Dimensionless Characteristics of the Dynamics of the Metal Cutting Process

L. M. SEDOKOV

(Presented by Professor Doctor A. M. Rozenberg)

The magnitude of the forces arising in the process of cutting metals is influenced by a large number of factors which, in one way or another, change the conditions under which this complex phenomenon proceeds. As a result of a number of theoretical and experimental works it has been established that, ultimately, the various conditions of the cutting process affect the magnitude of the forces in two ways: a) through a change in the shear angle and in the coefficient of friction on the rake face of the cutter; b) through a change in the magnitude of the tangential stresses in the shear plane.

The equation for determining the principal component of the cutting force, in general form, may be written as follows:

\[ P_1 = \tau k_1 ab. \tag{1} \]

Here

\[ \begin{aligned} \tau &\text{— tangential stresses in the shear plane;}\\ k_1 &\text{— dimensionless coefficient;}\\ a &\text{— chip thickness;}\\ b &\text{— chip width.} \end{aligned} \]

Analyzing the theoretical equations for determining the cutting force proposed by I. A. Time, K. A. Zvorykin, A. Briks, G. N. Loladze, A. M. Rozenberg, N. N. Zorev, and others, one may note that the authors give different expressions of the dependence for determining the magnitude of the tangential stresses and the coefficient \(k_1\).

Since the magnitude of the tangential stresses can be calculated from the physico-mechanical characteristics of the metal being machined, taking into account its strengthening in the course of the plastic deformation occurring during cutting, it is advisable to consider in detail the dimensionless quantity \(k_1\), which is numerically equal to the specific cutting force under the condition that the tangential stresses in the shear plane are equal to unity.

Let us call this basic dimensionless characteristic of the dynamics of the metal-cutting process the cutting coefficient. On the basis of the diagram of the forces acting on the rake face of the cutter, shown in Fig. 1,

\[ P_z = \tau \frac{ab}{\sin \beta_1}; \]

\[ P_1=\tau \frac{\cos(\eta-\gamma)}{\sin \beta_1 \cos(\beta_1+\eta-\gamma)}\,ab; \]

\[ k_1=\frac{\cos(\eta-\gamma)}{\sin \beta_1 \cos(\beta_1+\eta-\gamma)} =\operatorname{ctg}\beta_1+\operatorname{tg}(\beta_1+\eta-\gamma). \tag{2} \]

Thus, the cutting coefficient \(k_1\) is a function of the tool geometry \((\gamma)\), the conditions of friction on the rake face \((\mu=\operatorname{tg}\eta)\), and the degree of plastic deformation occurring, determined through the chip shrinkage and the shear angle \(\beta_1\).

Investigating the dynamic relations obtained, N. N. Zorev [1], A. M. Rozenberg, and L. A. Khvorostukhin [2, 3] obtained identical experimental relations between the coefficient of friction, chip shrinkage, and the rake angle of the cutter [4].

A direct relation between the shear angle and the coefficient of friction at a constant rake angle of the cutter, in our view, indicates the existence of a direct relation between the cutting force and the deformation of the metal in the cutting process. Therefore we set ourselves the task of finding an analytical expression for the relation of the cutting coefficient \(k_1\) to the degree of plastic deformation occurring during chip removal.

Fig. 1. Diagram of the forces acting on the rake face of a cutting tool.

Determination of the Direction of the Resultant Force

Below a graphical method is proposed for determining the direction of the resultant force from the known direction of the shear plane. From point \(O\) (Fig. 2), draw a line at an angle of \(45^\circ\) to the plane

Fig. 2. Diagram of the chip-formation process and graphical determination of the direction of the resultant force from the known direction of the shear plane.

shears. From an arbitrary point \(B\), lying in the plane of shears \(OB\), we draw a normal to this plane, which will intersect the previously drawn line at point \(C\). If now from point \(C\) we draw a normal to the front face, and from point \(B\) a parallel to it, then we obtain their point of intersection \(L\), and the line \(OL\), according to our assumption, must coincide with the direction of the resultant force.

