Chip Deformation in Force Cutting of Metals

P. N. OBUKHOV

(Presented by Prof., Doctor of Technical Sciences A. M. Rozenberg)

The process of cutting metals at large feeds differs from free cutting and ordinary constrained cutting in that the deformation of the metal during cutting, as a result of the presence of a second cutting edge parallel to the feed, substantially changes the process of chip formation. If the process of chip formation in ordinary constrained cutting may be considered analogous to the process of chip formation in free cutting and thereby all further conclusions simplified, then the process of chip formation when cutting with a tool having two cutting edges, at feeds equal to or greater than the depth of cut, differs substantially from the process of chip formation in free cutting. In this case, the study of the shape and position of the shear surface, and the distribution and measurement of the degree of deformation in the deformation zone during cutting, acquire great importance. The study of these questions is also necessary from the standpoint of the further introduction of the highly productive method of metal machining—force cutting with a tool of the design type of V. Kolesov, which is a particular form among the forms of tools with which it is possible to remove reversed or equal-sided chips.

Taking into account that in the literature up to the present time there are no sufficiently complete investigations of the questions of measuring chip shrinkage, the direction of chip flow along the rake face, and the position and shape of the shear surface in cutting with reversed and equal-sided chips, the present article will set forth the results of investigations of these questions.

General representation of the process of chip formation in cutting with a tool having two and three cutting edges

Figure 1 presents a diagram of the process of chip formation in free cutting. If chip widening is disregarded, then in this case the plane of shear may be considered to be the plane \(ABDC\), situated at the same angle \(\beta_1\) to the direction of cutting motion in any longitudinal section. In reality this will not be a plane, but rather a certain shear zone, or deformation zone, whose thickness is very small and amounts to tenths and even hundredths of a millimeter; this is indicated, for example, by T. N. Loladze [1] and other authors. If this zone is taken as a plane, then in the case of free cutting the area of the shear plane is determined very simply:

\(\dfrac{a \cdot b}{\sin \beta_1}\), where \(a\) is the thickness of the cut and \(b\) is the width of the cut. The direction of motion of the chip along the rake face will be perpendicular to the cutting edge \(BD\).

As is evident from the diagram, the shape of the shear plane can be judged from the line of intersection of the shear plane with the side surface of the layer being cut, \(BACD\), and from the cutting edge \(BD\), from which the shears begin.

In order to represent the possible shape of the shear surface when cutting with a two-edged cutter, special experiments on “cutting” plasticine were carried out.

Fig. 1

Fig. 2

From such a plastic material, chips of considerable size were cut off \((s \times t = 10 \times 10 \text{ mm})\), and by the method of stopping the cutting process, chip roots were obtained. Examination of the chip roots made it possible to represent the scheme of formation of a straight chip, as shown in Fig. 2, and of a reverse chip in Fig. 3.

An equilateral chip (Fig. 2) was obtained when the section of the cut \(ABCD\) was a rhombus, with side \(AB = BC\). Such a chip was called equilateral precisely because the cross-section of the cut in its formation is a square, or an equilateral rhombus.

Fig. 3

On the root of the “chip” made of plasticine, the boundary of the transition of the layer being cut into the chip \(AD'C\) was visible. The upper edge of the “chip,” \(D'D_2\), was formed by the transition of the edge \(D'D''\) of the layer being cut into the “chip.” The motion of the “chip” along the front face of the tool corresponded to the direction \(BD \perp AC\) (perpendicular to the diagonal of the section of the cut \(AC\)).

If, with such a cutting method, the deformation zone is very narrow and may be regarded as a surface, and if shears begin from the cutting edge itself, then it may be assumed that, in the case of formation of an equilateral chip, the shear surface will consist of two intersecting planes in the form of the triangles \(AD'B\) and \(BD'C\). In the case of formation of a chip of trapezoidal section (as occurs when cutting with feeds somewhat smaller and, conversely, with greater depths of cut), the shear surface will consist of two intersecting planes in the form of a triangle and a trapezoid. In this case, if a reverse chip is obtained (when cutting with \(\dfrac{s}{t} > \dfrac{1}{\sin \varphi}\), as shown in Fig. 3), then the triangular plane \(AD'B\) is located on the main cutting edge, and the trapezoidal plane \(BD'D_1'C\) on the auxiliary one; if, however, an ordinary chip is obtained, these planes from the cutting edges will change places.

