Kinematic Analysis of the Process of Continuous Chip Formation
Abstract Generated abstract
This paper presents a kinematic analysis of continuous chip formation in the cutting of ductile metals, treating the process as plastic shear deformation governed by the shear-plane angle, tool rake angle, and chip shrinkage. Using the Time-Zvorykin scheme with stated simplifying assumptions, it decomposes particle displacements in the removed layer and introduces a double displacement triangle from which relationships among shear angle, chip flow, tool motion, relative shear, and deformation geometry are derived. The analysis is further applied to graphical determination of cutting parameters and to reconstruction of metal grain shapes before and after deformation, while arguing that observed chip texture results from shear at the primary shear angle rather than sliding along the texture direction.
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PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV
Vol. 85 1957
KINEMATIC ANALYSIS OF THE PROCESS OF FORMATION OF CONTINUOUS CHIPS
L. M. SEDOKOV
(Submitted by Prof. A. M. Rozenberg, Doctor of Technical Sciences)
In machining steels and other ductile metals, continuous chips are usually formed, which indicates that the cutting process proceeds uniformly, without jumps, smoothly; and this circumstance makes it possible to study in detail the process of formation of this type of chip and, consequently, to reveal important regularities of the process of metal cutting.
The study of the process of formation of continuous chips must, first of all, consist in an analysis of the displacements of the various particles of the layer being removed in the process of plastic deformation occurring during the machining of metals by cutting.
However, the questions of the displacements of metal particles in the cutting process, i.e., the kinematics of the chip-formation process, have so far been developed only partially.
Meanwhile, an analysis of the displacements of the various points of the layer being removed, undergoing plastic deformation, could contribute considerably to success in the study of complex questions of cutting theory and would protect individual researchers from gross errors.
In the present article, on the basis of known and established propositions of cutting theory, a kinematic analysis of the process of formation of continuous chips is set forth.
Our task is made easier by the fact that Time [1] derived the relation between the displacements of the cutter in the direction of the cutting speed and the displacements of metal particles in the shear plane, and also by the fact that one of the chief questions of the kinematics of chip formation—the derivation of an expression for determining the magnitude of the relative shear—is given by investigators on the basis of this relation [3, 4, 5, 6 and 8].
The starting point for solving questions of the kinematics of the chip-formation process, in our view, should be the following proposition.
If the cutting process is a process of plastic shear deformation, then all displacements in the cutting process must occur by shears in a plane characterized by the angle \(\beta_1\). Consequently, the direction of displacement of each particle of the layer being removed in the given cutting process is determined by the angle \(\beta_1\). The magnitude of the displacement is determined by the degree of plastic deformation, i.e., also by the angle \(\beta_1\), but in combination with the rake angle of the cutting tool \(\gamma\).
Therefore, knowing the chip shrinkage and the rake angle of the tool, one can determine the direction and magnitude of the displacement of any particle of the layer being removed.
The path traversed by each point of the layer being cut off, and its component parts
Let us present the usual picture of the process of formation of continuous chips according to Time–Zvorykin (Fig. 1). When removing a layer of metal of thickness \(a\), machi-
a chip is formed having thickness \(a_1\). All points that, before deformation, lay on the outer side of the layer being removed, i.e., on the machined surface (points 1, 2, 3, 4, and 5 in Fig. 1), moving at an angle \(\beta_1\) to the direction of motion of the cutter, as a result of deformation will be on the outer (external) side of the chip, occupying positions \(1'\), \(2'\), \(3'\), \(4'\), and \(5'\) in Fig. 1.
Fig. 1
The points \(a\), \(б\), \(в\), \(г\), and \(д\), which lay on the inner side of the layer being removed, i.e., at a depth \(a\) from the machined surface, also as a result of plastic deformation, moving at an angle \(\beta_1\), will be on the inner, tool-side surface of the chip, occupying positions \(a'\), \(б'\), \(в'\), \(г'\), and \(д'\) in Fig. 1.
Point \(a\) will take position \(a'\) during the period in which the cutter moves the distance \(aO\). Point \(б\) is transferred to position \(б'\) during the period of motion of the cutter from \(б\) to \(O\), and so on.
This is how the displacement of points lying on the outer and inner sides of the layer being cut takes place.
Figure 2 shows the displacement of the points \(a\), \(б\), \(в\), \(г\), and \(д\), located at different distances from the outer side of the layer being removed during cutting.
Fig. 2
The displacement of two points \(a\) and \(д\) can be found easily, since point \(a'\) must lie on the outer side of the chip, and \(д'\) on the inner, tool-side surface of the chip.