It is clear that such a construction is formal, but it makes it possible to obtain an analytical relationship among all the dimensionless characteristics of the dynamics of the metal-cutting process; and a check against experimental materials over a wide range of variation of cutting conditions has given quite satisfactory results.

Analytical relationship of the dimensionless characteristics of the dynamics of the cutting process

Figure 3 shows a diagram of the forces acting on the front face of the cutter, constructed by the proposed method. From this diagram one can easily obtain the relationship between the coefficient of friction on the front face, the shear angle, and the front angle of the cutter,

\[ \mu = 1 - \operatorname{tg}(\beta_1 - \gamma), \tag{3} \]

\[ \operatorname{tg}\beta_1 = \frac{1 - \mu + \operatorname{tg}\gamma} {1 - \operatorname{tg}\gamma + \mu \operatorname{tg}\gamma}. \tag{4} \]

Fig. 3. Relationship between the shear plane and the direction of the forces acting on the front face of the cutting element.

According to the construction,

\[ \beta_1 + \eta - \gamma = \frac{\pi}{4} + \rho. \tag{5} \]

Consequently, the angle \(\rho\), additional to \(\pi/4\), can be expressed as follows:

\[ \operatorname{tg}\rho = \frac{\operatorname{tg}(\beta_1 - \gamma) - \operatorname{tg}^2(\beta_1 - \gamma)} {2 - \operatorname{tg}(\beta_1 - \gamma) + \operatorname{tg}^2(\beta_1 - \gamma)} \tag{6} \]

or

\[ \operatorname{tg}\rho = \frac{\mu - \mu^2}{2 - \mu + \mu^2}. \]

The obtained relationships mean that, for any combination of cutting parameters, but with the coefficient of friction on the front face constant, the sum of the angles \((\beta_1 + \eta - \gamma)\) and all the dimensionless characteristics of the dynamics of the metal-cutting process remain constant. The angle of friction and the angle \(\rho\) are functions only of the difference between the angles \(\beta_1\) and \(\gamma\). Their value will be the same for any values of \(\beta_1\) and \(\gamma\), if the difference between them remains unchanged. Therefore the difference \((\beta_1 - \gamma)\) should be regarded as one of the principal characteristics of the chip-formation process. One may introduce the concept of an action coefficient, which is related to the angle of action according to Time (see Fig. 3) and to the indicated difference of angles:

\[ \lambda = \operatorname{ctg}\beta = \operatorname{tg}(\beta_1 - \gamma). \tag{7} \]

In Fig. 4 the relationship is shown between the angles characterizing the chip-formation process.

On the basis of the force diagram (Fig. 1),

\[ P_z=\tau \kappa_2 ab. \]

\[ N=\tau \kappa_N ab \]

\[ F=\tau \kappa_F ab. \]

Fig. 4. Relationship between the angles characterizing the chip-formation process.

The values of the dimensionless characteristics \(\kappa_2\), \(\kappa_N\), and \(\kappa_F\), also taking into account (2), (3), and (5), can be analytically related only to the angles \(\beta_1\) and \(\gamma\):

\[ \kappa_2= \left[ \operatorname{ctg}\beta_1+\operatorname{tg}\left(\frac{\pi}{4}+\rho\right) \right] \operatorname{tg}\left(\frac{\pi}{4}+\rho-\beta_1\right), \tag{8} \]

\[ \kappa_N=k_1 \frac{ \cos\left(\frac{\pi}{4}+\rho-\beta_1+\gamma\right) }{ \cos\left(\frac{\pi}{4}+\rho-\beta_1\right) }. \tag{9} \]

\[ K_F = K_1 \frac{\sin\left(\dfrac{\pi}{4}+\rho-\varphi_1+\gamma\right)} {\cos\left(\dfrac{\pi}{4}+\rho-\varphi_1\right)} . \tag{10} \]

Figure 5 shows plots of the dependence of the indicated characteristics on chip shrinkage.