Thus, in “cutting” plasticine it was established that, from the configuration of the cutting edge of the tool, from the shape of the transverse section of the chip, and from the shape of the chip root, one may judge relatively the shape of the shear surface and, consequently, the deformation process itself.

It turned out that, when cutting copper, lead, and steel, the shapes of the chip roots obtained in cutting with the corresponding feeds, depths of cut, and corresponding cutters are similar to the roots of “chips” made of plasticine. Moreover, in these cases the boundary of transition of the layer being cut into the chip was visible even more distinctly.

It can be shown in the following way that the shear surfaces are sufficiently close to planes and may be taken as planes in all calculations connected with determining chip deformation and cutting forces. By special experiments it was proved that the direction of chip flow along the rake face of the cutter coincides with the perpendicular to the diagonal \(AC\) of the cross section of the cut. Therefore the plane passing through the line of intersection of the shear surfaces \(BD'\) and through the perpendicular \(BD\) may be called the chip-flow plane. If the shear surfaces are planes, then in any sections of the chip root parallel to the chip-flow plane one and the same values, constant for the given root, of the angles \(\beta_1\) between the direction of the cutting motion and the direction of the shear surface must be obtained.

This assumption was checked when cutting steel 12KhN3A. This steel was chosen as the work material because its high plastic properties facilitated obtaining, in cutting, purely continuous chips.

Cutting was carried out on a “Hasse Wrede” engine lathe, with a cutter tipped with hard alloy T15K6. To obtain chip roots, the method of instantaneous stopping of the cutting process was used by means of a “falling cutter” device. With a cutter having a rake angle \(\gamma = +8^\circ\), with an auxiliary cutting edge arranged parallel to the feed, at \(s = 2.40\ \mathrm{mm/rev}\) and \(t = 1\ \mathrm{mm}\), a reverse-chip root was obtained. The chip root obtained was embedded in babbitt, and then longitudinal polished sections were prepared in three sections parallel to the chip-flow plane. On all three polished sections the boundary of transition of the work material into the chip was clearly visible in the form of a very thin, narrow zone beginning at the cutting edge and extending to the point at which the machined surface passes into the outer surface of the chip.

Analysis of such polished sections makes it possible to regard the deformation zone as a plane or a surface.

The angles \(\beta_1\) measured on all the polished sections in all sections proved to be equal to \(15^\circ\). It may be assumed that it is precisely in planes parallel to the chip-flow plane that the shears occur as a result of which the chip texture is formed, and that these shears occur in identical directions in any section of the chip. This is proved not only by the equality of the angles \(\beta_1\) in any longitudinal section parallel to the chip-flow plane, but also by the following experiment.

If it is assumed that the shears in the deformation zone occur in different directions depending on the configuration of the cutting edges, then obviously individual grains in the chip will be oriented differently, and this can then be detected on a polished section of the chip cross section. A special experiment was carried out. On a heavy machine tool (main-drive motor power 40 kW) a thick, unequal-sided chip was removed with a cutter having two cutting edges. The chip was removed at \(s = 4.2\ \mathrm{mm/rev}\), \(t = 3.70\ \mathrm{mm}\), and \(v = 75\ \mathrm{m/min}\) from a billet of steel 12KhN3A. Under this regime the chip obtained was continuous, with barely noticeable traces of individual elements on the lateral sides. The cross section of the chip was approximately \(19\ \mathrm{mm}^2\). A polished section of the cross section of this chip was prepared, on which, after etching with a 5% solution of nitric ac—

...the fibers a texture was detected. The study of the texture led to the conclusion that the grains are uniformly and identically oriented over the entire cross section of the chip.

Assuming that the deformation zone can be represented as a surface consisting of several planes in the case of non-free cutting, and that the shape of the shear surface is determined by the configuration of the cutting edges of the tool and is related to the shape of the transverse section of the chip, it is not difficult to represent the process of chip formation in cutting by a tool with a transition cutting edge located at an angle $\varphi'$ to the direction of feed. A scheme of such a process is shown in Fig. 4.

Fig. 4

As follows from the scheme, the shape of the shear surface in this case is rather complex and consists of 4 intersecting planes: on the main cutting edge—$\triangle AB_1B$, on the transition cutting edge—$\triangle BB_1C$, and on the auxiliary cutting edge—the polygon $CB_1D_1FD$ and $\triangle DFE$.