The position of points б, в and г can be found by determining, for each point, the corresponding distance from the outer side of the chip, which will be proportional to the distance of the corresponding point from the outer side of the layer being removed before deformation.
In Fig. 2 the corresponding graphical construction is given. Points б′, в′ and г′ are found as the intersection of straight lines parallel to the front face and drawn at the corresponding distance from the outer side of the chip, with lines passing from the corresponding points of the cut layer (before deformation) at an angle \(\beta_1\) to the direction of the cutting speed. The position of points б′, в′ and г′ can be found more simply: since points б, в and г before deformation lay on one straight line \(a—O\), then after an equal amount of plastic deformation they must lie on one straight line \(a'—д'\). Consequently, points б′, в′ and г′ can be determined from the intersection of straight lines drawn from the corresponding points of the cut layer at an angle \(\beta_1\), with the straight line connecting point \(a'\) with point \(д'\).
In such a kinematic analysis we make a whole series of assumptions:
- All shears occur along the straight line \(OA\), and there is no chip thickening.
- The degree of plastic deformation throughout the entire thickness of the removed layer remains the same.
- The upper side of the chip still has contact with the front face of the tool, which over the entire length of contact has one and the same rake angle. There is no built-up edge.
- There is no so-called back-flowing upper layer in the chip [3].
The kinematic analysis proposed in the present work is constructed on the basis of the Time—Zvorykin chip-formation scheme, which should be considered approximate. In reality, shears occur not along the line \(OA\), but in a more or less substantial zone of plastic deformation [4]. However, the greater part of the plastic shears in cutting takes place somewhere near the line \(OA\) [5]. Therefore, with a known assumption, without making large errors, it may be considered that the shears occur in a certain narrow band—the zone of plastic deformation—which, for kinematic analysis, may be taken as the line \(OA\).
The degree of plastic deformation throughout the entire thickness of the removed layer may be considered the same, since investigators have not found any noticeable difference in hardness through the thickness of the chip, and hardness is a reliable indicator of the degree of plastic deformation.
Undoubtedly, the proposed kinematic analysis applies to work at high cutting speeds, which preclude built-up-edge formation.
The so-called back-flowing layer always occurs, but its thickness when working with ordinary feeds (\(a = 0.1\) mm per revolution and more) is comparatively small, and its influence may be neglected without any substantial errors.
Nevertheless, it should be noted that all subsequent conclusions from the kinematic analysis of the process of formation of a continuous chip will be correct only with due regard for the assumptions indicated above.
The subsequent conclusions can also be extended to cutting conditions characterized by built-up-edge formation, if one takes into account not the rake angle of the tool, but the rake angle formed by the built-up edge. Since the built-up edge is constantly appearing and disappearing, also changing its geometry, the kinematic calculations in this case will be of a very approximate character.
Continuing the kinematic analysis, let us consider the case when the metal particles before deformation were located at different distances from
the outer side of the layer being removed and at the same time lie on one line parallel to the front face (Fig. 3).
The points \(a, b, v, g\), and \(d\) of the layer being cut, after deformation in the chip, will take the positions \(a', b', v', g'\), and \(d'\).
Fig. 3
This case is the most interesting for kinematic analysis, since the displacement of metal particles from the points \(a, b, v, g\), and \(d\) into the positions \(a', b', v', g'\), and \(d'\) will correspond to one and the same displacement of the tool, equal to the distance \(dO\).
If, during the interval between two instants of cutting, the tool has traversed the distance \(l\) (Fig. 4), then the volume of metal removed during this time, \(1-a-b-5\), will take in the chip the position \(1'-a'-b'-5'\), while the points \(1, 2, 3, 4\), and \(5\), having traversed distances different in magnitude, will occupy respectively the positions \(1', 2', 3', 4'\), and \(5'\).
Fig. 4
For the study of the process of metal cutting as a process of plastic deformation, it is important to know the magnitude of the component parts of the displacement of the various metal particles during the cutting period.
Fig. 5
Fig. 6
For this purpose let us take two very close moments of cutting, characterized by the positions of the cutter I and II (Fig. 6). Let the path traversed by the cutter during the period of its displacement from position I to position II be denoted by \(\Delta l_0\). We shall choose two elementary layers located somewhere in the middle of the thickness of the layer being removed, and consider the magnitude and the component parts of the displacements of particles situated on the surface of these elementary layers.
