On the Coincidence of the Flow Curve in Cutting and Compression Processes
Abstract Generated abstract
This paper examines whether a single flow curve can describe plastic deformation in both metal cutting and compression, a question important for deriving cutting stresses and forces from static material tests. The authors compare brass L62 under low speed cutting and compression, using force measurements, chip deformation geometry, and hardness based stress estimates, and then study high speed cutting of steel 2Kh13 using chip root grids, microhardness fields, and computed strain intensity distributions. They find that under comparable temperature and strain rate conditions the cutting and compression flow curves coincide, but at high cutting speeds the relation between stress intensity and strain intensity generally differs because each material point experiences a changing deformation history, strain rate, and temperature.
Full Text
ON THE COINCIDENCE OF THE FLOW CURVE IN CUTTING AND COMPRESSION PROCESSES
V. A. GOVORUKHIN, G. L. KUFAREV
(Presented by the joint scientific seminar of the departments of machine tools and metal cutting and of machine-building technology)
The unity of the flow curve in cutting (plane deformation) and compression (a volumetric stress–strain state) substantially simplifies the determination of stresses and cutting forces from static tests of the material being machined.
There is no established, entirely definite opinion on the unity of the flow curves in cutting and compression. Some researchers believe that there is no single relationship between stresses and strains.
Thus, Prof. N. N. Zorev [1] determines the shear stresses in cutting by the formula:
\[ \tau_p = A \cdot \varepsilon^m \quad \text{at } \varepsilon = 2.5, \]
where
\[ \begin{aligned} \varepsilon &\text{— relative shear in cutting;}\\ m &\text{— coefficient of strain hardening of the material in compression;}\\ A &\text{— shear stress in compression at } \varepsilon = 1. \end{aligned} \]
Thus, he considers the stresses in cutting to be independent of the degree of deformation.
Prof. A. M. Rozenberg [2], taking as the basis of the derivation the hypothesis of equality of specific works at equivalent deformations in the processes of cutting and compression, arrives at the following expression for the shear stress in cutting:
\[ \tau_p = \frac{B \cdot \varepsilon^c}{1.5(c+1)}. \]
In this equation \(B\) and \(c\) are constants determined from a compression experiment. \(B\) is the stress \(\sigma\) obtained at \(\varepsilon = 1\); \(c\) is the tangent of the angle of inclination between the straight line \(\lg(\sigma)=f(\lg \varepsilon)\) and the positive direction of the \(\lg \varepsilon\) axis. Taking into account that for compression \(\sigma_i = B \cdot \varepsilon^c\), and also that \(\tau_{\max} = \frac{\sigma_i}{2}\), where \(\sigma_i\) is the stress intensity, we obtain
\[ \tau_p = \frac{2\tau}{1.5(c+1)}. \]
Thus, the shear stresses in cutting and compression will be equal only in the particular case when the strain-hardening coefficient \(c \approx 0.33\).
Other researchers regard the flow curves in cutting and compression as coinciding, but do so hypothetically.
To verify the coincidence of the flow curves in cutting and compression, the authors carried out two series of experiments. In the first series of experiments
the coincidence of the flow curves in compression and cutting was checked at a speed \(V = 40\) mm/min, when the temperature–rate conditions may be considered the same in each of these processes.
Brass L62 was taken as the material under investigation. To obtain the flow curve in compression up to \(l_i = 2.5\), the method described in [2] was used. It provides for successive regrinding of the compressed specimen in order to preserve the ratio of its diameter to its height, and also for the use, in order to reduce friction at the end faces, of lead foil coated with a colloidal mixture of graphite and glycerin. In these same experiments an experimental relation was established between the stress intensity \(\sigma_i\) and the hardness \(H_{200}\), measured on a PMT-3 instrument at various stages of specimen deformation.
The cutting specimen was a plate 3 mm wide. During the cutting process, the forces were measured with a dynamometer of VNII design. Variation of the final degree of deformation in the experiments was achieved by changing the rake angle of the tool and by using a lubricant. For the chips obtained, the strain intensity was calculated from the relation
\[ e_i = \frac{1}{\sqrt{3}}\,[\operatorname{ctg}\Phi + \operatorname{tg}(\Phi - \gamma)], \tag{1} \]
where
\[
\Phi
\]
is the angle determining the position of the conditional shear plane;
\[
\gamma
\]
is the rake angle of the tool.
