FORCE, DEFORMATION, AND STRESS IN METAL CUTTING

L. M. Sedokov

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

On the basis of the generally accepted scheme of chip formation and the commonly known simplifications, the cutting force and its components (without taking into account the forces on the rear face) may be calculated from the equation:

\[ P_i = \tau f_i(\beta_1; \gamma; r; \varphi)\,ab = \tau k_i ab . \tag{1} \]

Here \(\tau\) is the shear stress in the shear plane, as a characteristic of the resistance of the material being machined to plastic shear;

\(k_i\) is a dimensionless coefficient of the dynamics of the chip-formation process, representing a function of the geometric parameters of the cutting process, whose numerical value is equal to the corresponding force under the condition that the cross section of the cut is equal to unity and the shear stresses in the shear plane are also equal to unity;

\(\beta_1\) is the shear angle, which is determined from the chip compression and the rake angle of the cutting tool;

\(\gamma\) is the mean angle of friction on the rake face of the cutting tool;

\(r\) is the rake angle of the cutter;

\(\varphi\) is the angle in the plan of the cutter;

\(a\) and \(b\) are the thickness and width of the cut.

A significant difficulty in calculating the components of the cutting force by equations of type (1) is that, in addition to the geometry of the cutting tool \(r\) and \(\varphi\) and the shear angle \((\beta_1)\), it is also necessary to know the coefficient of friction on the rake face of the cutting tool, which varies within very wide limits: from 0.1 to 1.2 and higher.

On the basis of a kinematic analysis of the process of formation of a continuous chip [1], it is possible to establish an analytical relationship between the coefficient of friction—\(\mu\)—and the shear angle—\(\beta_1\), taking into account the rake angle \(r\) of the cutting tool:

\[ \mu = 1 - \operatorname{tg}(\beta_1 - r). \tag{2} \]

Equation (2) makes it possible to eliminate the friction angle from equations of type (1) and to represent all dimensionless chip-formation parameters—conditional specific forces—as functions only of the degree of deformation—\(\xi_1\)—and of the geometry of the cutting tool \(r\) and \(\varphi\) [1, 2]:

\[ P_i = \tau F(\beta_1; \gamma; \varphi)\,ab . \tag{3} \]

The angle between the shear plane and the direction of the equilibrating force, as experiments show [3, 4] and as follows from equation (2), is in the general case not equal to \(45^\circ\), but differs from it by some angle \(\rho\)

\[ \beta_1+\eta-\gamma=45^\circ+\rho . \tag{4} \]

The angle \(\rho\), additional to \(45^\circ\), can also be calculated analytically on the basis of equation (2). Figure 1 gives a graph of the dependence

\[ \tan \rho=\frac{\mu-\mu^2}{2-\mu+\mu^2} \]

Fig. 1. Relation between the angle \(\rho\) and the coefficient of friction on the rake face of the cutting tool.

of the angle \(\rho\) on the coefficient of friction on the rake face of the tool. Figure 2 shows graphs of the variation of dimensionless parameters of the mechanics of chip formation as a function of chip compression for \(\gamma=+10^\circ\).

For the analytical calculation of the magnitude of the shear stresses acting in the shear plane, as a characteristic of the resistance of the metal being machined to plastic deformation, it is necessary to analyze the deformed and stressed state of the metal being transformed into a chip.

Fig. 2. Dependence of the conditional specific forces on the rake face on chip compression for \(\gamma=+10^\circ\).

Deformed State and Characteristics of Plastic Deformation

Experimental studies in which the coordinate grid was measured before and after chip formation have shown that, in the cutting of metals, shear adjacent to compression takes place. The share of compression in this process is usually small. Table 1 gives the results of experiments by G. L. Kufarev [5].

Table 1

Characteristics of the deformed state

This means that the deformation in cutting metals is very close to plane deformation; that the minimax of the deformed state (the mean principal strain) is close to zero and amounts to less than 10% of the maximum elongation. When working at ordinary and high speeds this figure is probably still smaller. All this makes it possible, with sufficient reliability, to regard the cutting process as a process of plane deformation.

Fig. 3. Scheme of chip formation (a) and double displacement triangle (b).

In Fig. 3, a, a construction has been made on the indicated basis. When the cutter, over a differentially small period of time, travels the distance $dl_0$, the volume of the metal being removed, bounded by the parallelepiped with base $1aб5$, is transformed into a chip and, after the cutter has traveled some additional distance, will occupy the position $1' a' б' 5'$. All points that before deformation lay on one straight line parallel to the rake face (points $1—5$ or $a—б$), after deformation will also lie on one straight line ($1'5'$ or $a'б'$), which will form the angle $\beta_2$ with the direction of the cutting speed.