Determination of the Relative Shear

From the schemes considered for the process of formation of equal-sided and reverse chips, which are based on the proposition that shears in separate planes parallel to the chip-flow plane occur in the same direction and that therefore the angles $\beta_1$ in these planes are identical, one may conclude that the relative shears in these sections will also be identical, and their calculation can be carried out by the usual...

method [2]. To prove this, let us turn to Fig. 5, which shows the root of an equal-sided chip.

In this figure, \(q\) is the chip-flow plane; \(m\) is the plane in a section parallel to the chip-flow plane; \(acbea\) is the shear surface. The planes \(m\) and \(q\), when intersected with the shear surface, form the lines of shear direction, inclined at the same angle \(\beta_1\) to the direction of cutting motion. Since the main part of the chip deformation occurs in a very narrow deformation zone, which may be taken as the surface \(acbea\), then in section \(q\) the line of maximum shearing stresses may be considered to be the line \(ce\). The position of this line is determined by the angle \(\beta_1\). In this section let us isolate a certain volume of metal of width \(x\), in the form of a very thin plate located in this plane, in the form of the parallelogram \(cdfe\). Before deformation this parallelogram occupied the position \(cfge\), with point \(f\) lying on the edge \(cf\) of the machined surface. In the process of chip formation point \(f\) moved to point \(d\), lying on the outer crest of the chip, and point \(g\) moved to point \(f\). Then the relative shear in the given thin section will be

\[ \varepsilon = \frac{A}{x}, \]

where \(A\) is the length of the line \(ce\), or \(df\), or \(fg\).

Fig. 5

\[ A = x \operatorname{ctg} \beta_1 + x \operatorname{tg}(\beta_1 - \gamma), \qquad \text{whence} \]

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

The same can also be obtained in section \(m\):

\[ \varepsilon_1 = \frac{B}{x_1}, \]

where \(B\) is the length of the line \(c_1 l_1\), or \(d_1 f_1\), or \(f_1 g_1\).

\[ B = x_1 \operatorname{ctg} \beta_1 + x_1 \operatorname{tg}(\beta_1 - \gamma), \qquad \text{whence} \]

\[ \varepsilon_1 = \operatorname{ctg} \beta_1 + \operatorname{tg}(\beta_1 - \gamma). \]

Consequently, in the section \(m\), \(\varepsilon_1=\varepsilon\). This can also be obtained in any other section parallel to the plane of chip flow.

Consequently, to determine the relative shear \(\varepsilon\), it is necessary to know the angle \(\beta_1\) and the rake angle of the cutter.

Direction of chip flow along the rake face of the cutter

The direction of chip flow along the rake face of the cutter, which determines the position of the plane of chip flow and, consequently, determines the possibility of calculating the angle \(\beta_1\), depends to a considerable degree on the cutting conditions \(s\) and \(t\), and also on the configuration of the cutting edges of the cutter. If one proceeds from the relative role of the participation of each cutting edge of the cutter in the cutting process, then the direction of chip flow along the rake face of the cutter is determined as follows.

Let us call the chip-flow angle \(\psi\) the angle between the projection of the direction of motion of the chip along the rake face of the cutter onto the basic plane and the feed direction, and let us determine this angle for the simplest case of ordinary cutting with a cutter having rake angle \(\gamma=0^\circ\), with two cutting edges, one of which is located at an angle \(\varphi=0^\circ\) (Fig. 6).

Fig. 6

It was indicated earlier that, when cutting plastiline, it was observed that the direction of chip flow in all cases proves to be close to perpendicular to the diagonal of the area of cut. If one proceeds from the relative role of the cutting edges in the cutting process, then the direction of chip flow must be perpendicular to the diagonal of the area of cut, in the present case—to \(AM\). Then the simplest trigonometric calculations give:

\[ BO=s\cdot\cos\psi;\qquad BO=b'\cdot\cos(180-\varphi-\psi);\quad \text{whence:} \]

\[ s\cdot\cos\psi=b'\cdot\cos(180^\circ-\varphi-\psi). \]

Dividing both sides of the equation by \(b'\):

\[ \frac{s}{b'}\cos\psi=\cos\left[(180^\circ-\varphi)-\psi\right], \]

\[ \frac{s}{b'}= \frac{\cos(180^\circ-\varphi)\cos\psi}{\cos\psi} + \frac{\sin(180^\circ-\varphi)\sin\psi}{\cos\psi} = -\cos\varphi+\sin\varphi\cdot\tg\psi; \]

\[ \frac{s}{b'}\sin\varphi+\cos\varphi=\sin\varphi\cdot\tg\psi; \qquad \text{but } b'=\frac{t}{\sin\varphi},\ \text{then:} \]

\[ \frac{s}{t}\sin\varphi+\cos\varphi=\sin\varphi\cdot\tg\psi. \]