The path traversed by each point of the layer being removed in the process of cutting can be divided into three component parts.
- \(l_{c\partial}\)—the path traversed by a particle of metal due to the plastic deformation of that elementary part of the layer being removed which is enclosed between the two nearest particles under consideration \((1—2;\ 2—3;\ 3—4\ \text{and}\ 4—5)\).
Referring to Fig. 5, one may write
\[ l_{c\partial}^{1}=1'-1'', \]
\[ l_{c\partial}^{2}=2'-2'', \]
\[ l_{c\partial}^{3}=3'-3'', \]
\[ l_{c\partial}^{4}=4'-4''. \]
Point 5 does not move due to plastic deformation, since for it the elementary layer is equal to zero:
\[ l_{c\partial}^{5}=0. \]
The value \(l_{c\partial}\) in a given chip depends only on the thickness of the elementary layer.
2. \(l_{nn}\) is the path of simple displacement of a given metal particle due to plastic deformation of the entire remaining part of the layer being removed, situated below the elementary part of it under consideration.
Referring to Fig. 5, one may write
\[ l_{nn}^{1}=1''-1''', \]
\[ l_{nn}^{2}=2''-2''', \]
\[ l_{nn}^{3}=3''-3'''. \]
The total displacement of point 4 will consist only of the path traversed by this point due to plastic deformation of the underlying layer \((l_{c\partial})\) and the path corresponding to the further displacement of the whole chip \((l_{\partial n})\). Consequently, the simple displacement for point 4 is equal to zero.
The simple displacement for point 5 will also be equal to zero, since below point 5 there is no metal being transformed into chip.
3. \(l_{\partial n}\) is the path corresponding to the further displacement of the tool, when its motion has no influence on the relative arrangement of the metal particles in the chip and when all particles move along the front face of the tool with the same velocity, equal to the velocity of chip flow,
\[ l_{\partial n}^{1}=1-1''', \qquad l_{\partial n}^{2}=2-2''', \]
\[ l_{\partial n}^{3}=3-3''', \qquad l_{\partial n}^{4}=4-4'', \]
\[ l_{\partial n}^{5}=5-5', \]
\[ l_{\partial n}^{1}=l_{\partial n}^{2}=l_{\partial n}^{3}=l_{\partial n}^{4}=l_{\partial n}^{5}. \]
The total displacement of each point in the direction characterized by the angle \(\beta_{1}\), which is determined by the shrinkage of the chip, is
\[ l_{ob}=l_{c\partial}+l_{nn}+l_{\partial n}. \]
Point \(B\) (Fig. 7), located at the contact of two elementary layers of metal, as a result of plastic deformation of the lower layer will move to position \(B'\). Such displacement of point \(B\) causes simple displacement of the entire upper elementary layer, and consequently point \(A\) will occupy position \(A'\) only due to the deformation that has occurred in the lower layer. But the upper layer must also undergo plastic deformation. Therefore point \(A'\), as a result of plastic deformation of the upper layer, will occupy position \(A''\). In the actual cutting process the upper layer will be deformed first, and then the lower one, but this will not in the least change our reasoning:
\[ l_{c\partial}^{B}=l_{c\partial}^{A}, \]
\[ l_{ob}^{B}=l_{c\partial}^{B}, \]
\[ l_{ob}^{A}=l_{c\partial}^{A}+l_{nn}^{A}=l_{c\partial}^{A}+l_{c\partial}^{B}. \]
The quantity \(BB'\) will be the shear displacement \((l_{c0})\) for the lower layer and the simple displacement \((l_{nn})\) for the upper one.
It is not difficult to notice that \(A''B''=B'B''=A'B'=\Delta l_1\), where \(\Delta l_1\) is the displacement of the chip during the period of deformation of one elementary layer of metal.
Fig. 7
Double displacement triangle
From the displacement diagram of the elementary layers of metal in the cutting process (Fig. 7), one may single out the triangle \(A'', B'', B', B\), which in what follows we shall call the double displacement triangle (Fig. 8).
In this triangle \(B'B''\) is equal to the tool displacement \(\Delta l_0\), \(B'B''\) is the shear displacement \(\Delta l_{c0}\), and \(B''B'=A''B''\) is the displacement of the chip along the rake face \(\Delta l_1\) during the period of tool displacement by the amount \(\Delta l_0\).
The double oblique-angled triangle, reflecting the kinematics of the process of formation of a continuous chip, makes it possible to derive analytically the relation between the various characteristics of the process of plastic deformation in metal cutting.
