic deformation to the final stage of deformation.

Using the law of the compression polytrope,

\[ \sigma = \sigma_0\left(\frac{h_0}{h}\right)^n, \]

where \(\sigma_0, n\) are material characteristics;

$h_0, h$ are the initial and current height of the specimen under compression. Taking into account the known relation of the maximum relative shear to the maximum elongation under compression,

\[ (g_{\max})_{\text{comp}} = \frac{3}{2}\ln\frac{h_0}{h}, \]

the magnitude of the greatest shear stresses, averaged over the process of plastic deformation as arithmetic means, may be calculated from the following formula:

\[ (\tau_{\max})_{\text{avg}} = \frac{\sigma_0}{4}\left(1 + e^{\frac{2}{3}ng_{\max}}\right). \]

Here $e$ is the base of natural logarithms.

Then equation (5) may be written as follows:

\[ \tau = \frac{\sigma_0}{3}\left(1 + e^{\frac{2}{3}ng_{\max}}\right)\cos 2\rho . \tag{6} \]

Consequently, on the basis of the analysis given above of the stress state in the zone of plastic deformation during metal cutting, knowing the physicomechanical characteristics of the metal being machined ($\sigma_0, n$), the geometry of the cutting tool ($\gamma$), and the shrinkage of the chip, it is possible to calculate the direction of the shear plane and the direction of the plane of maximum shear stresses, the magnitude of the shear stresses in the shear plane, the dimensionless characteristics of the cutting process [4], the cutting force and its components.

Verification of equation (6) gives satisfactory results. It is also important to note here that the hitherto unclear actual decrease in the magnitude of the shear stresses with increasing nominal relative shear, which has repeatedly been recorded by researchers [5], is explained by equation (6). This equation includes the angle $\rho$, which in many cases increases with increasing relative shear, whereas the relative shear, for small values of the strain-hardening exponent, has only an insignificant effect on the magnitude of the shear stresses.

Moreover, if the deformed state during cutting is shear adjacent to compression, and the share of compression is significant, then the transition coefficient will be less than the value adopted in equation (5). This also explains the considerable scatter of experimental values of the shear stresses in the shear plane, which is often noted by researchers [5].

REFERENCES

1. Murashkin L. S. Fracture of metal in free cutting. Scientific and Technical Information Bulletin. Mechanical Engineering, No. 11, 1958. Published by the Leningrad Polytechnic Institute named after M. I. Kalinin.

2. Kudarev G. L. Experimental study of plastic deformation in metal cutting. Studies in Solid-State Physics, Academy of Sciences of the USSR, 1957.

3. Darkov A. V., Mitropolsky N. M., Shapiro G. S. Strength of Materials. Vysshaya shkola, 1959.

4. Sedokov L. M. Dimensionless characteristics of the dynamics of the metal-cutting process. Proceedings of the Tomsk Polytechnic Institute, vol. 96, issue 2, 1961.

5. Zorev N. N. Calculation of the Projections of the Cutting Force. Mashgiz, 1958.