Dividing all terms of the equation by \(\sin\varphi\), we obtain:

\[ \frac{s}{t}+\ctg\varphi=\tg\psi, \qquad \text{whence finally:} \]

\[ \psi=\operatorname{arc\,tg}\left(\frac{s}{t}+\ctg\varphi\right). \]

It is not difficult to obtain the same result for reverse and equilateral chips, when \(s\gg t\).

The dependence obtained was checked experimentally in cutting 12KhN3A steel with a tool having two cutting edges. The experiments were conducted on a DIP-300 lathe, with tools having $\varphi = 60^\circ$, $\varphi_1 = 0^\circ$, at a constant cutting depth $t = 1$ mm and various feeds $s = 0.21 \div 2.40$ mm/rev. The results of the experiments are presented in Fig. 7.

Fig. 7

In this graph the solid line shows the theoretical dependence, and the dashed line shows the experimental results. Each experimental point represents the mean value in a series of 10–12 experiments at different cutting speeds from 30 to 120 m/min. The same graph presents chip-flow angles calculated from the ratio of the cutting-force components

$$ \frac{P_y}{P_x}, $$

measured with a hydraulic dynamometer, and chip-flow angles at $\gamma = +10^\circ$, if this angle is ground on the tool not from one or another cutting edge, but in the direction of chip flow. In these experiments, up to a feed of $s = 1.15$ mm/rev ordinary chips were obtained, and at larger feeds—reverse chips.

As follows from the graph, the experimental results deviate from the theoretical curve by a maximum of 6%, for ordinary chips toward a decrease, and for reverse chips toward an increase in $\psi$. Evidently, the prevailing influence of the cutting edge that plays the greater role in the cutting process has some effect. However, such a small difference will not produce a large error in practical calculations.

It is very important that the chip-flow angles calculated from the ratio of the cutting-force components,

$$ \frac{P_y}{P_x}, $$

are very close to those measured in the experiment. This indicates that the deformation process determines the magnitude and direction of the resultant cutting force, and the position of its direction in the chip-flow plane (as evidenced by the experimental results) makes it possible to calculate the cutting forces in the horizontal plane, $P_x$ and $P_y$, from the chip-flow angle $\psi$.

It should also be pointed out that the rake angle $\gamma$ does not influence the chip-flow angle if it is ground on the tool in the direction of chip flow.

The formula for calculating the chip-flow angle will change somewhat in the case of cutting with a tool having a transition cutting edge. In this case,

the direction of chip flow likewise remains perpendicular to the diagonal of the area of cut. Trigonometric calculations analogous to the preceding ones lead to the following formula for computing the chip-flow angle $\psi$ in this case:

\[ \psi=\operatorname{arc\,tg}\left[\operatorname{tg}\varphi\left(1-\frac{a}{t}\sin\varphi'\right)+\frac{a}{t}\cos\varphi'+\frac{s}{t}\right], \]
where:

$\varphi$ — the principal plan angle,
$\varphi'$ — the plan angle of the auxiliary cutting edge,
$s$ — feed,
$t$ — depth of cut,
$a$ — length of the auxiliary cutting edge.

In the same way it is possible to determine chip-flow angles for the most varied geometry of the rake face of a cutting tool.

Chip widening

The chip removed in the process of ordinary cutting proves, as a rule, to be wider than the width of the cut; that is, chip deformation occurs also in the direction across the chip width. The same must take place in the process of force cutting. But if in ordinary cutting the width of the cut and the width of the chip are quite definite concepts, then in the process of force cutting, especially when the chip is of equal side, the concept of the width of the chip and of the cut, and also that of the widening process itself, proves to be indeterminate. Indeed, what is to be taken as the width of the chip, which in cross section represents a trapezoid or a triangle? Or what is to be taken as the width of the cut if the section of the layer being cut represents a parallelogram or a rhombus?

