Structure of the Chip in Cutting Plastic Metal

A. M. Rozenberg and A. N. Eremin

A chip removed from a plastic metal, having a continuous character, upon microscopic examination reveals a quite definite structure. If we schematically represent a longitudinal section through the chip, we obtain the view shown in Fig. 1.

The removed chip has, on the polished section, a layered structure. The impression is created that the chip consists of separate plates inclined to the direction of motion of the cutter at some constant angle \(\beta_2\). On approaching the rake face of the cutter, these layers in the chip bend in the direction opposite to the motion of the chip along the rake face, assuming a position almost parallel to the rake face of the cutter. This rotation of the layers is the result of the action of a considerable frictional force between the chip and the cutter, which impedes the movement of the chip and retards the metal of the chip that is in direct contact with the rake face of the cutter.

Fig. 1

The rectilinear layers of the chip are interrupted on passing into the main mass of the metal along a certain straight line located at an angle \(\beta_1\) to the direction of motion of the cutter.

In the chip one should distinguish two stages of plastic deformation. The first stage creates the chip layers inclined at the angle \(\beta_2\).

The second stage, under the influence of friction of the chip against the rake face, produces curvature of the layers in the immediate vicinity of the rake face of the cutter. The second deformation is superposed on the first, owing to which the chip zone,

adjacent to the front face of the cutter, proves to be considerably more deformed and to have a considerably higher hardness than the rest of the chip metal. The thickness of the secondary-deformation zone is very small and usually amounts to less than 10% of the total chip thickness.

Until now no unified opinion has been established concerning the mechanism of formation of the chip layers inclined at the angle $\beta_2$. Some authors consider that the direction of the layers is the direction of shear in the chip and that, consequently, the shear angle is the angle $\beta_2$. Others hold the view that the shear angle is $\beta_1$, while at the angle $\beta_2$ there are situated the axes of the metal grains elongated under the influence of shear deformation proceeding at the angle $\beta_1$ to the direction of motion of the cutter. This disagreement is of fundamental importance, since for a given chip shrinkage the degree of its deformation will be different depending on the direction in which we regard the shear as taking place (at the angle $\beta_1$ or at the angle $\beta_2$ to the direction of motion of the cutter). Thus, one cannot speak of the degree of plastic deformation of the chip, nor can one calculate it, without first establishing in what direction this deformation proceeds.

We consider that the shear deformation as a result of which the chip is formed proceeds at the angle $\beta_1$ to the direction of motion of the cutter. The chip layers situated at the angle $\beta_2$ are formed in the following way. In the formation of a chip by shear, a certain volume of metal before deformation, $bfde$ (Fig. 2), after deformation has assumed in the chip the position $bfhn$.

Fig. 2

The transformation of the figure $bfde$ into the figure $bfhn$ occurred by the shear of the upper side from the position $de$ to the position $hn$, with the lower side immobile. In this shear no change in the dimensions of the shifted upper side occurred, and $hn = de$.

Let us select in the metal before deformation a certain elementary volume of it, in the form of a cube, the lateral side of which is the square $obcd$, and let us place it so that the vertex $b$ coincides with the cutting edge of the cutter, while the vertex $d$ lies on the upper contour of the layer being cut off, $a$, and the sides $ob$ and $cd$ coincide with the direction of shear. Let this elementary volume represent a metal grain before its deformation. After shear deformation the upper base of the square $de$ will move to the position $hg$, and the point $d$, originally situated on the upper contour of the layer being cut off, will inevitably find itself at the point $h$, situated on the upper contour of the chip. The square $obcd$ will become the parallelogram $obgh$. The axis of symmetry of the square $bd$, originally situated at an angle of $45^\circ$ to the direction of shear, will become the diagonal of the parallelogram $bh$, situated at the angle $\psi$ to the direction of shear, with $\psi < 45^\circ$. A real microscopically small grain of round shape under such deformation is trans-

is transformed into an elongated plate with its longitudinal axis situated at some angle $\psi$ to the direction of shear. It is not difficult to express the angle $\psi$ as a function of the angles $\beta_1$ and $\gamma$.

\[ \operatorname{tg}\psi=\frac{bc}{ch}=\frac{bc}{cn+nh} =\frac{bc}{bc\cdot \operatorname{tg}(\beta_1-\gamma)+\dfrac{a}{\sin\beta_1}}, \]

\[ bc=od=\left(\frac{a}{\sin\beta_1}-ob\right)\operatorname{tg}\beta_1 =\left(\frac{a}{\sin\beta_1}-bc\right)\operatorname{tg}\beta_1, \]

\[ bc=\frac{a}{\sin\beta_1+\cos\beta_1}. \]

Upon substitution, after simple transformations we obtain

\[ \operatorname{ctg}\psi=1+\operatorname{ctg}\beta_1+\operatorname{tg}(\beta_1-\gamma). \tag{1} \]

Here $\gamma$ is the rake angle of the cutter,

$\beta_1$ is the angle determining the direction of shear during cutting.

The angle $\beta_1$ is related to the chip contraction by the dependence

\[ \operatorname{tg}\beta_1=\frac{\cos\gamma}{\zeta-\sin\gamma}, \]

where $\zeta$ is the chip contraction, which must be determined taking into account the widening of the chip in the process of its deformation.

To check our views on the role of the angles $\beta_1$ and $\beta_2$ in the process of chip formation, we carried out an experimental verification of formula (1). For this it was necessary to compare the value of $\psi$, calculated from formula (1), with its value measured directly on a chip section. To calculate the angle $\psi$ it is necessary to know the angle $\beta_1$, which can be determined for each given case from the chip contraction.