Fig. 5. Dependence of the coefficients \(K_1\), \(K_2\), \(K_N\), \(K_F\) on chip shrinkage.

Verification of the obtained relationships using experimental materials

The analytical expressions obtained, which relate the various dimensionless characteristics of the dynamics of the metal-cutting process, require careful and comprehensive experimental verification, as a result of which it is necessary to establish the difference between the actual and calculated values of the indicated quantities and the limits of practical application of the relationships obtained.

The relationship for the friction coefficient (3) was checked using the materials of a number of researchers \([3, 4, 5, 8, 9, 10 \text{ and } 11]\) over a wide range of variation in almost all cutting conditions. The results of the check showed that equation (3) satisfactorily describes the relationship between the coefficient of friction on the rake face, the shear angle, and the rake angle of the cutting element \([14]\). This relationship is also suitable for the case of plastic compression, when \(\gamma = 0\), the shear angle is \(45^\circ\), and the coefficient of friction on the end faces of the specimen is equal to zero. In the presence of friction on the end faces, the shear angle can be determined from equation (4).

A more or less significant scatter of the experimental points is explained, in our opinion, by possible errors in the experimental-

...in the experimental determination of the mean value of the coefficient of friction on the front face of the cutting element from three experimental values of the components of the cutting force, when the existing experimental errors may be superimposed.

Of great importance in the dynamics of the cutting process is relation (5). Many works have been devoted to identifying the relationship among the angles entering into it \([1, 5, 6, 7]\). Several formulas have been proposed, but none of them gives satisfactory agreement with experimental results over a wide range of variation of the parameters of the cutting process \([7]\). In Fig. 6, relation (5) is compared with the mean values of the angle between the shear plane and the direction of the resultant force according to the experiments of L. A. Khvorostukhin \([3]\), for front angles in the range from \(-10\) to \(+40^\circ\) in machining a number of steels, copper, and aluminum. From the graph presented, satisfactory

\[ \beta+\eta-\gamma = 45^\circ+\rho \]

\[ \tan \rho = \frac{\tan(\beta_1-\delta)+\tan^2(\beta_1-\delta)} {2-\tan(\beta_1-\delta)+\tan^2(\beta_1-\delta)} \]

Fig. 6. Calculated and mean actual value of the angle between the direction of the resultant force and the shear plane, according to the experiments of L. A. Khvorostukhin \([8]\).

\[ K_1=\operatorname{ctg}\beta_1+\tan\left(\frac{\pi}{4}+\rho\right) \]

Fig. 7. Calculated and actual values of the conditional specific cutting force according to the experiments of L. A. Khvorostukhin \([8]\), \(\lambda=0^\circ\).

agreement of the calculated and actual values is seen for all cases in which the action coefficient varies from \(-0.25\) to \(+0.7\).

The dependence for the cutting coefficient \(k_1\), which may be called the conditional specific cutting force, (2), taking (5) and (6) into account, has likewise been checked in detail against materials from a number of investigations carried out at different times and for different purposes. In Fig. 7 a comparison is given of the calculated and actual values of the cutting coefficient \(k_1\) according to the experiments of L. A. Khvorostukhin [3] in machining five steels.

In the case where an experimental dependence of the cutting forces and chip shrinkage on the cutting temperature is available, it is possible to compare the calculated and actual relation between the cutting coefficient \(k_1\) and the cutting temperature. Such a comparison, based on the materials of N. N. Zorev [1], is given in Fig. 8.

The materials presented demonstrate the reliability of the obtained dependencies for calculating the cutting coefficient. The difference between the calculated and actual values of this characteristic of the dynamics

Fig. 8. Calculated and experimental dependence of the conditional specific cutting force on the cutting temperature according to the experiments of N. N. Zorev (steel 5 and 35KhZMN) [6].

of the cutting process does not exceed 5–8%, that is, it lies entirely within the limits of possible experimental errors.