Let us consider the case of formation of an equal-sided chip when cutting with a tool with $\gamma=0^\circ$ and for this purpose turn to Fig. 2. The area of cut in this case is the rhombus $ABCD$ with diagonal $AC$. It may be assumed that, during cutting, the metal must widen uniformly in all free directions, proportionally to the thickness of the layer in the corresponding direction; moreover, during cutting this widening will be greatest at the lower base of the chip and will vanish as it approaches the chip vertex $D'$.

Fig. 8

In the given cutting case, the metal has the possibility of widening only in the directions along the cutting edges from vertex $B$, that is, in the directions $BA$ and $BC$. Consequently, it may be assumed that the widening in the direction $BA$ will be equal to the segment $AF$, proportional to $BA$ (Fig. 8), and in the direction $BC$ equal to the segment $CE$, proportional to the length of the cutting edge $BC$.

It is not difficult to see that the ratios

\[ \frac{BF}{BA}=\frac{BE}{BC}=\frac{FE}{AC}=\zeta_b, \]

(where $\zeta_b$ is the chip widening, always $>1$); that is, in any case the widening may be taken as the ratio of the width of the base of the real chip, which is easy to measure on the chip, to the diagonal of the area of cut, calculated from the cutting conditions $s$, $t$ and the geometry of the rake face of the cutter.

Determination of the Angle \(\beta_1\) and the Area of the Shear Surface

Taking into account the basic propositions considered in the preceding sections, the process of formation of an ordinary chip in cutting with a two-edged tool with a positive rake angle \(\gamma\) may be represented schematically as shown in Fig. 9. In this figure the hatched planes form the shear surface; the layer being cut is shown by a solid thick line, and the dash-dotted line shows the part of the chip of trapezoidal cross section flowing along the rake face of the tool. Under force-cutting conditions, in all cases the chip obtained is of rather considerable dimensions; therefore, without substantial error, for such a chip one can measure the width of the lower base \(A_1C_1' \) or \(A_2C_1'\) and its height \(O_1O'\) or \(O_3'O_1'\). Denote \(A_1C_1=b\), and \(O_1O'=a_1\). Knowing these dimensions, as well as \(s\), \(t\), \(\varphi\), \(\gamma\), it proves possible to determine geometrically the angle \(\beta_1\) and the area of the shear surface \(Q\) occurring under the given cutting condition.

Fig. 9

In accordance with the designations given in Fig. 9, the derivation of the formulas may be presented briefly in the following order:

\[ \beta_1=\beta_1' + \gamma; \]

\[ \beta_1'=\frac{BO_1}{a_1};\quad BO_1=BO_2-O_1O_2;\quad O_1O_2=a_1\operatorname{tg}\gamma;\quad BO_2=\frac{BO_2'}{\cos\gamma}. \]

From \(\triangle BO_2'C_0\):

\[ BO_2'=\frac{s}{\sin\psi\,(\operatorname{ctg}\varphi+\operatorname{ctg}\psi)} \]

and then:

\[ BO_1=\frac{s}{\sin\psi\cdot\cos\gamma\cdot(\operatorname{ctg}\varphi+\operatorname{ctg}\psi)}-a_1\cdot\operatorname{tg}\gamma, \]

finally:

\[ \operatorname{tg}\beta_1' = \frac{s}{a_1\cdot\sin\psi\cdot\cos\gamma\cdot(\operatorname{ctg}\varphi+\operatorname{ctg}\psi)} -\operatorname{tg}\gamma. \]

Thus, the angle \(\beta_1\) is determined. For the subsequent calculations the angles \(\beta_2'\) and \(\beta_3'\) will be needed. The formulas for determining them are obtained very simply:

\[ \operatorname{tg}\beta_2'=\frac{EO_1}{a_1}; \qquad EO_1=BO_1\cdot\sin(\varphi+\psi), \]

then

\[ \operatorname{tg}\beta_2' = \frac{BO_1\cdot\sin(\varphi+\psi)}{a_1} = \operatorname{tg}\beta_1'\cdot\sin(\varphi+\psi). \]

\[ \operatorname{tg}\beta_3'=\frac{FO_1}{a_1}; \qquad FO_1=BO_1\cdot\sin\psi, \]

then

\[ \operatorname{tg}\beta_3' = \frac{BO_1\cdot\sin\psi}{a_1} = \operatorname{tg}\beta_1'\cdot\sin\psi. \]

Let us now derive equations for calculating the area of the shear surface \(Q\).

