SOME ISSUES IN THE MECHANICS OF CHIP FORMATION IN METAL CUTTING

M. F. POLETIKA

(Presented by the scientific-technical seminar of the Department of Machine Tools and Metal Cutting)

For the theoretical development of questions concerning wear of the cutting tool and the machinability of metals by cutting, for the analysis of the influence of cutting conditions on the strength of the cutting part of the tool, and also for the solution of a number of other pressing problems, it is necessary to further develop the mechanics of the cutting process, chiefly in that part of it which concerns contact phenomena on the rake face of the tool, their connection with chip formation, and also their dependence on various factors: the properties of the work and tool materials, the geometric parameters of the tool, the elements of the cutting regime, and other factors. Most of the problems connected with this are still very far from being completely solved, and the results of our investigations set forth below are intended partly to remedy this deficiency.

Fig. 1.

One of the most important problems, to the consideration of which we shall now turn, is the problem of the distribution of contact stresses on the rake face of the cutter.

Let us turn to Fig. 1, which gives the cutting-zone scheme as it is understood at present. In the middle of this zone lies the plastic region \(EFHCOD\), compressed between two elastically stressed regions \(SEDOT\) (the workpiece) and \(FKLCH\) (the chip). The plastic region itself is usually divided into two parts: the zone of primary shear deformations (the main plastic region) \(EFHOD\), and the zone of secondary deformations \(OHC\) in the contact layer of the chip.

The tool material \(NOW\) is elastically stressed, and therefore the nature of the contact between the chip and the tool is twofold. On the segment \(OC\), an elastic body is in contact with a plastic one (the plastic segment of contact); on the segment \(CB\), there is contact between two elastic bodies (the elastic segment of contact).

As experimental investigations [1] show, as the cutting speed increases, the dimensions of the main plastic region decrease considerably. The thickness of the plastic region in the contact layer of the chip also decreases, although the ratio between the length \(OC\) of the plastic segment of contact and the total contact length \(OB\) remains approximately constant. Naturally, as the thickness of the plastic regions decreases, the role of the elastic regions in the distribution of stresses in the cutting zone increases.

If one does not set the aim of an exact investigation of the stress-strain state inside the plastic region, but confines oneself only to determining the general stress field, in particular the contact stresses on the rake face, then, in a sufficiently general mathematical formulation, the cutting problem may be considered as an elastoplastic contact problem. However, the complex nature of the boundary conditions does not permit this problem to be solved in its general formulation even with the use of the usual simplifying assumptions (the problem is static and plane, the temperature is constant at all points of the cutting zone, the material is isotropic, etc.). Additional restrictions have to be introduced.

We proposed that

1) the tool is a rigid (absolutely hard) body;

2) throughout the entire cutting zone the material is stressed only elastically;

3) the coefficient of friction of the chip against the tool is constant at all points of contact.

Fig. 2.

Since the point of application of the force acting on the rake face of the tool (the center of pressure of the chip on the tool) is always located not at the middle of the contact length, but closer to the tool tip, the contact problem under consideration can be likened to the problem of pressure on a certain elastic region by a rigid punch loaded by an eccentrically applied force or, what is the same, by a centrally acting force and a moment. Under the action of the moment the punch rotates, elastically deforming the base, as a result of which, beginning from a certain magnitude of the moment, the sole of the punch separates from the base (Fig. 2, a). A completely analogous picture is observed in cutting, when the tool, elastically deforming the chip and the entire cutting zone, separates from the chip at a certain point \(B\) (Fig. 2, b).

For the solution of the problem posed, the mathematical apparatus of the theory of functions of a complex variable was used. In view of the complex

forms of the elastic region under investigation (the cutting zone), we used a conformal mapping of the half-plane onto this region. In the particular case of a zero rake angle (only this case makes it possible to express the final results in terms of elementary functions)

Fig. 3.

the function effecting the conformal transformation is expressed as follows:

\[ z=\omega(s)=\frac{a_1}{\pi i}\left[ -\frac{\sqrt{(s-1)(s-d_0)}}{s} +\ln \frac{\sqrt{s-d_0}+\sqrt{s-1}}{\sqrt{s-d_0}-\sqrt{s-1}} -\frac{1+d_0}{2\sqrt{d_0}}\ln \frac{\sqrt{s-d_0}+\sqrt{d_0(s-1)}}{\sqrt{s-d_0}-\sqrt{d_0(s-1)}} \right], \tag{1} \]

where

\[ z=x+iy \]

is the complex coordinate in the original region,

\[ s=\xi+i\eta \]

is the complex coordinate in the transformed region,

and, in accordance with Fig. 3:

\[ a \]

is the thickness of cut,

\[ a_1 \]

is the chip thickness.