It is not difficult to see that the perpendicular dropped from point \(B''\) to the direction of the cutting velocity will represent the thickness of the elementary layer before deformation, \(\Delta a\), while the perpendicular dropped from point \(B'\) to the rake face (to the line \(A''B''B'\)) will be the thickness of the elementary layer after deformation.
The angle \(\beta_1\), defining the shear plane, in our case \(B'B''\), can be found from the relation
\[ \tg \beta_1 = \frac{\Delta l_1 \cos \gamma}{\Delta l_0-\Delta l_1 \sin \gamma} = \frac{\cos \gamma}{\dfrac{\Delta l_0}{\Delta l_1}-\sin \gamma}. \]
Taking into account that the ratio of the length of the tool path to the corresponding length of the chip path is equal to the shrinkage \(\xi\), we obtain
\[ \tg \beta_1=\frac{\cos \gamma}{\xi-\sin \gamma}. \tag{1} \]
In a similar way one can also find the magnitude of the angle \(\beta'_2\), which determines the locus of points in the chip that, before deformation, lay on one line parallel to the front face of the tool
\[ \tg \beta'_2 = \frac{2\cos\gamma}{\xi - 2\sin\gamma}. \tag{2} \]
Fig. 8
From the double oblique-angled triangle of displacements all relations and characteristics of the chip-formation process can be determined.
The magnitude of the shear path in the deformation of a layer of thickness \(\Delta a\)
\[ \Delta l_{c\partial} = \frac{\Delta a}{\sin \beta_1}. \]
Hence the ratio of the magnitude of shear to the thickness of the layer being removed is
\[ \frac{\Delta l_{c\partial}}{\Delta a} = \frac{1}{\sin \beta_1}. \tag{3} \]
The magnitude of the path of the tool required for complete deformation of the layer is
\[ \Delta l_0 = \Delta a \frac{\cos(\beta_1 - \gamma)}{\cos\gamma \sin\beta_1}. \]
Hence the ratio of the path of the tool required for complete deformation of the layer to the thickness of this layer is
\[ \frac{\Delta l_0}{\Delta a} = \frac{\cos(\beta_1 - \gamma)}{\cos\gamma \sin\beta_1}. \tag{4} \]
Analogously, the ratio of the shear length to the path of the chip for one and the same layer can be derived as
\[ \frac{\Delta l_{c\partial}}{\Delta l_1}=\frac{\cos \gamma}{\sin \beta_1}. \tag{5} \]
The ratio of the shear length to the length of the cutter path necessary for the complete deformation of the layer and, consequently, for carrying out this shear is
\[ \frac{\Delta l_{c\partial}}{\Delta l_0}=\frac{\cos \gamma}{\cos(\beta_1-\gamma)}. \tag{6} \]
The thickness of the elementary layer after deformation can be determined as follows:
\[ \Delta a_1=\Delta a \frac{\cos(\beta_1-\gamma)}{\sin \beta_1}, \]
\[ \Delta a_1=\Delta l_{c\partial}\cos(\beta_1-\gamma). \]
Whence the ratio of the layer thickness after deformation to the thickness of this layer before deformation (shrinkage) is
\[ \zeta=\frac{\Delta a_1}{\Delta a}=\frac{\cos(\beta_1-\gamma)}{\sin \beta_1} \tag{7} \]
and the ratio of the shear path to the layer thickness after deformation is
\[ \frac{\Delta l_{c\partial}}{\Delta a_1}=\frac{1}{\cos(\beta_1-\gamma)}. \tag{8} \]
The magnitude of the relative shear is the ratio of the magnitude of the displacement to the height of the sheared layer.
Since it is assumed that all shears occur in the plane characterized by the angle \(\beta_1\), below which there are no shears, the plane \(OA\) should be regarded as the fixed base at the moment of shear of the elementary layer.