\[ Q=Q_{A_1D'O'B}+Q_{BO'C}. \]

\[ Q_{A_1D'O'B} = \frac{A_1B+D'O'}{2}\,EO'; \qquad A_1B=AB\cdot\zeta_b. \]

If we assume that \(\angle A_0BA\simeq \angle C_0BC\simeq\gamma\), then

\[ AB=\frac{A_0B}{\cos\gamma} = \frac{b'}{\cos\gamma} = \frac{t}{\sin\varphi\cdot\cos\gamma}; \qquad A_1B=\frac{t\cdot\zeta_b}{\sin\varphi\cdot\cos\gamma}. \]

\[ D'O'=DO_2=DC-O_2C; \qquad DC=AB=\frac{t}{\sin\varphi\cdot\cos\gamma}. \]

From \(\triangle BO_2C\):

\[ O_2C=\frac{s}{\sin\varphi\cdot\cos\psi\,(\operatorname{ctg}\varphi+\operatorname{ctg}\psi)}, \]

and then

\[ D'O' = \frac{t}{\sin\varphi\cdot\cos\gamma} - \frac{s}{\sin\varphi\cdot\cos\gamma\,(\operatorname{ctg}\varphi+\operatorname{ctg}\psi)}; \qquad EO'=\frac{a_1}{\cos\beta_2'}. \]

Then we finally obtain:

\[ Q_{A_1D'O'B} = \left[ t(\zeta_b+1) - \frac{s}{(\operatorname{ctg}\varphi+\operatorname{ctg}\psi)} \right] \frac{a_1}{2\cdot\sin\varphi\cdot\cos\gamma\cdot\cos\beta_2'}. \]

The area of the triangle \(BO'C\) is determined simply:

\[ Q_{BO'C} = \frac{BC_1\cdot FO_1}{2}; \qquad BC_1=\frac{s\cdot\zeta_b}{\cos\gamma}; \qquad FO_1=\frac{a_1}{\cos\beta_3'}. \]

\[ Q_{BO'C}=\frac{s\cdot a_1\cdot \zeta_b}{2\cdot \cos\gamma\cdot \cos\beta_3'}. \]

The formula for determining the total area of the shear surface after substitution and transformations will be:

\[ Q=\frac{a_1}{2\cos\gamma}\left\{\frac{1}{\sin\varphi\cdot \cos\beta_2'}\left[t(\zeta_b+1)-\frac{s}{(\operatorname{ctg}\varphi+\operatorname{ctg}\psi)}\right]+\frac{s\cdot \zeta_b}{\cos\beta_3'}\right\}. \]

Fig. 10

In an analogous manner, the relations for a reverse chip are also derived. The scheme of formation of such a chip is shown in Fig. 10, and the final form of the formulas will be as follows: \(\beta_1=\beta_1'+\gamma;\)

\[ \tan\beta_1'=\frac{t^1}{a_1\cdot \sin\psi\cdot \cos\gamma}-\tan\gamma; \]

\[ \tan\beta_2'=\tan\beta_1'\cdot \sin(\varphi+\psi); \]

\[ \tan \beta_3'=\tan \beta_1'\cdot \sin \psi; \]

\[ Q=\frac{a_1}{2\cos \gamma} \left\{ \frac{1}{\sin \varphi} \left[ \frac{t\cdot \zeta_b}{\cos \beta_2'} - \frac{t'\cdot \sin(\varphi+\psi)}{\sin \psi\cdot \cos \beta_3'} \right] + \frac{s(\zeta_b+1)}{\cos \beta_3'} \right\}. \]

Fig. 11

In these formulas \(t'\) is a dimension that takes into account the influence of the radius of the workpiece on the position of the boundary of transition of the cut layer \(O'D_2'\) into the chip. This is shown schematically in Fig. 11, from which it is evident that if the cut layer had no curvature, then the line of transition of the machined surface into the chip would be projected at point \(K\), and not at point \(O'\), as actually occurs. From this figure \(t'\) is very simply determined:

\[ t'=r\cdot \sin\left(\arccos \frac{a_1}{r}\right)-(r-t). \]

Thus, from the external parameters of the chip and the cutting conditions, the angle \(\beta_1\) and the area of the shear surface in force cutting with a tool without a transitional cutting edge are calculated. In the case of formation of an equal-sided chip, the formulas may be used both for an ordinary chip and for a reverse chip.