Using the equations of the plane theory of elasticity in complex representation, we solved the stated problem as a particular case of a linear conjugation problem [2]. The following expression was obtained for the contact pressure at any point along the length of contact between the chip and the cutter:

\[ p(y)=p(\xi)=- \frac{2N\sqrt{m d_0}\,(A_1m-A_3d_0)\sqrt{\xi-m}} {a_1\,[2mA_1(\sqrt{m d_0}-m)+A_3(m-d)]\sqrt{1-\xi}} . \tag{3} \]

In this expression the coefficients \(A_1\) and \(A_2\) are determined by the formulas

\[ A_1=\frac{a_1}{\pi}\left(2\arctan \sqrt{\frac{1-d_0}{d_0-m}}-\pi\right), \tag{4} \]

\[ A_2=\frac{a_1\sqrt{-m}}{\pi}\ln \frac{\sqrt{m(d_0-1)}+\sqrt{d_0-m}} {\sqrt{m(d_0-1)}-\sqrt{d_0-m}}, \tag{5} \]

and the remaining quantities entering into it denote:

\(y\)—the ordinate of the current point in the region under study (Fig. 3);

\(\xi\)—the abscissa of the current point in the transformed region;

\(m\)—the value of this abscissa at the point (more precisely, at the image of the point) of chip separation from the tool;

\(N\)—the normal force on the rake face of the tool.

The quantity \(\xi\) can be determined for each current value of \(y\); for this purpose one can use expression (1), transforming it for the case \(\eta=0\), \(s=\xi\) (\(\eta\) is the ordinate in the mapped region). It is more convenient, however, to use another formula for the same purpose:

\[ \psi=\frac{a'}{\pi c}\left[ \ln\frac{\sqrt{1-\xi}+\sqrt{d_0-\xi}} {\sqrt{1-\xi}-\sqrt{d_0-\xi}} -\frac{\sqrt{(1-\xi)(d_0-\xi)}}{\xi} -\frac{2}{\xi}\operatorname{arc\,tg}\sqrt{\frac{d_0-\xi}{d_0(\xi-1)}} \right], \tag{6} \]

in which

\(\zeta\)—chip shrinkage,

\(c\)—length of chip contact with the tool,

and \(\psi\)—the relative distance (in fractions of the contact length) of the current point from the tool tip. In terms of the ordinate \(y\), the quantity is expressed as follows:

\[ \psi=\frac{a-y}{c}. \tag{7} \]

Formula (6) can easily be presented in the form of a nomogram.

Simultaneously with the theoretical study, an experimental investigation was carried out of the distribution of contact stresses on the rake face of the tool; for this purpose we used the polarization-optical method of stress analysis. The procedure for its application and the results obtained have already been reported earlier [3]. As follows from Fig. 4, the theoretical and experimental curves of contact pressure are quite close to one another. The theoretical diagram of the tangential contact stress is not given in Fig. 4, since it obviously will not coincide with the corresponding experimental curve. The assumption of constancy of the coefficient of friction along the contact length, which does not noticeably affect the accuracy of the solution for contact pressure, in the present case leads to large deviations.

Fig. 4.

Analysis and generalization of the theoretical results, of the experimental data obtained by the polarization-optical method in cutting various materials (lead, cadmium, copper), and also of the data

special experiments carried out at high cutting speeds on steel with a cutter having a limited length of the rake face show that the fundamental character of the contact-stress diagrams remains unchanged when the cutting conditions are varied within fairly wide limits.

The character of the contact-stress diagrams is rather closely connected with the relative length of the plastic region of contact (\(OC\) in Fig. 1). Within this region the tangential stress \(\tau_t\) is approximately constant and must be somewhat higher than the yield point of the chip material. The normal stress on the region \(OC\) must be sufficiently large for the plastic state to be maintained in the contact layer.