Consequently, the height of the sheared layer must be taken as the length of the perpendicular dropped from point \(B''\) to the shear line \(B'B''\). Numerically, the height of the layer will be equal to
\[ \Delta l_1\cos(\beta_1-\gamma)=\Delta a\frac{\cos(\beta_1-\gamma)}{\cos\gamma}. \]
Then the magnitude of the relative shear is
\[ \varepsilon=\frac{\Delta l_{c\partial}}{\Delta l_1\cos(\beta_1-\gamma)}, \tag{9} \]
but
\[ \Delta l_{c\partial}=\Delta l_1\cos(\beta_1-\gamma)[\operatorname{ctg}\beta_1+\operatorname{tg}(\beta_1-\gamma)]. \]
Substituting the value for \(\Delta l_{c\partial}\), we finally obtain
\[ \varepsilon=\operatorname{ctg}\beta_1+\operatorname{tg}(\beta_1-\gamma). \tag{10} \]
Using the established relation between shrinkage and the angles \(\beta_1\) and \(\gamma\), the magnitude of the relative shear can be expressed through the shrinkage and the rake angle as
\[ \varepsilon=\frac{\zeta^2-2\zeta\sin\gamma+1}{\zeta\cos\gamma}. \tag{11} \]
Further, taking into account that
\[ \Delta l_{cd}=\frac{\Delta a}{\sin \beta_1} \]
and
\[ \Delta l_1=\frac{\Delta a}{\cos \gamma}, \]
the magnitude of the relative displacement can be obtained in the following form:
\[ \varepsilon=\frac{\cos \gamma}{\sin \beta_1 \cos(\beta_1-\gamma)} . \tag{12} \]
Using the derived relations, the connection between relative displacement and shrinkage can likewise be written in the following form:
\[ \varepsilon=\zeta \frac{\cos \gamma}{\cos^2(\beta_1-\gamma)} . \tag{13} \]
A graphical comparison of the magnitude of shrinkage and relative displacement for one and the same chip is given in Fig. 9, where \(\operatorname{tg}\alpha=\zeta\), and \(\operatorname{tg}\varphi=\varepsilon\).
Fig. 9
Thus, using the double oblique-angle triangle of displacements, one can obtain an analytical expression for all the geometrical characteristics of the chip-formation process, and also determine the connection between them (1—13).
An additional characteristic of the chip-formation process—the angle \(\beta'_2\), which determines the geometrical locus of points in the chip that before deformation lay on one line parallel to the rake face of the cutting tool, in our view makes it possible to represent in greater detail the actual picture of this complex process.
Using the angle \(\beta'_2\) and knowing the orientation of the grains before deformation, one can determine the texture angle in the chip. As calculations have shown, the numerical value of the angle \(\beta'_2\) for cutting with tools having a comparatively small rake angle coincides very closely with the magnitude of the texture angle in the chip \(\beta_2\), determined by the formulas of a number of authors. Therefore, when working with cutters having rake angle \(\gamma=+15^\circ \div -10^\circ\), for deter-
of determining the angle \(\psi\) between the texture in the chip and the shear plane, along with the formulas of Prof. A. M. Rosenberg [6]
\[ \operatorname{ctg}\psi=\varepsilon+1 \]
and N. N. Zorev [4]
\[ \operatorname{ctg}\psi=\frac{\varepsilon}{2}+\sqrt{\frac{\varepsilon^{2}}{4}+1} \]
the following formula may be proposed, following from relations (2) and (10)
\[ \operatorname{ctg}\psi=\varepsilon+\operatorname{tg}(\beta_{1}-\gamma), \tag{14} \]
where
\[ \psi=\beta'_{2}-\beta_{1}\approx\beta_{2}-\beta_{1}. \]
A New Variant of the Derivation of A. M. Rosenberg’s Theoretical Equation for Determining the Cutting Force
The scheme of action of forces on the front face of the tool may be represented in the following form (Fig. 10).
[Diagram with labels: \(\Delta l_{c0}\), \(P_a\), \(P_h\), \(P_i\), \(\omega\), \(\Delta l_0\), \(N\), \(F\), \(\eta\), \(\Delta l_1\).]
Fig. 10
In the direction of the cutting velocity acts the force \(P_1\), and in the perpendicular direction the force \(R_h\). In the shear plane acts the force \(P_\tau\), and in the perpendicular direction the force \(P_a\). Along the front face acts the friction force \(F\), and in the perpendicular direction the normal force \(N\).
When the cutting process is carried out over a certain interval of time, the cutting work expended is
\[ A_p=P_1\Delta l_0. \]
The cutting work is expended on the work of plastic deformation and on the work of the friction force
\[ A_\tau=P_\tau\Delta l_{c0}, \]
\[ A_F=F\Delta l_1. \]
Consequently,
\[ P_1\Delta l_0=P_\tau\Delta l_{c0}+F\Delta l_1. \]
The forces \(R_h\), \(P_a\), and \(N\) do no work, since they are directed perpendicular to the corresponding displacements.