Here it is also important to note that the steels assigned by N. N. Zorev [5] to different groups according to their influence on the coefficient of friction gave good agreement with the single calculated values (see Fig. 8).

The results of the verification carried out for the obtained dynamic dependencies, the materials of which are presented in this article only in part because of limitations on the length of the article, show that, in the entire diversity of cutting conditions, no boundaries have been found for the practical application of these dependencies, with the exception of micro-speeds, which are used only under laboratory conditions.

Cutting coefficient and relative displacement

As was shown above, the cutting coefficient is determined with sufficient accuracy by the formula

\[ k_1=\operatorname{ctg}\beta_1+\operatorname{tg}\left(\frac{\pi}{4}+\rho\right). \]

This coefficient may be regarded as the specific work of cutting under the condition that the tangential stresses in the shear plane are always equal to unity. Taking into account the investigations of A. M. Rosenberg and A. N. Eremin [4], this conditional specific work of cutting may be represented as consisting of two parts: a) the conditional specific work of plastic deformation \((k_\varepsilon)\), and b) the conditional specific work of friction \((k_\mu)\),

\[ k_1 = k_\varepsilon + k_\mu. \]

\[ k_\varepsilon + k_\mu = \operatorname{ctg}\beta_1 + \operatorname{tg}\left(\frac{\pi}{4}+\rho\right). \]

As A. M. Rosenberg and A. N. Eremin [4] showed, the specific work of plastic deformation in cutting is

\[ A w_p = \tau \varepsilon = \tau[\operatorname{ctg}\beta_1+\operatorname{tg}(\beta_1-\gamma)]. \]

Consequently,

\[ k_\varepsilon=\operatorname{ctg}\beta_1+\operatorname{tg}(\beta_1-\gamma)=\varepsilon. \tag{11} \]

Then the conditional specific work of friction can be determined from the relation

\[ k_\mu=\operatorname{tg}\left(\frac{\pi}{4}+\rho\right)-\operatorname{tg}(\beta_1-\gamma). \tag{12} \]

This expression can also be obtained on the basis of a kinematic analysis of the formation of continuous chips [12].

Figure 9 presents the dependence of \(k_1\), \(k_\varepsilon\), and \(k_\mu\) on the magnitude of the relative shear. It turned out that the coefficient \(k_\mu\) tends toward unity. Subsequent verification using experimental materials likewise showed that, for a given rake angle of the tool, the value of the relative shear practically does not occur below that value which corresponds to a substantial decrease in \(k_\mu\).

Fig. 9. Dependence of the coefficients \(K_1\), \(K_\varepsilon\), and \(K_\mu\) on the magnitude of the relative shear.

Consequently, the conditional specific work of friction on the rake face of the tool is almost always equal to unity. Then we may write

\[ k_1=k_\varepsilon+k_\mu=\varepsilon+1. \tag{13} \]

Since expression (13) is a simplification of previously obtained dependences for the cutting coefficient, it was additionally checked in detail against investigators’ data and received very satisfactory confirmation [13]. In Fig. 10, as an example, results are given for checking the indicated dependence using the data of Gordon and Klushin [11]. Dependence (13) agrees well with the experimental results for all cutting regimes giving a change in relative shear from 2.0 to 11.0.

Fig. 10. Calculated \(K_1=(\varepsilon+1)\) and actual values of the conditional specific cutting force according to the experiments of M. B. Gordon and M. I. Klushin [16].

On the basis of the foregoing, we recommend the following equation for calculating the principal component of the cutting force:

\[ P_1=\tau(\varepsilon+1)ab. \tag{14} \]

Conclusion

In the present work, analytical expressions have been obtained for the principal dimensionless characteristics of the dynamics of the cutting process, which are quite satisfactorily confirmed by numerous experiments carried out over a broad range of cutting conditions.