The position of point \(C\), which separates the plastic region of contact from the elastic one, depends on the friction conditions on the rake face. As is known, in the usual case of dry friction the actual area of contact of dissimilar rubbing surfaces is always smaller than the nominal area of their contact and depends on the magnitude of the contact pressure, increasing as it rises. In cutting, the contact pressure near the tool tip is so great that the contact here is continuous; the actual contact area is equal to the nominal one. As the distance from the tip increases, the contact pressure decreases and, beginning at some point \(K\) (Fig. 1), the actual contact area becomes smaller than the apparent area and continues to decrease up to point \(B\), where the chip separates from the cutter; this is confirmed by direct experiment [1, 4].

Under the conditions of the high frequency of the contacting surfaces characteristic of the cutting process, the friction process in the region of discontinuous contact must be characterized by an increased role of adhesion and be accompanied by the formation and rupture of bonds between the surfaces of the chip and the cutter. In this case the bonds arise at the peaks of surface irregularities (protrusions), while rupture (shearing) occurs predominantly in the softer material (the chip).

Let us choose within the region \(KB\) a small area \(\Delta A_n\) and denote by \(\Delta A_f\) the actual contact area within the boundaries of this area. Then from the equilibrium conditions we must obtain

\[ \tau_c \Delta A_f = \tau_t \Delta A_1, \tag{8} \]

where \(\tau_c\) is the tangential stress necessary for the rupture of the bonds that have arisen directly at the places of joining of dissimilar metals;

\(\Delta A_1\) is the total (within the limits) area of the cross sections of the protrusions of the chip surface at those places where shearing occurs;

\(\tau_t\) is the tangential stress acting in the above-mentioned sections.

The stresses \(\tau_t\) and \(\tau_c\) do not depend (or depend only very weakly) on the coordinates of the selected area. The quantity \(\Delta A_f\), however (for a chosen, in each case constant, value of \(\Delta A_n\)), will increase if we move from the separation point toward the tool tip. In this case the right-hand side of equality (8) must also increase, and hence so must the area \(\Delta A_1\).

For a sufficiently large value of the real contact area \(\Delta A_f\), the total section of the protrusions becomes equal to the nominal contact area \(\Delta A_n\), and equality (8) then takes the form

\[ \tau_c \Delta A_f = \tau_t \Delta A_n . \tag{9} \]

Physically this means that in the given contact region the force of adhesion of the dissimilar metals is sufficient for the plastic defor-

formation spread into the depth of the softest of them—the chip metal. Condition (9) corresponds to point \(C\), lying on the boundary of the plastic portion of the contact.

Transform equation (9) as follows:

\[ \frac{\tau_m}{\tau_c}=\frac{\Delta A_{\phi}}{\Delta A_{\mathrm{n}}}. \tag{10} \]

The right-hand side of the expression obtained is the ratio between the areas of real and nominal contact and, as such, is a function of the contact pressure at point \(C\):

\[ \frac{\Delta A_{\phi}}{\Delta A_{\mathrm{n}}}=f(p_k), \tag{11} \]

which henceforth we shall call the limiting contact pressure.

From comparison of expressions (10) and (11) it follows that

\[ p_k=F\left(\frac{\tau_m}{\tau_c}\right), \tag{12} \]

that is, the limiting pressure is a function of the strength of the chip material \((\tau_m)\) and the strength of the bonds \((\tau_c)\), and consequently must depend on all those factors which, either directly or indirectly, can affect these quantities.

The limiting pressure is an important factor determining the other contact characteristics and, first of all, the length of contact between the chip and the tool. The latter follows from the two important circumstances noted above: the constancy of the relative length of the plastic portion of the contact \((c_{\mathrm{p}}/c=\mathrm{const})\) and the very weak dependence on cutting conditions of the law of distribution of contact pressures on the rake face. It is then easy to show that an increase in the limiting pressure will always cause a decrease in the contact length, and conversely.

Fig. 5.