Determination of the Longitudinal Shrinkage of the Chip

At present, the longitudinal shrinkage of the chip \(\zeta_e\) is calculated from the ratio of the length of the cut to the length of the removed chip, \(l_o/l_1\), proceeding from the equality of the volumes of the cut layer and the chip, whose weight will be the same. The weight of the cut layer may be expressed by the formula: \(G=l_o\cdot s\cdot t\cdot d_{yo}\), where:

\(G\) — weight of the cut layer,
\(l_o\) — length of the cut layer,
\(d_{yo}\) — specific weight of the metal being machined.

Hence the length of the cut may be found:

\[ l_o=\frac{G}{s\cdot t\cdot d_{yo}}\ \text{mm}. \]

The weight \(G\) is found by weighing the removed chip. Consequently, the longitudinal shrinkage of the chip will be

\[ \zeta_e=\frac{l_o}{l_1}=\frac{G}{e_1\cdot s\cdot t\cdot d_{yo}}, \]

where:

\(l_1\) — length of the removed chip.

But to determine the angle \(\beta_1\), it is necessary to know not the longitudinal shrinkage of the chip, but the shrinkage by thickness, that is, \(a_1/a_o\), where \(a_1\) is the chip thickness, and \(a_o\) is the cut thickness; otherwise a correction for chip widening must be introduced. Let the volume of the cut layer be \(v=l_o\cdot a_o\cdot b_o\ \text{mm}^3\), and the volume of the chip \(v=l_1\cdot a_1\cdot b_1\), where

\(b_o\) — width of the cut layer,
\(b_1\) — width of the chip.

Then \(l_o \cdot a_o \cdot b_o = l_1 \cdot a_1 \cdot b_1\), and if \(\dfrac{b_1}{b_o}=\zeta_b>1\) (widening), then

\[ \frac{a_o}{a_1}=\frac{l_o}{l_1 \cdot \zeta_b}=\frac{\zeta_e}{\zeta_b}, \]

in other words, the shrinkage in thickness will be less than the longitudinal shrinkage by the amount of chip widening \(\zeta_b\).

Then

\[ \operatorname{tg}\beta_1= \frac{\cos\gamma}{\dfrac{a_1}{a_o}-\sin\gamma} = \frac{\cos\gamma}{\dfrac{\zeta_e}{\zeta_b}-\sin\gamma}. \]

Thus, the longitudinal shrinkage of the chip and the angle \(\beta_1\) are determined for the case of ordinary non-free cutting, when the feed is considerably less than the depth of cut.

When cutting with a tool having two cutting edges, the thickness of the cut \(a_o\) and the chip thickness \(a_1\) assume an indeterminate expression, since they are variables in different longitudinal sections, although their ratio \(\dfrac{a_1}{a_o}\) remains constant. Therefore it is obvious that the given method for determining the longitudinal shrinkage of the chip in force cutting will be formal and will not reflect the actual picture of the process.

It was indicated above that the angles \(\beta_1\) in different longitudinal sections of the chip when cutting at large feeds will be identical, and a method was given for calculating these angles when cutting with a two-edged tool. Knowing the angle \(\beta_1\), one can determine the ratio \(\dfrac{a_1}{a_o}\), which will also be the same in any section of the chip, by the formula:

\[ \operatorname{tg}\beta_1= \frac{\cos\gamma}{\dfrac{a_1}{a_o}-\sin\gamma}, \qquad \text{whence} \qquad \frac{a_1}{a_o}=\frac{\cos\gamma}{\operatorname{tg}\beta_1}+\sin\gamma. \]

This is the second variant for calculating the longitudinal shrinkage of the chip.

Finally, from the presented deformation diagrams one can obtain a third variant of the formula for determining \(\zeta_l\). If the longitudinal shrinkage of the chip is the ratio of the cross section of the chip to the cross section of the cut, then for an ordinary chip one can write:

\[ \zeta_l= \frac{\text{cross section of the chip}}{s \cdot t} = \frac{\text{area } A_2D'O_1'C_1'}{s \cdot t} \quad (\text{Fig. }9) \]

or

\[ \zeta_l= \left(A_1C_1+D_1'O_1'\right) \frac{O_3'O_1'}{2 \cdot s \cdot t}, \]

where: \(A_1C_1=b_1\) (chip width), and \(O_3'O_1'=O_1O'=a_1\) (chip thickness). As a result of certain additional constructions and trigonometric calculations, one can determine \(D_1'O_1'\) as:

\[ D_1O_1'= \frac{\sin(\varphi+\psi)}{\sin\varphi} \left[ t-\frac{s}{(\operatorname{ctg}\varphi+\operatorname{ctg}\psi)} \right], \]

then:

\[ \zeta_l= \left\{ b_1+ \frac{\sin(\varphi+\psi)}{\sin\varphi} \left[ t-\frac{s}{(\operatorname{ctg}\varphi+\operatorname{ctg}\psi)} \right] \right\} \frac{a_1}{2 \cdot s \cdot t}. \]