Such a construction makes it possible to find the numerical value of the necessary characteristics of the deformation of the metal taking place. This is very conveniently done on the basis of the double displacement triangle [1], which is shown in Fig. 3, b and is indicated by dashed lines in Fig. 3, a.

Over a differentially small period of deformation into the chip, a narrow layer of metal lying along the rake face of the cutting tool will be transformed—

of the cutting tool, and it will be located in the chip as a narrow strip that forms, with the direction of the cutting velocity, the angle \(\beta_2\). The narrow strip of metal being transformed into the chip may, in the limiting case, be replaced by a single fiber whose length before deformation was equal to \(r\):

\[ r=\frac{a}{\cos\gamma}. \tag{5} \]

After deformation this fiber will have the length \(r'\), which, according to the calculation scheme in Fig. 3,b, is determined by the following relation:

\[ r'=\frac{2a}{\sin\beta_2}. \tag{6} \]

Consequently, the absolute deformation of this fiber is

\[ \Delta r=r'-r=a\left(\frac{2}{\sin\beta_2}-\frac{1}{\cos\gamma}\right), \]

the nominal relative deformation is

\[ \frac{\Delta r}{r}=\frac{r'}{r}-1=\frac{2\cos\gamma}{\sin\beta_2}-1 \tag{7} \]

and the logarithmic (true) elongation strain is

\[ \delta_1=\ln\frac{r'}{r}=\ln\frac{2\cos\gamma}{\sin\beta_2}. \tag{8} \]

Taking into account that the deformation is plane, the true shear can be calculated from the formula

\[ g_{\max}=2\delta_1=2\ln\frac{2\cos\gamma}{\sin\beta_2}. \tag{9} \]

On the basis of the constructions shown in Fig. 3,a, the maximum true shear can be expressed in another way:

\[ g_{\max}=2\ln\frac{\cos(\beta_1-\gamma)}{\sin(\beta_2-\beta_1)}, \tag{10} \]

\[ g_{\max}=2\ln\frac{\cos(\beta_1-\gamma)}{\sin\beta_1\cos(\beta_2-\gamma)}. \tag{11} \]

The last three formulas (9), (10), and (11), obtained on the basis of a kinematic analysis of the process of formation of a continuous chip and of the established interrelation of the geometric parameters of chip formation, are identical.

The geometric locus in the chip of points which, before deformation, lay on one straight line parallel to the rake face of the cutting tool will also be a straight line forming the angle \(\beta_3\) with the cutting-velocity vector. It may be determined on the basis of a kinematic analysis of the process of formation of a continuous chip [6]:

\[ \tg\beta_2=\frac{2\tg\beta_1\cos\gamma}{\cos\gamma-\tg\beta_1\sin\gamma}. \tag{12} \]

Figure 4 gives a graph of relation (12), from which the direction of the texture in the chip, coinciding with the direc—

by the pressure of the greatest elongation, if the shear angle and the rake angle of the cutting tool are known.

The magnitude of the maximum true shear differs from the magnitude of the relative shear, which is usually calculated in the theory of metal cutting as

\[ \varepsilon = \operatorname{ctg}\beta_1 + \operatorname{tg}(\beta_1 - \gamma). \tag{13} \]

The quantity calculated by formula (13) is a conventional relative shear. It is related to the true shear, calculated from logarithmic elongations, as is known, by the following relations:

\[ g_{\max} = 2 \operatorname{ar\,sh}\frac{\varepsilon}{2}, \tag{14} \]

\[ g_{\max} = \ln\left(1+\frac{\varepsilon^2}{2}+\varepsilon\sqrt{\frac{\varepsilon^2}{4}+1}\right). \tag{15} \]

The verification carried out showed that calculation by formulas (9), (10), and (11), on the one hand, and calculation by formulas (13), (14), and (15), on the other, give coinciding results, which once again confirms the correctness of the derivation and of the final formulas (9), (10), and (11) for calculating the true relative shear in metal cutting.

Fig. 4. Relation between the angles \(\beta_2\) and \(\beta_1\).
(In the graph: vertical axis \(\operatorname{tg}\beta_2\), horizontal axis \(\operatorname{tg}\beta_1\); curves for \(\gamma=+20^\circ\), \(\gamma=+10^\circ\), \(\gamma=0^\circ\), and \(\gamma=-10^\circ\).)

As calculations show, at large deformations the amount of chip shrinkage approaches the conventional relative shear, while the difference between the conventional and true (logarithmic) shears becomes very substantial.

Analysis of formulas for determining the texture angle in the chip

Several formulas have been proposed for calculating the texture angle in the chip. The authors of the indicated formulas make various assumptions and suppositions. Therefore, their analysis and comparison will help refine the mechanism of chip formation.

Below is given a table of formulas, and in Fig. 5 the corresponding graphical constructions.