Another most important contact characteristic is the mean tangential contact stress \(q_F\). N. N. Zorev showed that, in machining steels, this stress for a given work material is practically independent of cutting speed and temperature (if the built-up-edge formation zone is not taken into account). It follows from our experiments that the independence of the stress \(q_F\) from a whole series of factors (apart from the strength of the work material) is observed when cutting the most diverse materials (copper, bronze, brass, aluminum, titanium, cadmium, etc.) and, consequently, is a general law of chip formation. With increasing strength of the work material, the value of \(q_F\) also increases. As can be seen from Fig. 5, the dependence of the mean shear stress on the actual strength limit proved to be almost single-valued.

Since the friction force \(F\) can be determined as the product of the stress \(q_F\) by the area of contact between the chip and the cutter, then, for a constant width of cut, by virtue of what has been said above it must be proportional to the contact length \(C\).

The link connecting the contact characteristics \((c, q_F, q_N, \psi)\) with the parameters of the chip-formation process proper \((\beta_1, \zeta)\) is the state of stress, common to the chip and to the adjoining regions of the workpiece. When chip formation is idealized as a process of successive shears occurring along a single shear plane, the role of such a connecting link is played by the action angle \(\omega\), which determines the direction of the resultant \(R_{\mathrm{pr}}\) of the forces on the rake face of the cutter. It is assumed here that the dependence between the action angle \(\omega\) and the shear-plane angle \(\beta_1\) is single-valued, and it is especially emphasized [5] that the scalar magnitude of the resultant \(R_{\mathrm{pr}}\) has no significance in the relation under consideration, but is only a function of the degree of deformation in the principal plastic zone \((\beta_1, \zeta)\).

As N. N. Zorev has shown, for steels [6] there is in fact no single-valued relation between the angles \(\omega\) and \(\beta_1\). The shear-plane angle \(\beta_1\), in addition to the angle \(\omega\), also depends on three factors: the mechanical properties of the material being machined, the cutting speed, and the rake angle of the cutter. Having carried out experiments with 19 metals being machined (mostly nonferrous), we not only found the same thing for other metals as well, but also showed that the number of factors determining the angle \(\beta_1\) is considerably greater than three. This circumstance, together with certain other experimental facts on which we shall not dwell here, leads to the conclusion that the scheme with a single shear plane is unsuitable for explaining the mechanism of the relations between the contact characteristics on the rake face and the chip contraction (or the angle \(\beta_1\)).

Fig. 6.

Taking into account the actual outlines of the principal plastic zone (Fig. 1) and temporarily neglecting the presence of the zone of secondary deformation in the contact layer of the chip, one may represent the chip and the cutter as a single rigid body—a “die” (Fig. 6), under the pressure of which-

a plastic zone arises in the workpiece. Naturally, the dimensions of this zone, and hence the chip shrinkage, must depend on the force pressing the “punch” against the base. An increase in the force \(R_{\text{pr}}\), as well as its rotation in the direction of increasing the angle \(\varphi\), will lead to an increase in chip shrinkage.

The proposed analogy with the action of a “punch,” if used as a computational scheme, will lead to more complicated mathematical relationships than those that follow from the scheme with a single shear plane. However, it better reflects the actual stressed state in the cutting zone than the scheme with a single plane. In particular, this analogy clearly shows that chip shrinkage depends not only on the direction of the resultant \(R_{\text{pr}}\), but also on its scalar magnitude. As follows from Fig. 7, the independent influence of the latter on chip shrinkage is confirmed by direct experiments.

Fig. 7.

The considerations set forth above concerning the interrelation between the various characteristics of the chip-formation process, if supplemented by an analysis of the mechanism of the influence of external independent factors on chip formation, lead to the scheme of interaction of factors in the cutting process presented in Fig. 8. Turning to consideration of this scheme, we note that, because of the complexity of the stress-strain state and of the physical regularities in cutting, it is not capable of reflecting all the relationships that occur in the real process, but illustrates only the principal ones.

External independent factors are divided by us into five groups: properties of the work material, properties of the tool material, geometrical parameters of the tool, and elements of the cutting conditions. Among the principal contact characteristics we include the limiting contact pressure \(p_k\), the cutting temperature, the length of contact \(c\) of the chip with the cutter, the mean tangential contact stress \(q_F\), and the friction force \(F\) on the rake face. Chip contraction and the principal component of the forces applied to the rake face are taken as the main parameters of the chip-formation process proper.

Fig. 8.