It should be borne in mind that here:

\(b_1\) — chip width, mm,
\(a_1\) — chip thickness, mm,
\(s\) — feed, mm/rev,
\(t\) — depth of cut, mm,
\(\varphi\) — plan angle,
\(\psi\) — angle of chip flow along the rake face.

The quantities \(a_1\) and \(b_1\) can be measured on the chip itself with calipers with sufficient accuracy, if the burrs and ridges are filed down to half their size.

In an analogous way one can obtain an expression for determining the longitudinal shrinkage of a reverse chip:

\[ \xi_l = \left\{ b_1+ \left[ s-\frac{t'\cdot \sin(\varphi+\omega)} {\sin\varphi\cdot \sin\psi} \right]\sin\psi \right\} \frac{a_1}{2\cdot s\cdot t}. \]

For an equal-sided chip both formulas are applicable.

An experimental verification was carried out of the chip shrinkages calculated by all three variants in cutting steel 12KhN3A. The carbide tool T15K6 had the geometry: \(\varphi=60^\circ\), \(\varphi_1=0^\circ\), \(\gamma=+10^\circ\). The depth of cut in all experiments was 1 mm, and the feeds had the values 0.21—1.20—2.40 mm/rev, i.e., ordinary, equal-sided, and reverse chips were obtained.

The results of the experiments are summarized in Table 1. As follows from the table, there is some difference in the shrinkages calculated by the different variants. Thus the shrinkage calculated by the weight of the chip for reverse chips is somewhat smaller than by the formulas. This is evidently explained by an error in calculating the shrinkage by weight, since traces of individual elements were nevertheless noticeable on all chips, which indicates the not purely continuous character of the chip; with a somewhat greater length, a smaller shrinkage is obtained.

Table 1

Note: For the equal-sided chip, the fraction indicates: in the numerator, \(\zeta_1\), calculated by the formula for an ordinary chip; in the denominator, by the formula for a reverse chip.

But this difference is small, and the results may be considered to agree satisfactorily. A somewhat greater difference is obtained in the shrinkages of equal-sided chips calculated according to variants I or II and III; moreover, variants II and III differ quite insignificantly.

Conclusion

1. The processes of chip formation in force cutting may be regarded as proceeding by means of shears in individual layers of metal as they pass through the deformation zone.

2. The shear zone, which is the zone of maximum tangential stresses, is a very narrow region which, in the case of cutting by a tool with several cutting edges, may be represented as part of the lateral surface of a pyramid or truncated pyramid constructed on the cutting edges of the tool.

3. The shear surface has a quite definite shape in accordance with the configuration of the cutting edges of the tool, and a direction characterized by identical angles \(\beta_1\) in different longitudinal sections of the chip parallel to the plane of chip flow, which accounts for the equality of chip shrinkage in these sections.

4. The shape of the shear surface determines the transverse section of the chip and the direction of chip flow along the rake face of the tool, which in the general case may be taken as perpendicular to the diagonal of the cut area.

5. The relative shears \(\varepsilon\) in the formation of ordinary equal-sided and reverse chips may be calculated by the usual method.

6. The deformation schemes presented make it possible to determine the angle \(\beta_1\) and the areas of the shear surface from the external parameters of the chip and the cutting conditions.

1. It proves possible to calculate the longitudinal shrinkage of the chip by means of the three proposed variants.

2. In all cases, the widening of the chip is the ratio of the lower base of the chip’s cross section to the diagonal of the cutting section.

REFERENCES

1. T. N. Loladze, Chip Formation in Metal Cutting. Mashgiz, 1952.

2. A. M. Rozenberg and A. N. Eremin, Plastic deformation of the chip in metal cutting. Proceedings of the Tomsk Polytechnic Institute, vol. 75, 1954.

3. P. N. Obukhov, Investigation of the process of force cutting of steels. Author’s abstract of dissertation, Tomsk, 1955.