The stress intensity was determined from the formula
\[ \sigma_i = \frac{\tau}{\sqrt{3}}. \tag{2} \]
The shear stress was calculated in this case from the cutting forces measured by the dynamometer and from the area of the conditional shear plane determined in the experiment. The experimental points obtained in this way were plotted on the flow curve obtained in compression—the solid line (Fig. 1). The agreement of the experimental results in cutting
Fig. 1. Relation between stress intensity and strain intensity in compression and cutting processes at microspeed
and compression makes it possible to conclude that, under conditions of constancy of the temperature–rate regime, there exists a single flow curve.
Fig. 2. Distribution of microhardness and strain intensity in the chip root (steel 2Kh13, \(V = 56.5\) m/min, \(S = 0.31\) mm/rev, \(t = 6\) mm, \(\gamma = 17^\circ\))
In the second series of experiments, the cutting of two disks ground against one another was carried out at speeds from 26 to 150 m/min with a transverse feed \(S = 0.31\) mm/rev and a rake angle of the cutter \(\gamma = 17^\circ\). A graduation grid was applied to one of the disks, formed by concentric circles with a spacing of 0.05 mm and radial rays, the central angle between which was 5 minutes. To obtain the chip roots, an explosive-action device was used. Steel 2Kh13 was taken as the material being machined.
The strain intensity in the chip-formation zone was calculated on the Minsk-1 digital computer by a procedure specially developed for this type of coordinate grid. In Fig. 2a, as an example, the distribution of strain intensity for one of the roots is shown \((V = 56.5\ \text{m/min})\).
The stress-intensity field was determined from the distribution of hardness values in the chip-formation zone (Fig. 2v) and from the relation \(\sigma_i = f(H_{200})\) obtained in compression experiments. It is considered reliably proven that this relation does not depend on the type of stress state [3, 4, 5]. Superposing the fields of strain intensity and stress intensity makes it possible to determine both of these characteristics for any point of the plastically deformed zone in the cutting process, and therefore also to construct the flow curve.
Additional experiments with measurement of forces in the cutting process made it possible to obtain the dependence \(\sigma_i\)—\(e_i\), determining \(e_i\) and \(\sigma_i\) respectively from formulas (1) and (2).
In Fig. 3, the solid line shows the flow curve in compression, and the dashed line shows the relation \(\sigma_i\)—\(e_i\) in cutting at high speeds, obtained from formulas (1) and (2).
Fig. 2. Relation between stress intensity and strain intensity in compression and high-speed cutting processes
However, this relation cannot be called a flow curve. Numerous experiments and, in particular, the experiments of N. N. Zorev [1] indicate that this relation may be arbitrary depending on the intensity of the effects in the given particular process, the strain rate, which increases stresses, and temperature, which lowers stresses. The black points in Fig. 3 depict the flow curves for three cutting cases \((V = 26.5—56.5—91\ \text{m/min})\), obtained by superposing the fields of strain intensity and hardness. It can be seen here that the flow curves in cutting and compression coincide only in the particular case when the point of final deforma-
... during cutting lies on the compression curve. In all other cases, the curve of the increase in stresses with increasing strain during cutting differs from that for compression.
In this connection it should be borne in mind that each of the points of the flow curves obtained in cutting has its own combination, inherent only to it, of temperature and strain rate. As each material point passes through the chip-formation zone in the experiments described, the strain rate at first increased, reaching values of
\[ \dot e_i = 6000 \ \frac{1}{\sec} \]
(root No. 79), then decreased to zero, while the temperature continuously increased.
Thus, the deformation history of each material point of the layer being cut is extremely complex; i.e., the absence of coincidence between the flow curve in compression and the relation \(\sigma_i - e_i\) in cutting is natural.
REFERENCES
-
N. N. Zorev. Problems in the Mechanics of the Metal-Cutting Process, Mashgiz, 1956.
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Metal Cutting and the Tool, ed. A. M. Rozenberg. Mashinostroenie, 1964.
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L. A. Khvorostukhin. Calculation of Forces in High-Speed Cutting on the Basis of the Physico-Mechanical Characteristics of Metals. Dissertation, Tomsk, 1953.
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A. M. Rozenberg, L. A. Khvorostukhin. “Hardness and Stress in a Plastically Deformed Body,” ZhTF, vol. XXV, issue 1, 1955.
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A. M. Rozenberg, A. N. Eremin. Elements of the Theory of the Metal-Cutting Process, Mashgiz, 1956.