The structure of the relationships among the four contact characteristics \(p_k\), \(q_F\), \(c\), and \(F\) has already been discussed above. Let us now consider the mechanism by which the fifth characteristic—the mean contact temperature (cutting temperature)—exerts its influence.

The heat arising in the zone of contact between the chip and the cutter is due in its origin to two processes: the process of deformation in the plastic region \(OHC\) (Fig. 1) and the process of friction on the rake surface of the cutter. The region \(OHC\) has a relatively large volume; the heat generated in it is distributed more or less uniformly, and no sharp increase in temperature at individual points is observed.

In the process of friction proper, the source of heat generation is a thin surface layer within the actual contact. Investigations show [7] that the temperature in this layer can reach very high values, close to the melting temperature of the more fusible of the contacting materials.

Since, in ordinary friction, the actual contact area is always smaller than the nominal one and contact occurs at individual surface asperities, the microvolumes of the asperities are the first to be affected by this high temperature. The asperities—especially their layers immediately adjacent to the contact surface—are heated so strongly that their hardness decreases, possibly even to the point of melting.

In cutting, analogous phenomena take place in the region of intermittent contact, in particular in the vicinity of point \(C\) (Fig. 1). This means that, with increasing temperature, the shear resistance \(\tau_c\) of the contacts that have formed will decrease. In accordance with formula (12), the limiting pressure will then increase, while the contact length and the friction force, on the contrary, will decrease.

The diagram in Fig. 8 reflects the many-sided influence on the cutting process of the properties of the work and tool materials, as well as of the surrounding medium. Their thermophysical properties (mainly thermal conductivity) affect chip formation through the cutting temperature; chemical properties act directly through the limiting pressure; the mechanism of the influence of mechanical properties is the most complex.

The fact that the chemical properties of the work and tool materials exert a significant influence on the cutting process has been established by us for the first time and is of fundamental importance. The physical nature of this influence is due to the specific conditions of friction in cutting (cleaned surfaces, high temperatures and contact pressures), owing to which the friction process is accompanied by intensive adhesion (seizure) of the contacting surfaces. Therefore, the characteristics of friction in cutting must depend on the elementary forces of interaction between the surfaces of the cutter and the chip and, consequently, on the chemical properties of their materials relative to one another.

Work in the field of friction and seizure of dissimilar metals [8, 9] shows that the intensity of scoring and seizure during friction is directly dependent on the ability of the contacting metals to form solid solutions with one another and to enter into chemical compounds. It is the higher, the greater this ability.

It is impossible to verify the validity of these conclusions as applied to the cutting process by varying the work materials, since, in selecting materials with different chemical properties, we inevitably change the entire complex of the other properties of the work material, including its mechanical properties, which exert an independent and substantial influence on chip formation. However, the desired result can be obtained by using different tool materials.

We carried out experiments on turning two work materials—lead and copper—with cutters made of various tool materials (high-speed steel R9, hard alloys VK8, T5K10, T15K6, beryllium bronze BrB2, titanium alloy VT-2, tungsten, steels U12 and G13). Lead is a chemically inert metal, forming neither solid solutions nor chemical compounds with most metals. Therefore, as we expected, when cutting lead no influence of the chemical properties of the tool material on chip formation is observed. The chip compression changes insignificantly, being almost a single-valued function of the thermal conductivity of the tool material.

When machining chemically active copper, the picture changes sharply. As is easily seen from Figs. 10 and 9, the influence of the tool material on the characteristics of the cutting process is extremely great. The transition from a tungsten cutter to a bronze one entails an increase in chip compression by almost four times, and in the friction force by approximately 8 times.

It is interesting to note that the thermal conductivity of tungsten is considerably higher than that of alloy VT-2. Therefore, if the influence of the contact chemical properties of the pair were absent, then, taking into account the difference in the rake angles of the tungsten and bronze tools, the curves corresponding to the former should have been located much higher than we see in the graphs.

Fig. 9.

Fig. 10.

Thus, under certain conditions that ensure the effective manifestation of chemical properties in contact, the influence

the tool material on the characteristics of the cutting process becomes very substantial. The mechanism of this influence (as also of the influence of the chemical properties of the material being machined) is determined by the structure of expression (12). The more actively the surfaces of the chip and the tool interact, the greater the value of the stress $\tau_c$ required for the destruction of intermetallic contacts, and the lower the limiting contact pressure. A reduction of this pressure entails an increase in the length of contact of the chip with the tool, the friction force, and, ultimately, the chip shrinkage.