Here the angle \(\psi\) is the angle between the texture in the chip and the shear plane,

\[ \psi = \beta_2 - \beta_1. \]

Formulas (16) and (17) were obtained from the deformation scheme of a cube standing with one face on the shear plane (Fig. 5, a). All shears occur at the angle \(\beta_1\). If the face of the parallelogram formed is taken as the direction of the texture, formula (16) is obtained; if its diagonal is taken, formula (17) is obtained.

Table 2

Formulas for determining the texture angle in the chip

It has been proved [14] that, in the general case, neither the side nor the diagonal of the parallelogram coincides with the direction of the most elongated fiber under shear. The orientation of the initial cube according to this scheme does not depend on the rake angle of the tool, which should also be regarded as a shortcoming of formulas (16) and (17).

Fig. 5. Determination of the texture angle in the chip according to Merchant (16), according to Rozenberg and Eremin (17), according to Glebov, Kuzyushin, Zorev, Nekrasov (18), according to Apraksin (20), and according to the author’s proposal (19).

A more substantiated scheme of texture formation is that of S. F. Glebov [9], using which V. V. Kuzyushin [10], N. N. Zorev [11], and S. S. Nekrasov [12] obtained formula (18). A circle, which is the projection of a sphere as the most probable shape of a grain in the equilibrium structure of the metal being machined, is transformed into an ellipse. Formula (18) gives the direction of the most elongated fiber of the initial sphere.

However, the scheme of shear of a sphere is analogous to the scheme of shear of a cube discussed above. If one finds the most elongated fiber of the original cube, this leads to formula (18). Consequently, formula (18) can be obtained from the scheme of A. M. Rozenberg and A. N. Eremin [8], whose principal shortcoming is its indifference to the rake angle of the tool.

V. I. Apraksin [13] proposed an entirely new scheme for the formation of the texture in the chip (Fig. 5,б), whose positive feature is the use of narrow elementary strips of thickness $dl$ (path differential) and height $a$ (depth of cut), which corresponds to the methods of the engineering theory of plasticity [15]. V. I. Apraksin’s formula can be transformed into formula (20).

Formula (19) was obtained on the basis of a kinematic analysis of the process of formation of a continuous chip [6]. The direction of the texture in the chip is taken to coincide with the geometric locus of material points in the chip which, before deformation, lay on one straight line parallel to the rake face of the cutting tool. The difference between our scheme and that of V. I. Apraksin [13] is that we isolate a narrow strip that is not perpendicular to the cutting speed, but parallel to the rake face, as the volume of metal transformed into chip over a differentially small path of the cutter.

Stress State

The elementary volume of metal transformed into chip is acted upon by the stresses $\sigma_x$, $\sigma_y$, $\sigma_z$ and $\tau_{xy}$ (Fig. 6). When considering stresses on planes parallel to the shear plane, the stress $\sigma_z$ may be arbitrary.

In the general case, for each material particle lying in the shear plane, the stress state will be its own. Neglecting the variation of the stress state along the shear plane, or replacing it by an average one, we shall have

Fig. 6. For analysis of the stress state.

by changes in the stress state along the shear plane, or replacing it by the average one, we shall have

\[ \sigma_{1,3}=\frac{1}{2}\left\{(\sigma_x+\sigma_y)\pm\sqrt{(\sigma_x-\sigma_y)^2+4\tau_{xy}^2}\right\}. \]

or

\[ \sigma_{1,3}=\frac{1}{2}\tau_{xy}\left\{(\alpha_x+\alpha_y)\pm\sqrt{(\alpha_x-\alpha_y)^2+4}\right\}, \]

where

\[ \alpha_x=\frac{\sigma_x}{\tau_{xy}}=\tan(\beta_1+\eta-\gamma)=\tan(45^\circ+\rho), \]

\[ \alpha_y=\frac{\sigma_y}{\tau_{xy}}. \]

If one takes \(\rho=0\) and \(\alpha_y=\alpha_x\), then the stressed state along the shear plane will correspond to uniaxial compression (with \(\sigma_z=0\)), while the shear stresses in the shear plane will be maximal. One may arrive at such a conclusion by regarding the stressed state in chip formation as shear with hydrostatic compression [16], which is numerically equal to the shear stress in the shear plane,

\[ (\sigma_x=\sigma_y=\sigma_z=\tau_{xy}). \]

In the general case \(\rho\ne 0\). Then

\[ \sigma_{1,3}=\tau_{xy}\{\tan(45^\circ+\rho)\pm 1\}. \]

If the condition \(\alpha_y=\alpha_x\) is retained, the stressed state will be biaxial compression; moreover, the ratio of the second stress at small \(\rho\) (Fig. 1) will be close to zero, and the shear stresses in the shear plane will remain maximal.