It is known that
\[ P_\tau=\tau \frac{ab}{\sin \beta_1}, \]
\[ F=P_1\frac{\sin \eta}{\cos \omega}. \]
Substituting these values and dividing by \(\Delta l_0\), we obtain
\[ P_1=\tau \frac{ab}{\sin \beta_1}\cdot \frac{\Delta l_{cb}}{\Delta l_0} +P_1\frac{\sin \eta}{\cos \omega}\cdot \frac{\Delta l_1}{\Delta l_0}. \]
Taking into account (6), (7), and (12), we finally obtain
\[ P_1=\frac{\tau \varepsilon}{1-\dfrac{\sin \eta}{\zeta \cos \omega}}\cdot ab. \tag{15} \]
The expression obtained is the theoretical equation of A. M. Rosenberg for determining the principal component of the cutting force [5].
Comparison of Different Cutting Regimes
For the purpose of an in-depth study of the cutting process, it is often necessary to compare some cutting regimes with others. But since different characteristics of the chip-formation process often by no means coincide with the corresponding characteristics of another process, comparison becomes impossible.
When the cutting speed is changed, for example, the coefficient of friction, chip shrinkage, relative shear, and the angle of the shear plane will change. A change in, say, only one feed will lead to a change in the cutting temperature, coefficient of friction, shrinkage, and almost all other characteristics of the chip-formation process.
Therefore, before comparing different cutting processes it is necessary to establish an additional condition for comparison.
In studying force relationships, for example, it may become necessary to compare cutting processes in which the magnitude of the relative shear is the same.
In other cases, it may be necessary to take as the basis for comparison the condition of constancy of shrinkages, etc.
On the basis of a kinematic analysis of the process of formation of a continuous chip, one can construct graphs which may, to a considerable extent, prove useful for comparing different cutting regimes.
Figs. 11, 12, and 13 present graphs characterizing cutting processes under the condition of constant chip shrinkage; Figs. 14 and 15, for the condition of constant angle \(\beta_1\); and Figs. 16 and 17, for the condition of constant relative shear.
Increasing the tool rake angle from \(-20^\circ\) to \(+60^\circ\) at a constant shrinkage \(\zeta=2\) (Fig. 11), the angle determining the shear plane—\(\beta_1\)—will at first increase somewhat, and then, having reached approximately \(30^\circ\), will fall almost to its initial value of \(22^\circ\). For shrinkage \(\zeta=3\) this oscillation
Fig. 11
Visible labels in Fig. 11: \(\xi = 2\); \(\gamma\); \(60^\circ\), \(50^\circ\), \(40^\circ\), \(30^\circ\), \(20^\circ\), \(10^\circ\), \(0^\circ\), \(-10^\circ\), \(-20^\circ\); \(10^\circ\), \(20^\circ\), \(30^\circ\); \(\beta_1, \beta'_2\); \(\beta'_2\), \(\beta_1\).
Fig. 12
Visible labels in Fig. 12: \(\xi = 3\); \(\gamma\); \(60^\circ\), \(50^\circ\), \(40^\circ\), \(30^\circ\), \(20^\circ\), \(10^\circ\), \(0^\circ\), \(-10^\circ\), \(-20^\circ\); \(10^\circ\), \(20^\circ\), \(30^\circ\), \(40^\circ\); \(\beta_1, \beta'_2\); \(\beta'_2\), \(\beta_1\).
Fig. 13
Visible labels in Fig. 13: \(\xi = 4\); \(\gamma\); \(60^\circ\), \(50^\circ\), \(40^\circ\), \(30^\circ\), \(20^\circ\), \(10^\circ\), \(0^\circ\), \(-10^\circ\), \(-20^\circ\); \(10^\circ\), \(20^\circ\), \(30^\circ\); \(\beta_1, \beta'_2\); \(\beta'_2\), \(\beta_1\).
\[ \beta_1 = 30^\circ \]
\[ \gamma \]
\[ 50^\circ \quad 40^\circ \quad 30^\circ \quad 20^\circ \quad 10^\circ \quad 0^\circ \quad -10^\circ \quad -20^\circ \]
\[ \beta_2' \qquad \beta_1 \]
Fig. 14
\[ \beta_1 = 15^\circ \]
\[ \gamma \]
\[ 60^\circ \quad 40^\circ \quad 20^\circ \quad 0^\circ \quad -20^\circ \]
\[ \beta_2' \qquad \beta_1 \]
Fig. 15.
will be smaller (Fig. 12), and for $\zeta=4$ a change in the rake angle of the tool from $\gamma=-20^\circ$ to $+50^\circ$ will cause a change in the angle $\beta_1$ only within the limits of $3^\circ$ (Fig. 13).