As a result of analysis of the interrelation of these characteristics, a dependence of the cutting coefficient on the degree of plastic deformation, common to any cutting conditions, has been obtained (13), as well as a new, very simple equation for determining the principal component of the cutting force (14). According to this equation, the cutting force depends on the resistance of the machined metal to plastic deformation (through the magnitude of the tangential stresses in the shear plane at the given degree of plastic deformation of the metal transformed into chip); on the degree of plastic deformation (the magnitude of the relative shear); and on the volume of the removed layer of metal when the cutter travels a path equal to unity.

If the chip compression and the rake angle of the cutting element are known, and consequently the degree of plastic deformation \((\varepsilon)\) and the shear angle \(\beta_1\), then, on the basis of the dependences obtained in the present work, it is possible to calculate:

1. The average value of the coefficient of friction on the rake face, by equation (3) or from the graphs in Fig. 4.

2. The angle between the shear plane and the direction of the resultant force, from dependences (5) and (6) or from the graphs in Figs. 4 and 6.

3. The angle between the direction of the cutting speed and the direction

the resultant force according to relations (5) and (6) and the graphs in Figs. 4 and 6.

1. The cutting coefficient \(\kappa_1\), entering into the equation for determining the cutting force in the general form (1), from relations (2) and (13) or from the graphs in Figs. 5 and 9.

2. The coefficient \(\kappa_2\), entering into the equation for determining the horizontal component of the cutting force, from relation (8) or from the graph in Fig. 5.

3. The coefficient \(\kappa_N\), entering into the equation for determining the normal force on the rake face of the cutting element, from relation (9) or from the graph in Fig. 5.

4. The coefficient \(\kappa_F\), entering into the equation for determining the friction force on the rake face, from relation (10) or from the graph in Fig. 5.

Having calculated the magnitude of the tangential stresses in the shear plane from the physicomechanical characteristics of the metal being machined, and knowing the cross section of the chip and the corresponding dimensionless characteristics of the dynamics of the cutting process, one can calculate with sufficient reliability the magnitudes of the forces acting on the rake face of the cutting element, namely:

a) the principal component of the cutting force—\(P_1\);

b) the horizontal component of the cutting force—\(P_2\);

c) the normal force—\(N\);

d) the friction force—\(F\);

e) the force acting in the shear plane—\(P_\tau\);

f) the force acting normal to the shear plane—\(P\alpha\); and

g) the resultant force \(R\).

REFERENCES

1. Zorev N. N. Investigation of Elements of the Mechanics of the Cutting Process, Mashgiz, 1952.

2. Rozenberg A. M. and Khvorostukhin L. A. “Equation for the Cutting Force in High-Speed Turning of Steel,” Vestnik mashinostroeniya, No. 1, 1954.

3. Khvorostukhin L. A. Calculation of Forces in High-Speed Cutting on the Basis of Physicomechanical Characteristics of Metals. Author’s abstract of dissertation, Tomsk, 1953.

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

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

6. Merchant M. E., “Mechanics of the Metal Cutting Process,” Journal of Applied Physics, No. 6, vol. 16, 1945.

7. Shaw M. C., Cook N. H. and Finnie J., “The Shear—Angle Relationship in Metal Cutting,” Transactions of the ASME, vol. 75, No. 2, 1953.

8. Zimin Yu. P. Investigation of the Process of High-Speed Cutting of Steels. Author’s abstract of dissertation, Tomsk, 1948.

9. Obukhov P. N. Investigation of the Process of Power Cutting of Steels. Author’s abstract of dissertation, Tomsk, 1955.

10. Eremin A. N. The Physical Essence of Phenomena in Cutting Steels, Mashgiz, 1951.

11. Klushin M. I. Metal Cutting, Mashgiz, 1953.

12. Sedokov L. M. Kinematic Analysis of the Process of Formation of Continuous Chips. Izvestiya Tomskogo politekhnicheskogo instituta, vol. 85, 1957.

13. Sedokov L. M. Equation for Calculating the Cutting Force, Tomsk, 1956.

14. Sedokov L. M. “On Calculating the Coefficient of Friction in Metal Cutting,” Stanki i instrument, No. 2, 1958.