The more probable assumption is that the linear stressed state is preserved. This will mean:

\[ \sigma_1=0;\qquad \sigma_3=2\tau_{\max}=\frac{2\tau_{xy}}{\cos 2\rho}, \]

\[ \sigma_{x,y}=\tau_{xy}\left(\frac{1}{\cos 2\rho}\mp \tan 2\rho\right), \]

\[ \tau_{xy}=\tau_{\mathrm{cut}}=\tau_{\max}\cos 2\rho. \tag{21} \]

Calculation of shear stresses

It is known [17] that, in the region of large plastic deformations, the relation between stresses and deformation is approximated sufficiently reliably by a linear function:

\[ \tau_{\max.\mathrm{comp}}=\tau_0(1+2\alpha g_{\max}), \tag{22} \]

where

\(\tau_{\max.\mathrm{comp}}\) is the maximum shear stress under compression up to the relative shear \(g_{\max}\);

\(\tau_0\) is the yield limit of the metal under the condition that dependence (22) is also valid in the region of small deformations;

\(2\alpha\) is the strain-hardening characteristic of the metal, numerically equal to the ratio of the maximum shear stresses at deformation \(g_{\max}=1\) to the conditional yield limit \(\tau_0\).

When a strain-hardening metal is machined, stresses will act in the shear plane whose numerical value will be greater than the yield limit in shear and less than the stresses corresponding to the final stage of deformation [1,18].

The average stresses in the compression process from the initial to the final stage, on the basis of (22), can be calculated as follows:

\[ \tau_{\mathrm{avg.comp}}=\tau_0(1+\alpha g_{\max}). \tag{23} \]

To pass to the cutting process it is necessary to introduce a coefficient \(B\), which takes into account the difference in deformed states [14]:

\[ B=\frac{2}{3}\left[2+\frac{\delta_2}{\delta_3}\right]. \tag{24} \]

Here \(\delta_2\) is the mean principal elongation, which in plane deformation is equal to zero.

Then, taking (21) into account, we shall have:

\[ \tau_{\mathrm{cut}}=B\tau_0(1+\chi g_{\max})\cos 2\varphi . \tag{25} \]

Or, simplifying: \(\delta_2=0\) and \(\varphi=0\),

\[ \tau_{\mathrm{cut}}=\frac{4}{3}\tau_0(1+\chi g_{\max}). \tag{26} \]

Consequently, the cutting force and its components can be determined from the equations

\[ P_i=\frac{4}{3}\tau_0(1+\chi g_{\max})k_iab. \tag{27} \]

LITERATURE

1. L. M. Sedokov. Stresses and deformation in the process of metal cutting. Report at the All-Union Scientific and Technical Conference on Problems of Metal Cutting. Proceedings, No. 1, Moscow, 1963.

2. L. M. Sedokov. Dimensionless characteristics of the dynamics of the cutting process. Proceedings of the Tomsk Polytechnic Institute, vol. 92, issue 2, 1961.

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

4. N. N. Zorev. Problems in the Mechanics of the Metal-Cutting Process. Mashgiz, 1956.

5. G. A. Kufarev. Experimental study of plastic deformation in metal cutting. In: Studies in the Physics of the Solid State. Academy of Sciences of the USSR, 1957.

6. L. M. Sedokov. Kinematic analysis of the process of continuous-chip formation. Proceedings of the Tomsk Polytechnic Institute, vol. 85, 1957.

7. M. E. Merchant. Mechanics of the Metal Cutting Process. Journal of Applied Physics, No. 6, Vol. 16, 1945.

8. A. M. Rozenberg, A. N. Eremin. Chip structure in cutting ductile metal. Proceedings of the Tomsk Polytechnic Institute, vol. 75, 1954.

9. S. F. Glebov. Mechanism of plastic deformation in cutting steel. Machine Tools and Tooling, No. 7, 1949.

10. V. V. Kuzyushin. Plastic deformation in cutting steel. Machine Tools and Tooling, No. 4, 1951.

11. N. N. Zorev. Investigation of Elements of the Mechanics of the Cutting Process. Mashgiz, 1942.

12. S. S. Nekrasov. On the maximum value of the specific cutting pressure. Bearing, No. 2, 1953.

13. V. I. Apraksin. Kinematics of the cutting zone. Scientific and Technical Information Bulletin of the Leningrad Polytechnic Institute, No. 2, 1962.

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

15. E. P. Unksov. Engineering Theory of Plasticity. Mashgiz, 1959.

16. M. I. Klushin. Determination of the contact zone between the chip and the front surface of the tool and of the pressure in this zone. Machine Tools and Tooling, No. 3, 1960.

17. P. Bridgman. Studies of Large Plastic Deformations and Fracture. Foreign Literature Publishing House, Moscow, 1955.

18. L. M. Sedokov. Calculation of shear stresses in the process of metal cutting. Proceedings of Higher Educational Institutions. Mechanical Engineering, No. 10, 1963.