To measure the angle $\psi$ directly on the chip section, it would have been necessary to bring the cutting process to an instantaneous stop, cut the chip together with the metal to which it is connected out of the blank, make a longitudinal cut through the chip and metal, and prepare a polished section. Such a procedure involves a number of difficulties, since, first of all, at high cutting speeds it is very difficult to produce a truly instantaneous stopping of the cutting process. Therefore, in determining $\psi$ in the chip, we proceeded in a different way. The cutting process was not interrupted; instead, a chip produced under certain conditions of interest to us was taken, its longitudinal cut was made, and a polished section was prepared, which had the form schematically shown in Fig. 3.

With the aid of a microscope having crosshairs and a degree scale in the eyepiece, the angle was measured between the direction of the layers and the tangent to the side of the chip that had slid along the rake face of the cutter.

It is not difficult to see that

\[ \psi=90^\circ-\xi+\gamma-\beta_1. \]

Such a comparison of the angles $\psi$, calculated by formula (1) and measured on the chip, was carried out by us for a wide range of rake angles, from $\gamma=+30^\circ$ to $\gamma=-11^\circ$, over a wide range of cutting speeds from 30 to 400 m/min and chip contractions from $\zeta=2$ to $\zeta=6.85$. For all these measurements the chips were taken under such conditions that no built-up edge formed on the cutter, which could distort the magnitude of the rake angle and introduce inaccuracies into the calculations.

The results of the experiments are summarized in a graph (Fig. 4), on which the curve is drawn according to equation (1), while the points are plotted from the results of direct measurement of the angle $\xi$.

On the graph, for each angle $\gamma$ one should have drawn its own curve $\psi = f(\beta_1)$, but all these curves are so close to one another that they have been replaced by a single one. The graph shows that the points obtained as the result of

Fig. 3

measurement are distributed uniformly around the theoretical curve. The greatest deviation does not exceed $5^\circ$ and can be explained partly by measurement errors, and partly by the fact that in each chip both the shrinkage and the angles $\beta_1$ and $\psi$ fluctuate within certain limits; we determined the angle $\beta_1$ from the mean shrinkage and therefore obtained the mean value of the angle $\beta_1$, whereas the angle $\psi$ was determined by measuring particular (and not mean) values of the angle $\xi$.

Visible labels in Fig. 4: vertical axis $\psi^\circ$; horizontal axis $\beta_1^\circ$.

Legend in Fig. 4:

• ○ $\gamma=-10^\circ$, $v=67\text{–}400$ m/min
• △ $\gamma=15^\circ$, $v=104\text{–}191$ m/min
• □ $\gamma=10^\circ$, $v=30$ m/min
• ◊ $\gamma=20^\circ$, $v=69$ m/min
• × $\gamma=0^\circ$, $v=41\text{–}386$ m/min
• ρ $\gamma=30^\circ$, $\{\,v=68\text{–}164$ m/min; [[unclear: symbol]] $=0.0011\text{–}0.092\,\}$

Fig. 4

The good arrangement of the experimental points relative to the theoretical curve confirms the correctness of our conception that shear in the cutting process proceeds at the angle $\beta_1$ to the direction of the cutting speed, while the angle $\xi$ determines the arrangement of the axes elongated by the process of shear of the metal grains.

Turning to Fig. 4, one can see that the angle $\psi$ increases with an increase in $\beta_1$, i.e., with a decrease in chip shrinkage, i.e., with a decrease in the degree of deformation of the chip, since at a constant rake angle, with a decrease in

the chip shrinkage decreases invariably, and so does the degree of its deformation. Deformation in the chip will be absent if \(\psi=45^\circ\).

Using the equation

\[ \operatorname{ctg}\psi = 1 + \operatorname{ctg}\beta_1 + \operatorname{tg}(\beta_1-\gamma) \]

one can decide at what shrinkage the least-deformed chip will be obtained. For this purpose we take the first derivative \(\dfrac{d\psi}{d\beta_1}\), set it equal to zero, and find

\[ \frac{d\psi}{d\beta_1} = -\frac{1}{\sin^2\beta_1} + \frac{1}{\cos^2(\beta_1-\gamma)} =0, \]

whence

\[ \operatorname{tg}\beta_1=\frac{\cos\gamma}{1-\sin\gamma}. \]

This equality will be valid if \(\zeta=1\). Thus, the greatest value of the angle \(\psi\), i.e., the least deformation of the chip, can be obtained if the chip shrinkage \(\zeta\) is equal to unity. For all other values of shrinkage, both greater and less than unity, the degree of deformation of the chip will increase.

Deformation in the chip will be absent if \(\psi=45^\circ\), i.e., if \(\operatorname{ctg}\psi=1\), i.e., if

\[ \operatorname{ctg}\beta_1+\operatorname{tg}(\beta_1-\gamma)=0. \]

Expressing \(\beta_1\) through \(\zeta\) and \(\gamma\), we obtain

\[ \zeta^2-2\zeta\sin\gamma+1=0. \]

This equality, for \(\zeta \geq 1\), can be satisfied only by the values \(\zeta=1\) and \(\gamma=90^\circ\).

Thus, on the basis of equation (1) we have shown that in a chip formed by the process of shear, deformation will be absent only in the case when \(\zeta=1\) and \(\gamma=90^\circ\).

In all real cases (\(\gamma<90^\circ\)) the chip will always be deformed. The least degree of chip deformation will occur at \(\zeta=1\). For \(\zeta \gtrless 1\) the degree of chip deformation increases.