It is considerably easier to determine experimentally how the mechanical properties of the material being machined affect the process of chip formation than to do so for its chemical properties. For this purpose we selected beryllium bronze BrB2 as the material being machined, imparting different hardnesses to it by means of heat treatment. The experiments showed that a change in the mechanical properties causes a change in all the characteristics of chip formation: contact length, mean contact stresses, friction force, and chip shrinkage. As an illustration, in Fig. 11 we give the results of measuring chip shrinkage when turning bronze of three hardnesses.

Fig. 11.

In accordance with the scheme in Fig. 8, the nature of the effect of the mechanical properties of the material being machined on the cutting process is determined by three lines of influence. First, with an increase in the strength properties of the machined metal, the shear stress $\tau_m$ in the contact zone of the chip increases, and, according to formula (12), the limiting contact pressure increases. The contact length in this case will decrease, thereby reducing the magnitude of the friction force.

Second, simultaneously with the stress $\tau_t$, the mean tangential contact stress $q_F$ will also increase, which will, on the contrary, increase the friction force. Experiments show that the first line of influence dominates in this case, and the friction force decreases as the hardness of the material being machined increases.

Finally, the mechanical properties of the material being machined have a direct effect on the behavior of the material within the primary plastic zone. Returning to the analogy with the action of a punch (Fig. 6), one may assert that, for the same value of the vector of the resultant force, the dimensions of the plastic zone under the punch, and consequently also the chip compression, will be the greater, the softer the material being machined.

The influence of speed on the chip-formation process also occurs through several channels. The mechanism of this influence has already been discussed in the literature [1], and we shall not dwell on it here.

As for the influence of the elements of the cross section of the cut on the cutting process, it is twofold. On the one hand, both the depth of cut and the feed affect the mean temperature of contact between the chip and the tool: as they increase, the latter rises. On the other hand, each of these parameters plays the role of a geometric factor: with increasing feed, the thickness of the cut and the contact length increase, while the depth of cut, through the width of the cut, determines the magnitude of the friction force.

The plan angle of the tool performs an approximately analogous role; in addition to the cutting temperature, it determines the shape of the transverse section of the cut, that is, the ratio between the thickness and the width of the cut.

The rake angle is one of the most important external factors. This angle determines the orientation of the contact zone on the rake face with respect to the primary plastic zone and thereby the degree of constraint of flow in this zone. Geometrically, the rake angle, together with the friction force, determines the magnitude and direction of the resultant, that is, the vector $R_{\text{pl}}$, as is shown in the diagram.

REFERENCES

1. N. N. Zorev, Problems in the Mechanics of the Process of Cutting Metals. Mashgiz, Moscow, 1956.

2. N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity. Publishing House of the Academy of Sciences of the USSR, Moscow–Leningrad, 1949.

3. M. F. Poletika and M. Kh. Utёshev, “Investigation of the cutting process by the polarization-optical method,” Izvestiya TPI, vol. 113, 1964.

4. T. N. Loladze, Wear of the Cutting Tool. Mashgiz, Moscow, 1958.

5. A. M. Rozenberg and A. N. Eremin, Elements of the Theory of the Process of Cutting Metals. Mashgiz, Moscow–Sverdlovsk, 1956.

6. N. N. Zorev, “Some problems of the science of metal cutting and the mechanics of the cutting process,” in Problems of Metal Cutting, Publishing House of the Moscow House of Scientific and Technical Propaganda named after F. E. Dzerzhinsky, Moscow, 1963.

7. Friction and Boundary Lubrication. Ed. Prof. I. V. Kragel’skii, IL, Moscow, 1953.

8. L. F. Coffin, “A study of the sliding of metals with particular reference to atmosphere,” Lubrication Engineering, 1956.

9. A. E. Roach, C. L. Goodzeit, and R. P. Hunnicutt, “Scoring characteristics of thirty eight different elemental metals in high-speed sliding contact with steel,” Trans. ASME, vol. 78 (1956), No. 8.