This means that the greater the shrinkage, the less significant is the influence of the tool geometry ($\gamma$) on the position of the shear plane ($\beta_1$). Therefore, in the presence of large shrinkages it is possible to compare high-speed cutting conditions with conditions characterized by built-up edge formation (the actual $\gamma$ is variable).
The angle $\beta'_2$, which determines the geometrical locus of points in the chip that, before deformation, lay on one line parallel to the rake face of the tool, will, for shrinkage $\zeta=2$, increase continuously with increasing rake angle of the tool. For $\zeta=4$ it will already begin to decrease from $\gamma=+20^\circ$ onward.
Fig. 16
For large shrinkages, the limits of variation of the angle $\beta'_2$ will narrow considerably: for $\zeta=4$ it will vary within the limits from $21^\circ$ to $28^\circ$, whereas for shrinkage $\zeta=2$ the angle $\beta'_2$ will increase from $34^\circ$ ($\gamma=-20^\circ$) to $80^\circ$ ($\gamma=60^\circ$).
Consequently, the location of metal particles in the chip that, before deformation, lay on one line parallel to the rake face of the tool will, for large shrinkages, depend little on the magnitude of the rake angle of the cutting tool. Conversely, for small shrinkages this indicator of chip formation will be highly sensitive to changes in the tool geometry (its rake angle).
For cutting conditions that give the same position of the shear plane ($\beta_1=\mathrm{const}$), when the rake angle of the tool $\gamma$ is changed from $-20^\circ$ to $+60^\circ$, the chip shrinkage will increase for some time and then begin to decrease; the magnitude of the relative shear will continuously decrease, while the angle $\beta'_2$ will continuously increase (Figs. 14 and 15). Moreover, the limits of variation of the indicated characteristics of chip formation will be smaller for smaller values of the angle $\beta_1$.
Figures 16 and 17 present kinematic schemes of chip formation under the condition that the magnitude of the relative shear is constant, i.e., the degree of plastic deformation of the metal. From the graph in Fig. 16 it is seen that a relative shear \(\varepsilon=1\) can be obtained only by using a tool having a rake angle \(\gamma\) greater than \(38^\circ\). For larger values of the relative shear, this range expands toward smaller rake angles.
Changing the rake angle \(\gamma\) from \(0\) to \(60^\circ\), while maintaining the magnitude of the relative shear \((\varepsilon=2)\), the angle \(\beta_1\) will decrease from 45 to \(19^\circ\); the chip shrinkage will increase up to \(\gamma=+50^\circ\), and then decrease somewhat.
What is characteristic here is the fact that a wide change in the rake angle of the tool in the range of considerable positive rake angles (\(\gamma\) greater than \(20^\circ\) for \(\varepsilon=2\)) does not cause substantial changes in \(\beta_1\) and \(\zeta\). And this is all the more true the greater the magnitude of the relative shear.
Fig. 17
All these graphs should be combined into one general graph, by means of which all the basic geometrical characteristics of the chip-formation process can be determined. Such a graph is presented in Fig. 18. Let us give two examples of its use.
Example 1. When working with a cutter having \(\gamma=15^\circ\), at high cutting speeds a chip shrinkage \(\zeta=2.5\) was obtained. It is required to determine the angle \(\beta_1\) and the relative shear \(\varepsilon\).
From point \(B\) we draw a line corresponding to the rake angle \(\gamma=15^\circ\) and obtain the point of intersection with the circle of constant shrinkage \(\zeta=2.5\). Through this point and point \(O\) we draw a straight line to the scale of the angle \(\beta_1\), and thereby determine the angle of the shear plane \(\beta_1=23^\circ 30'\). The segment of the line corresponding to \(\beta_1=23^\circ 30'\), enclosed between the curves \(\varepsilon=2\) and \(\varepsilon=3\), is divided approximately in half by the point of intersection (somewhat closer to the line \(\varepsilon=2\)); consequently \(\varepsilon=2.45\).
Example 2. When working with a cutter having rake angle \(\gamma=-5^\circ\), at high speeds it is necessary to obtain a relative shear \(\varepsilon=4.0\). What will the chip shrinkage and the angle \(\beta_1\) be in this case?
From point \(B\) we draw a line corresponding to \(\gamma=-5^\circ\). We find the point of intersection of this line with the curve of constant relative shear \(\varepsilon=4.0\). Through the point of intersection and point \(O\) we draw a straight line determining the angle \(\beta_1=15^\circ 30'\), and along the circumference we proceed to the shrinkage scale, on which we find \(\zeta=3.55\).
This graph can also be used for cutting conditions in the built-up-edge formation zone to determine the actual rake angle (the rake angle created by the built-up edge), if we could determine, by means of a microscope or microsection, the value of the angle $\beta_1$; then, knowing the chip shrinkage and the angle $\beta_1$, it would be easy to find from the graph the value of the actual rake angle.
Fig. 18
Having found from the graph the corresponding characteristics of the chip-formation process, we automatically obtain the principal diagram of the chip-formation process, on the scale of the graph, for the given specific conditions.
Forms of grains before deformation and after deformation
In order to understand the process of cutting metals, beginning with Usachev [7], the metallographic method of investigation has been widely applied. In microphotographs of chip roots, the change in the shape of grains in the process of plastic deformation occurring during chip removal is clearly visible. Comparison of the forms of grains before deformation and after deformation makes it possible to study the cutting process in detail. However, very often the grains of the metal before deformation do not have a strictly identical shape, which leads to a variety of grain forms after deformation. This greatly complicates the investigation, since we do not know what shape the given grain had before deformation.
Using kinematic analysis of the chip-formation process, it is possible, more or less accurately, to determine from the shape of a grain located in the chip the shape that it had before deformation, and conversely, if the rake angle of the tool with which the given cutting process was carried out and the chip shrinkage are known.
Figs. 19, 20, 21 give the graphical construction of the grain shape after deformation, if before deformation the grain had, in longitudinal section (in the section of the microsection), the form of a square, a rhombus, and a circle. The construction method is simple and requires no explanation.
Fig. 19
Fig. 20
Figure 22 gives an example of how one should find the shape of a grain before deformation if its shape after deformation, the rake angle, and the angle \(\beta_1\), which determines the shear plane, are known.
Fig. 21
Such a method can be applied for any cutting conditions, including the cutting process accompanied by built-up edge formation, if, in constructing it, one takes the actual rake angle formed by the built-up edge.
Fig. 22
This method cannot be applied to grains located very close to the cut surface of the chip, since there an additional deformation will take place in a plane parallel to the rake face; nor can it be applied for very large deformations, when the grain in the chip acquires an excessively elongated shape.
Conclusion
The double displacement triangle (Fig. 8), obtained as a result of the kinematic analysis of the process of formation of a continuous chip, makes it possible simply to obtain an analytical expression for any geometric parameters of the cutting process and their relationships, and also to determine the relative position in the chip of any material point of the layer being removed.
On the basis of the kinematic analysis carried out for the process of formation of a continuous chip, it has been shown that, provided all points of the layer being removed are displaced at the angle \(\beta_1\), the grain of the layer being removed after deformation forms a texture whose direction makes an angle \(\beta_2\) with the velocity vector. This proposition completely refutes the statement by the authors of the textbook Metal Cutting [9] that “plastic deformation of the material within the element itself is accompanied by sliding of individual layers of plastically deformed material at a certain angle \(\Theta_1\)” (in our notation \(\beta_2\)). There is no such sliding, and the texture angle \((\beta_2)\) is formed only as a result of shears occurring at the angle \(\beta_1\).
REFERENCES
- Time I. A.—A Memoir on the Planing of Metals. St. Petersburg, 1877.
- Klushin M. I.—Metal Cutting. Mashgiz, 1953.
- Loladze T. N.—Chip Formation in the Cutting of Metals. Mashgiz, 1952.
- Zorev N. N.—Investigation of Elements of the Mechanics of the Cutting Process. Mashgiz, 1952.
- Rozenberg A. M. and Eremin A. N.—Theoretical equation of the cutting force, Vestnik Mashinostroeniya, No. 8, 1953.
- Rozenberg A. M. and Eremin A. N.—The structure of the chip in cutting a plastic metal. Proceedings of the Tomsk Polytechnic Institute named after S. M. Kirov, Order of the Red Banner of Labor, vol. 75, Tomsk, 1954.
- Usachev Ya. G.—Phenomena occurring in the cutting of metals. Proceedings of the Petrograd Polytechnic Institute, 1915.
- Kuzyushin V. V.—Measurement of plastic deformation in the cutting of metals. Problems in the Theory of Machining Metals by Cutting, Mashgiz, 1954.
- Granovsky G. I., Gurov P. P., Krivoukhov V. A., Larin M. N., Malkin A. Ya.—Metal Cutting. Mashgiz, 1954.